Recent Progress in Inequalities
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centrefor Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 430
Recent Progress
in Inequalities
edited by
G. V. Milovanovic
University of Nis,
Faculty of Electronic Engineering,
Nis, Yugoslavia
Springer Science+Business Media, LLC
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-4945-2
DOI 10.1007/978-94-015-9086-0
ISBN 978-94-015-9086-0 (eBook)
Printed on acid-free paper
All Rights Reserved
© 1998 Springer Science+Business Media New York
Originally published by Kluwer Academic Publishers in 1998
Softcover reprint of the hardcover 1st edition 1998
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
inc\uding photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
This Volume is Dedicated to
Professor Dragoslav s. Mitrinovic
(1908 - 1995)
Table of Contents
Preface ...................................................................
xi
Life and Inequalities: D. S. Mitrinovic (1908-1995)
G. V. Milovanovic ........................................................
1
Publications of D. S. Mitrinovic
R. Z. Djordjevic and R. R. Janic
11
Invited Papers
Complex Polynomials and Maximal Ranges: Background and Applications
V. V. Andrievskii and S. Ruscheweyh .....................................
31
Exact Classical Polynomial Inequalities in Hp for 0 ~ p ~ 00
V. V. Arestov ............................................................
55
Vietoris's Inequalities and Hypergeometric Series
R. Askey .................................................................
63
Inequalities for Norms of Intermediate Derivatives and Some Their
Applications
V. F. Babenko .................................... . . . . . . . . . . . . . . . . . . . . . . .
77
Table of Inequalities in Elliptic Boundary Value Problems
C. Bandie and M. Flucher ................................................
97
A Catalogue of Help and Help-type Integral and Series Inequalities
M. Benammar, C. Bennewitz, M. J. Beynon, B. M. Brown, N. G. J. Dias,
W. D. Evans, W. N. Everitt, V. G. Kirby, and L. L. Littlejohn ........... 127
Remarks of the Jackson and Whitney Constants
B. Bojanov ............................................................... 161
On the Application of the Peano Representation of Linear Functionals
in Numerical Analysis
H. Brass and K.-J. Förster ............................................... 175
Inequalities Due to T. S. Nanjundiah
P. S. Bullen .............................................................. 203
Marcinkiewicz-Zygmund Inequalities: Methods and Results
D. S. Lubinsky ........................................................... 213
Shapiro's Inequality
A. M. Fink ............................................................... 241
Bernstein Type Inequalities for Rational Functions With Prescribed Poles
N. K. Govil and R. N. Mohapatra ........................................ 249
vii
viii
TABLE OF CONTENTS
Some Generalisations and Refinements of the Hardy Inequality
H. Heining, A. Kufner, and L. E. Persson ................................ 271
Discrete Inequalities of Wirtinger's Type
G. V. Milovanovic and 1. Z. Milovanovic ................................. 289
Convexity Properties of Special Functions and Their Zeros
M. E. Muldoon ........................................................... 309
Inequalities in Circular Arithmetie: A Survey
Lj. D. Petkovic and M. S. Petkovic ....................................... 325
Properties of Isometries and Approximate Isometries
Th. M. Rassias ........................................................... 341
Inequalities for the Zeros of an Orthogonal Expansion of a Polynomial
G. Schmeisser ............................................................ 381
Error Inequalities for Discrete Hermite and Spline Interpolation
P. J. Y. Wong and R. P. Agarwal ........................................ 397
Contributed Papers
An Inequality Concerning Symmetrie Functions and Some Applications
D. Andrica and L. Mare .................................................. 425
A Note on the Second Largest Eigenvalue of Star-like Trees
F. K. Bell and S. K. Simic ............................................... 433
Refinements of Ostrowski's and Fan-Todd's Inequalities
M. Bjelica ................................................................ 445
On the Stability of the Quadratie Functional Equation and Related Topies
S. Czerwik ............................................................... 449
A Diriehlet-type Integral Inequality
W. N. Everitt ............................................................ 457
On the Hyers-Ulam-Rassias Stability of Mappings
P. Gavruta ............................................................... 465
Functions With Quasieonvex Derivatives
V. Govedarica and M. Jovanovic ......................................... 471
Local Approximation by Quasi-polynomials
Yu. Kryakin ............................................................. 475
Logarithmic Concavity of Distribution Functions
M. Merkle ................................................................ 481
Sharpening of Cauchy Inequality
Z. Mijalkovic and M. Mijalkovic .......................................... 485
TABLE OF CONTENTS
ix
A Note on the Least Constant in Landau Inequality on a Finite Interval
A. Yu. Shadrin ........................................................... 489
Some Inequalities Involving Harmonie Numbers
M. S. Stankovic, B. M. Dankovic, and S. B. Trickovic .................... 493
Inequalities for Polynomials in L o Norm
E. A. Storozenko ......................................................... 499
Some Inequalities for Altitudes and Other Elements of Triangle
M. R. Ziiovic and M. R. Stevanovic ...................................... 505
Author Index ............................................................. 511
Preface
This volume is dedieated to Professor Dragoslav S. Mitrinovic (1908-1995), one
of the most accomplished masters in the domain of inequalities. Inequalities are
everywhere and play an important and significant role in almost all subjects of
mathematies including other areas of sciences. Professor Mitrinovic often used to
say: "There are no equalities, even in the human life, the inequalities are always
met".
Inequalities present a very active and attractive field of research. As Richard
Bellman has so elegantly said at the Second International Conference on General
Inequalities (Oberwolfach, July 30 - August 5, 1978): "There are three reasons for
the study of inequalities: praetieal, theoretieal, and aesthetie." On the aesthetie
aspects he said: "As has been pointed out, beauty is in the eyes of the beholder.
However, it is generally agreed that eertain pieees of musie, art, or mathematies
are beautiful. There is an eleganee to inequalities that makes them very attraetive. "
A great progress in inequalities was made by seven Oberwolfach conferences on
inequalities with the corresponding seven volumes under the title General Inequalities 1 - 7, published by Birkhäuser (1978, 1980, 1983, 1984, 1987, 1992, and
1997), as weIl as by several other international conferences dedieated to inequalities. One of these conferences was held in 1987 at the University of Birmingham,
England, under the auspices of the London Mathematical Society, and dedieated
to the work of G. H. Hardy, J. E. Littlewood and G. P6lya in writing the book
Inequalities, whieh was first published by the Cambridge University Press in 1934.
This book has to be counted as one of the outstanding achievements in mathematical scholarship in this century, as said Norrie Everitt in the Preface of the volume
Inequalities - Fifty years on /rom Hardy, Littlewood and P61ya (Marcel Dekker,
1991). Norrie said also: "Of great intrinsie interest, indeed, faseination, the book
has proved an invaluable referenee work for more than fifty years, and a souree
of lasting inspiration to workers in the vineyard of inequalities." Until the early
sixties only this classieal work intended to transform the field of inequalities from
a collection of isolated formulas into a systematie discipline. Since that time, other
books on inequalities have appeared, especially two Springer's: Inequalities (1971)
by E. F. Beckenbach and R. Bellman, and Analytic Inequalities (1970) by D. S.
Mitrinovic. After the classieal Inequalities by Hardy, Littlewood and P6lya, this
MitrinoviC's famous work is the most referred to books in the field of inequalities.
Mitrinovic was interested in all kinds of inequalities, from elementary inequalities, geometrie inequalities, inequalities with means, inequalities in analysis and
approximation theory, including inequalities in number theory. In collaboration
with fellow colleagues he produced several books in different subjects concerning
inequalities during the last ten years of his life. Five of them have been published by Kluwer: Means and Their Inequalities (1988) with P. S. Bullen and
P. M. Vasic, Reeent Advanees in Geometrie Inequalities (1989) with J. E. Pecaric
and V. Volenec, Inequalities Involving Functions and Their Integrals and Derivatives (1991) with J. E. Pecaric and A. M. Fink, Classieal and New Inequalities in
xi
xii
PREFACE
Analysis (1993) with J. E. Pecaric and A. M. Fink, Handbook 0/ Number Theory
(1986) with J. Sandor and B. Crstici, and one book was published by World Scientifie: Topics in Polynomials: Extremal Problems, Inequalities, Zeros (1994) with
G. V. Milovanovic and Th. M. Rassias.
In order to provide a multi-disciplinary forum of discussion in mathematics and its
applications in which the essentiality of inequalities is highlighted, a new journal
with title Journal 0/ Inequalities and Applications is just started this year by
Gordon and Breach Science Publishers.
An International Memorial Conference dedicated to the late Professor D. S. Mitrinovic was held at the Faculty of Electronic Engineering, University of NiS, Yugoslavia, from June 20-22, 1996. This conference was organised by the foHowing
institutions: The Serbian Scientific Society (Belgrade), The Mathematics Institute
of Serbian Academy of Sciences and Arts (Belgrade ), Faculty of Electrical Engineering (Belgrade), and Faculty of Electronic Engineering (Nis). There were 93
participants from 17 countries and the work on the conference was organised in
the foHowing three sections: Recent Progress in Inequalities, Advances in Mathematical Analysis, and Topics in Mathematics with Applications. More than 140
authors sent their survey and contributed papers to the Program Committee. After a refereeing process, a number of selected papers on inequalities are included
in this volume. Ten members of the Editorial Board of the Journal o/Inequalities
and Applications appear as authors in this volume.
This book is divided into three sections: An introduction to the life and scientific
work of Professor Mitrinovic, Invited Papers, and Contributed Papers. In each
section the papers appear in alphabeticalorder according to the initial of the last
name of the first-named author. An author index is also included at the end of
the book.
Lastly, I wish to express my warmest thanks to all of the scientists who contributed
to this volume, as weH as to all of my coHeagues from the Department of Mathematics, University of Nis, who helped in the preparation of this volume. The
financial support for preparing this book is given by Nis Assembly. It is, also, a
pleasure to acknowledge the superb assistance that the staff of Kluwer Academic
Publishers provided.
Nis, June 1997
Gradimir V. Milovanovic
LIFE AND INEQUALITIES: D. S. MITRINOVIC
(1908-1995)
GRADIMIR V. MILOVANOVIC
Faculty 0/ Electronic Engineering, Department 0/ Mathematics, P.G. Box 73,
18000 Nis, Yugoslavia
1. Biographical Data
Professor Dragoslav S. Mitrinovic, the famous scientist, a modest man, teacher and
a model of many generations, died on April 2, 1995. He was born in Smederevo,
Serbia, on June 23, 1908, as the first child of Svetislav and Marija Mitrinovic.
His sister Ruzica (1909-1993) was the second and the last child in the Mitrinovic
family. Their father, a known judge, died when Dragoslav was seven, so that he
was forced to fight for his living himself. He received elementary and secondary
education in Pristina and Vranje. In 1932 he graduated mathematics at the Faculty
of Philosophy, University of Belgrade. The next year, as a student of Professor
Mihailo Petrovic - Alas, he defended his Ph. D. thesis in the field of Differential
equations entitled "Investigations 0/ an important differential equation 0/ the first
order".
FIG. 1. Dragoslav with his sister
Ruzica (from 1913)
FIG. 2. D. S. Mitrinovic as a student
(from 1929)
G. V. Milovanovic (ed.), Recenl Progress in Inequalilies, 1-10.
© 1998 Kluwer Academic Publishers.
2
G. V. MILOVANOVIC
In 1933 he got married to Olga Sretenovic (1910-1996). Olga was also a mathematician and she worked as a secondary school teacher. Their sons, Svetislav
(1934) and Mihailo (1945), are the university professors.
FIG . 3. Prof. Mihailo Petrovic - Alas
(1868-1943)
FIG. 4 . Dragoslav and Olga
(from 1933)
Until 1946 D. S. Mitrinovic worked as a secondary school teacher. He spent some
time as a researcher at the Paris University. His ID-cards from that period are
shown in Figures 5 and 6.
FIG. 5. University immatriculation card
FIG. 6 . Card for the National Library
During this period Mitrinovic published about 50 scientific papers, mainly on
differential equations.
2. Professional Career
Mitrinovic started his university career in Skoplje, Macedonia, as an Associate
Professor at the Philosophical Faculty. It took hirn only five years (1946-1951) to
found the Skoplje School of Mathematics. At the Philosophical Faculty he founded
LIFE AND INEQUALITIES: D. S. MITRINOVIC (1908-1995)
3
the Department of Mathematics and two mathematieal journals ( "Zbornik radova
Filozo/skog /akulteta u Skoplju" in 1948 and "Bilten drustva matematicara i fizicara
Makedonije" in 1950). The first mathematieal research papers in Macedonia were
done by Professor Mitrinovic. His persistent work resulted in the foundation a
rieh professional mathematical library there and in a wide exchange of scientific
publications with foreign countries. At the beginning, all the lecturing in Skoplje
was performed by two mathematicians only. It was at that time that a core of
scientific workers was formed in Skoplje, which is today one of the recognised
scientific centers. A number of Ph.D. theses were defended, mainly under the
supervision of Professor Mitrinovic. Thanks to his scientific contribution he was
elected the member of the Macedonian Academy 0/ Science and Art.
From 1951 to his retirement in 1978 Professor Mitrinovic taught at the Faculty
of Electrical Engineering, University of Belgrade, and in 1953 he was elected the
Head of the Department of Mathematics. During his long period of teaching he
supported young and talented mathematicians, students of his faculty (to whom
mathematies would be their future profession), gave them instructions for their
scientific research, made them get to know the scientific references he knew so weIl
and helped them publish their results in the country and abroad. He made his
collaborators work as hard as he practised himself. He encouraged the progress
and success of all his assistants. He founded the weIl-known Belgrade School of
Functional Equations, Differential Equations and Inequalities. He was also the
founder of the Publications 0/ the Faculty 0/ Electrical Engineering, Series: Mathematics and Physics, which soon became the worldwide renown journal. Numerous
world weIl-known and outstanding mathematicians published their papers in the
Publications. This journal is available in many university libraries all over the
world.
Soon after foundation of the first faculties in Nis in 1960, Professor Mitrinovic
founded another school of mathematies. In the period between 1965 and 1975
he was the Head of the Department of Mathematics at the Faculty of Electronie
Engineering, University of Nis. He supported the development of any field in
mathematies, encouraged his collaborators and assistants, introduced them into
new fields he himself didn't work in and was in touch with developed centers all
over the world. His collaborators appreciated and accepted such approach of his.
Thanks to all this, the Nis School of Mathematics soon grew into a powerful center
of Approximation Theory, Inequalities and Numerical Mathematics, without any
problems and separations which are characteristie for this country.
Professor Mitrinovic was a very communieative person. He maintained epistolary
relationship with numerous world respectable mathematicians. He was a longtime member of the American Mathematical Society, SocieU Mathematique de
France and one of the founders of the Serbian Scientific Society. His social activity on the professional plan is also noteworthy. He was the founder of the Mathematical documentation center 0/ the Society 0/ mathematicians and physicists 0/
Serbia, the Vice-president 0/ the Union 0/ societies 0/ mathematicians and physicists 0/ Yugoslavia, the President 0/ the Society 0/ mathematicians and physicists
0/ Macedonia, the President 0/ the Commission tor mathematics 0/ the Federal
G. V. MILOVANOVIC
4
Council lor the coordination 01 scientijic research, a M ember and the President
01 the corresponding commission in Serbia, the Vice-president 01 the Commission
lor text-books, not to mention several other duties within the framework of the
University. For a long time, Professor Mitrinovic was a member of the Editorial
Board of East European Series "Mathematics and Its Applications" in the Kluwer
Academic Publishers.
FIG . 7. S:Milojkovic, D. S.Mitrinovic, R . i . Djordjevic, and G . V. Milovanovic
(Poree, 1975)
Mitrinovic was a prolific writer of many university books as weIl as significant
monographs of high scientific level, published by the world's most famous publishing houses. His monograph Analytic Inequalities (with P. M. Vasic) published in
1970 by Springer Verlag, had a very powerful influence on the development of this
field in Yugoslavia and abroad. Many generations of students and mathematicians
studied from Professor MitrinoviC's books. His name on the covers always signified
high standards and a rigorous mathematical style.
3. Scientific Work in Inequalities
The scientific work of Professor Mitrinovic and his contributions in mathematics
can be classified into the following areas:
1. Differential equations;
2. F\mctional equations;
3. Inequalities;
4. Other fields.
His work in the first two areas (Differential and Functional equations) has been
described in [10] (see also [1-4] and [6-7]). Beside more than one hund red papers
on differential equations and more than thirty papers on functional equations, he
LIFE AND INEQUALITIES: D. S. MITRINOVIC (1908-1995)
5
published three text-books on differential equations. His starting papers on functional equations from fifties were important for developing a well-known Belgrade
School of Functional Equations as well as the appearing of his "Mathematics Problem Book", Vol. II1 (1960), with several interesting open problems related to the
classieal functional equations. These problems were a "glue" for young mathematicians and for the most talented students.
We mention that 7 mathematicians took their Ph. D. theses in differential equations with Professor Mitrinovic: B. S. Popov (1952), I. Bandic (1958), D. PerCinkova (1963), I. Sapkarev (1964), J. D. Keckic (1970), P. R. Lazov (1977), and B.
Piperevski (1982). Also, Professor Mitrinovic gave seven Ph. D. theses in functional equations: D.Z. Djokovic (1963), K. Milosevic-Rakocevic (1963), P. M. Vasic
(1963), R. Z. Djordjevic (1966), R. R. Janic (1968), I. Stamate (1971), and B. Zaric
(1975). A niee review on these theses has just been written by Professor B. D.
Crstici (see [3]).
The last and the greatest MitrinoviC's passion in mathematies was the one called
- 1nequalities. He was involved in all kinds of inequalities. He often used to say:
"There are no equalities, even in the human li/e, the inequalities are always met".
Until early sixties only the classieal work 1nequalities by Hardy, Littlewood, and
P6lya, appeared in 1934, intended to transform the field· of inequalities from a
collection of isolated formulas into a systematie discipline. Professor A. M. Fink
(Iowa State University) even said: "I had not considered inequalities as a research
subject, even though I owned a copy 0/ Hardy, Littlewood, and P6lya 's "lnequalities". 1nequalities were a sidelight to my research in differential equations. But
through Pro/essor MitrinoviC's book "Analytic 1nequalities" from 1970 and his correspondence with me, I saw the richness 0/ the subject 0/ inequalities, the care he
took to ascribe intellectual ideas to their real sources, and his personal integrity in
writing about the subject."
MitrinoviC's interest in inequalities started very early considering some inequalities
for elementary symmetrie functions (1959). His work can be classified into the
following areas:
1. Elementary inequalitiesj
2. Geometrie inequalitiesj
3. Means and their inequalitiesj
4. Analytie inequalitiesj
5. Inequalities and extremal problems with polynomialsj
6. Various partieular inequalitiesj
7. Inequalities in number theory.
To each of these areas Mitrinovic devoted at least one monograph. At this point
we could cite Professor Diek Askey, who told: "He was a collector 0/ interesting
and important older mathematical results. This resulted in a number 0/ books
which have /ew i/ any rivals. When an inequality arises, as it often does in my
work or in letters /rom others asking about one, the first place I look is in the
books 0/ Mitrinovic. There are /ew with his dedication to preserving interesting
6
G. V. MILOVANOVIC
mathematies. Fortunately, he did not write all 0/ his books alone, so he helped
train others to /ollow in his /ootsteps. May they earry on his legaey 0/ service to
the eommunity 0/ mathematicians around the world."
1. Mitrinovie started with elementary inequalities in 1959. Very soon in 1964 he
published (in eooperation with E. S. Barnes, D. C. B. Marsh and J. R. M. Radok)
the book entitled "Elementary Inequalities" (P. Noordhoff, Groningen). This tutorial text and problem eolleetion is designed to introduee the student, at undergraduate or senior high school level, to the elementary properties of inequalities.
Considerably enlarged version of this book appeared in Polish in 1972, with P. M.
Vasie and R. R. Janie as eo-authors. Among many elementary inequalities treated
by Professor Mitrinovie we mention only those with elementary symmetrie funetions (Tk = (Tk(Xl, .•. ,xn ). If 1 ~ k ~ n -1 and 0 ~ v ~ k -1, Mitrinovie proved
that
(~V(Tk_V)2 - (~v(Tk_v+1)(~v(Tk_v_l) ~ 0,
where ~ is the standard forward differenee operator. Also he proved the following
implieation for 1 ~ p ~ v,
2. Several papers Mitrinovie devoted to the geometrie inequalities. In 1969 the
book "Geometrie Inequalities" (Groningen), written by O. Bottema, R. Z. Djordjevie, R. R. Janie, D. S. Mitrinovie, and P. M. Vasie, was appeared. The book is very
appreciated and has been mueh quoted in the mathematiealliterature. It eontains
about 400 varied geometrie inequalities related to the elements of figures in the
plane (triangles, quadrilaterals, n-gons, circles) and 225 authors are cited in it.
After the appearanee of this book (ealled "Bible of Bottema" in the Canadian
journal Crux Mathematieorum), during the period from 1969-1986 a large number of papers and problems eoneerning geometrie inequalities were published in
mathematieal journals and this inspired Professor Mitrinovie to eompile an eneyclopedie work "Recent Advanees in Geometrie Inequalities" (Kluwer, 1989) jointly
with J. E. Pecarie and V. Volenee. This book eontains several thousands ofinequalities, not only for elements of figures in the plane, but also for elements of figures in
space and hyperspaee (tetrahedra, polyhedra, simpliees, polytopes, spheres). This
book is a good base for the various synthesis of apparently uneonneeted results
about geometrie inequalities, and also represents a rieh souree book for obtaining
some deeper and essential generalisations.
3. Mitrinovie also devoted several papers to the means and their inequalities.
His main eollaborator in this field was P. M. Vasic (1934-1996). Unifying the
results proved by W. N. Everitt [Amer. Math. Monthly 70 (1963), 251-255] and
Mitrinovie and Vasie [Univ. Beograd. Pub!. Elektrotehn. Fak. Ser. Mat. Fiz. No
159 - No 170 (1966), 1-8], H. W. MeLaughlin and F. T. Metealf [Pacifie J. Math.
22 (1967), 303-311] obtained some interesting inequalities for means of order r.
Later, Mitrinovie and Vasie (1968) proved even more general results whieh eontain
inequalities of MeLaughlin and Metealf. In 1966 Mitrinovie and Vasie introdueed
one method, so-ealled A-method, for getting inequalities. This method ean be
summarised as follows:
LIFE AND INEQUALITIES: D. S. MITRINOVIC (1908-1995)
7
(1) Start with an inequality which can be proved by the theory of maxima and
minima;
(2) In a convenient manner introduce one or more parameters into the function
from which that inequality was obtained;
(3) Find the extreme values of such a function, treating the parameters as
fixed.
In this way an inequality involving one or more parameters is obtained. Assigning
conveniently chosen values to those parameters, one may obtain various inequalities whose forms bear no similarity to the original. This method often unifies
isolated inequalities and yields known inequalities as special cases. Using this
method Mitrinovic and Vasic obtained many interesting inequalities with means.
As a top in this field is certainly the monograph "Means and Their Inequalities"
written on 459 pages by D. S. Mitrinovic, P. S. Bullen and P. M. Vasic and published
by Kluwer in 1988.
4. The most important MitrinoviC's work on inequalities inequalities appeared in
the Mathematical Analysis. He considered many important dassical inequalities
induding their variations and generalisations. Especially, we mention his work on
the Steffensen inequality from 1969, as wen as a joint paper with P. M. Vasic on an
integral inequality ascribed to Wirtinger. In 1974 Mitrinovic and Vasic published
one important paper on history, variations and generalisations of the Chebyshev
inequality and the quest ion of so me priorities.
In 1965 Mitrinovic published the book "Nejednakosti" in Serbian on 240 pages.
Five years later, a grandiose work appeared by Springer Verlag - "Analytic Inequalities." Talking on MitrinoviC's contribution in mathematics, Professor P. S.
Bullen says: "Du ring his long and active life Professor Mitrinovic not only did
much original work in various jields, although mainly in inequalities. In addition
he became famous for research into the obscure origins of many famous results.
However his most abiding contribution are three. The famous book, done with the
collaboration of Professor Vasic, "Analytic Inequalities". It is, after the classic
"Inequalities" by Hardy, Littlewood and P6lya, the most referred to book in the
jield of inequalities. "1)
The complete material of this book is divided into three parts. In the first part
("Introduction") an approach to inequalities is given, while the main attention is
devoted to convex functions. The second and main part ("General Inequalities")
consists of 27 sections, each of which is dedicated to a dass of inequalities of
l)Further, Bullen says: "I have called the Publikacije Elektrotehnickog Fakulteta Univerziteta u
Beogradu, serija Matematika i Fizika "his journal" and it was so in a very real sense. It is an
essential tool for working in the field of inequalities, and the almost complete run that I have is
one of my most valuable possessions in the my mathematical library. I only wish that it were
complete. Finally there are the many students Professor Mitrinovic brought along and who are
now carrying on his work all over the world. I mention Professors Vasic, Pecaric as being the
ones that I know best, but there are many others as any perusal of "his journal" will show. I
think it is no exaggeration to say that they are keeping hirn alive, and will continue to do so for
many years to come."
8
G. V. MILOVANOVIC
importanee in Analysis. FinaHy, the third part ("Particular Inequalities") gives a
eoHection of various inequalities.
5. The monograph "Topics in Polynomials: Extremal Problems, Inequalities, Zeros", written by G. V. Milovanovic, D. S. Mitrinovic, and Th.M. Rassias, and published by World Scientifie, eontains some of the most important results on the analysis of polynomials and their derivatives. Besides the fundamental results, which
are treated with their proofs, the book also provides an aeeount of the most reeent
developments eoneerning extremal properties of polynomials and their derivatives
in various metrics with an extensive analysis of inequalities for trigonometrie sums
and algebraic polynomials, as weH as their zeros. Many extremal problems of
Markov, Bernstein, Nikolskil, and Turan type were eonsidered. The inequalities
are given for various domains, various norms and for various subclasses of polynomials, both algebraie and trigonometrie. Some 1200 referenees have been cited,
including preprints. Professor T. Erdelyi in his review on this book in the Journal of Approximation Theory (Vol. 82 (1995), 471-472) says: "The topics are
tastefully selected and the results are easy to find. Although this book is not really
planned as a textbook to teach /rom, it is excellent for self-study or seminars. This
is a very useful reference book with many results which have not appeared in a book
form veto It is an important addition to the literature. " Professor E. W. Cheney
in Mathematics of Computation (Vol. 65 (1996), 438-439) eoncludes his review
by words: "The book is written in a gentle style: one can open it anywhere and
begin to understand, without encountering unfamiliar notation and terminology.
It is strongly recommended to individuals and to libraries. " (see also the reviews
written by Professor N. K. Govil in Mathematical Reviews (95m: 30009) and by
H. M. Srivastava in Zentralblatt für Mathematik (848-147)).
6. The third part of MitrinoviC's monograph "Analytic Inequalities", which is
entitled "Particular Inequalities", represents a eolleetion of various inequalities,
more or less closely intereonneeted. This 200-pages part includes diserete inequalities, inequalities with algebraie and trigonometrie functions and polynomials, inequalities with exponential, logarithmic and gamma funetions, as weH as integral
inequalities and inequalities in the eomplex domain. Many of these results belong to Professor Mitrinovic. Besides extensions and generalisations, Mitrinovic
always wanted to link various isolated inequalities and find their eommon souree.
Reeently he published by Kluwer two monographs with sueh results: "Inequalities
Involving Functions and Their Integrals and Derivatives" and "Classical and New
Inequalities in Analysis" (jointly with J. E. Pecaric and A.M. Fink). In his reeent
papers, mainly written jointIy with Pecaric, various particular inequalities were
eonsidered (Erdös-MordeH's and related in~qualities of Gauss-Winekler, inequalities for polygons, some trigonometrie inequalities, Neuberg-Pedoe and Oppenheim
inequalities, Steffensen's inequality, some determinantal inequalities, inequalities
of Godunova and Levin, Ozeki's inequalities, Lebed's inequality, inequalities of
Hilbert and Widder, Masuyama's inequality, ete.).
7. The last MitrinoviC's monograph was the "Handbook of Number Theory", written jointly by J. Sandor and B. Crstici and published by Kluwer this year (1996).
Unfortunately, after the manuseript was finished and during its preparation for
LIFE AND INEQUALITIES: D. S. MITRINOVIC (1908-1995)
9
printing, Professor Mitrinovic died, not having the chance to see his last work in
libraries. The aim of this book was to systematise and to present in an easily accessible framework the most important results from some parts of Number Theory,
which are expressed by inequalities or by estimates. The book focuses on the most
important arithmetic functions in Number Theory, together with various generalisations, analogues and extensions of such functions, and also properties of some
functions related to the distribution of the primes and of the quadratic residues
and to partitions, etc. We note that the "yeast" for this Handbook was the previous book "1nequalities in Number Theory" published in 1978 by Mitrinovic and
M.S. Popadic (Naucni Podmladak, University of Nis).
We mention also that 4 mathematicians took their Ph. D. theses in inequalities
with Professor Mitrinovic: Lj. R. Stankovic (1975), I. B. Lackovic (1975), G. V.
Milovanovic (1976) and I. Z. Milovanovic (1980).
MitrinoviC's scientific interest was also in the other fields as Bernoulli's and Stirling's numbers and polynomials (31 papers), as weIl as in complex analysis, special
functions, orthogonal polynomials, summation formulas, abstract algebra, etc. Especially, we mention the monograph "The Cauchy Method ot Residues - Theory
and Applications" in two volumes, written jointly with J. D. Keckic and published
by Kluwer. The first volume, which appeared in 1984, is the only book that covers
all known applications of the calculus of residues. They range from the theory of
equations, theory of numbers, matrix analysis, evaluation of real definite integrals,
summation of finite and infinite series, expansions of functions into infinite series
and products, ordinary and partial differential equations, mathematical and theoretical physics, to the calculus of finite differences and difference equations. On
the other hand, the second volume (appeared in 1993) is devoted to new results
in this field. Also, it contains some special contributions written by various authors and they are based mainly on their own research work. They include topics
as the generalised value of an improper integral, numerical evaluation of definite
integrals, inclusive calculus of residues, polynomials orthogonal on a semicircle in
the complex plane, and an interesting generalisation of the residue theorem to
distribution.
***
The total bibliography of Professor Mitrinovic contains 372 units, including 278
scientific papers and 30 other papers, as weIl as 16 monographs, 35 text-books, and
13 other books (see [5]). There are over 40 scientists who received their doctoral
degrees by Professor Mitrinovic. He enabled his collaborators and doctorands
to use his huge scientific documentation in which he kept old, rare and valuable
papers, systematically collected over the past years, and pedantically arranged into
fields. His collaborators were frequently surprised by his familiarity with references
in topics that were not in his immediate circle of interest. In his last years, he
used to give whole folders of precious papers to his visitors as a present, saying:
"1 do not have any more time tor that".
Professor Mitrinovic devoted his whole life to mathematics. He led a modest
life. His works will remain a long lasting value and will be cited in mathematical
10
G. V. MILOVANOVIC
literature for a long time. He will remain in the memory of his numerous associates
and students as a truly exceptional man they could leam a lot of from.
References
1. M. Bertolino and P. M. Vasic, Professor Dragoslav S. Mitrinovic, Univ. Beograd. Pub!.
Elektrotehn. Fak. Ser. Mat. Fiz. No. 602 - No. 633 (1978), 3-7.
2. B. Crstici, Sur les contributions du Prof. D. S. Mitrinovic
la theorie des inegalites,
Dragoslav S. Mitrinovic - Life and Work, University "Kiril and Metodij" - Faculty of Mathematics - Skopje, Skopje, 1980, pp. 83-97.
3. _ _ _ , About some doctoral thesis directed by Professor Dragoslav S. Mitrinovic in the
domain of functional equations, Scientific Review 21-22 (1996), 15-22.
4. D. Dimitrovski, Dragoslav S. Mitrinovic, Dragoslav S. Mitrinovic - Life and Work, University "Kiril and Metodij" - Faculty of Mathematics - Skopje, Skopje, 1980, pp. 1-12.
(Macedonian)
5. R. Z. Djordjevic and R. R. Janic, Publications of D. S. Mitrinovic, This Volume, pp. 11-27.
6. J. D. Keckic, Contribution of Professor D. S. Mitrinovic to Differential Equations, Univ.
Beograd. Pub!. Elektrotehn. Fak. Ser. Mat. Fiz. No. 602 - No. 633 (1978), 17-46.
7. ___ , Contribution of Professor D. S. Mitrinovic to functional equations, Dragoslav
S. Mitrinovic - Life and Work, University "Kiril and Metodij" - Faculty of Mathematics
- Skopje, Skopje, 1980, pp. 67-82.
8. M. Merkle, IN MEMORIAM - Professor Dragoslav S. Mitrinovic (1908-1995), Univ. Beograd. Pub!. Elektrotehn. Fak. Ser. Mat. 6 (1995), 3-5.
9. G. V. Milovanovic, IN MEMORIAM - Prof. dr Dragoslav S. Mitrinovic, Politika (May 9,
1995 & August 8, 1995).
10. ___ , Dragoslau S. Mitrinovic (1908-1995) - Life and Scientific Work, Scientific Review
21-22 (1996), 1-13.
a
PUBLICATIONS OF D. S. MITRINOVIC
RADOSAV Z. DJORDJEVIC
University 0/ Nis, Faculty 0/ Electronie Engineering, Department 0/ Mathematies,
P. O. Box 79, 18000 Nis, Yugoslavia
RADOVAN R. JANIC
University 0/ Belgrade, Faculty 0/ Eleetrical Engineering, Department 0/ Mathematies,
P. O. Box 95-54, 11120 Belgrade, Yugoslavia
Professor Dragoslav S. Mitrinovic (1908-1995) was active researeher during his
life span for whole 65 years, from 1931 until1995. During that time he wrote by
himself or jointly with other authors 16 monographs, 35 university textbooks (with
many expanded and revised editions), as weH as 13 other important mathematical
books.
In the same time, Professor Mitrinovic published alone or jointly 275 papers in
distinguished scientific journals. He presented 3 papers on the international mathematical congresses and conferences, 30 professional papers, and many ordinary
journalistic papers.
This Bibliography is a complete survey of all published papers by Professor Mitrinovic and consists of two parts: Books and Papers. The section Books contains
the survey of published Monographs, Text-Books, and Other Books, and section
Papers the published Journal Papers, Conference Papers, and Other Papers. The
survey of journalistic papers is not presented.
Taking in consideration the whole creative work of Professor Mitrinovic, the editors
of this Bibliography concluded that Professor Mitrinovic published over 25,000
printed pages of mathematical text, with no counting the reprints. That means,
he wrote during his 65 years long working life span on the average more than one
printed page a day, and even a three pages a day, taking in account the reprints,
that have been, by rule, revised and expanded.
BOOKS
Monographs
1. Nejednakosti. Gradevinska knjiga, Beograd, 1965.
2. (with P. M. Vasic, R. Z. Djordjevic and R. R. Janic) Geometrijske nejednakosti.
Zavod za izdavanje udzbenika, Beograd, 1966.
3. (with O. Bottema, R. Z. Djordjevic, R. R. Janic and P. M. Vasic) Geometrie Inequalities. Wolters - Noordhoff Publishing, Groningen, 1969.
4. (with P. M. Vasic) Sredine. Zavod za izdavanje udzbenika, Beograd, 1969.
5. (with P. M. Vasic) Analytie Inequalities. Springer Verlag, Berlin - Heidelberg - New
York,1970.
11
G. V. Milovanovic (ed.), Recent Progress in lnequalities, 11-27.
© 1998 Kluwer Academic Publishers.
12
R. Z. DJORDJEVIC AND R. R. JANIC
6. (with P. M. Vasic) Analiticke nejednakosti. Gradevinska knjiga, Beograd, 1970.
7. (with M. S. Popadic) Inequalities in Number Theory. Naucni podmladak, Nis, 1978.
8. (with P. S. Bullen and P. M. Vasic) Sredine isa njima povezane nejednakosti. Univ.
Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz. ']\[2600 (1978).
9. (with J. D. Keckic) Cauehy Method 0/ Residues. D. Reidel Publishing Company,
Dordrecht - Boston - Lancaster, 1984.
10. (with P. S. Bullen and P. M. Vasic) Means and Their Inequalities. D. Reidel Publishing Company, Dordrecht - Boston - Lancaster - Tokio, 1988.
11. (with J. E. Pecaric and V. Volenec) Reeent Advanees in Geometrie Inequalities.
Kluwer Academic Publishers, Dordrecht - Boston - London, 1989.
12. (with J. E. Pecaric and A. M. Fink) Inequalities Involving Funetions and Their Integrals and Derivatives. Kluwer Academic Publishers, Dordrecht - Boston - London,
1991.
13. (with J. D. Keckic) The Cauehy Method 0/ Residues. Theory and Applieations. Vol.
2. Kluwer Academic Publishers, Dordrecht - Boston - London, 1993.
14. (with J. E. Pecaric and A. M. Fink) Classieal and New Inequalities in Analysis.
Kluwer Academic Publishers, Dordrecht - Boston - London, 1993.
15. (with G. V. Milovanovic and Th. M. Rassias) Topie in Polynomials: Extremal Problems, Inequalities, Zeros. World Scientific Publishing Co., Singapore - New Jersey
- London - Hong Kong, 1994.
16. (with J. Sandor and B. Crstici) Handbook 0/ Number Theory. Kluwer Academic
Publishers, Dordrecht - Boston - London 1996.
Text-Books
1. Matematicki problemi. Sveska prva. Stamparija "Davidovic" Pavlovica i druga,
Beograd, 1931.
2. Zbirka zadataka iz matematike za studente tehnickih i prirodno-matematickih /akulteta. Znanje, Beograd, 1954.
3. Zbornik matematickih problema sa prilozima i numerickim tablieama, I. Nolit, Beograd, 1957. [New editions: 1958, 1962]
4. Metod matematicke indukeije. Nolit, Beograd, 1957. [Newedition: 1958]
5. (with D. Mihailovic and J. Ulcar) Zbornik matematickih problema sa prilozima i
numerickim tablicama, II. Naucna knjiga, Beograd, 1958. [Newedition: 1960]
6. (with D. Mihailovic and P. M. Vasic) Linearna algebra. Polinomi. Analiticka geometrija. Gradevinska knjiga, Beograd, 1959. [New editions: 1962, 1966, 1968,
1971, 1973, 1975, 1978, 1979, 1983, 1985, 1988. 1990]
7. (with J. Ulcar and V. Devide) Zbornik matematickih problema sa prilozima i numerickim tablicama, III. Naucna knjiga, Beograd, 1960.
8. (with J. Ulcar, P. Dimik and I. Sapkarev) Zbirka zadaci po visa matematika za
studenti na tehnicki /akulteti. Univerzitet - Skopje, Skopje, 1961.
9. Zbirka zadataka iz matematike za prvi stepen nastave na /akultetima. N aucna knjiga,
Beograd, 1962.
10. Matematicka indukeija. Binomna /ormula. Kombinatorika. Zavod za izdavanje
udzbenika, Beograd, 1963. [New editions: 1970, 1980, 1990]
11. Matematika za prvi stepen nastave na /akultetima u obliku metodicke zbirke zadataka
sa resenjima. Gradevinska knjiga, Beograd, 1964.
12. (with D. Z. Dokovic) Speeijalne Junkeije. Gradevinska knjiga, Beograd, 1964.
13. (with E. S. Barnes, D. C. B. Marsh and J. R. M. Radok) Elementary Inequalities.
Publishing Noordhoff Ltd., Groningen, 1964.
PUBLICATIONS OF D. S. MITRINOVIC
13
14. (with E. S. Barnes and J. R. M. Radok) Functions of a Complex Variable. Publishing
Noordhoff Ltd., Groningen, 1965.
15. (with R. B. Potts) Elementary Matrices. Publishing Noordhoff Ltd., Groningen,
1965.
16. (with J. H. Michael) Calculus of Residues. Publishing Noordhoff Ltd., Groningen,
1966.
17. (with D. Z. Dokovic) Polinomi i matrice. Gradevinska knjiga, Beograd, 1966. [New
editions: 1975, 1986, 1991]
18. Matematika u obliku metodicke zbirke zadataka sa resenjima, 1. Gradevinska knjiga,
Beograd, 1967. [New editions: 1971, 1973 , 1978, 1982, 1986, 1989]
19. Matematika u obliku metodicke zbirke zadataka sa resenjima, 11. Gradevinska knjiga,
Beograd, 1967. [New editions: 1972, 1977, 1982, 1987, 1989]
20. Kompleksna analiza. Gradevinska knjiga, Beograd, 1967. [Neweditions: 1971, 1973,
1977, 1981, 1988]
21. (with J. VIcar and R. S. Anderssen) Differential Geometry. Wolters - Noordhoff
Publishing, Groningen, 1969.
22. (with J. D. Keckic) Algebra - Zbirka problema iz kombinatorike, polinoma i jednacina. Naucna knjiga, Beograd, 1969.
23. (with D. D. Adamovic) Nizovi i redovi - Definicije, stavovi, zadaci, problemi. Naucna knjiga, Beograd, 1971. [New editions: 1980, 1987, 1990]
24. (with R. R. Janic) Uvod u specijalne junkcije. Gradevinska knjiga, Beograd, 1972.
[Neweditions: 1975, 1986]
25. Matematika u obliku metodicke zbirke zadataka sa resenjima, 111. Gradevinska knjiga, Beograd, 1972. [New editions: 1976, 1984, 1988J
26. (with D. D. Tosic and R. R. Janic) Specijalne junkcije - Zbomik zadataka i problema.
Naucna knjiga, Beograd, 1972. [Neweditions: 1978, 1986, 1990]
27. Matrice i determinante - Zbomik zadataka i problema. Naucna knjiga, Beograd,
1972. [New editions: 1975, 1980, 1986, 1990]
28. (with J. D. Keckic) Kompleksna analiza - Zbomik zadataka i problema. Naucna
knjiga, Beograd, 1972. [Neweditions: 1979, 1985, 1989]
29. (with P. M. Vasic) Diferencijalne jednacine - Zbomik zadataka i problema. Naucna
knjiga, Beograd, 1972. [New editions: 1978, 1986, 1990]
30. (with J. D. Keckic) Jednacine matematicke fizike. Gradevinska knjiga, Beograd,
1972. [New editions: 1978, 1985]
31. Predavanja 0 redovima. Gradevinska knjiga, Beograd, 1974. [New editions: 1980,
1986, 1989]
32. Predavanja 0 diferencijalnim jednacinama. Minerva, Subotica - Beograd, 1976.
[New edition: 1983]
33. (with J. D. Keckic) Matematika 11 - Redovi, diferencijalne jednacine, kompleksna
analiza, Laplaceova transformacija. Gradevinska knjiga, Beograd, 1981. [New editions: 1987, 1989]
34. (with J. D. Keckic) Complex Analysis. Exercises and Problem Manual. Naucna
knjiga, Beograd, 1990.
35. (with D. D. Tosic) Matematika u obliku metodicke zbirke zadataka sa resenjima, IV.
Gradevinska knjiga, Beograd, 1987. [New edition: 1990]
Other Books
1. Savremene tendencije u nastavi matematike. Nolit, Beograd, 1957.
2. Referati 0 srednjoskolskim udibenicima iz matematike. Nolit, Beograd, 1957.
3. Vainije nejednakosti. Nolit, Beograd, 1958.
14
R. Z. DJORDJEVIC AND R. R. JANIC
4. (with D. C. B. Marsh) Problemi iz elementame teorije brojeva. Zavod za izdavanje
udzbenika, Beograd, 1966.
5. (with P. M. Vasic and R. R. Janic) Elementame nierownosci. PaIistowe wydawnictwo naukowe, Warszawa, 1972.
6. (with J. D. Keckic) Gauchyev racun ostataka sa primenama. Naucna knjiga, Beograd, 1978. [Newedition: 1991.)
7. (with J. D. Keckic) Metodi izracunavanja konacnih zbirova. Naucna knjiga, Beograd, 1984. [New edition 1990.)
8. (with J. E. Pecaric) Diferencijalne i integraine nejednakosti. Naucna knjiga, Beograd, 1988.
9. (with J. E. Pecaric) Srednje vrednosti u matematici. Naucna knjiga, Beograd, 1989.
10. (with J. E. Pecaric) Hölderova i srodne nejednakosti. Naucna knjiga, Beograd, 1990.
11. (with J. E. Pecaric) Monotone Junkcije i njihove nejednakosti. Naucna knjiga,
Beograd, 1990.
12. (with J. E. Pecaric) Giklicne nejednakosti i ciklicne Junkcionalne jednacine. Naucna
knjiga, Beograd, 1991.
13. (with J. E. Pecaric) Nejednakosti i norme. Naucna knjiga, Beograd, 1991.
PAPERS
Journal Papers
1. Novi slucaji integrabiliteta jedne diferencijalne jednacine prvog reda. Glas Srpske
Akademije 154 (1933), 145-170. [Nouveaux cas d'integrabilite d'une equation differentielle du premier ordre. BuH. Aead. Sei. Math. Nat. Belgrade 1 (1933), 107-117.]
2. Sur les lignes geodesiques d'une classe des surfaces. Publ. Math. Univ. Belgrade
3 (1934), 167-170.
3. Remarque sur une equation differentielle du premier ordre. Publ. Math. Univ. Belgrade 3 (1934),171-174.
4. Sur l'equation differentielle des lignes asymptotiques. Publ. Math. Univ. Belgrade
3 (1934),175-178.
5. Novi integrabilni oblici jedne znacajne diferencijalne jednacine prvog reda. Glas
Srpske Akademije 163 (1934), 47-55. [Nouvelles form es integrables d'une equation
differentielle importante du premier ordre. BuH. Aead. Sei. Math. Nat. Belgrade,
2 (1935), 61-65.]
6. Investigations 01 an differential equation of the first order. Ph. D. Thesis, Beograd,
1935. [Defended: Oetober 24, 1933]
7. 0 diferencijalnoj jednacini ravnih krivih, ciji je luk data funkcija potega i polamog
ugla. Glas Srpske Akademije 165 (1935), 155-161. [Sur l'equation differentielle
des curbes planes dont l'arc est une fonction donnee des coordonnee polaires. BuH.
Aead. Sei. Math. Nat. Belgrade 2 (1935), 245-246.]
8. Prilog integraljenju izvesne klase algebarskih diferencijalnih jednacina prvog reda.
Glas Srpske Akademije 165 (1935), 165-170. [Gas d'integrabilite d'une certaine
classe d'equations differentielles algebriques du premier ordre. BuH. Aead. Sei. Math.
Nat. Belgrade 2 (1935), 247-248.]
9. Gontribution a l'integration de l'equation differentielle de J. Liouville. Publ. Math.
Univ. Belgrade 4 (1935), 149-152.
10. Sur certaines trajectoires algebriques planes de genre zero, un et deux. Publ. Math.
Univ. Belgrade 4 (1935), 153-160.
11. Remarques sur les lignes asymptotiques et sur lignes de courbure. Prak. Aead.
Athenön 10 (1935), 480-483.
PUBLICATIONS OF D. S. MITRINOVIC
15
12. Parabole ci parametre rationnel. Mathesis, 49 (1935), 369.
13. Novi oblik Lagrange-Serretove primedbe 0 diferencijalnim jednacinama. Glas Srpske
Akademije 110 (1936),369-179. [Nouvelle forme de la remarque de Lagrange-Serret
relative aux equations differentielles ordinaires. BuH. Aead. Sei. Math. Nat. Belgrade 3 (1936), 37-39.]
14. Prilog teoriji prvih integrala diferencijalnih jednacina. Glas Srpske Akademije 113
(1936), 19-22. [Contriboution ci la theorie des integrales premieres d'equations
differentielles. BuH. Aead. Sei. Math. Nat. Belgrade 3 (1936), 33-35.]
15. 0 integraciji jedne vaine diferencijalne jednacine prvoga reda. Glas Srpske Akademije 113 (1936),77-117. [Sur l'integration d'une equation differentielle importante
du premier ordre. BuH. Aead. Sei. Math. Nat. Belgrade 3 (1936), 7-19.]
16. Transformation et integration d'une equation differentielle du premiere ordre. Publ.
Math. Univ. Belgrade 5 (1936), 10-12.
17. Sur les lignes de courbure des surfces reglees ci plan directeur. Publ. Math. Univ.
Belgrade 5 (1936), 100-102.
18. Un problem sur les fonctions analytiques. Rev. Math. Union Interbalkan. 1 (1936),
53-57.
19. Equation differentielle des asymptotiquaes et equation des cordes vibrantes qui s 'y
rattache. Rev. Math. Union Interbalkan. 1 (1936), 135-137.
20. Remarque sur les surfaces de translation. Prak. Aead. Athenön 11 (1936), 356-359.
21. Sull'integrazione dell'equatione differenziale del tipo di Abel. Rend. Reale Istit. Lombardo sei. lett. (2) 69 (1936), 203-208.
22. Asymptotiques d'une classe des surfaces. Bull. Aead. Royal Belgique (5) 22 (1936),
948-950.
23. Asimptotiques d'une classe des surfaces et equations differentielles lineaires du second ordre s'y rattachant. BuH. Aead. Royal Belgique (5) 22 (1936), 1047-1049.
24. Sur l'emploi de la partie reelle et de la partie imagunaire des fonctions analytiques
dans l'integration des equations differentielles. Töhoku Math. J. 42 (1936),179-184.
25. Theoreme sur les lignes asymptotiques. Mathesis 50 (1936), 367-368.
26. Integration d'une equation differentielle du premier ordre et polynomes d'Hermite
qui s'y rattachent. Rev. Ciene. (Lima) 38 (1936), 123-127.
27. Sur une equation differentielle du premier ordre intervenant dans divers problemes
de geometrie. C. R. Aead. Sei. Paris 204 (1937), 1706-1708.
28. Sur l'equation differentielle des lignes geodesiques des surfaces spirales. C. R. Aead.
Sei. Paris 205 (1937), 1194-1196.
29. Un probleme sur les lignes asymptotiques et la methode de l'integration logique des
equations differentielles de Jules Drach. C. R. Aead. Sei. Paris 205 (1937), 13581360.
30. Sur une equation differentielle du premier ordre intervenant divers problemes de
geometrie. Bull. Sei. Math. (2) 61 (1937),323-325.
31. Un probleme sur les lignes asymptotiques d'une classe de surfaces. BuH. Aead.
Royal Belgique (5) 23 (1937), 378-380.
32. (with R. Godeau) Sur certaines surfaces dont les lignes asymptotiques se determinent par quadratures. Mathesis 51 (1937), 115-116.
33. Istraiivanja 0 asimptotskim linijama povrsina. Glas Srpske Akademije 175 (1937),
45-69. [Recherches sur les lignes asymptotiques. Bull. Aead. Sei. Math. Nat. Belgrade 4 (1938), 105-120.]
34. Sur l'equation differentielle des lignes de courbure. Publ. Math. Univ. Belgrade
6-1 (1938), 32-35.
16
R. Z. DJORDJEVIC AND R. R. JANIC
35. Theoremes relatifs d l'equation differentiell de Riccati. C. R. Aead. Sei. Paris
206 (1938), 411-413.
36. Problem geometriques ou interviennent diverses equations differentielles. C. R.
Aead. Sei. Paris 206 (1938), 568-570.
37. Sur une formule d'Analyse. Rev. Ciene. (Lima) 40 (1938), 449-452.
38. Sur une slasse d'equations differentielles. BuH. Sei. Math. (2) 62 (1938), 36-41.
39. Sur une probleme de Darboux. BuH. Seet. Sei. Aead. Roumaine 20 (1938), 23-25.
40. Abelove diferencijalne jednaeine viseg reda. Glas Srpske Akademije 178 (1939), 4547. [Equations differentielles d'Abel d'ordre superier. BuH. Aead. Sei. Math. Nat.
Belgrade 5 (1939), 25-31.]
41. Problem 0 asimptotskim linijama pravolinijskih povrsina eije resenje zavisi od Riccatieve diferencijalne jednaeine. Glas Srpske Akademije 178 (1939), 161-165. [Problem, dont la solution depend d 'une equation de Riccati, relatif aux asymptotiques
d'une surface reglee. BuH. Aead. Sei. Math. Nat. Belgrade 5 (1939), 89-92.]
42. 0 jednoj klasi diferencijalnih jednaeina prvoga reda na koje se nailazi u raznim problemima geometrije. Glas Srpske Akademije 181 (1939), 133-168. [Sur une classe
d'equations differentielles du premier ordre que l'on rencontre dans divers problem
de Geometrie. BuH. Aead. Sei. Math. Nat. Belgrade 6 (1939), 99-120.]
43. Nekoliko stavova 0 Riccatievoj diferencijalnoj jednaeini. Glas Srpske Akademije
181 (1939), 171-236. [Quelques propositions relatives d l'equation differentielle de
Riccati. BuH. Acad. Sei. Math. Nat. Belgrade 6 (1939), 121-156.]
44. Theoreme sur l'equation de Riccati. C. R. Aead. Sei. Paris 208 (1939), 156-157.
45. Sur le probleme de Beitrami: deformer une surface reglee de telle manier que l'une
de ses courbes, assignee d l'avance, devienne planne. BuH. Sei. Math. (2) 63 (1939),
99-105.
46. Sur l'integration d'une equation lineaire aux derivees partielles. C. R. Aead. Sei.
Paris 210 (1940), 783-785.
47. Remarque sur certaines equations aux derivees partielles. Atti Reale Istit. Veneto
Sei. Lett. Arti 99 (1940), parte 11, 357-360.
48. Veza izmeilu diferencijalne jednaeine drugog reda i jedne linearne integraine jednaeine tipa Volterra. Glas Srpske Akademije 185 (1941), 281-288. [Correspondance
entre l'equation differentielle du second ordre et une equation integrale de Volterra.
BuH. Aead. Sei. Math. Nat. Belgrade 7 (1941), 191-195.]
49. 0 jednoj linearnoj parcijalnoj jednaeini. Glasnik Mat.-Fiz. Astr. 1(1946), 168-181
& 209-226.
50. Sur un procede fournissant des solutions d'une equation aux differences finies rattachee d la theorie des coefficients de Stirling. BuH. Aead. Royal Belgique (5) 33
(1947), 244-247.
51. Sur une classe d'equation differentielles d'ordre superier. BuH. Aead. Royal Belgique (5) 33 (1947), 521-526.
52. Apropos d'une Note de D. Pompeiu relative d l'equation de Riccati. BuH. Seet. Sei.
Aead. Roumaine 30 (1947), 256-263.
53. 0 Stirlingovim brojevima. Fac. Philos. Univ. Skopje. Seet. Sei. Nat. Annuaire
1 (1948), 49-89.
54. 0 transformaciji jedne diferencijalne jednaeine. Fac. Philos. Univ. Skopje. Seet. Sei.
Nat. Annuaire 1 (1948), 97-109.
55. Sur une equation differentieie lineaire du second ordre transformable en elle-meme.
C. R. Aead. Sei. Paris 228 (1949), 1188-1190.
56. 0 jednoj determinanti Escherichova tipa. Fae. Philos. Univ. Skopje. Seet. Sei. Nat.
Annuaire 2 (1949), 135-139.
PUBLICATIONS OF D. S. MITRINOVIC
17
57. 0 algebarskim iracionalnim jednacinama. Fae. Philos. Univ. Skopje. Seet. Sei. Nat.
Annuaire 2 (1949), 141-159.
58. 0 jednoj klasi Riccatievih jednacina koje su invarijantne u odnosu na jednu smenu
funkeije. Fae. Philos. Univ. Skopje. Seet. Sei. Nat. Annuaire 2 (1949), 165-182.
59. 0 jednoj diferencijalnoj jednacini drugoga reda koja se pojavljuje u jednom problemu
matematicke fizike. Fae. Philos. Univ. Skopje. Seet. Sei. Nat. Annuaire 2 (1949),
187-193.
60. Postupak za formiranje kriterijuma integrabiliteta linearnih diferencijalnih jednacina ciji koeficijenti imaju oblike unapred date. Fae. Philos. Univ. Skopje. Seet. Sei.
Nat. Annuaire 2 (1949), 207-237.
61. Sur un cas de reductibilite d 'equations dijerentielles lineaires. C. R. Aead. Sei. Paris
230 (1950), 1130-1132.
62. Mise en correspondance d 'un probleme non resolu de theorie de I'elasticite avec un
probleme resolu par Darboux et Drach. C. R. Aead. Sei. Paris 231 (1950), 327-328.
63. Sur un procede fournissant des equations differentielles lineaires integrabIes d'un
type assigne d'avance. Aead. Serbe. Sei. Pub!. Inst. Math. 3 (1950), 227-234.
64. Primedba 0 determinantama Escherichova tipa. Bull. Soe. Math. Phys. Maeedoine
1(1950), 5-20.
65. (with I. Vidav) 0 jednoj diferencijalnoj jednacini. Bull. Soe. Math. Phys. Maeedoine
1 (1950),21-27.
66. Povodom Görtlerovih rezultata 0 linearnoj diferencijalnoj jednacini drugoga reda.
Fae. Philos. Univ. Skopje. Seet. Sei. Nat. Annuaire 3 (1) (1950), 1-19.
67. 0 operacijama max imin. Fae. Philos. Univ. Skopje. Seet. Sei. Nat. Annuaire
3 (4) (1950),1-10.
68. 0 diferencijalnoj jednacini jednog vainog problema teorije i prakse elasticiteta. Fae.
Philos. Univ. Skopje. Seet. Sei. Nat. Annuaire 3 (5) (1950), 1-22.
69. 0 jednoj neodreaenoj diferencijalnoj jednacini. Fae. Philos. Univ. Skopje. Seet. Sei.
Nat. Annuaire 3 (6) (1950), 1-16.
70. Sur une propriete des operations max et min. C. R. Aead. Sei. Paris 232 (1951),
286-287.
71. Sur une equation differentielle indeterminee intervenant dans un problem important
de l'Elasticite. C. R. Aead. Sei. Paris 232 (1951),681-683.
72. Sur certaines relations de l'algebre des ensembles. C. R. Aead. Sei. Paris 232 (1951),
617-918.
73. Sur un procede d'integration d'une equation de Monge. C. R. Aead. Sei. Paris
232 (1951), 1334-1336.
74. Tre6a metoda integracije Nemenyi-Truesdellove jednacine. Bull. Soe. Math. Phys.
Maeedoine 2 (1951), 17-20.
75. Sur I 'equation differentielle d 'un problem de K uhelj. Bull. Soe. Math. Phys. Maeedoine 2 (1951), 31-34.
76. Sur la solution de Ribaud de I 'equation de Fourier. Bull. Soe. Math. Phys. Maeedoine 2 (1951),105-107.
77. Sur equation differentielle de Laplace. Bull. Soe. Math. Phys. Maeedoine 2 (1951),
109-112.
78. Sur un operateur differentiel. La Revue seientifique, Paris 89 (1951), 44.
79. On an equation of Nemenyi and Truesdell. J. Washington Aead. Sei. 41 (1951), 123.
80. Sur une equation differentielle indeterminee du second ordre. Bull. Aead. Royal
Belgique (5) 37 (1951), 227-228.
81. Sur une equation fonctionnelle. C. R. Aead. Sei. Paris 237 (1953), 550-551.
18
R. Z. DJORDJEVIC AND R. R. JANIC
82. Sur une equation differentielle du premier ordre. Jber. Deutsch. Math.-Verein. Abt.
2, 58 (1955), l.
83. Sur l'equation differentielle d'Emden generalisee. C. R. Acad. Sci. Paris 241 (1955),
724-726.
84. Sur l'equation differentielle d'un problem d'Hydrodynamique. C. R. Acad. Sci. Paris
241 (1955), 1708-1710.
85. Sur l'equation differentielle d'un problem de teehnique etudie par R. Gran Olsson.
Norske Vid. Selsk. Forh. (Trondheim) 28 (1955), 171-175.
86. Sur le determinant de Stern generalise. Bull. Soc. Math. Phys. Serbie 1 (1955),
153-160.
87. Complements au Traite de Kamke. Note lI. Bull. Soc. Math. Phys. Serbie 1 (1955),
161-164.
88. 0 nekim neodreaenim difereneijalnim jednacinama. Bull. Soc. Math. Phys. Serbie
7(1955),171-178.
89. Complements au Traite de Kamke. Note I. Jber. Deutsch. Math.-Verein. Abt. II
58 (1956), 58-60.
90. Neke formule 0 apsolutnim vrednostima realnih brojeva. Bull. Soc. Math. Phys.
Macedoine 7 (1956), 39-41.
91. Sur l'equation differentielle de Somerfeld pour la stabilite hydrodynamique. C. R.
Acad. Sci. Paris 242 (1956), 2287-2289.
92. Nouvelles formules relatives aux polynomes de Legendre. C. R. Acad. Sei. Paris
243 (1956), 1387-1389.
93. Su un determinante e sui numeri di Stirling ehe vi si eollegano. Boll. Uno Mat. Ital.
(3) 11 (1956), 93-96.
94. Complements au Traite de Kamke. Note III. Boll. Uno Mat. Ital. (3) 11 (1956), 168171.
95. Inegalites po ur derivees des polynomes de Legendre. Boll. Uno Mat. Ital. (3) 11
(1956),172-177.
96. Probleme sur les progressions arithmetiques. Boll. Uno Mat. Ital. (3) 11 (1956), 256257.
97. Complements au Traite de Kamke. Note IV. Glasnik Mat.-Fiz. Astr. 11 (1956),
7-10.
98. Sur eertaines equations aux derivees partielles cl deux fonctions ineonnues. Bull.
Soc. Math. Phys. Serbie 8 (1956), 3-6.
99. Sur eertaines relations restant valables si I 'on permute les operateurs y intervenant.
Bull. Soc. Math. Phys. Serbie 8 (1956), 15-22.
100. Neke formule koje se odnose na Legendreove polinome. Univ. Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz. N~ 1 (1956), 1-20. [Some formulas eoneerning the Legendre polynomials. National Bureau of Standards, Boulder, Colorado 1960, 27pp.]
101. Sur un proeede fournissant des equations fonetionnelles dont les solutions eontinues
et differentiables peuvent etre determinees. Univ. Beograd. Publ. Elektrotehn. Fak.
Sero Mat. Fiz. N~ 5 (1956), 1-8.
102. Sur une question d'analyse diophantienne. Univ. Beograd. Publ. Elektrotehn. Fak.
Sero Mat. Fiz. N~ 6 (1956), 1-4.
103. Sur quelques formules somatoires. Univ. Beograd. Publ. Elektrotehn. Fak. Sero Mat.
Fiz. N~ 7 (1956), 1-8.
104. Sur une demonstration dans l'algebre de Dubreil. Univ. Beograd. Publ. Elektrotehn.
Fak. Sero Mat. Fiz. N~ 10 (1956), 1-3.
105. Complements au Traite de Kamke. Note V. Univ. Beograd. Publ. Elektrotehn. Fak.
Sero Mat. Fiz. N~ 11 (1957), 1-10.
PUBLICATIONS OF D. S. MITRINOVIC
19
106. Sur une equation lineaire aux derivees partielles a coetJicients constants. Math. Gaz.
41 (1957), 41-43.
107. Formulles sur valeurs absolues des nombre reels. Eiern. Math. 12 (1957), 111-112.
108. (with R. S. Mitrinovic) Sur les polynomes de Stirling. BuH. Soe. Math. Phys. Serbie
10 (1958), 43-49.
109. 0 Stirlingovim brojevima prve vrste i Stirlingovim polinomima. Univ. Beograd.
Pub!. Elektrotehn. Fak. Sero Mat. Fiz. J\{2 23 (1959), 1-19.
110. 0 Macmillanovoj modifikaciji Gauss-Chioovog postupka za izracunavanje determinanata. Univ. Beograd. Pub!. Elektrotehn. Fak. Sero Mat. Fiz. J\{2 25 (1959), 1-8.
111. Complements au 'Jlraite de Kamke. VI. Univ. Beograd. Pub!. Elektrotehn. Fak. Sero
Mat. Fiz. J\{2 27 (1959), 1-4.
112. 0 nekim nejednakostima. Univ. Beograd. Pub!. Elektrotehn. Fak. Sero Mat. Fiz.
J\{ 2 29 - J\{2 32 (1959), 1-4.
113. Sur les nombres de Bernoulli d'ordre superieur. BuH. Soe. Math. Phys. Serbie
11 (1959), 23-26.
114. Primedba i problem 0 jednoj linearnoj diferenccijalnoj jednacini. BuH. Soe. Math.
Phys. Serbie 11 (1959), 213-214.
115. Nouvelles formules relatives aux nombres de Stirling. C. R. Aead. Sei. Paris 248
(1959), 1754-1756.
116. A sumation formula. Math. Gaz. 43 (1959), 44.
117. A theorem on prime numbers. Math. Gaz. 43 (1959), 125.
118. Equivalence of two sets of inequalities. Math. Gaz. 43 (1959), 126.
119. Problem sur les progressions arithmetiques. Math. Gaz. 43 (1959), 126.
120. (with R. S. Mitrinovic) Tableaux qui fournissent des polynomes de Stirling. Univ.
Beograd. Pub!. Elektrotehn. Fak. Sero Mat. Fiz. J\{2 34 (1960), 1-23.
121. (with R. S. Mitrinovic) Sur le nombres de Stirling et les nombres de Bernoulli d'ordre
superieur. Univ. Beograd. Pub!. Elektrotehn. Fak. Sero Mat. Fiz. N2 43 (1960), 163.
122. Sur une formule concernant les nombres de Bernoulli d'odre superieur. BuH. Soe.
Math. Phys. Serbie 12 (1960), 21-23.
123. Equation algebriques a parametres. BuH. Soe. Math. Phys. Serbie 12 (1960), 25-26.
124. (with D. Z. Dokovic) Sur une relation de recurrence concernant les nombres de
Stirling. C. R. Aead. Sei. Paris 250 (1960), 2110-2111.
125. Sur une relation de recurrence relative aux nombres de Bernoulli d'ordre superier.
C. R. Aead. Sei. Paris 250 (1960), 4266-4267.
126. (with K. Slipicevic) Sur lequation d'Emden. Mathesis 69 (1960), 74-75.
127. Problemes sur une equation differentielle. Mathesis 69 (1960), 223-224.
128. Une hupothese sur les nombres de Stirling de premiere espece. Mathesis 69 (1960),
334-336.
129. (with D. Z. Dokovic) Sur une classe d'equations fonctionnelles cycliques. C. R.
Aead. Sei. Paris 252 (1961), 1090-1092.
130. (with D. Z. Dokovic) Sur une classe itendue d'equations fonctionnelles. C. R. Aead.
Sei. Paris 252 (1961),1717-1718.
131. Sur une classe de nombres relies aux nombres de Stirling. C. R. Aead. Sei. Paris
252 (1961), 2354-2356.
132. (with D. Z. Dokovic) Sur quelques equations fonctionnelles. C. R. Aead. Sei. Paris
252 (1961), 2982-2984.
133. (with D. Z. Dokovic) Sur certaines equations fonctionnelles. Univ. Beograd. Pub!.
Elektrotehn. Fak. Sero Mat. Fiz. J\{2 51-N2 54 (1961), 9-16.
20
R. Z. DJORDJEVIC AND R. R. JANIC
134. (with R. S. Mitrinovic) Bur une classe de nombres se rattachant aux nombres de
Btirling. (Appendice: Table des nombres de Btirling.) Univ. Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz. N~ 60 (1961), 1~62.
135. (with D. Z. DokoviC) Bur certaines equations fonctionnelles dont les solutions peuvent etre determinees. Univ. Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz. N~ 61
~N~ 64 (1961), 1~11.
136. (with D. Z. Dokovic) Bur quelques equations fonctionnelles. Publ. Inst. Math. Belgrade 1 (15) (1961), 67~73.
137. (with D. Z. Dokovic) Bur un operateur se rattachant dune classe d 'equations fonctionnelles. Publ. Inst. Math. Belgrade 1 (15) (1961), 75~80.
138. (with S. B. Presic) Bur une equation fonctionnelle cyclique d'odre superieur. Univ.
Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. N~ 10~N~16 (1962), 1~2.
139. (with S. B. Presic) Une classe d'equations fonctionnelles homogenes du second degre.
Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. N~ 10~N~16 (1962), 3~6.
140. (with R. S. Mitrinovic) Tableaux d'une classe de nombres relies au nombres de Btirling. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. N~ 11 (1962), 1~77.
141. Dopune Kamkeovom delu. VII. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat.
Fiz. N~ 18~N~83 (1962), 16~18.
142. (with S. B. Presic) Bur une equation fonctionnelle cyclique non lineaire. C. R. Aead.
Sei. Paris 254 (1962), 611 ~613.
143. (with D. Z. Dokovic) Propriete d'une matrice cyclique et ses applications dune
equation fonctionnelle. C. R. Aead. Sei. Paris 255 (1962), 3109~3110.
144. (with D. Z. Dokovic) Propriete d'une matrice cyclique et ses applications. Publ.
Inst. Math. Belgrade 2 (16) (1962), 53~54.
145. Bur une inegalite algebrique. Univ. Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz.
N~ 84~N~ 91 (1963), 3~7.
146. Bur une note de Co§nitif, relative aux trajectoires isogonales des famillies de cercles.
Univ. Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz. N~ 84~N~ 91 (1963),8.
147. Comptements au Traite de Kamke. VIII. Univ. Beograd. Publ. Elektrotehn. Fak.
Ser. Mat. Fiz. N~ 84~N~ 91 (1963), 19~20.
148. (with D. Z. Dokovic) Certaines inegalites ou intervient la fonction puissance. Univ.
Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz. N~ 100 (1963), 1~1O.
149. Jedan jednostavan postupak za odreaivanje osa simetrije i metrickih elemenata konusnih preseka. Univ. Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz. N~ 101 ~
N~ 106 (1963), 9~13.
150. (with D. D. Adamovic and D. Z. Dokovic) Formule de decomposition d'une fraction
rationnelle en elements simples suivie de quelques applications. Univ. Beograd.
Publ. Elektrotehn. Fak. Sero Mat. Fiz. N~ 101 ~ N~ 106 (1963), 17~20.
151. (with R. S. Mitrinovic) Tableaux d'une classe de nombers relies aux nombers de
Btirling. II. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. N~ 101~N~ 108
(1963), 1~77.
152. (with D. Z. Dokovic) Complements au Traite de Kamke. IX. Univ. Beograd. Publ.
Elektrotehn. Fak. Ser. Mat. Fiz. N~ 101 ~N~ 108 (1963), 78~79.
153. Bur les lignes asymptotiques. Univ. Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz.
N~ 115~N~ 121 (1963), 1~4.
154. Bur certaines equations fonctionnelles lineaires d plusieurs fonctions inconnues.
Univ. Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz. N~ 115~N~ 121(1963), 5~12.
PUBLICATIONS OF D. S. MITRINOVIC
21
155. Equation /onctionnelle Li /onctions inconnues dont toutes ne dependent pas du meme
nombre d'arguments. Univ. Beograd. Pub!. Elektrotehn. Fak. Sero Mat. Fiz. ]\f!? 115
- ]\f!? 121 (1963), 29-30.
156. (with P. M. Vasic and S. B. Presic) Sur une equation /onctionnelle du second degre.
Pub!. Inst. Math. Belgrade 3 (17) (1963), 57-60.
157. (with P. M. Vasic) Gomplements au TI-aite de Kamke. X. Pub!. Inst. Math. Belgrade
3 (17)(1963), 61-68.
158. (with P. M. Vasic) Quelques equations /onctionnelles cycliques non lineaires Li proprietes curieuses. Pub!. Inst. Math. Belgrade 3 (17)(1963), 105-114.
159. Equations /onctionnelles lineaires paracycliques de premiere espece. Pub!. Inst.
Math. Belgrade 3 (17) (1963), 115-128.
160. (with D. Z. Dokovic) Sur une equation /onctionnelle. C. R. Aead. Sei. Paris 257
(1963), 2388-2391.
161. Equation /onctionnelle cyclique generalisee. C. R. Aead. Sei. Paris 257 (1963),29512952.
162. Sur les equations fonctionnelles lineaires paracycliques de seconde espece. Glasnik
Mat.-Fiz. Astr. 18 (1963), 177-182.
163. (with Z. R. Pop-Stojanovic) About integrals expressible in terms 0/ hyperelliptic
integrals. Glasnik Mat.-Fiz. Astr. 18 (1963), 235-239.
164. (with S. B. Presic and P. M. Vasic) Sur deux equations fonctionnelles cycliques non
lineaires. BuH. Soe. Math. Phys. Serbie 15 (1963), 3-6.
165. Formule exprimant les nombres de Gotes Li l'aide de nombres de Stirling. BuH. Soe.
Math. Phys. Serbie 15 (1963), 13-16.
166. Sur une equation fonctionnelle binome. C. R. Aead. Sei. Paris 258 (1964), 55775580.
167. (with P. M. Vasic) 0 jednoj ciklicnoj homogenoj funkcionalnoj jednacini drugoga
reda. Mat. Vesnik 1 (16) (1964), 1-7.
168. Sur un critere po ur determiner le rang d'une matrice. Mat. Vesnik 1 (16) (1964),
50-51.
169. Sur une formule concernant les derivees des polynomes de Legendre. Mat. Vesnik
1 (16) (1964), 51.
170. (with P. M. Vasic) Gomplements au TI-aite de Kamke. XI. Mat. Vesnik 1 (16) (1964),
181-185.
171. Equation fonctionnelle cyclique generalisee. Pub!. Inst. Math. Belgrade 4(18)(1964),
29-41.
172. (with P. M. Vasic) Equations fonctionnelles lineaires generalisees. Pub!. Inst. Math.
Belgrade 4 (18) (1964), 63-76.
173. A simple procedure for the determination 0/ the axes of symetry and metrical elements of the conics. BoH. Uno Mat. !tal. 19 (3) (1964), 208-215.
174. (with R. S. Mitrinovic) Tableaux d 'une classe de nombers relies aux nombers de
Stirling. V. Univ. Beograd. Pub!. Elektrotehn. Fak. Sero Mat. Fiz. ]\f!?132-]\f!?142
(1965), 1-22.
175. Sur deux quastions de priorite relatives aux nombres de Stirling. Univ. Beograd.
Pub!. Elektrotehn. Fak. Sero Mat. Fiz. ]\f!? 132 - ]\f!?142 (1965), 23-24.
176. Gongruence ou interviennent des polynomes homogenes. Univ. Beograd. Pub!. Elektrotehn. Fak. Sero Mat. Fiz. ]\f!? 143-]\f!? 155 (1965), 1-2.
177. Limitations en module d'une fonctions homographique sur un cercle. Univ. Beograd.
Pub!. Elektrotehn. Fak. Sero Mat. Fiz. ]\f!? 143 -]\f!? 155 (1965), 3-4.
178. Inegalites impliquees par le systeme des egalites a+b+c = p, bc+ca+ab = q. Univ.
Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz. ]\f!? 143 -]\f!? 155 (1965), 5-7. [On
22
R. Z. DJORDJEVIC AND R. R. JANIC
a system of equalities and inequalities. Math. Gaz. 49 (1965), 228-229.]
179. (with D. Z. Dokovic) Note bibliographique sur une formule relative aux fonctions
de Legendre. Univ. Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz. J'{.'? 143J'{.'?155 (1965), 13-15.
180. (with D. D. Adamovic) Sur une inegalite elementaire ou interviennent des fonctions
trigonometriques. Univ. Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz. J'{.'? 143J'{.'? 155 (1965), 23-34.
181. Inequalities of R. Rado type for weighted means. Publ. Inst. Math. Belgrade 6 (20)
(1966),105-106.
182. An inequality eoneerning the arithmetie and geometrie means. Math. Gaz. 50 (1966),
310-311.
183. (with P. M. VasiC) Nouvelles inegalite pour les moyennes d'ordre arbitraire. Univ.
Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz. J'{.'? 159 -J'{.'? 170 (1966), 1-8.
184. (with P. M. Vasic) Une classe d'inegalites ou interviennent les moyennes d'ordre
arbitraire. Univ. Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz. J'{.'?159-J'{.'?170
(1966), 9-14.
185. (with D. D. Adamovic) Complement ci l'article "Sur une inegalite elementaire ou
interviennent des fonetions trigonometriques". Univ. Beograd. Publ. Elektrotehn.
Fak. Sero Mat. Fiz. J'{.'? 159-J'{.'? 170 (1966),31-32.
186. (with R. S. Mitrinovic) Tableaux d'une classe de nombers relies aux nombers de
Stirling. VII. Univ. Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz. J'{.'? 172J'{.'? 173 (1966), 1-26.
187. (with R. S. Mitrinovic) Tableaux d'une classe de nombers relies aux nombers de
Stirling. VIII. Univ. Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz. J'{.'? 172N.'? 173 (1966), 27-53.
188. (with P. M. Vasic) Une classe d'inegalites. Mathematica (Cluj) 8 (31) (1966), 325328.
189. (with P. M. Vasic) Complements au Traite de Kamke. XII. Des eriteres d'integrabilite de l'equation differentielle de Riecati. Univ. Beograd. Publ. Elektrotehn.
Fak. Sero Mat. Fiz. J'{.'? 175 - J'{.'? 179 (1967), 15-21.
190. (with R. S. Mitrinovic) Table des nombres de Stirling de seeonde espeee. Univ.
Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz. J'{.'? 181- J'{.'? 196 (1967), 1-16.
191. Certain inequalities involving elementary symetrie funetions. Univ. Beograd. Publ.
Elektrotehn. Fak. Sero Mat. Fiz. N.'? 181-J'{.'? 196 (1967),17-20.
192. Some inequalites involving elementary symetric funetions. Univ. Beograd. Publ.
Elektrotehn. Fak. Sero Mat. Fiz. J'{.'? 181-J'{.'? 196(1967), 21-27.
193. (with P. M. Vasic) Proprietes d'un rapport ou interviennent les moyennes generalisees. Univ. Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz. J'{.'? 181-J'{.'? 196
(1967), 29-33.
194. (with P. M. Vasic) Monotonost kolicnika dve sredine. Univ. Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz. J'{.'? 181- J'{.'? 196 (1967), 35-38.
195. An old inequality rediseovered by Wilf. Univ. Beograd. Publ. Elektrotehn. Fak. Sero
Mat. Fiz. N.'?181-J'{.'?196 (1967), 39-40.
196. (with P. M. Vasic) 0 jednoj kvadratnoj funkeionalnoj jednacini. Univ. Beograd.
Publ. Elektrotehn. Fak. Sero Mat. Fiz. J'{.'? 21O-J'{.'? 228 (1968),1-9.
197. Inequalities coneerning the elementary symetric funetions. Univ. Beograd. Publ.
Elektrotehn. Fak. Sero Mat. Fiz. N.'? 210-J'{.'? 228 (1968),17-19.
198. (with P. M. Vasic) Generalisation d'un procede fournissant des inegalites du type
de Rado. Univ. Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz. J'{.'? 210-J'{.'? 228
(1968), 27-30.
PUBLICATIONS OF D. S. MITRINOVIC
23
199. (with P. M. Vasic) Inegalies du type de Rado concernant des /onctions symetriques.
Univ. Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz. N~ 210-N~ 228 (1968),
31-34.
200. (with P. M. Vasic) Generalisation d'une inegalite de Henrici. Univ. Beograd. Publ.
Elektrotehn. Fak. Ser. Mat. Fiz. N~210-N~228(1968), 35-38.
201. (with P. M. Vasic) Inegalites po ur les /onctions symetriques elementaires. Univ.
Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. N2 210-N~ 228 (1968),39-42.
202. (with P. M. Vasic) Dopune Kamkeovom delu. XIII. 0 kriterijumima integrabilnosti Riccatieve jednacine. Univ. Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz.
N~ 210-N~ 228 (1968),43-48.
203. (with P. M. Vasic) Une inegalite generale relative aux moyennes d'ordre arbitraire.
Univ. Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz. N~ 210-N~ 228 (1968),
81-85.
204. The Steffensen inequality. Univ. Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz.
N~247-N~273(1969), 1-14.
205. A cyclic inequality. Univ. Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz. N~ 247N~ 273 (1969), 15-20.
206. Lignes asymptotiques d'une classe des sur/aces. Univ. Beograd. Publ. Elektrotehn.
Fak. Sero Mat. Fiz. N~ 247 -N~ 273 (1969), 53-56.
207. Sur quelques equations aux derivees partielles ci deux /onctions inconnues. Univ.
Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. N2 247 -N~ 273 (1969),57-60.
208. (with R. S. Mitrinovic and S. S. Turajlic) A table 0/ coefficients /01' numerical
differentiation. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. N~ 247:N~ 273 (1969),115-122.
209. (with P. M. Vasic) An integral inequality ascribed to Wirtinger, and its variations
and generalizations. Univ. Beograd. Publ. Elektrotehn. Fak. Spr. Mat. Fiz. N~ 247N~ 273 (1969), 157-170.
210. (with J. D. Keckic) From the history 0/ nonanalytic /unctions. Univ. Beograd. Publ.
Elektrotehn. Fak. Sero Mat. Fiz. N~ 274-N~ 301 (1969), 1-8.
211. On geodesic lines 0/ a class 0/ sur/aces. II. Univ. Beograd. Publ. Elektrotehn. Fak.
Sero Mat. Fiz. N~ 302 - N~ 319 (1970), 29-31.
212. (with J. D. Keckic) From the history 0/ nonanalytic /unctions. II. Univ. Beograd.
Publ. Elektrotehn. Fak. Sero Mat. Fiz. N~ 302-N2 319 (1970),33-38.
213. (with P. M. Vasic) Sur une equation /onctionnelle non-lineaire. Univ. Beograd.
Publ. Elektrotehn. Fak. Ser. Mat. Fiz. N~412-N2460(1973), 3-10.
214. (with P. M. Vasic) History, variations and generalizations 0/ the Cebisev inequality
and the question 0/ some priorities. Univ. Beograd. Publ. Elektrotehn. Fak. Sero
Mat. Fiz. N~ 461-N~ 497 (1974), 1-30.
215. (with P. M. Vasic) The centroid method in inequalities. Univ. Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz. N~ 498 - N2 541 (1975), 3-16.
216. (with J. D. Keckic) Complements au Traite de Kamke. XIV. Applications o/the variation 0/ parametres method to nonlinear second order differential equations. Univ.
Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz. N2 544-N~ 576 (1976),3-7.
217. (with P. M. Vasic) On a theorem 0/ W. Sierpinski concerning mean. Univ. Beograd.
Publ. Elektrotehn. Fak. Sero Mat. Fiz. N~ 544-N2 576 (1976), 113-114.
218. (with P. M. Vasic) Addenda to the monograph "Analitic inequalities". I. Univ. Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz. N2577-N~598(1977), 3-10.
219. (with I. B. Lackovic and M. S. Stankovic) Addenda to the monograph "Analitic
inequalities". II. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. N~ 634N~ 677 (1979), 3-24.
24
R. Z. DJORDJEVIC AND R. R. JANIC
220. On the univalence 0/ rational functions. Univ. Beograd. Publ. Elektrotehn. Fak.
Sero Mat. Fiz. N~ 577 -:N~ 598 (1979), 221-227.
221. (with G. Kalajdzic) On an inequality. Univ. Beograd. Publ. Elektrotehn. Fak. Sero
Mat. Fiz. :N~ 678 -:N~ 715 (1980), 3-9.
222. (with J. D. Keekic) On a binomial functional equation and some related equations.
Univ. Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz. :N~ 716-:N~ 734(1981),3-10.
223. (with J. D. Keekic) Variations and generalizations 0/ Clairaut's equations. Univ.
Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz. :N~ 716-:N~ 734 (1981), 11-21.
224. (with I. B. Lackovic) Hermite and convexity. Aequationes Math. 28 (1985),229-232.
225. (with J. E. Peearic) Note on O. Bottema's inequality /or two triangles. C. R. Math.
Rep. Aead. Sei. Canada 8 (1986), 141-144.
226. (with J. E. Peearic) Erdös-Mordell's and related inaqualities. C. R. Math. Rep.
Aead. Sei. Canada 8 (1986), 381-386.
227. (with J. E. Pecaric) On the Erdös-Mordell inequality /or a polygon. J. College Arts
Sei. Chiba Univ. 19 (1986), 3-6.
228. (with J. E. Pecaric) Note on the Gauss- Winckler inequality. Anz. Österreich. Akad.
Wiss. Math. Natur. Kl. 123 (1986), 89-92.
229. (with J. E. Pecaric) An inequality /or a polygon. Zbornik Fak. za pomorstvo (Kotor)
11-12 (1985/86), 73-74.
230. (with J. E. Pecaric) On some applications 0/ Hermite's interpolation polynomial. C.
R. Math. Rep. Aead. Sei. Canada 9 (1987), 55-58.
231. (with J. E. Peearic) The generalized Fermat-Torricelli point and the generalized
Lhuilier-Lemoine point. C. R. Math. Rep. Aead. Sei. Canada 9 (1987), 95-100.
232. (with J. E. Peearic and V. Volenee) History, variations and generalizations 0/ the
Möbius-Neuberg theorem and the Möbius-Pompeiu theorem. Bull. Math. Soe. Roum.
Sei. 31 (79) (1987), 25-38.
233. (with J. E. Peearic and W. Janous) Some trigonometrical inequalities. Rad Jugoslav. Akad. Znan. Umjet. 428 (1987), 103-127.
234. (with J. E. Pecaric) Inequality between the sides 0/ triangles with given areas. Obue.
po matematika N~ 4 (1987), 39-40.
235. (with J. E. Peearic) Generalizations 0/ the Jensen Inequality. Österreich. Akad.
Wiss. Math. Natur. Kl. Sitzungsber. 11. 196 (1987), 21-26.
236. (with J. E. Pecaric) On a method due to R. Bellman. Österreich. Akad. Wiss. Math.
Natur. Kl. Sitzungsber. 11. 196 (1987), 399-402.
237. (with J. E. Pecaric) About the Neuberg-Pedoe and the Oppenheim inequalities. J.
Math. Anal. Appl. 129 (1988), 196-210.
238. (with J. E. Pecaric) On the Bellman generalization 0/ Steffensen's inequality. III. J.
Math. Anal. Appl. 135 (1988), 342-345.
239. (with J. E. Peearic, C. Tanaseseu and V. Volenee) Inequalities involving R, rand s
/or speciel triangles. Rad Jugoslav. Akad. Znan. Umjet. 435 (1988), 75-106.
240. (with J. E. Peearic and V. Volenee) On the polar moment 0/ inertia inequality. Rad
Jugoslav. Akad. Znan. Umjet. 435 (1988), 107-110.
241. (with J. E. Pecaric) Remarks on some determinantal inequalities. C. R. Math. Rep.
Aead. Sei. Canada 10 (1988), 41-45.
242. (with J. E. Pecaric) Determinantal inequalities 0/ Jensen's type. Anz. Österreich.
Akad. Wiss. Math. Natur. Kl. 125 (1988), 75-78.
243. (with J. E. Peearic) Generalizations 0/ two inequalities 0/ Godunova and Levin.
L'Aeademie Polonaise des seiences. Bull. Sero Sei. 36 (1988), 645-648.
244. (with J. E. Peearic) On two lemas 0/ N. Ozeki. J. College Arts Sei. Chiba Univ.
21 (1988), 107-110.
PUBLICATIONS OF D. S. MITRINOVIC
25
245. (with J. E. Pecaric) Unified treatment of some inequalities for mixed means. Österreich. Akad. Wiss. Math. Natur. Kl. Sitzungsber. H. 197 (1988), 391-397.
246. (with G. V. Milovanovic and Th. M. Rassias) On some extremal problems for algebraic polynomials in L r norm. In: Generalized Functions and Convergence (Katowiee, 1988). World Seientifie, Singapore, 1990, 343-354.
247. (with J. E. Pecaric) A general integral inequality for the derivative of an equimeasurable rearrangement. C. R. Math. Rep. Aead. Sei. Canada 11 (1989), 201-105.
248. (with J. E. Pecaric) On two-plaee eompletely monotone functions. Anz. Österreich.
Akad. Wiss. Math. Natur. Kl. 126 (1989), 85-88.
249. (with J. E. Pecaric and V. Volonee) An elementary method for maximizing of some
funetions. Bull. Math. Soe. Sei. Math. R. S. Roumanie (N.S.) 34 (82) (1990),37-47.
250. (with J. E. Pecaric) History, variations and generalizations of the Cebisev inequality
and question of some properties. II. Rad Jugoslav. Akad. Znan. Umjet. 450 (1990),
139-156.
251. (with J. E. Pecaric) Note on a dass of functions of Godunova and Levin. C. R.
Math. Rep. Aead. Sei. Canada 12 (1990), 33-36.
252. (with J. E. Pecaric) On an extension of Hölder's inequality. Boll. Uno Mat. !tal. A
(7) 4 (1990), 405-408.
253. (with J. E. Pecaric) On Bernoulli's inequality. Facta Univ. Sero Math. Inform.
5 (1990), 55-56.
254. (with J. E. Pecaric) Interpolations of determinantal inequalities of Jensen's type.
Tamkang J. Math. 22 (1990),39-42.
255. (with J. E. Pecaric) Remarks on the paper "A note on Everitt type integral inequality". Tamkang J. Math. 21 (2) (1990), 169-170.
256. (with J. E. Pecaric) A note on an inequality with noneonjugate parameters. Österreich. Akad. Wiss. Math. Natur. Kl. Sitzungsber. H. 199 (1990), 155-160.
257. (with J. E. Pecaric) On inequalities of Hilbert and Widder. Proe. Edinburgh Math.
Soe. 34 (2) (1991), 411-414.
258. (with J. E. Pecaric, V. Volenee and J. Chen) Addenda to the monograph "Recent Advanees in Geometrie Inequalities". I. J. Ningbo Univ. Nat. Sei. Engin. 4 (2) (1991),
79-145.
259. (with J. E. Pecaric) Two integral inequalities. Southeast Asian BuH. Math. 15 (2)
(1991), 153-155.
260. (with J. E. Pecaric) On some inequalities for monotone funetions. Boll. Uno Mat.
!tal. B (7) 5 (1991), 407-416.
261. (with J. E. Pecaric) On an inequality of G. K. Lebed. Makedon. Akad. Nauk. Umet.
Oddel. Mat.-Tehn. Nauk. Prilozi 12 (1) (1991), 13-19.
262. (with J. E. Pecaric) On an identity of D. Z. Dokovic. Makedon. Akad. Nauk. Umet.
Oddel. Mat.-Tehn. Nauk. Prilozi 12 (1) (1991), 21-22.
263. (with J. E. Pecaric) Comments on an inequality of M. Masuyama. SUT J. Math.
27 (1991), 89-91.
264. (with J. E. Pecaric) On a problem of Sendov involving an integral inequality. Math.
Balkaniea (N.S.) 5 (1991), 258-260.
265. Mihailo Petrovic and Appelrot's theorem. Univ. Beograd. Publ. Elektrotehn. Fak.
Sero Mat. 2 (1991). 95-99.
266. (with J. E. Pecaric) On eompletely monotonie sequenees. Anz. Österreich. Akad.
Wiss. Math. Natur. Kl. 128 (1991), 63-67.
267. (with J. E. Pecaric and L. E. Persson) On a general inequality with applieations. Z.
Anal. Anwendungen 11 (1992), 285-290.
R. Z. DJORDJEVIC AND R. R. JANIC
26
268. Remarks on the paper "Inequalities related to generalized means" by V. Laohakosol
and P. Ubolsri. Makedon. Akad. Nauk. Umet. Oddel. Mat.-Tehn. Nauk. Prilozi
13 (1) (1992), 5-8.
269. (with J. E. Pecaric) Remark on Pachpatte's generalization of Hardy's inequality.
Indian J. pure appl. Math. 23 (2) (1992), 129-130.
270. (with J. E. Pecaric) Some propperties of Bernstain polynomials. Makedon. Akad.
Nauk. Umet. Oddel. Mat.-Tehn. Nauk. Prilozi 14 (2) (1993), 39-44.
271. (with J. E. Pecaric) Jensen's inequality for some non-convex functions. Makedon.
Akad. Nauk. Umet. Oddel. Mat.-Tehn. Nauk. Prilozi 14 (2) (1993), 45-47.
272. (with G. V. Milovanovic and Th. M. Rassias) On some Thran's extrem al problems
for algebraic polynomials. In Topics in Polynomials of One and Several Variables
and Their Applications: A Mathematical Legaci of P. L. Chebyschev (1821-1894)
(Th. M. Rasias, H. M. Srivastava, A. Yanushauskas, eds.). World Seientifie, Singapore, 1993, 403-433.
273. (with J. E. Pecaric) An application of the Chebyshev integral inequality. In Topics
in Polynomials of One and Several Variables and Their Applications: A Mathematical Legaci of P. L. Chebyschev (1821-1894) (Th. M. Rasias, H. M. Srivastava,
A. Yanushauskas, eds.). World Scientifie, Singapore, 1993, 457-461.
274. (with J. E. Pecaric) Bemoulli's inequality. Rend. Cire. Mat. Palermo 42 (2) (1993),
317-337.
275. Sur la forme de I 'integrale generale de quelques equations differentielles du premier
ordre. Univ. Beograd. Publ. Elektrotehn. Fak. Sero Mat. 6 (1995), 8-11.
Conference Papers
1. Sur l'etude des lignes courbure en coordonnes tangentielles. Comptes rendus de
seanees et eonferenees de la Societe mathematique de Franee, Paris, 1937, 32.
2. Organizacija naucnog rada i priprema naucnih kadrova u oblasti matematike. In
Premier congres mathfmaticiens et physiciens de la RPF Yougoslavie (Bled, 1949),
Vol. 2. Naucna knjiga, Beograd, 1950, 175-187.
3. (with J. E. Pecaric, S. J. Bilehev, E. A.Velikova) On an inequality of O. Kooi. Proe.
17th Conf. of the Union of Bulgar. Math. 1988, 566-568.
Other Papers
1. Matematicke grupe za ucenike srednjih skola. BuH. Soe. Math. Phys. Macedoine
2 (1951), 57-64.
2. Nov referativni casopis za matematiku. BuH. Soe. Math. Phys. Serbie 6 (1954),
267-270.
3. Beleska 0 delatnosti Mihaila Petrovica u oblasti diferencijalnih jednacina. BuH. Soe.
Math. Phys. Serbie 7 (1955), 125-127.
4. Sur quelques identites elementaires. Elem. Math. 10 (1955), 65.
5. Mihailo Petrovic - Biografske zabeleske i uspomene. Nauka i priroda 8 (1955), 276284.
6. Kako prici matematici? Razgovori 0 ucenju matematike. Matematicko-fizicki list
za ucenike srednjih skola 6 (1955/56), 41-43.
7. Kakvu predspremu iz matematike ocekuju tehnicki fakulteti od buducih studenata.
Nastava matematike i fizike 5 (1956), 4-8.
8. Osvrt na prve kvalifikacione ispite iz matematike na tehnickim fakultetima u Beogradu. Nastava matematike i fizike 5 (1956), 260-267.
PUBLICATIONS OF D. S. MITRINOVIC
27
9. Legendreovi polinomi i Besselove Jv.nkcije. In S. FempI: Redovi. Zavod za izdavanje
udzbenika, Beograd, 1960, 173-220.
10. Prilozi za biografiju Muhaila Petrovica. BuH. Soc. Math. Phys. Serbie 12 (1960),
143-175.
11. (with C. Stanojevic) Uvoaenje u elemente apstraktne algebre. In Uvoaenje mladih u
naucni rad, I. Zavod za izdavanje udzbenika, Beograd, 1961, 21-30.
12. 0 algebarskim iracionalnim jednacinama. In Uvoaenje mladih u naucni rad, I. Zavod
za izdavanje udzbenika, Beograd, 1961, 143-156.
13. Znacaj i uloga matematike danas. In Izabrana poglavlja iz matematike, I. Zavod za
izdavanje udzbenika, Beograd, 1961, 5-12.
14. Hurvitzovi polinomi. In Izabrana poglavlja iz matematike, I. Zavod za izdavanje
udzbenika, Beograd, 1961, 221-231.
15. Mihailo Petrovic. In Izabrana poglavlja iz matematike, I. Zavod za izdavanje udzbenika, Beograd, 1961, 233-236.
16. (with D. Z. Dokovic) Ciklicne Jv.nkcionalne jednacine. In Izabrana poglavlja iz
matematike, 11. Zavod za izdavanje udzbenika, Beograd, 1962, 5-23.
17. (with D. Z. Dokovic) Neki nereseni problemi u teoriji funkcionalnih jednacina. In
Neki nereseni problemi u matematici. Zavod za izdavanje udzbenika, Beograd, 1963,
153-168.
18. Jedan pogled na razvoj matematike u Srbiji. In Uvoaenje mladih u naucni rad, III.
Zavod za izdavanje udzbenika, Beograd, 1963, 77-83.
19. (with R. S. Mitrinovic) Tableaux d'une classe de nombers relies aux nombers de
Stirling. III. Posebna izdanja Matematickog instituta u Beogradu, Beograd, 1963,
1-200.
20. (with R. S. Mitrinovic) Tableaux d'une classe de nombers relies aux nombers de
Stirling. IV. Posebna izdanja Matematickog instituta u Beogradu, Beograd, 1964,
1-115.
21. (with R. S. Mitrinovic) Tableaux d'une classe de nombers relies aux nombers de
Stirling. VI. Posebna izdanja Matematickog instituta u Beogradu, Beograd, 1966,
1-52.
22. Zivot Mihaila Petrovica. In Mihailo Petrovic: Covek-Filozof-Matematicar. Zavod
za izdavanje udzbenika, Beograd' 1968, 1-32.
23. 0 jednoj nejednakosti. In Mihailo Petrovic: Covek-Filozof-Matematicar. Zavod za
izdavanje udzbenika, Beograd, 1968, 93-96.
24. 0 jednoj diferencijalnoj jednacini. In Mihailo Petrovic: Covek-Filozof-Matematicar.
Zavod za izdavanje udZbenika, Beograd, 1968, 97-100.
25. Mihailo Petrovic i Stirlingovi brojevi. In Mihailo Petrovic: Covek-Filozof-Matematicar. Zavod za izdavanje udzbenika, Beograd, 1968, 113-116.
26. Pionir nase matematicke nauke. In Uvoaenje mladih u naucni rod. IV. Zavod za
izdavanje udzbenika, Beograd, 1969, 177-179.
27. Jedan postupak za obrazovanje nejednakosti. In Uvoaenje mladih u naucni rad. VI.
Zavod za izdavanje udzbenika, Beograd, 1969, 59-63.
28. (with P. M. Vasic) A-metod. In Uvoaenje mladih u naucni rad. VI. Zavod za izdavanje udzbenika, Beograd, 1969, 64-71.
29. (with B. S. Popov) Joie Ulcar - In memoriam. In Uvoaenje mladih u naucni rod.
VI. Zavod za izdavanje udzbenika, Beograd, 1969, 203-209.
30. Zapaianja 0 univerzitetskoj nastavi i naucnom rodu u Moskvi. In Uvoaenje mladih
u naucni rad. VI. Zavod za izdavanje udzbenika, Beograd, 1969, 226-229.
Invited Papers
COMPLEX POLYNOMIALS AND MAXIMAL RANGES:
BACKGROUND AND APPLICATIONS
VLADIMIR V. ANDRIEVSKII
Institute /or Applied Mathematics and Mechanics 0/ the National Ukrainian
Academy 0/ Sciences, Rozy Luxemburg 74, Donetsk 340114, Ukraine
STEPHAN RUSCHEWEYH
Mathematisches Institut, Universität Würzburg, D-97074 Würzburg, FRG
Abstract. This survey is dedicated to the discussion of the various aspects of the notion
of maximal polynomial ranges. These are the unions of ranges of polynomials restricted
by a geometrical condition. The theory of maximal ranges in essentially constructive
and permits in many cases the identification of extremal functions. It thereby leads to
a unified approach to many old and new inequalities for polynomials. We also discuss
the relation of this concept to the approximation of conformal maps in the unit disk by
univalent polynomials.
1. The Concept of Maximal Ranges
The notion of maximal ranges of polynomial spaces has been introduced in [5];
however, it only generalises and unifies various classicp.! concepts and related results in the geometry of complex polynomials (and, in fact, other spaces of analytic
functions). Indeed, as we shall see soon, the maximal range concept itself is embedded in a more general approach to solve linear extremal problems for spaces of
analytic functions in the unit disk lD> with constraints to their images. We use two
examples to point out the general nature of these problems.
Example 1. Let P be a complex polynomial of degree at most n, and assume
that P(O) = 0, P(z) :I 1 for z EID>. Then a well-known classical result says that
(1)
where -< denotes subordination. In a less precise version this means
P(lD»
c Q(lD»,
where Q(z):= 1- (1 + z)n.
This relation obviously describes completely the possible (=maximal) range of all
(normalised) polynomials with range in C \ {1}. It is surprising that one single
polynomial gives the maximal range.
1991 Mathematics Subject Classification. Primary 30ClO, 30G35j Secondary 41AI0.
Key woms and phrases. Maximal rangesj Complex polynomialsj Univalent polynomialsj Conformal mappings.
31
G. V. Milovanovic (ed.), Recent Progress in lnequalities, 31-54.
© 1998 Kluwer Academic Publishers.
32
V. V. ANDRIEVSKII AND S. RUSCHEWEYH
Example 2. Let P be a complex polynomial of degree at most n, and assurne
that P(O) = 0, ReP(z) > -1/2 for z EID>. A problem is to describe the range of
the functional P(r), 0 < r :::; 1, fixed. It is clear that this range increases with
n, and that the limit for n -+ 00 is the disk {z/(l - z) : Izl:::; r}. The explicit
solution for fixed degree n is apparently not known for r < 1 (and can probably
not be given in explicit terms; for details see our discussion of this problem in
Appendix A.8). For r = 1 the complete and explicit solution of this (maximal
range) problem will be obtained as well.
In both cases, the admissible function sets consist of a linear subspace of the
space of analytic functions in ii), with the additional restriction that the images of
the unit disk are contained in some given domain (the punctured complex plane
in Example 1, and a half plane in Example 2). The aim is to describe further
properties of these admissible sets, for instance ranges of linear functionals defined
on them.
To be more precise, let 11. be some linear space (over C or IR) of analytic functions
in ii), and let 0 C C be a domain with 0 E O. Then we define
11.(0) := {I E 11. : 1(1D» CO}.
If Ais a non-constant linear functional on 11. (with respect to the same field) , then
the description of A(1I.(0)) and of the extremal functions 1 E 11.(0) (i.e., those for
which
(2)
AU) E 8A(1I.(0))
holds) is the principal aim of the method described in this article.
If the set 11.(0) happens to be complete, i.e.,
(3)
1 E 11.(0)
'* (TI lxi:::; 1) I(xz) E 11.(0),
a very natural condition in our context, then the special functional AU) := 1(1)
leads to the maximal range problem, namely the description of the set
01l:=
U I(ID»·
/E1l(O)
Clearly, this latter definition is also meaningful if (3) is not fulfilled.
A general theory to treat the above mentioned problems will be discussed in Section 2. It turns out that strong necessary conditions for the extremal functions
as in (2) hold in a similar fashion as the well-known Kolmogorov conditions from
approximation theory (compare [21]). These conditions are even sufficient if the
domain 0 is a convex set and the range in question is compact. For the special
cases that 11. is the set P~ of complex polynomials P of degree (at most) n with
P(O) = 0, and that the functional corresponds to the maximal range problem
(see above), these Kolmogorov type conditions can be translated into much more
concrete statements, describing the geometry of the extremal polynomials. This
COMPLEX POLYNOMIALS AND MAXIMAL RANGES
33
is particularly striking in the case of convex domains 0, where it turns out that
every arc of the boundary of the maximal range, which does not meet the boundary of 00, lies in the boundary of the range of a single extremal polynomial. The
intriguing arc-conjecture states that this might be the case without restrictions on
the domain o. A striking example for this conjecture is Example 1 from above.
Section 3 is devoted to a non-convex case where the arc-conjecture holds. This
observation leads to a very general subordination theorem for polynomials, which
exhibits rather curious and unexpected geometrical properties of complex polynomials. It is shown that an important subcase of the famous Smale conjecture on
the critical values of polynomials is covered by this result. It is interesting that
the maximal range problem (and the present knowledge concerning its solution)
has not only been the guide to discover this subordination theorem, but is also
used as a tool in its proof (which cannot be explained in this survey).
Our next task (see Section 4) is the description of a new approach to the approximation of conformal maps of lD> onto domains 0 by means of complex univalent
polynomials. The maximal range theory implies that the extremal polynomials
for a given 0 are always univalent, and they seem to approximate the corresponding conformal mapping 'from inside'. This kind of question has been discussed in
great detail in the papers [3-4], [13-14] where sharp estimates for approximations
of the following type are derived: find the asymptotically best (smallest) numbers
c(O, n) such that there exists a univalent polynomial P E P~ with
f ({z : Izl < 1- c(O,n)}) C P(lD» C felD»~ = 0,
where f is univalent in lD> with f(O) = O. We use geometrical properties of 0 to
estimate the numbers c(O, n).
This theme, viewed from a different angle, is continued in Section 5, where we try
to obtain estimates for the Hausdorff distance of an and oOn (the boundary of
the maximal range with respect to n and the space P~), and thereby a measure
for the possible rate of approximation to 0 by polynomial ranges, in terms of the
geometry of O.
In the Appendix of this survey we shall apply the maximal range theory to a large
number of concrete situations (domains 0 and polynomial spaces), including halfplanes, strips, slit-domains (single and double), rectilinear quadrilaterals, and the
interior and exterior of circular discs. In all of these cases a more or less explicit
description of the extremal polynomials can be given, and as a consequence of
these results we obtain a number of new inequalities for complex polynomials
which frequently generalise and sharpen previously known ones. In these cases the
arc-conjecture holds always true. The Appendix closes with abrief account to the
solution of the problem mentioned in Example 2 above (0 < r < 1).
2. The General Theory
The theory presented in this section can be found in the three articles [5-6], [9].
We make use of the definitions in Section 1. Let 0 be some domain with 0 E 0,
and for f E H(O) we define
rj := {z E 0lD> : fez) E an},
34
V. V. ANDRIEVSKII AND S. RUSCHEWEYH
which under our assumptions is a compact set. If A is a non-constant linear
functional on 1l then we say that 1 E 1l(O) satisfies the Kolmogorov condition if
(4)
J '" 0,
{ rmaxRe
[z!'(z)u(z)] ~ 0,
zEr,
u E 1l A ,
where 1l A denotes the subspace of 1l(O) whose elements u satisfy A(U) = 0. We
then have the following two basic results.
Theorem 1. Let 11. be linear over C, A a non-constant complex linear functional
over 1l, and 0 a domain. 1/ ~ := A(1l(O)) is a compact set and 1 E 1l(O), then
AU) E a~ only i/ 1 satisfies (4).
It is not known to which extent the above mentioned condition is not only necessary, but also sufficient. Theorem 2 describes a general case where this is so.
Theorem 2. Let 11. be linear over IR, A a non-constant complex or real functional
over 1l, linear with respect to IR, and 0 a convex domain. 11 ~ := A(1l(O)) is a
compact set and 1 E 1l(O), then AU) E a~ il and only i/I satisfies (4).
If in Theorem 2 the set ~ happens to be line segment, then a~ is understood to
consist of the endpoints of this segment.
From now on we make the general assumptions that
(i) 1l(O) is compact and complete (compare (3)).
(ii) Ais an evaluation functional, i.e., AU) = I(r) for some r E (0,1].
In this case we generally have
~ := A(1l(O)) =
U 1(JI)r ),
JE1i(O)
with JI)r := {z : Izl < r}. If 0 is not simply connected then the knowledge of ~
may not be sufficient for some purposes. Instead, one may want to have the more
precise information obtained if the elements of 1l(O)) are considered as mappings
into the universal covering of 0 (which can always be assumed to be of hyperbolic
type in this context). If F maps this covering conformally onto JI), then the set
U FU(JI)r))
~ *:=
JE1i(O)
carries much more information than~. Fortunately, Theorem 1 extends to this
situation.
Theorem 3. Let 11. be linear overC, and1l(O), A, ~* be as above. 1/1 E 1l(O),
then AU) E a~* only i/I satisfies (4).
We now turn to the case 1l = P~ (see Section 1), and note that the general
assumptions made above are automatically fulfilled. We also restriet our interest
to the maximal range case, i.e., to the functional A(P) = P(I), and we use
On :=
U P(JI))
PE'P~(O)
COMPLEX POLYNOMIALS AND MAXIMAL RANGES
35
as abbreviation for this maximal range. For obvious reasons we now rest riet the
notation extremal polynomial to those P E p~(n) for which
The elements of r p are ealled points 01 contact, although they are aetually preimages of eontaet points of P(l!)) and an. The next theorem is fundamental for
the maximal range theory.
Theorem 4. For every point w E an n \ an there exists at least one extremal
polynomial P E p~(n) such that w = P(l). Moreover, every extremal polynomial
P with P(l) E an n \ an satisfies the lollowing conditions:
1° P' has all 01 its zeros on al!) \ {I}. Let ei1/Ji , j = 1, ... , n - 1, denote these
zeros, ordered as lollows: 0 < 'l/Jl s:; ... s:; 'l/Jn-l < 27r.
2° There exist at least n points 01 contact ei(}i, j = 1, ... , n (multiplicities
counted) such that
(5)
3° 11 n is simply connected then P is univalent in l!).
This ean be remarkably refined if n is a eonvex domain.
Theorem 5. 11 n is a convex domain, then in addition to 1°-3° in Theorem 4
we have:
4° 11 w is as above, then there is a unique extremal polynomial P E p~(n)
such that w = P(l).
5° 11 ()l, ()n /rom Theorem 4 are chosen that way that no () in [0, ()l) u (()n, 27r]
corresponds to a point 01 contact then the arc
(6)
is a connected component 01 an n \ an.
The arc-conjecture states that Theorem 5 holds without the restrietion to eonvex
domains (but applying the interpretation of maximal ranges on universal eoverings
as mentioned above, if n is not simply eonneeted).
The fundamental properties 1° and 2° in Theorem 4 are eonsequenees of Theorem 1 and a general alternant theorem which is of independent interest and is
therefore stated here. Let P n denote the set of polynomials of degree s:; n.
Theorem 6. Let R(z) =
m
TI (z-Zj) and let H"I- 0 be a compact subset olal!)\{l}.
j=l
Write A := H n {Zl, ... ,zm} and denote by Vj the multiplicity 01 the zero Zj 01
R(z). Assume there exists no polynomial V E Pm such that V(l) = 0 and
(7)
ReR(()V(() < 0,
(E H\A,
36
(8)
V. V. ANDRIEVSKII AND S. RUSCHEWEYH
V(k) (Zj) =0,
k=0,1, ... ,vj-1,
Re [R(Vj) (Zj)V(Vj) (Zj) ] < 0,
Zj E A.
Then Zj E 8lDl, j = 1, ... ,m, and no connected subarc of 8lDl \ H contains more
than one element 01 {I, ZI, . . . , zm}.
The extension of Theorem 4 to the universal covering interpretation has not yet
been fully developed. It is very likely, however, that this can be done with the
corresponding conclusions.
3. Subordination of Polynomials
The result presented in this section was suggested by the arc-conjecture, and is a
partial verification of it, see [8]. It implies a very surprising property of complex
polynomials and their critical values.
The idea is as follows. Assurne that P E P~ is univalent in lDl, and has all zeros
of its derivative on 8lDl (they are simple). Let (1, ... , (n-1 be these zeros, and
consider the domain 0 := C \ {P(l), ... ,P(n-J}}. If the arc-conjecture were
true, then one could readily deduce, that On = P(lDl). Therefore, if Q E P~ is
some other polynomial, which does not have any ofthe points P(l),'" , P(n-J}
in its image of lDl, then consequently Q E P~(O), and therefore Q(lDl) C P(lDl).
Dynamically speaking: the range of Q cannot go beyond the range of P without
covering first one of P's critical values P(J}, ... , P(n-1).
The theorem which eventually emerged from these considerations is much more
general and implies the case it originated from: a partial verification of the arcconjecture. P does not have to be univalent, and the location of the critical points
of P can be much less restrictive.
Theorem 7. Let n ~ 2 and assume that PEPn has all its critical points (j,
j = 1, ... ,n - 1, in D. Let Q E P n satisfy P(O) = Q(O) and
P(j)iQ(lDl),
j=l, ... ,n-1.
Then Q -< P, and, in particular, Q(lDl) C P(lDl).
Let PEPn satisfy the conditions of Theorem 7, and be univalent. Then Theorem 7
can be looked at as a weakened form of the sufficient condition for a polynomial
Q to be subordinate to P: instead of assuming Q(lDl) C P(lDl) it suffices to require
P(j) i Q(lDl) , j = 1, ... ,n -1. The following example illustrates this kind of
interpretation.
Theorem 8. Let P E P~ be such that
P(z) :f:. n - 1 exp ( 27rij ),
n
n-1
j = 1, ... , n - 1,
Z E lDl.
Then P -< Z - zn In, and, in particular,
IP(z)1 ~ n + 1, Z E lDl.
n
In connection with his fundamental investigations concerning the global behaviour
of the Newton method, Smale [27] posed the following conjecture:
COMPLEX POLYNOMIALS AND MAXIMAL RANGES
37
Conjecture. For any P E P n and any z E C we have
min
{x:P'(x)=O}
IP(Z) - P(x) I:::; 1P'(z)l.
Z - X
It is not difficult to see that this conjecture has the following equivalent formulation:
Conjecture. For any P E P~ and all zeros 0/ its derivative in C \ JO) we have
min
(9)
{x:P'(x)=O}
IP(x) I:::; IP'(O)I·
x
The following application of Theorem 7 is a partial verification of Smale's conjecture (9).
Theorem 9. The relation (9) holds i/ all zeros 0/ P' are on 8JO).
Proof. We mayassume that P'(O) i:- 0, and even P'(O) = 1. If (9) were false for
P, then IP«()I > 1 in the critical points (. By Theorem 7 this implies pz -< P, for
some p > 1 and therefore IP'(O)I ~ p > 1, a contradiction. 0
Concerning the application of Theorem 7 to the maximal range problem, corresponding to the introductory remark of this section, we note that it is easy to
construct univalent polynomials with all zeros of the derivative on 8JO). For instance, following a result of Suffridge [29], we have that
1 II (1 z n-1
P(z) =
o j=1
te ia ;) dt
is univalent in JO) if
n 2:1 :::;min{laj-ak+27rml: 1:::;j:::;k:::;n-1,mEZ}.
One can generalise Theorem 7 from polynomials P to certain meromorphic functions in C whose image curves of 8JO) satisfy restrictions to the speed of tangent
rotation. This generalisation sheds some light on the deeper background of Theorem 7, but leads too far away from our present theme. We refer to the original
paper [8].
4. Conformal Mappings and Subordinated Univalent
Polynomials
4.1. GENERAL ESTIMATES
Let 0 be a simply connected domain, 0 E 0, and On its maximal range. Let / be
a conformal mapping of JO) onto 0 with /(0) = O. The main theme of this section
is to relate important properties of On to /. We make use of the notation
Is(z) := 1«1 - s)z),
The fundamental result is (see [3]):
z E JO),
0:::; s < 1.
38
V. V. ANDRIEVSKII AND S. RUSCHEWEYH
Theorem 10. There exists a universal constant Co > 1 with the following property: for each simply connected 0 and n ~ 2Co there exists a (univalent) P E P~(O)
such that fco/n --< P --< f. In particular,
fco/n(lJ)) C On C O.
The proof of Theorem 10 is constructive and the polynomial P is derived from f
by means of an integral transform. This approach is essentially due to Dzyadyk
[10], and uses the generalised Jackson kerneis
Sin mt/2) 2(k+1)
Imk(t)=amk ( sint/2
'
where k, m E N, and the coefficients amk are determined by the condition
-2
1
7r
1'"
_".
Imk(t) dt = 1.
Then let m ~ 2, and for Izl < 1 - l/m we define
Tm,(z)
,~ 2~ j Im,(t) 2~i
-".
J {~~ [
1- (1-
1<I=l-l/m
(e~;; ~ z
t'1
d( dt
Note that the functions T mk are algebraic polynomials of degree less or equal to
(k + l)(m -1) - 1. Our proof shows that for Co large enough (but independent of
o and n), and large n, the choice
P(z):=T ,6((I-:)z), m:=[~],
m
yields P as claimed in Theorem 10.
4.2. UNBOUNDED DOMAINS
For unbounded domains we can get an even more precise conclusion which, on the
other hand, also implies that the bound in Theorem 1 (Le., eo/n) is of the right
order. We shall use the foHowing notation:
IIgll := sup Ig(z)l·
zED
C2
Theorem 11. There exist two positive universal constants Cl,
with the following property: for each unbounded, simply connected domain 0 and each n ~ 2eo
we have
(10)
~ sup Iwl ~
Cl Ilfl/nil
c21Ifl/nll·
wEO ..
Theorem 11 relates On to a conformal mapping f of lJ) onto O. It may be of
interest to have the same information using just the geometry of O. Theorem 12
below is a step in this direction. It deals with unbounded convex domains only,
but the result should be true for other unbounded domains as weH.
Let 0 be an unbounded convex domain, 0 E 0, and let g be a ray in 0, starting
out from the origin. For each r > 0 let lo(r) be that arc of 0 n {w : Iwl = r}
which meets g. Note that
._ Ilo(r)1
()o(r ) .,
r
where Ilo(r)1 denotes the length of lo(r), is non-increasing with r.
COMPLEX POLYNOMIALS AND MAXIMAL RANGES
39
Theorem 12. There are universal constants bj > 0, j = 1,2,3, such that for each
o as described above, and for n > b1 , we have
b2 x n ~ sup Iwl ~ b3 x n ,
(11)
wEnn
where X n is the unique solution of
'Ir
r Ilo(r)1
dr
logn,
ldo
n
=
and do := dist(O, aO).
4.3. BOUNDED DOMAINS
In eontrast to the ease of unbounded domains we ean hope to get better bounds
in Theorem 10 (instead of eo/n). We wish to diseuss the smallest value of s =
s(n,O) > 0 so that the relation
(12)
fs --< P --< f
holds in the sense of Theorem 10, and (in this subsection) we always assurne that
o is bounded, and that f is not a polynomial itself (Le., s > 0 for large n.)
Theorem 13. There exists a universal constant C3 with the following property:
for n > 3 and if 0 < s(n, 0) < l/n then
This result enables us to show that Theorem 10 is sharp even for a wide elass
of bounded domains. To simplify things we confine ourselves to domains with
quasi-eonformal boundaries (quasi-disks) (eompare [1], [12]).
Let 0 be a Jordan domain, 0 E 0, and z E ao. For r > 0 sufficiently small (Le.,
r < 10 = 10(0)) we denote by ')'(z, r) C 0 an are of the cirele {( : I( - zl = r} that
separates z from 0, Le., ')'(z, r) has non-empty interseetion with every Jordan are
in 0 whieh joins 0 and z. If this are is not uniquely determined, then we ehoose
one of those for which the remaining eonneeted eomponent of 0\ ')'(z, r) containing
the origin is as large as possible.
Theorem 14. Let 0 be a quasi-disk such that there exists a point z E ao with
(13)
1
lim ( - log x +
x~o
R~O
'Ir
lR I ( )I
dt )
= 00.
Rx ')' z, t
Then s(n, 0) ~ C4/n where C4 depends on 0 only.
As an example for (13) assurne there is a cireular sector with center at z, radius 15
and opening ß'Ir, 1 < ß < 2, in C \ O. Then
I')'(z, t)1 ~ (2 - ß)'lrt,
0 ~ t ~ 15,
V. V. ANDRIEVSKII AND S. RUSCHEWEYH
40
l .,...-.,.-:-;- >
and therefore
R
dt
Rz 1'Y(z, t)1
1
- (ß - 2)11"
log x,
so that (13) is fulfiIled. Hence, if the boundary of 0 has in at least one point
something like an acute (interior) angle, then the order given in Theorem 10 cannot
be improved.
Another interesting consequence of Theorem 10 for bounded domains comes from
an application of a distortion theorem due to Lavrentiev [18]. Let
O(t) := {z E 0 : dist (z, ao) > t},
t > O.
Theorem 15. There exists a constant Cs = cs(O) > 0 such that lor n > 2 the
maximal range On contains a simply connected component 0/0 (cs(logn)-1/2).
4.4. BOUNDS FOR THE CONSTANTS
Although the proof of Theorem 10 is constructive, estimate for the constant Co
obtained this way is prohibitively large. Greiner [13] made numerous refinements
to the method and gave a reasonable upper bound. Also, since the maximal range
for the Koebe domain C\ (-00, -1/4] is known (compare Section 6), a good lower
bound has been found. This and the other results in this subsection are in [14].
Theorem 16. For the (best possible) constant Co in Theorem 10 we have 11" $
Co< 73.
Better results can be obtained for domains with certain general geometrie properties. It turns out that the Cesaro means of analytic functions in JI)) are partieularly
suitable in this context. Unfortunately, the approximative properties of these
means are not yet so weIl understood as for the generalised Jackson kerneis used
for the present proof of Theorem 10.
00
00
k=O
k=O
We recall that for I(z) = E akzk, g(z) = E bkZk the Hadamard-convolution is
00
defined as (f * g)(z) := E akbkzk. For given I, holomorphie in JI)) with power
00
k=O
series I(z) = E akzk, the n-th Cesaro mean of order 0: ~ 0 is defined by
k=O
n (n+o<-k)
n-k
O'n0«1 ,z)..- '"'
L.J (n+o<)
k=O
n
ak Zk .
Using the abbreviation O'~(z) for O'~ (1/(1- z),z) one may also write
* O'~)(z) .
Lewis [19] has shown that the Cesaro means 1 * O'~ of order 0: ~ 1 are univalent
O'~(f, z) = (f
in JI)), in fact close-to-convex, for 1 convex (i.e., univalent and with I(JI))) convex).
The given lower bound for 0: is sharp with respect to this property.
COMPLEX POLYNOMIALS AND MAXIMAL RANGES
41
Theorem 17. For 1 convex the Gesaro means 1 *a;:: of order 0: 2: 1 are univalent
in!Dl and fulfil
l(a+l)!(n+a+1) -<
1 * a;:: -< f
for all n E N. If 1(!Dl) is a half-plane, then, for no n E N and no 0: 2: 1, the
number (0: + l)/(n + 0: + 1) can be replaced by any smaller one.
Using this and another case of explicit knowledge of a special maximal range we
deduce the following result, similar to Theorem 10.
Theorem 18. Let n be convex and 1 a conformal mapping!Dl -t n with 1(0) = 0,
and n E N. Then there exists a univalent polynomial P of degree ~ n such that
hin -< P -< f.
(14)
The constant 2 is best possible.
It is due to Egervary [11] (for 0: = 3) and Ruscheweyh [24] (for 0: > 3) that 1 * a;::
are convex univalent for these values of 0: if 1 is convex. Therefore, we have the
following second consequence of Theorem 17.
Theorem 19. Let 1 be convex and n E N. Then there exists a convex univalent
polynomial P of degree ~ n such that
(15)
14!n -< P -< f.
This is a first step into the general quest ion of how good one can approximate
conformal maps with certain geometrical properties by polynomials with the same
property (in this case: convexity).
5. Hausdorff Distance Between an and an n
In this section we measure the distance between a domain and its maximal ranges
(depending on n). The suitable measure for this purpose is the HausdorfI distance,
which is defined as follows: for a set Ace let U(<5, A) denote the set of points
which have a distance ~ <5 from A. Then
T(A,B):= inf{<5 : Be U(<5,A)
and
Ac U(<5,B)}
is the HausdorfI distance of the sets A, B. It is a metric on the space of closed
sets in <C.
We are interested in the cases of domains n for which there exists 0: > 0 with
(16)
We note first that for a domain with a cusp (inner angle zero at a boundary point)
(16) will generally not hold for any 0: > O. We shall not discuss this in detail, but
point out typical cases: for any increasing function h E G[0,2] with h(O) = 0 and
h(2) < 'Ir define
nh := {z = -1 + reH/ : 0< r < 2
and
181 < rh(r)} .
V. V. ANDRIEVSKII AND S. RUSCHEWEYH
42
Then for any a > 0
lim T (anh, an~) na: = 00.
n-+oo
If we exclude this kind of cusps then we come naturally to the notion of John
domains (compare [20], [23]). A (not necessarily simply connected) domain n is
called a John domain if there is a constant Cl = Cl (n) > 0 such that any of its
points ( can be connected with the origin 0 by an arc I = 1((,0) C n which has
the following property: if 1((, z) is the subare of I connecting ( and z then
dist (z, an) ~ Clll((, z)l, z E I.
However, for our purpose it is advantageous to use another, equivalent definition
[2] of John domains in terms of quasiconformal mappings. After Gehring [12], a
bounded Jordan domain G is called a k-quasidisk, 0 ~ k < 1, if any conformal
mapping of ID> onto G has a K -quasiconformal extension homeomorphism of C onto
itself, where K = (1 + k)/(1 - k). The lens-shaped domains G = G(k,8), 8 > 0,
0< k < 1, which are symmetrie with respect to both, the real and the imaginary
axis, bounded by two circular ares with vertices in ±8 and interior angles of 11"(1- k)
are k-quasidisks [12]. As shown in [2, Thm. I] n is a John domain Hf there exist
constants k E [0,1), C2 > 0, Ca > 0 such that for each zEn there is a k-quasidisk
G zen satisfying
(17)
z E aG z,
dist(aGz,an) ~ Ca dist (z,an).
diamG z ~ C2,
Theorem 20. Let n be a John domain satisfying (17). Then there exists C4 =
C4(n) such that
(18)
T(aO, an n ) ~ C4 n k - l ,
nE N.
To get an idea of the quality of the estimate (18) we look at domains with piecewise
smooth boundaries. A smooth Jordan curve L is called Dini smooth if the angle
ß(s) of the tangent to L (parametrised using the arc-Iength s) satisfies
Iß(S2) - ß(sdl ~ h(S2 - sd,
where h is an increasing function with
1
1 h(x) dx
o
x
0< SI < S2,
< 00.
Theorem 21. Let Zo E L := an and assume that for some f, with 0 < f <
1/4 diam L, the set L n {z : Iz - Zo I < f} consists of two Dini smooth arcs joining
at zo, where they form an inner angle (with respect to n) a1l", 0 < a < 1. Then
for any univalent polynomial P E p~(n)
dist (zo, P(Ö)) ~ csIIP'IIa:/(a:-l),
and, consequently,
(19)
with constants Cs > 0, Cs > 0, which are independent of n.
Note that for the lens-shaped domains n := G(k,8) a combination of (18) and
Corollary 16 below yields the better result
(20)
cs n k - l ~ sup dist (z, nn) ~ C4 n k - l •
zE80
43
COMPLEX POLYNOMIALS AND MAXIMAL RANGES
Appendix. Explicit Maximal Ranges
Up to now we discussed the maximal range problem (and related problems) always
under the normalising assumption that 0 E 0, and that P E P~(O). There is
obviously no harm in choosing another point as a point of reference instead of the
origin. The theory developed in Section 2 remains valid (with obvious necessary
changes). We shaH use Wo E 0 as point of reference and deal with polynomials
P E P~O(O) := {P E P n : P(O) = Wo
and
P(ID» CO} .
A.1. HALF PLANES
This is the basie situation. We choose 0 as the right half plane, Wo = 1. We
are then dealing with polynomials with positive real part in ID> and P(O) = 1,
a frequently studied dass of polynomials, related to non-negative trigonometrie
polynomials. For this dass Fejer has shown that Re P( z) ~ n + 1, z E ID>, and that
this is sharp (in z = 1) only for the Fejer polynomials
~n+l-k k
Fn(z) = 1+2L.." n+l z.
k=1
Fejer's result and Theorem 5 immediately imply that Fn is univalent and On =
co(Fn (ID>)), where co stands for the convex huH of a set. More precisely we have:
Theorem 22. an n = ')'1 U ')'2, where
_ .{ Fn (it/»
.
271' }
e
. 14>1 ~ n + 1 '
')'1 -
')'2
= {it : t E~, Itl ~ cot
5
FIG . 1: Maximal range for n = 4 and a half-plane
From this we can draw two immediate consequences:
(n : 1) }.
44
V. V. ANDRIEVSKII AND S. RUSCHEWEYH
Corollary 1. Let PE P n satisfy P(O) = 1 and ReP(z) ~ 0 in lI)). Then
IImP(z)1 $ cot
(2n: 2)' z
E lI)).
1/, in addition, ReP(zo) = 0 tor some Zo E alI)), then
(21)
IImP(zo)1 $ cot
(n: 1) .
Both bounds are sharp.
Since F n has positive coefficients our result also immediately implies that IP(z)1 $
n + 1, z E lI)), for the polynomials of Corollary 1. This was an earlier refinement
of Fejer's theorem due to Holland [16]. Fig. 1 shows the curve F4 ( e il/l) and the
maximal range n4 .
A.2. STRIP DOMAINS
We consider strip domains in the normalised form n = {z : IRezl < I}, Wo = A E
( -1, 1). Correspondingly we use the space
P~ := {P E P n : P(O) = A}
to study the maximal ranges n~. It is clear from Theorem 5 that, for n, A fixed,
there can be (up to rotations in the argument) only one or two extremal polynomials, and that in the case of two extremal polynomials they must be (essentially)
conjugate to each other. These extremal polynomials can be constructed in terms
of the Fejer polynomials as folIows. Let
2
Gn(z) = - 1 (Fn(z) - 1),
n+
and, for -1 < A $ 0,
S~(z) := A + (~(A) - [~(A)]) Gn (Z[I«'\»)z) +
[1<(>,)]-1
L Gn(Zk Z),
k=O
where ~(A) := (A + l)(n + 1)/2, Zk := e 21rik /(n+1) = eil/l k , and [x] stands for the
largest integer $ x.
Theorem 23. For nE N, -1< A $ 0, we have
n~ = co (S~(l!))) + cj) ,
where + cj means that the complex conjugate set should be added. Furthermore,
n,\n = -n-'\
n ,
The portion 0/ an~ \
0 < A < 1.
an in the upper hai/-plane is given by S~(eil/l), 0< <P < <PI.
45
COMPLEX POLYNOMIALS AND MAXIMAL RANGES
FIG . 2: Maximal ranges for n = 4 and a strip domain
Note that for A elose to -1 or 1, we have essentially a half-plane situation, since
the maximal range cannot reach the opposite boundary component.
Among other applications, Theorem 23 has the following corollaries.
Corollary 2. Let P E P n satisfy P(O) = A E (-1, 0] and IReP(z)1 < 1 in]!)).
Then
2
111m Pli:::; n+ 1
([K(A))-l
{;
cot
(2k2n ++ 12 ) + (K.(A) - [K.(A)]) cot (2[~n(A)]+ +2 1 ))
11"
11"
•
This bound is sharp for S~ at z = e i1T /(n+l). Clearly, the bounds for A E (0, 1) are
the same as for -A.
Corollary 3. Let PE P n satisfy P(O) E (-1 , 1) and IReP(z)1 < 1 in]!)). Then
(22)
2 [(n+l)/2)
111m Pli :::; - cot
n+ 1 k=l
L
This bound is sharp for S~ at z = e i1T /(n+1) .
(2k- 2
- 211"·
1 )
n+
V. V. ANDRIEVSKII AND S. RUSCHEWEYH
46
Corollary 3 is known; different proofs are due to Szegö [30] and Mulholland [22].
However, the form of the extremal polynomials, and their dose relation to the
Fejer polynomials is a new ingredient.
In the context of Corollary 3 it is natural to look at
n*·n·-
U n"
n'
"E( -1,1)
for instance in order to decide whether (22) holds for P instead of Im P. It turns
out that this problem is better studied directly in terms of Theorem 2, using
P~ := {P E P n
: P(O) ER}.
Theorem 24. We have
n~ =
n~,
n odd,
{ co ( n:!(n+1) u n;;1/(n+1))
n even.
With respect to the previously mentioned quest ion we now get:
Corollary 4. Under the assumptions 01 Theorem 23 we have
(23)
IIPII ::;
(2k - 1 )
2 [(n+l)/2]
-L
cot - - {
n +1
2n + 2
k=l
n odd,
'Ir
max {Is~/(n+l) (ei"') I : n:
'
1< < n 2: I}' n even.
<P
For even n the bounds in (23) are slightly larger than those in (22). For some
numerical results see [5]. It is not very likely that the bounds for n even can be
given more explicitely than in (23).
A.3. CIRCULAR DOMAINS
Let n be a circular domain, Le., the interior or exterior of a dosed disko We
normalise the situation as folIows: let Wo = 1 and
np = {
(24)
][)lp := {z
_
: Izl < p},
I.C \ ][)lp,
p> 1,
O<p<l.
Note that the cases with p > 1 belong to the disks, while 0 < P < 1 corresponds to
exterior of a dosed disko We find that for fixed n in each of these cases (essentially)
only one extremal polynomial exists, and that this depends in a simple way of p.
Note that this extends Example 1 from Section 1, where the exterior of a disk has
degenerated to a punctured plane. The polynomial is
Qp (Z2) := ~ z2n+3 ~ {z-n-1 T
n
n +1
dz
n+1
(p-1/(n+l) 1 + z2) }
2z'
where Tn is the Chebyshev polynomial of degree n. Note that Q~ is indeed a
polynomial in P~. In the exterior case we find:
COMPLEX POLYNOMIALS AND MAXIMAL RANGES
47
Theorem 25. For 0< p < 1, nE N, we have
(25)
n~
= exp (eo(log( Q~ (11))))) ) .
Depending on p, n the sets n~ ean be simply or doubly eonnected. In any ease, the
boundary eonsists of sub-ares of Q~(all))) and the boundary of n. This situation
is much better understood if we consider the images of the polynomials involved
on the universal eovering of n. It is dear from the form of Theorem 25 that
it is essentially a deseription of the maximal ranges of z -+ log P(z), eompare
Theorem 3. This carries much more information than the description of n~ as
a plane set; for instanee, it permits to estimate the maximal argument of such
polynomials P.
Corollary 5. Let P E P n satisfy P(O) = 1 and IP(z)1 > p, zEll)). Then
I argP(z)1 :5 (n + 1) areeos(pl/(n+1) eos 2n:
2) - i, zEll)).
This estimate is sharp for P = Q~.
The other direct condusion from Theorem 25 is
Corollary 6. Let P E P n satisfy P(O) = 1 and IP(z)1 > p, zEll)). Then
This estimate is sharp for P = Q~.
An earlier proof of Corollary 6, independently of Theorem 25, was already given
in [25].
For the interior case we obtain:
Theorem 26. Let p > 1, n > 1. 1f p < Pn := (cos7rj(n + 1))-n-l, then n~ is
the interior Jordan domain bounded by 71 + 72, where
72 = { pe'
: 101<
01 := 2 arceos(pl/(n+1) eos
_7r_).
n+l
'(J
and
For p ~ Pn we have n~ = n p •
Here n~ is a convex subset of
n +1 }
7r - -2-01
,
n We mention a few applications of Theorem 26.
p•
Corollary 7. Let n > 1, PE P n , P(O) = 1, and IIPII :5 p. Then, tor zEll)), we
have
ReP(z) ~
[
7r ]-n-l '
pTn+1 (p-l/(n+1») , l:5p:5 cosn+
1
{
-p,
These bounds are sharp.
elsewhere.
48
V. V. ANDRIEVSKII AND S. RUSCHEWEYH
Corollary 8. Let P E P n , P(O) = 1, and
IIPII :::; [cos
2n:
2
r
n
-
1
Then Re P(z) > 0 in [J). The bound given in (26) is best possible.
Corollary 9. Let P(z) E P n with P(O) = 0, P'(O) = 1, and
IIP'II:::; [cos
2n: 2r
n
-
1
Then P is univalent in [J). The bound tor IIP'II is best possible.
Corollary 10. Let PE P n , IIPII :::; 1, and P(I) = & E [0,1). Then
IP(O)I:::; [cos a:c~~&] n+l
The bound is sharp.
Corollary 10, for & = 0, is due to Lachance, Saff, and Varga [17], while the remaining cases have been established in [26]. However, Corollary 7 is stronger than
Corollary 10.
A.4. ONE-SLIT DOMAINS
The case of one-slit domains with Wo in the direction of the slit has been solved
in [7], and with other Wo in [15]. In the first case we can restrict the discussion to
the domain
n:=C\[I,oo),
(27)
wo=O.
Here, and in many other situations a special system of univalent polynomials play
a central röle, which were introduced by Suffridge [28].
n
(28)
P(z; j) =
L Ak,j zk,
j = 1, ... ,n,
k=l
where
. kjn
Slll-A . _ n-k+l
n+l
k,J n
. jn'
Slll-n+l
Theorem 27. Par n as in (27) we have
k,j=I, ... ,n.
3n- } .
le-nl < n+l
As an application we get
COMPLEX POLYNOMIALS AND MAXIMAL RANGES
49
Corollary 11. 1f P E P~(O), 0 = C \ (1,00) then for z E][)
n +1
7f
IImP(z)1 ~ -2- cot n + 2'
7f
cos 2 _ _ cot 2 _7f_ < ReP(z) <
3 2n + 2
2n + 2 -.
7f
.
7f
sm 2n + 2 sm 2n + 2
IP(z)1 ~ cot 2 n: 2·
All bounds are best possible.
In the case of other one-slit domains we may use the normalised situation
oa :=C\{it: t2:a},
wo=l,
aEIR.
We recall Corollary 1, in particular (21), to conclude that if a < - cot7f/(n + 1)
we are essentially back in the half-plane situation and Theorem 22 applies. In the
remaining cases we have
Theorem 28. 1f a 2: - cot 7f / (n + 1) then the only extremal polynomial (up to
rotations) for O~ is
P(z) = Fn(z)
z(2i-yb + ia(8 + 1) - "( + l)«znH - 1)(z 28 - 1) - (n + l)(z - 1)(z8 - 1))
+
(n + l)(z - 1)2(z8 -1)2
'
with,,(=exp(i7f/(n+1)), 8="(2, b=Va2 +1. Wehave80~="(1+"(2, where
"(1
= {p (e it ) : q~ t ~ n 2; I}'
"(2
= {it : a ~ t ~ P(e 2i7r /{nHl)} ,
with
-47f
q._ { n + l'
37f
-cot-<a
n+ 1 - ,
7f
7f
37f
arccota - 27f - - - , - cot - - < a < - cot - - .
n+1
n+1n+1
Clearly this result has various corollaries. We mention only one which extends
Fejer's theorem mentioned in Section Al.
Corollary 12. Let P E P n be such that P(O) = 1 and P(][)) c C \ {it : t 2: a}
where a 2: - cot 7f /(n + 1). Then
n (Ja
+7f + acot -7
-f)
sin--
1
- -+2
2
1
n+1
n+1
-1
~ ReP(z)
+
n 1 (
~ -
2
va
2
+7f 1 + a cot --1
7f
+ 1) .
sin-n+
n+1
V. V. ANDRIEVSKII AND S. RUSCHEWEYH
50
A.5. TWO-SLIT DOMAINS
In this seetion we diseuss the ease of the domains
oa := C \ (( -00, -al U [1,00)),
Wo = 0,
for a > O. It follows from the previous seetion that if a is too small or too big,
then the maximal range looses eontaet with the one or the other of the slits, and
we are baek in the one-slit situation. Taking this into aeeount we see that we ean
restrict our attention to the eases where
-2
1l'
2
1l'
eot 2n + 2 < a < eot 2n + 2 .
To state our result define "1 := 1l' /(n + 1), and, for r = 1, ... ,n - 1:
1
2
r1l'
ar := tan 2n + 2 .
fr(B) := eos B - eos (r"1) ,
We also reeall the Suffridge polynomials from (28).
Theorem 29. Let a E [ar, ar+1), r E {I, ... ,n - I}. Then there is exactly one
a E (0,1] with
h r ( (h) = h r ( ( 2 ) = a : 1 ,
B1 E ( - n: 1 ' 0] ,
B2 E (n n; 1 ' 1l']
for which
is satisfied. Then
(30)
aP(z; r) + (1 - a)P (e i1T /(n+1) Z', r + 1)
P( )
E p~(oa),
z := aP(ei1h;r) + (l-a)P(e i1T /(n+1)+lh;r+ 1)
and
(31)
80~ \80 a
'0
r - 2
r +2 }
= { P(e'):
--1l' < B < --1l' +ej.
n+1
n+1
Note that the eases a = a r are very simple sinee then a = 1, B1 = 0 and the
polynomials (30) are typically real. As the special ease of probably greatest interest
we diseuss the one with a = 1 (symmetrie slit):
Corollary 13. Let 0 = C\ ((-00, -1] U [1,00)). Then for n odd, we have
(32)
P(z) = P (z; (n + 1/2) E pO(O)
P (1; (n + 1)/2)
n
and
(33)
80 n \ao={p(e iO ) :
~-n2:1 <B<~+n2:1} +ej.
For n even, the extremal polynomial and the maximal range On are described
by (30), (31), with a = 1, a = 1/2, r = [(n + 1)/2], and h(Bt) = -1, B1 E
(-1l'/(n + 1),0).
COMPLEX POLYNOMIALS AND MAXIMAL RANGES
51
Corollary 14. Let n be as in Corollary 13, n odd. Then for P E p~(n) we have
in ][))
1
IReP(z)l:$ -•- - , ;7rp -
(34)
sm-n+l
and
IP(z)1 :$ n; 1 .
(35)
This bound is sharp for P as given in (32) at z = ie- ill"/(n+l), i, respectively.
Note that this result holds, in particular, for typically real polynomials P E p~
with IP(x)1 :$ 1, x E (-1,1).
A.6. SECTORS
We now look at infinite angular sectors with an opening of Cl:7r, and with Wo on
the bisector. This can be normalised as
(36)
n'" := {z E C : Re z+ cot Cl:27r 11m zl < I},
Wo = 0,
°< <
Cl:
1.
It turns out that for n odd we will have one extremal polynomial for the maximal
range, and for n even there will be two of them.
To state the result we define
(37)
tk := 7r
1- Cl: + 2k
l'
n+
S(1)(z) := Tn+l(z) - 2nz
k = 0, 1 ... ,n,
II (Z2 - cos2 t;) ,
n/2-1
k=O
S(2)(Z) := Tn+l(z) - 2nz
ii:
(Z2 _ cos2
t;) ,
k=n/2+1
!!
(n-l)/2
S(3)(z) := Tn+l(z) and for j = 1,2,3,
2n
(z2 _ cos2
t;) ,
52
V. V. ANDRIEVSKII AND S. RUSCHEWEYH
Theorem 30. Let 00< be as in (36). Then
00< = { co (p{l)(JD)) U p(2)(JD))) , n even,
n
co (p(3) (JD))) ,
n odd.
The form of the extremal polynomials does not seem to admit a niee explicit bound
for sup Re O~ or inf Re O~. We have, however,
Corollary 15. Let 00< be as in (36). Then there exist constants Cj = cj(a) > 0
such that
Cl ~ nO«I- sup(ReO~)) ~ C2,
and
(38)
Note that an estimate corresponding to (38), but for sup 10nl, can be derived from
Theorem 12.
A.7. RHOMBS
We are looking at rhombs symmetrie to the real and the imaginary axis, and with
the origin as reference point:
(39)
00< := {z E C : IRe zl + cot a2'1r 11m zl < I},
Wo = 0,
0 < a < 1,
whieh have an interior angle of mr at the corner in z = 1. A complete explicit
solution of the O~ problem is so far only available for n odd. We restrict ourselves
to this case. The formalism is similar to the one in the previous section.
We define tk as in (37) and set
p
g(z) :=
For n = 4p + 1 let
II (Z2 - cos tk)' p:= [i] .
2
k=O
S{l)(z):= cos(7r 1- a) t(-I)k
2g(Z)costk
,
2
k=O
g'(costk)(Z2 - cos2 tk)
p(l)(z) := z(n+1)!2 [S{l)(w) and for n = 4p + 3:
n:
1 S(l)' (w)] ,
w = ~ (z + ~),
S(2)(z) ._ cos(7r 1- a) ~(_I)k
2g(z)z
.2
L...J
g'(costk)(z2-cos2tk)'
k=O
p(2)(z) := z(n+1)!2 [S(2)(W) -
n:
1 S(2)' (w)] ,
w = ~ (z + ~),
Of course, p(2) depends on a. If P* denotes the corresponding polynomial for the
parameter 1 - a instead of a, then we set
p(3)(z) := i cot('Ir 1 ~ a)p*(z).
COMPLEX POLYNOMIALS AND MAXIMAL RANGES
53
Theorem 31. Let n" be as in (39) and n odd. Then
n" _ { co (p(1) (lDl)) ,
n -
n = 4p+ 1,
co (P(2)(lDl) U p(3) (lDl)) , n = 4p+3.
The discussion of the maximal distance of n~ from an" is closely related to the fact
that n" has always one corner with an opening ~ 7r /2, where the approximation
is bad (compare Section 5). Therefore we restrict our attention to the distance
from the point z = 1 to the boundary of n~. Here we get
Corollary 16. For 0< a < 1 there exists a constant C5 = c5(a) such that
A.8. HALF-PLANES AGAIN
As pointed out in the discussion of Example 2 in the first section, the theory
developed in Section 2 also permits the study of problems like the determination
of the sets
~n,r := U {P(lDlr ) : P E P n , P(O) = 1, ReP(z) > 0 for z E lDl},
0< r < 1.
Indeed, Theorem 32 below gives at least an indirect solution to this problem, which
can be used to get the desired information numerically. For details we refer to [5].
n
Theorem 32. Let 0 ~ () < 27r. I/ Qe,r (z) = z TI (z - ei</>i) satisfies the equation
j=l
j = 1, ... ,n,
and i/ P E P n with P(O) = 1 is given by
2
R P ( i</» = IQe,r (ei</» 1
e
e
IIQe,rll~'
then P(r) E a~n,r. Furthermore, each boundary point 0/ ~n,r can be obtained this
way.
References
1. L. V. Ahlfors, Leetures on Quasieonformal Mappings, Van Nostrand, Princeton, N.J., 1966.
2. V. Andrievskii, Approximation of harmonie functions in eompaet sets in C, Ukrain. Mat.
Zh. 45 (1993), 1467-1475.
3. V. Andrievskii and St. Ruscheweyh, Maximal polynomial subordination to univalent funetions in the unit disk, Constr. Approx. 10 (1994), 131-144.
4. _ _ , The maximal range problem for a bounded domain (to appear).
5. A. C6rdova and St. Ruscheweyh, On maximal ranges of polynomial spaees in the unit disk,
Constr. Approx. 5 (1989), 309-328.
54
V. V. ANDRIEVSKII AND S. RUSCHEWEYH
6. ___ , On maximal polynomial mnges in circular domains, Complex Variables Theory
Appl. 10 (1988), 295-309.
7. ___ , On the maximal mnge problem for slit domains, Proc. of a Conference, Valparaiso,
1989, Lect. Notes Math. 1435, Springer, Berlin - Heidelberg - New York, 1990, pp. 33-44.
8. ___ , Subordination of polynomials, Rocky Mountain J. Math. 21 (1991), 159-170.
9. _ _ _ , On the univalence of the extremal polynomials in the maximal mnge problem (to
appear).
10. V. K. Dzyadyk, Introduction to the Theory of Uniform Approximation of Functions by
Polynomials, Nauka, Moscow, 1977. (Russian)
11. E. Egervary, Abbildungseigenschaften der arithmetischen Mittel der geometrischen Reihe,
Math. Z. 42 (1937), 221-230.
12. F. W. Gehring, Chamcteristic Properties of Quasidisks, Les Presses de l'Universite de
Montreal, 1982.
13. R. Greiner, Zur Güte der Approximation Schlichter Abbildungen durch Maximal Subordinierende Polynomfolgen, Diplomarbeit, Würzburg, 1993.
14. R. Greiner and St. Ruscheweyh, On the approximation of univalent functions by subordinate
polynomials in the unit disk, Approximation and Computation (R.V.M. Zahar, ed.), ISNM
119, Birkhäuser, Boston, 1994, pp. 261-27l.
15. C. Günther, St. Ruscheweyh, and L. Salinas, New maximal mnges (to appear).
16. F. Holland, Some extremum problems for polynomials with positive real part, Bull. London
Math. Soc. 5 (1973), 54-58.
17. M. Lachance, E. B. Saff and R. S. Varga, Inequalities for polynomials with a prescribed zero,
Math. Z. 168 (1979), 105-116.
18. M. Lavrentiev, Sur la continuite des fonctions univalentes, C.R. Acad. Sei. USSR 4 (1936),
215-217.
19. J. Lewis, Applications of a convolution theorem to Jacobi polynomials, SIAM J. Math. Anal.
10 (1979), 1110-1120.
20. O. Martio and J. Sarvas, Injectivity theorems in plane and space, Ann. Acad. Sei. Fenn., Ser.
A I Math. 4 (1978), 383-40l.
21. G. Meinardus, Approximation of Functions: Theory and Numerical Methods, Springer, New
York,1967.
22. H. P. Mulholland, On two extremal problems for polynomials in the unit circle, J. London
Math. Soc. 31 (1956), 191-199.
23. Chr. Pommerenke, Boundary Bevaviour of Conformal Maps, Springer, New York, 1992.
24. St. Ruscheweyh, Geometrie properties of Cesaro means, Resultate Math. 22 (1992), 739748.
25. St. Ruscheweyh and R. S. Varga, On the minimum moduli of normalized polynomials with
two prescribed values, Constr. Approx. 2 (1986), 349-369.
26. St. Ruscheweyh and K. J. Wirths, On an extremal problem for bounded polynomials, Approx.
Theory Appl. 1 (1985), 115-125.
27. S. Smale, The fundamental theorem of algebm and complexity theory, Bull. Amer. Math.
Soc. 4 (1981), 1-36.
28. T. J. Suffridge, On univalent polynomials, J. London Math. Soc. 44 (1969), 496-504.
29. ___ , Starlike functions as limits of polynomials, Advances in Complex Function Theory
(Maryland, 1973/1974), Lect. Notes Math. 505, Springer, Berlin - Heidelberg - New York,
1976, pp. 164-203.
30. G. Szegö, On conjugate trigonometrie polynomials, Amer. J. Math. 65 (1943), 532-536.
EXACT CLASSICAL POLYNOMIAL INEQUALITIES
IN Hp FOR O<p<oo
VITALII V. ARESTOV
Ural State University, Lenin Avenue, 51, 620083 Ekaterinburg, Russia
Abstract. This paper is devoted to the exact Bernstein, Szegö and Zygmund inequalities
for trigonometrie polynomials (on the realline) and for algebraic polynomials on the unit
disk in the complex plane, as weH as to some more general inequalities.
1. Let Tn be the set of trigonometrie polynomials
n
(1.1)
fn(t) = ~o + '~::)akcoskt+bksinkt)
k=l
of degree n with complex coefficients. The known and often used Bernstein inequality [6]
(1.2)
Ilf~llc ~ nllfnllc,
fn E Tn,
holds in the set Tn; here IIflic is the uniform norm IIflic = max{lf(t)l: t E
[0, 27r]}. Bernstein proved this inequality in 1912 with the constant 2n. In 1914,
M. Reisz [12] got this inequality with the best constant n using the known Reisz interpolation formula for derivatives of trigonometrie polynomial. As a consequence
of (1.2), the exact inequality
(1.3)
holds for any r :::: 1. Different generalisations of these inequalities are known in
the literature. In 1928 Szegö [14] proved that the exact inequality
(1.4)
IIf~ cosa + 1~ sinallc ~ nllfnlb
fn E Tn,
holds for any real a, where
n
ln(t) = ~)bk cos kt - ak sin kt)
k=l
1991 Mathematics Subject Classijication. Primary 26C05, 26D05, 42A05j Secondary 41A44.
Key words and phrases. Polynomial inequalitiesj Normj Best constantj Algebraic polynomialsj
Trigonometrie polynomials.
The research supported by the Russian Foundation for Fundamental Research (Project 96-0100122).
55
G.v. Milovarwvic (ed.). Recent Progress in Inequalities. 55--62.
© 1998 Kluwer Academic Publishers.
V. V. ARESTOV
56
is the conjugate polynomial for In. In partieular, this inequality implies Bernstein
inequalities (1.2) - (1.3) and the inequality
(1.5)
for derivatives of order r ~ 1 of the conjugate trigonometrie polynomial.
In 1933 Zygmund [15, Vol. II, eh. x, (3.25)] proved the following statement. Let
a function r.p be downward convex and nondecreasing on the half-line [0,00); then
(1.6)
211"
1211"
1o r.p(1 COSQ/~(t) + sinQl~(t)l)dt ~ 0 r.p(nl/n(t)l)dt,
In E Tn.
Putting r.p(u) = u P , P ~ 1, we obtain the inequality
(1.7)
in the space L p with the norm
(1.8)
1 1 21"
) l/p
1I/IILp = ( 2'1r 0 II (t)IP dt
It follows from (1. 7) that the inequalities
(1.9)
(1.10)
are valid for any natural rand 1 ~ p ~ 00. All the above-mentioned inequalities
are exact and they reduce to equalities for polynomials In (t) = a cos nt + b sin nt.
In what follows we consider the functional II . IIp for 0 ~ p ~ 00. In the case
o < P < 00 we assume that the functional is defined by (1.7). For extreme values
of p we put
(1.11)
1I/IIL oo = lim 1I/IILp = 1I/IIc = max{l/(t)l: tE [0,2'1r]}
P-++OO
and
(1.12)
1I/IILa = lim II/IIL = exp
P-++O
p
(211 211" log I/(t)ldt) .
'Ir
0
Studying direct and inverse theorems of approximation theory in the space L p ,
o < P < 1, Ivanov [9], Storozenko, Krotov, Osvald [13] proved in 1975 that the
inequality
(1.13)
EXACT CLASSICAL POLYNOMIAL INEQUALITIES IN Hp FOR 0 ::; p ::; 00
57
holds for 0 < P < 1 with some constant cp •
These works initiated a number of results on refinement of constant Cpo In 1979
Arestov [1] (see [2] for details) proved that for all p, 0 ~ p < 1, the constant cp
in inequality (1.13) is equal to 1, and so, the inequality (1.9) holds for all p ~ o.
With respect to inequality (1.10) this is not a such case.
We denote by Kp(n, r) the least constant in the following inequality
fE Tn·
(1.14)
The previous results of Szegö and Zygmund can be written as
Kp(n,r) = n r
for
1 ~ p ~ 00.
The below-mentioned results of this section are proved in our paper [5].
Theorem 1. For any 0 ~ p ~ 00 and for integer n ~ 0 and r ~ 0 the following
inequalities
(1.15)
hold.
The last inequality means that the constant Kp(n, r) as function of parameter p
achieves its maximum at p = o.
Theorem 2. For all integer n ~ 0 and r ~ 0 and for p = 0 the extremal polynomial in inequality (1.14) is
and, as a consequence,
Theorem 3. 1f r ~ n log 2n then
(1.16)
for all p ~ o.
The equality (1.16) holds for all p ~ 0 if and only if it holds for p = O. The
condition r ~ nlog2n is only sufficient for Ko(n,r) = n r . This equality holds if
and only if all 2n zeros of the concrete polynomial h~) are real. The computer
calculations show that the condition r ~ n - 1 is probably necessary and sufficient.
58
V.V.ARESTOV
Theorem 4. For a fixed rand for n -+ 00 the following relation is valid
Ko(n, r) = 4En ,
Cn
= n + o(n).
Thus, for a fixed r the constant Ko(n, r) as function of n increases essentially more
quickly than Kp(n,r) = n r for 1;::; p;::; 00.
2. The previous inequalities for trigonometrie polynomials can be written as corresponding inequalities for algebraic polynomials on the unit disk (more exactly
on the unit circle) in the complex plane. Other interesting inequalities are also
valid for algebraie polynomials on the unit diskj Bernstein and Szegö inequalities
are among the most famous. Let P n be the set of all algebraic polynomials Pn of
degree n with complex coefficients. It is convenient to write it in the form
(2.1)
The following inequality, obtained by Szegö [14] in 1928,
(2.2)
holds in the set P n , where IIPnlloo = max{IIPn(z)lI: Izl = I}. Obviously, (2.2)
implies the well-known Bernstein inequality
(2.3)
obtained by himself in 1926 (see cite7). It is clear that the inequalities
(2.4)
IIDrpnll oo ;::; nrll RePnll oo ,
(2.5)
IIDrPnll oo ;::; nrlIPnlloo
are valid parallel with (2.2) and (2.3) for any integer r ~ 1, where Dr is the k-th
power of the operator
d
D=z-.
dz
The inequality (1.9) includes the inequality
(2.6)
for 1 ;::; p < 00 and one more general inequality
(2.7)
for 1 ;::; p < 00 and r ~ 1 j in these inequalities
EXACT CLASSICAL POLYNOMIAL INEQUALITIES IN Hp FOR 0::::; p ::::; 00
59
Inequalities (2.2)-(2.7) are exact and they reduce to equalities only for polynomials
cz n .
In the author's papers [1-2] it was shown that the inequality (2.6), and as a
consequence, the inequality (2.7) are valid for 0 :::; p < 1 too; here the functional
IlPlip is defined by
IIPlip = (2~
1
27r
IP(eit)IPdt) l/p ,
IlPlio = P-++O
lim IlPlip = exp
0< p < 00,
(21 r log
7r J-7r
IP(eit)ldt) .
Thus, Bernstein inequality (2.5) can be extended (with the constant n T ) to the
spaces Hp for all values of p, 0:::; p :::; 00. With respect to Szegö inequality (2.2),
this is not such a case.
Let ICp(n, r) be the best constant in the following inequality
(2.8)
for 0 :::; p :::; 00 and r ~ O. The constant ICp(n, r) depends on p even for 1 :::; P :::; 00.
Szegö result can be written as the equality ICoo(n, r) = n T • It is easy to check that
IC 2 (n, r) = ../2nT • The statements below in this section are proved in the author's
paper [4].
Theorem 5. For any 0 :::; p :::; 00 and for any integer n ~ 0 and r ~ 0 the
following inequalities
(2.9)
n T :::; ICp(n, r) :::; ICo(n, r)
hold.
So, the constant ICp(n, r) as function of p takes its maximum at p = O.
Theorem 6. For all integer n ~ 0 and r ~ 0 and for p = 0 the extremal polynomial in the inequality (2.8) is given by
and, as a consequence,
Using this theorem it is easy to show that the estimate
(2.10)
is valid for all values of parameters n, rj according to the next theorem this
estimate is exact for large values of parameter r.
60
V. V. ARESTOV
Theorem 1. If
r~
log2n
n'
log-n-1
then
(2.11)
Theorem 8. FOT a fixed rand fOT n -+ 00 the following asymptotic equality is
valid
ICo(n, r) = 4e", Cn = n + o(n).
3. Most of the preceding statements are the consequences of the author's inequalities [2-3] for Szegö composition of algebraic polynomials. The polynomial
AnPn(z) =
(3.1)
t (~)
AkCkZ k
k=O
is called the Szegö composition of polynomials
An(z) =
(3.2)
t (~)
Ak Zk
k=O
and
(3.3)
A lot of papers are devoted to the properties of this operation (see [11, Vol. II,
Sec. V], [10, eh. IV] and references there). For fixed An equality (3.1) defines a
linear operator in Pn ; we shall denote this operator by the same symbol An as the
polynomial (3.2).
Let ~ be the set of all functions cp on (0,00) such that cp is nondecreasing on (0,00),
absolutely continuous on any finite segment of (0, 00) and the function ucp'(u) does
not decrease on (0,00). For example, the functions log u and u P for < P < 00
satisfy these conditions. In the author's paper [3] the following statement was
proved.
°
Theorem 9. FOT any function cp E ~ and for any polynomials An, Pn E P n the
following inequality
(3.4)
1
2
1<
cp(I(AnPn)(eit)l) dt ~
1
2
1<
cp(IIAn 1I0IPn (eit)l) dt.
holds.
By P~ we denote a subset of polynomials from P n such that all their n roots lie in
the closed unit disk Izl ~ 1. Similarly, let P;:O be the subset ofpolynomials PE P n
with all roots outside the disk Izl < 1. Finally, the subset P~ n P;:O of polynomials
from P n such that all their n roots lie on the unit circle will be denoted by P~.
The known Jensen formula for meromorphic functions (see, for example, [11, Vol.
I, Sec. III, Thm. 175] implies that if An E P~, then IIAnil o = IAnl, and if An E P;:O,
then IIAnil o = IAol. Thus, Theorem 9 includes the following statement, although it
was obtained earlier [2].
EXACT CLASSICAL POLYNOMIAL INEQUALITIES IN Hp FOR 0 ~ p ~ 00
61
Theorem 10. If An E P~ or An E P~, then for any junction cp E ~ in P n the
exact inequality
(3.5)
holds with the constant
If one of the following conditions holds
then inequality (3.5) reduces to an equality for the following polynomials
(a, bE C),
respectively.
It was proved in [8], that under certain supplementary restrictions on cp and An
only polynomials (3.7) are extreme in inequality (3.5).
For function cp{ u) = log u E ~ inequality (3.4) can be written as
(3.8)
this inequality was written earlier in some different form in [7, Thm. 7]. It is easy to
see that inequality (3.8) reduces to an equality for the polynomialll'n(z) = (l+z)n.
Given a fixed An for 0 :::; p :::; 00, we denote by Np(A n ) the least constant in the
inequality
IIAnPnll p :::; Np{An)llPnllp, Pn E Pn.
As it was noted just now, No (An) = IIAnll o. By Theorem 9 the estimate
is valid.
Let us remind for comparison the well-known fact
Under conditions of Theorem 10 on the polynomial An, the equality
holds for all values of p, 0:::; p :::; 00.
62
V. V. ARESTOV
References
1. V. V. Arestov, On inequalities of S. N. Bernstein for algebraie and trigonometrie polynomiais, Soviet Math. Dok!. 20 (1979), 600-603.
2. ___ , Integral inequalities for trigonometrie polynomials and their derivatives, Izv. AN
SSSR Ser. Mat. 45 (1981), 3-22 (Russian) [Eng!. Trans.: Math. USSR-Izv. 18 (1982), 1-17].
3. ___ , Integral inequalities for algebraie polynomials on the unit eircle, Mat. Zametki 48
(1990), 7-18. (Russian)
4. ___ , On one Szego inequality for algebraie polynomials, Trudy Inst. Mat. Mekh. (Ekaterinburg) 2 (1992), 27-33.
5. ___ , The Szego inequality for derivatives of a eonjugate trigonometrie polynomial in Lo,
Math. Notes 56 (1994), 1216-1227.
6. S. N. Bernstein, Sur I'ordre de la mailleure approximation des fonctions eontinues par des
polynomes de degre donne, Memoires de l'Academie Royale de Belgique (2) 4 (1912), 1-103.
7. _ _ _ , Ler;ons sur les proprietes extremales et la meilleure approximation des fonctions
analytiques d'une variable ree!e, Collection Borei, Paris, 1926.
8. N. G. de Bruijn and T. A. Springer, On the zeros of eomposition-polynomials, Neder!. Akad.
Wetensch. Proe. 50 (1947), 895-903 [= Indag. Math. 9 (1947), 406-414].
9. V. I. lvanov, Some inequalities for trigonometrie polynomials and their derivatives in different metrics, Mat. Zametki 18 (1975), 489-498. (Russian)
10. M. Marden, The Geometry of the Zeros of a Polynomials in a Complex Variable, Math.
Survey, No. 3, Amer. Math. Soc., New York, 1949.
11. G. Polya and G. Szegö, Problems and Theorems in Analysis, Vols. land II, Springer Verlag,
Berlin, 1976.
12. M. Riesz, Eine trigonometrische Interpolationsforme! und einige Ungleichungen für Polynome, Jahresbericht der Deutschen Mathematiker-Vereinigung 23 (1914), 354-368.
13. E. A. Storozenko, V. G. Krotov and P. Osval'd, Direct and converse theorems of Jackson
type in LP spaces, 0< p < 1, Mat. Sb. (N.S.) 98 (140) (1975),395-415. (Russian)
14. G. Szegö, Über einen Satz des Herrn Serge Bernstein, Schrift. Königsberg. Gelehrten
Gesellschaft. 5 (1928), 59-70.
15. A. Zygmund, Trigonometrie Series, Vols. land II, Cambridge Univ. Press, Cambridge,
1965.
VIETORIS'S INEQUALITIES AND
HYPERGEOMETRIC SERIES
RlCHARD ASKEY
University 0/ Wisconsin-Madison, Madison, WI53706, U.S.A.
Abstract. The inequalities of Vietoris have been a good source of problems and new
results. Some of these are outlined, and a hypergeometric sum suggested by one of the
problems is evaluted.
1. Fejer. There are few serious collectors of mathematies. D. S. Mitrinovic
was one of them. I became aware of his collecting ability when [24] appeared.
In addition to many familiar inequalities, there were some I had not seen before.
One, in partieular, was a great surprise. To explain why, and to show a bit about
what this inequality of Vietoris suggests, is the aim of this paper.
At the start of this century, Fejer [14] showed that inequalities for trigonometrie
polynomials can be very useful. In partieular, his work on (C, 1) summability of
Fourier series is based on the inequality.
(1.1)
This is a discrete extension of
(1.2)
l
Z
sintdt = 1- cosx ~ O.
The obvious extension of (1.2) is
n
(1.3)
LSin k8,
k=l
but this is not nonnegative for all 8, 0 ~ 8 ~ 1C'. For
sin8 + sin28 = sin8(1 + 2cos8)
1991 Mathematics Subject Classification. Primary 26D05, 33C05, 33C20j Secondary 26D15.
Key words and phrases. Inequalitiesj Hypergeometric series.
Supported in part by NSF grant DMS-9300524.
63
G. V. Milovanovic (ed.), Recent Progress in lnequalities, 63-76.
© 1998 Kluwer Academic Publishers.
R.ASKEY
64
Fejer was aware that a result like (1.1) implies much more. Summation by parts
gives
(1.4)
when
(1.5)
ak ~ akH ~ 0, k
= 0, 1, ... ,n - 1.
This suggests that while the sum in (1.3) is not nonnegative, there might be a
useful substitute for it if a decreasing sequence ak is introduced as a multiplier.
Fejer considered the sequence ak = l/k and conjectured that
(1.6)
~sink(}
~ k
> 0,
0<(}<1r.
k=l
Fejer gave the history of this inequality in [17]. He had mentioned this conjecture
to E. Landau, and then received proofs from T. H. Gronwall (October 22, 1910)
and D. Jackson (December 19, 1910). These proofs were published as [21] and
[22].
There are many proofs of (1.6), including some which are referenced or outlined
in [24]. Also see [23] for some other references. One striking fact which I thought
was important is that the sequence ak = l/k seemed to be sharp. To see this,
divide (1.6) by sin () and let () -+ o. The result is the series
which vanishes when n is even.
Imagine my surprise when looking through section 3.5 of [24] and coming across
a result of Vietoris [28] which shows that this inequality is not sharp.
2. Vietoris. While trying to find a proof of the Fejer-Jackson-Gronwall inequality
(1.6) which he liked, Vietoris discovered an extension. I am not sure how he
discovered this result, but here is one way to motivate it. One can try to be more
greedy then Fejer was. For the series with two terms, the best you can do to obtain
a nonnegative sum is
sin () + ~ sin 2(} = sin (}(1 + cos 8) ~ o.
Any positive number less than ~ can be used, but a partial summation shows that
~ gives a stronger result. Nothing larger than ~ can be used. Fejer extended the
sequence 1, ~ to
111
1, 2' 3' 4'
VIETORIS'S INEQUALITIES AND HYPERGEOMETRIC SERIES
h
65
Given the terms 1, ~,
then ~ is the best that can be used, as was seen earlier
by dividing by sin 0 and letting 0 approach 7r. What Vietoris did was to replace
by a larger number. If a monotone decreasing sequence is desired, then the best
choice for the coefficient of sin 30 in
t
sin 0 + ~ sin 20 + a sin 38
is a = ~. Vietoris (28) showed that this series is positive for 0 < 0 < 7r. Then, for
the next coefficient, the best choice is ~ . ~, which works. Again there is equality
when the series is divided by sin 0 and 0 is taken to be 7r.
In general, Vietoris proved the following:
Theorem 1. I/
(2.1)
and
(2.2)
2ka2k S (2k - 1)a2k-l,
k = 1,2, ... , [nj2),
then
n
(2.3)
L ak sin kO > 0, 0 < 0 <
7r,
k=l
and
n
(2.4)
L ak cos kO > 0, 0 < 0 <
7r.
k=O
The critical case is when there is equality in (2.2) and in the remaining inequalities
in (2.1). Then
(2.5)
where
(2.6)
(ah = a(a + 1)··· (a + k - 1) = r(k + a)jr(a).
The general case follows from this special case via summation by parts.
After I saw this inequality, Iwanted to make sure that attention was called to
it. One way to do this was to have the review in Mathematical Reviews mention
this inequaity as a highlight. It seemed likely that Ralph Boas would be asked
to review (24), so I wrote hirn and mentioned this inequality, and suggested he
mention it if he were asked to write the review. He replied immediately that he
had been asked to write the review, and asked if Iwanted to write it instead of
R. ASKEY
66
hirn. I said no, but would be willing to write a joint review with hirn. This is how
this joint review was written for Mathematical Reviews.
I sent a copy of our review to John Steinig. He replied that Mathematical Reviews
had not published a review of Vietoris's paper [28]. He was right. The Executive
Editor checked the files and found that it had been sent out for review three times,
but returned unreviewed each time. I wrote a review, and included an application
to show that the inequalities (2.3) and (2.4) are useful. This review was turned
down on the grounds that the paper had appeared long before, and they did not
have the resources to go back and fill in all the missing reviews. The editor also
said that he found rny application more interesting than Vietoris's inequalities, so
I should publish the application. The review of it would call attention to Vietoris's
paper. He was wrong about the relative importance. Exactly how wrong will be
seen in the next section.
3. Askey and Steinig. I decided I did not have the energy to fight for this
review, so contacted John Steinig and suggested we write a joint paper. He had
rnentioned that part of Vietoris's argument could be simplified. While writing the
paper we came up with greater simplifications, much more interesting applications,
and were able to irnbed these inequalities into a problem of quadrature studied
by Fejer, P6lya, and Szegö. See [9]. The earlier application I had mentioned
was dropped and I have forgotten what it was. The new work was much more
interesting than that application. However, I still find Vietoris's observations and
his inequalities more interesting than anything I have been able to do with his
inequalities.
Here is a setting for these inequalities.
Vietoris observed that
(3.1)
so that
(3.2)
(1 + e- i9 )(1 - e- 2i9 )-1/2 =
L
00
Ck e- ik9 ,
k=O
with
(3.3)
Taking real and imaginary parts gives
(3.4)
1
()
("2 cot "2)
1/2
=
L
00
k=1
Ck sin k() =
L
00
Ck cos k().
k=O
Then Vietoris showed that the partial sums of both series are positive for 0 < () <
7r. It is the formulas in (3.4) which can be used to explain one general problem
which leads to the inequalties Vietoris proved.
VIETORIS'S INEQUALITIES AND HYPERGEOMETRIC SERIES
67
A natural setting for this problem is Jacobi polynomials p~o:,ß)(x). They can be
defined as
(3.5)
When a,ß > -1, these are orthogonal:
m ",n,
(3.6)
where
m=n,
.(o:,ß) _
Jn
-
r(n + a + l)r(n + ß + 1)
(2n + a + ß + 1) r(n + l)r(n + a + ß + 1)
2o:+ ß+1
There are four special cases which reduce to more familiar functions. When x =
cosO, a = ß = -1/2 gives a multiple of cosnO; a = ß = 1/2 gives a multiple of
sin(n + I)O/sinO; a = -ß = 1/2 gives a multiple of sin(n + ~)O/sin(O/2); and
a = -ß = -1/2 gives a multiple of cos(n + ~)O/ cos(O/2). See Szegö [27, Chapter
IV] for results on Jacobi polynomials.
One problem which leads to Vietoris's inequalities is to form the series
(3.7)
(1 - x)-1' (1 + x)-6 '"
L akP~a,ß) (x),
00
k=O
where
The question is for which a,ß,,,/,8 are the partial sums of (3.7) positive for -1 <
x < 1 or nonnegative for -1 ~ x ~ 1.
When
(3.8)
111
a = ß = -"2' "/ = 4"' 8 = -4"' x = cosO,
the series (3.7) becomes the eosine series in (3.4).
When
(3.9)
1
3
1
a=ß="2' "/=4"' 8=4"' x=cosO,
the series (3.7) reduces to the sine series in (3.4).
Another special case comes from the generating function for Legendre polynomials,
the case a = ß = 0 of (3.5). This generating function is
(3.10)
(1 - 2xr + r 2)-1/2 =
L Pk(x)r k .
00
k=O
R.ASKEY
68
Then r = 1 gives
T 1/ 2(1_ X)-1/2 =
(3.11)
L Pk(X).
00
k=O
Fejer [15] proved that
-1< X< 1.
(3.12)
More generally,
(3.13)
For -1/2 < 0 < 0 and -1 < 0 < -1/2 respectively, Fejer [16] and Szegö [26] have
proven that
-1< x < 1.
(3.14)
Fejer's first sum (1.1) arises from the following.
(3.15)
Then formally
(3.16)
f
k=O
r k sin(k + ~)(} = (1 + r) sin((}/2) .
2
1- 2rcos(} + r 2
f
k=O
sin(k + ~)(}
1
sin((}/2) - 1 - cos(}·
While this series diverges, it is the formal orthogonal expansion of
(3.17)
1
1 - x '"
00
p~1/2,-1/2)(X)
L p(-1/2,1/2)(I)
k=O
k
which exists since (1 - x)-l(1 - X)1/2(1 + X)-1/2 is integrable on (-1,1). The
partial sums of the series (3.16) are just those Fejer looked at, and are given as
(1.1) after (1.1) is divided by sin()/2.
Other examples arise from a quadrature problem. This occurs when 'Y = 0,8 = ß.
See [17, Theorem 15.2.4] for the connection with this quadrature problem. In [3],
the cases o,ß ~ 0,0 + ß ~ 1, and 0 = ß + 1, 1/2 ~ 0 ~ 1 were shown to give
positive partial sums. Earlier, the case 0 = ß, -1 < 0 ~ 3/2 had been shown to
lead to positive partial sums. See [5] for the final intervals and references to the
VIETORIS'S INEQUALITIES AND HYPERGEOMETRIC SERIES
69
earlier results. For 0: > 3/2 or ß > 3/2, positivity fails when 'Y = 0:,8 = ß. See
Szegö [27].
4. Hypergeometrie functions. With positivity having been shown in some
special cases, it is worthwhile recording some intermediate calculations. First, it
is easy to find the coefficients in (3.7).
Jacobi polynomials can be given as hypergeometric series. For instance,
(4.1)
p(a,ß)( )
x
k
= (o:+l)k F (-k,k+O:+ ß +l.I-X)
k!
2
0: + 1
1
'2
'
where
(4.2)
This can be used in
to show that
(4.4)
a
= F (-k, k + 0: + ß + 1,0: - 'Y + 1.1) x
0: + ß - 'Y - 8 + 2,0: + 1 '
(2k + 0: + ß + 1)(0: + ß + Ihf(o: + ß + 2)f(0: - 'Y + l)f(ß - 8 + 1)
x ~~--~~~~~~~~~~~~~~--~~~~--~~
(0: + ß + 1)(ß + l)k21'H f(0: + l)f(ß + l)f(o: + ß - 'Y - 8 + 2)
k
3
2
All of the cases which were treated above had coefficients which are products
rather than a sum of products. This tells us that the 3F2 can be evaluated for
many different choices of the parameters.
There are a number of sums of 3F2S which are responsible for being able to sum
this series. One is Watson's sum:
(4.5)
F (
32
f(!)f(c
+ !)f(tllli)f(
c-a-btl)
)
a, b,C
•1 _
2
2
2
2
(a+b+l)/2,2c'
-f(~)f(~)r(c+12a)f(c+12b)·
Another sum is the Pfaff-Saalschütz sum:
(4.6)
F (
3
2
.1) _(C)k(C
(c -
-k, a, b
c, a + b + 1 - k - c'
-
a)k(c - b)k
a - bh .
This leads to the summation of the 3F2 when 8 = 0, and so the case (3.14) as well
as (3.17). Both of these sums are included in [10] and a number of other books on
special functions.
R. ASKEY
70
The cases Vietoris treated seem to be dose to Watson's sumo They come from
p. (
(4.7)
32
a, b,c
(a+b+2)/2,2c;
1) .
However, if you know much about hypergeometric series, you see that this miss is
large. The parameters in (4.5) and (4.7) are the same except in one place, where
they differ by 1/2. If they differed by one, there would be a good chance they were
related. It is differing by one which is responsible for another of the sums. The
case Cl: = ,"(, ß = 8 gives
(4.8)
3
p.
2
(-k, k+ ++1,2ß + 1,1.1)
,.
Cl:
Cl:
This is one away from two other series which can be summed. If the 1 is replaced
by 2 or the 2 is replaced by 1, the series reduces to a 2Fl' and these can be summed
by Gauss's sum,
.1) _
r(c)r(c - a - b)
F ( a,b
(
) (
).
c , - fc-afc-b
(4.9)
21
Two hypergeometric series are said to be contiguous if they have the same parameters except for one place, and differ by 1 in this place. For 3F2 's, aseries at x = 1
and any two contiguous to it are linearly related. This has been known for quite
a long time, but the fundamental relations were first worked out by Wilson [29]
and Raynal [25].
It is possible to transform (4.5) and (4.7) into 3F2 's which are contiguous. The
following transformation formula is dassical,
(4.10)
D
3L"2
(a, b, c 1)
.
d, e'
e- a-
r(d)f(e)f(d +
b - c)
x
- f(a)f(d + e - a - b)f(d + e - a - c)
d-ae-ad+e-a-b-c )
x 3 F2 ( d
'
, b, d +e-a-c ; 1 .
+e-a-
See Bailey [10]. A direct proof can be given by integrating Euler's transformation
(4.11)
2 F1 (
a,d b.,x) -_ (1 _ X )d-a-b 2 F1 (d - a,d d - b.,x)
with respect to xC-1(I - xy-c-l on (0,1), and then iterating the resulting 3F2
transformation.
The series in formula (4.5) transforms to
(4.12)
.1)
p. ((b + 1 - a)/2, 2c - a, c + (1 - a - b)/2
c+(b-a+I)/2,2c+(I-a-b)/2'
32
VIETORIS'S INEQUALITIES AND HYPERGEOMETRIC SERIES
71
and (4.7) transforms to
F ((b + 2 - a)/2, 2c - a, c + 1 - (a + b)/2 .1)
c+l+(b-a)/2,2c+l-(a+b)/2'
(4.13)
32
These series as written are not contiguous, but consider their essential structure.
The series (4.12) have the form
3
F (a + 1 -cx,ß"
.1)
ß, a + 1 - "
2
with a = 2c - a. These parameters can be paired, a numerator and adenominator
together, so that their sum is the same. The parameter a is paired with 1, which
comes from n! in the series. Such series are said to be well-poised.
The series (4.13) are
3
F (
2
a -1,ß"
.1)
a + 1 - ß, a + 1 - , '
with a = 2c - a. In this case, ß and , can be paired with a + 1 - ß and a + 1 - ,
but, to be well-poised, a - 1 would have to be paired with 2. Thus it is one too
small to be well-poised. Such series have arisen before. Bressoud [11] observed
that some series which are one off of being well-poised could be summed using
work he did jointly with Agarwal and Andrews [1] and with Goulden [12].
Bressoud dealt with basic hypergeometric series, but his results pass to a limit
when q approaches 1 to give hypergeometric series results. In particular, one of
his results implies
(4.14)
3 F2 (
.1)
-2n, b, c
2 - b - 2n, 2 - c - 2n '
(-2n)n(2 - 2n - b - c)n (1 - b - 2n)(1 - c - 2n)
(1 - 2n - b)n(1 - 2n - c)n
(1 - b)(1 - c)
However, we need (4.14) when the series does not terminate.
It is possible to show that this series can be summed when -2n is replaced by a
real number a as long as there is another terminating parameter. WhippIe showed
that
(4.15)
D
(
4E 3
.1) _(8)m(8-b-c)m
(8-b)m(8-c)m
a,b,c,-m
" - b"
"
,
u
,u-c,u+m
-
x F. (
54
x
(8-a)/2, (8-a+l)/2,b,c,-m .1)
8-a,8/2,(8+1)/2,b+c+I-8-m'
.
See [10, 4, 6(2)]. When a = 8 - 2, the 5F4 essentially reduces to a 4F3, so that
(4.16)
4 F3 (
8-2,b,c,-m
8 - b, 8 - c, 8 + m
;1) (8)m(8-b-c)m
x
(8 - b)m(8 - c)m
=
F. (
X54
1,3/2, b, c, -m
1)
2,8/2,(8+1)/2,b+c+I-8-m;
.
R. ASKEY
72
When c = 8/2, the 4F3 becomes the 3F2 we want, but with a termination condition
we do not want. The series on the right becomes a 4F3, and it is balanced, Le. the
sum of the numerator parameters plus 1 is the sum of the denominator parameters.
This series is
(4.17)
where c = 3/2 + a - m - b.
Formula (4.17) can be rewritten as
E
(3/2h-l (ah-l (-m)k-l
k=l
(l)k (b )k-l (C)k-l
~ (1/2)k(a - l)k( -m - l)k
= k=l
~
k!(b - l)k(c - l)k
(b - l)(c - 1)
. (1/2)(a - 1)( -m - 1)
_ 2(b-1)(c-1) [Po (1/2,a-1,-m-1. 1) -1]
- (a - 1)( -m - 1) 3 2
b - 1, c - l '
.
The series is balanced, and so can be summed by (4.6). To get the same type
of series when the non-poised term is the terminating parameter, sum the series
backwards, as Bressoud did.
However, this still does not give the nonterminating series. There is a natural way
to try to sum this series. 3F2'S at x = 1 satisfy 3-term contiguous relations, and the
fundamental ones were given by Raynal [25] and Wilson [29], as was mentioned
earlier. These can be iterated, so that a 3F2 at x = 1 and two others whose
parameters differ by integers are linearly related. Thus
(4.18)
3
(4.19)
3
(4.20)
Po (
a,b,c
1)
a + 1 - b, a + 1 - c ;
,
Po (
ab, c
a + 1 - b, a + 1 - c ;
2
2
3
1,
1,
Po ( a b, c .
2
a - b,a - c'
1) ,
1)
are linearly related. Since
(4.21)
3 F2
(a+1_ab~~c+1_C;l)
_ r(a - 1- b)r(a + 1- c)f(~ + l)f(~ + 1- b - c)
- f(~ + 1- b)f(~ + 1- c)f(a + l)f(a + 1- b - c)
VIETORIS'S INEQUALITIES AND HYPERGEOMETRIC SERIES
73
[10,3.1(1)], if the coefficient of (4.19) does not vanish, we have the sum of (4.19).
To give this relation, we combine two identities given by Wilson [29]. These are
(4.22) (a-l) [3F2 (ad~~c;l) -3 F2 (a-/~b,c;I)]
= (d -1) [3F2 (ad-_l1~~c;
1) -
3F2 (a -d~~b,c;
1)]
and
(4.23)
(d-a)(d-b-1)(d-c-1) [3 F2(a-1,b,c';1) -3 F2(a-_1,b'C;1)]
d,e
d 1,e
+(d - 1)(e -1)(d + e - a - b - c - 1) [3F2 (da_-1~~b,-c1; 1) - 3F2 (a;_1i~~c j 1)]
-(a - 1)bc 3F2 ( a ;;!l~'eC,
;1) = O.
The last changes to
(d - a)(d - b -1)(d - c -lhF2 (a -d~~b,c;
1)
+(d - 1)(e - l)(d + e - a - b - C -lhF2 (da_-l~~b,-cl;
1)
= (d - I)A(a, b, c, d, ehF2 ( ad-~.\~~ c;
1)
where
A(a,b,c, d, e) = [d2 +e 2 +de+ab+ac+bc- (a+b+c)(d+e) +2a+b+c- 2d - 2e+ 1].
Use (4.22) to replace the 3F2 on the right. Then
(d - a)(d - b -1)(d - c -lhF2 (a -/~b,c;
1)
+ (d - 1)(e - 1)(d + e - a - b - c - IhF2 (da_-l~~b,-\;
1)
=A(a,b,c,d,e) [(a-l hF2 (ad~~c) + (d-ahF2 (a-/~b,c;I)].
A calculation shows that
A(a, b, c, a + 1 - b, a + 1 - c) = (a - b - c)(a + 2 - 2b - 2c).
Then
(4.24) 2(1 - b)(l - c)(a - b - chF2 (a + t_-b~~b~cl_ c; 1)
= (a - 1)(a - b - c)(a + 2 - 2b - 2chF2 (a + 1 _ab~~c+ 1 _ c; 1)
- (a - b)(a - c)(a + 1 - 2b - 2chF2 ( a -bI, b, c ;
a- ,a-c
1) .
74
R. ASKEY
Using (4.21) we have
F! (
3
2
1,
ab, c
a + 1 - b, a + 1 - c ;
1)
_
(a - l)r(a + 1 - b)r(a + 1 - c)r(a/2 + 1)r(a/2 + 2 - b - C)
- (b - l)(c - 1)r(a/2 + 1 - b)r(a/2 + 1 - C)r(a + l)r(a + 1 - b - C)
r(a + 1 - b)r(a + 1 - c)r(a/2 + 1/2)r(a/2 + 3/2 - b - C)
(b - l)(c - 1)r(a/2 + 1/2 - b)r(a/2 + 1/2 - C)r(a)r(a + 1 - b - C) .
Notiee that two terms occur here, while in the terminating case only one occured.
If a = 2n, the second term vanishes because of l/r( -n) = 0, n = 0,1, ... , while
the first can be evaluated by letting a by -2n - c and letting c --+ 0.
When a = -2n -1, the first term vanishes and the second can be evaluated by a
similar limit.
5. Conclusion. Other uses of Vietoris's inequalities are given in [9]. One of
these is a theorem which shows that all the zeros of a specific dass of trigonometrie
polynomials have real zeros whieh can be separated from each other. In the present
paper it led me to find a new hypergeometrie sum whieh is likely to be useful in
other contexts. I hope to use this sum to discover other instances where the partial
sums of (3.7) are positive. Of course, the partial sums of (3.7) are not the only
interesting polynomials. Various Cesaro means also lead to interesting results. See
[8] for one instance, and [20] for the most complete results. There are other series
which have positive partial sums. One of these,
L p1
n
(5.1)
a ,ß) (x)/
p1
ß ,a) (1),
k=O
has been especially fruitful. The cases ß = 0, a a nonnegative integer were used by
deBranges in his proof of the Bieberbach conjecture. There are some expository
papers whieh give more on these problems. See [4], [6]. These papers deal with
work of about 20 years ago. Very important recent work has been done by Gavin
Brown and a number of coworkers. The first of these is [13]. Most of this work
is still unpublished, so the best source will be Mathematieal Reviews in a few
years. In partieular, Brown, Koumandos and Wang have completely determined
when (5.i) with a = ß is nonnegative for all -1 :5 x :5 1 and n = 0,1, .... The
inequality Szegö [19] proved for the integral of Bessel functions
1
z
where
r a Ja(t) dt ~ 0,
1
x > 0,
i 'Q'2 r-a J-a(t) dt =
a ~ a,
°
with ja,2 the second positive zero of Ja(t), is the determining factor. This is a
remarkable result.
VIETORIS'S INEQUALITIES AND HYPERGEOMETRIC SERIES
75
References
1. A. K. Agarwal, G. E. Andrews and D. M. Bressoud, The Bailey lattice, J. Indian Math. Soc.
57 (1987), 57-73.
2. R. Askey, Positive Jacobi polynomial sums, Töhoku Math. J. 24 (1972), 109-119.
3. ___ , Positivity of the Gotes numbers for some Jacobi abseissas, Numer. Math. 19 (1972),
46-48.
4. ___ , Positive quadrature methods and positive polynomial sums, Approximation Theory
V (C. K. Chui, L. L. Schumaker and J. D. Ward, eds.), Aeademic Press, Orlando, 1986,
pp. 1-29.
5. R. Askey and J. Fitch, Positivity of the Gotes numbers for some ultraspherical abscissas,
SIAM J. Numer. Anal. 5 (1968), 199-201.
6. R. Askey and G. Gasper, Positive Jacobi sums. JI, Amer. J. Math. 98 (1976), 709-737.
7. ___ , Inequalities for polynomials, The Bieberbach Conjeeture (A. Baernstein 11, D.
Drasin, P. Duren, and A. Marden, eds.), Proc. Symp. Oceasion of Proof, Amer. Math. Soe.,
Providenee, 1986, pp. 7-32.
8. R. Askey and H. Pollard, Absolutely monotonic and completely monotonic functions, SIAM
J. Math. Anal. 5 (1974), 58-63.
9. R. Askey and J. Steinig, Some positive trigonometric sums, Trans. Amer. Math. Soe. 187
(1974), 295-307.
10. W. N. Bailey, Hypergeometric Series, Cambridge, 1935 [Reprinted by Hafner, New York,
1964].
11. D. M. Bressoud, Almost poised basic hypergeometric series, Proc. Indian Aead. Sei. (Math.
Sei.) 97 (1987), 61-66.
12. D. M. Bressoud and I. E. Goulden, Gonstant term identities extending the q-Dyson theorem,
Trans. Amer. Math. Soe. 291 (1985), 203-228.
13. G. Brown and E. Hewitt, A class of positive trigonometrie sums, Math. Ann. 268 (1984),
91-122.
14. L. Fejer, Sur les fonetions bornees et integrables, C.R. Acad. Sei. Paris 131 (1900), 984-987
[Reprinted in [18, Vol. I, pp. 37-41]].
15. ___ , Sur le developpement d 'une fonction arbitraire suivant les fonctions de Laplace,
C.R. Acad. Sei. Paris 146 (1908), 224-227 [Reprinted in [18, Vol. I, pp. 319-322]].
16. ___ , Ultrasphärikus polynomok összegerlil, Mat. es Fiz. Lapok 38 (1931), 161-164 [Reprinted in [18, Vol. 11, pp. 418-420; German transl. 421--423]].
17. ___ , Eigenschaften von einigen elementaren trigonomeschen Polynomen, die mit der
Flächenmessung auf der Kugel zusammenhängen, Comm. semin. math. de l'univ. de Lund,
tome suppl. dedie a. Marcel Riesz, 1952, pp. 62-72 [In [18, Vol. 11, pp. 801-810]].
18. ___ , Gesammelte Arbeiten, I, 11 (P. Thran, ed.), Birkhäuser Verlag, Basel, 1970.
19. E. Feldheim, with a note by G. Szegö, On the positivity of certain sums of ultraspherical
polynomials, J. d'Anal. Math. 11 (1963), 275-284 [Reprinted in G. Szegö, Colleeted Papers,
Vol. 3, Birkhäuser Verlag, Boston, 1982, pp. 821-830].
20. G. Gasper, Positive sums of the classical orthogonal polynomials, SIAM J. Math. Anal. 8
(1977), 423-447.
21. T. H. Gronwall, Über die Gibbssche Erscheinung und die trigonometrischen Summen sinx+
~ sin 2x + ... + ~ sin nx, Math. Ann. 72 (1912), 228-243.
22. D. Jackson, Über eine trigonometrische Summe, Rend. Cire. Mat. Palermo 32 (1911), 257262.
23. G. V. Milovanovic, D. S. Mitrinovic and Th. M. Rassias, Topics in Polynomials: Extremal
Problems, Inequalities, Zeros, World Seientific, Singapore - New Jersey - London - Hong
Kong, 1994.
24. D. S. Mitrinovic, Analytic Inequalities, Springer Verlag, Berlin - Heidelberg - New York,
1970.
25. J. Raynal, On the definition and properties of generalized 3 - j symbols, J. Math. Phys. 19
(1978), 467--476.
76
R. ASKEY
26. G. Szegö, Ultrasphaerikus polinomok összegerol (On the sum 01 ultraspherical polynomials),
Mates Fiz. Lapok 45 (1938), 36-38 [Reprinted in G. Szegö, Collected Papers, Vol. 2, 1982,
pp. 700-702].
27. G. Szegö, Orthogonal Polynomials, 4th edition, Amer. Math. Soc., Providence, R.I., 1975.
28. L. Vietoris, Über das Vorzeichen gewisser trigonometrischer Summen, S. B. Öst. Akad. Wiss.
167 (1958), 125-135 [Anzeigen Öst. Akad. Wiss. (1959), 192-193].
29. J. Wilson, Three-term contiguous relations and some new orthogonal polynomials, Pade and
Rational Approximation (E. B. Saff and R. S. Varga, eds.), Academic Press, New York, 1977,
pp. 227-232.
INEQUALITIES FOR NORMS OF INTERMEDIATE
DERIVATIVES AND SOME THEIR APPLICATIONS
VLADISLAV F. BABENKO
Dnepropetrovsk State University, Dnepropetrovsk, Ukraine
Abstract. This survey is devoted to inequalities of Landau-Hadamard-Kolmogorov type
for norms of intermediate derivatives of some classes of functions. Some general schemes
for obtaining inequalities and their generalisations are presented. Inequalities for derivatives of half-integer orders and their applications in approximation theory, as weil as the
inequalities of Hörmander type on the half-line, are also considered.
1. Introduction
Let G be a Lebesgue rneasurable subset of ~m such that J-LG > O. We consider
spaces Lp(G), 0 ~ p ~ 00, of aB rneasurable functions f : G -+ ~ such that (in the
case J-LG < 00)
Ilfllo = IIfIIL o(G) := exp {J-L~
IIfll p = IlflILp(G) := {J-L~
fa log If(t)1 dt} <
fa If(t)IP dt}
I/p
< 00,
00,
0 < P < 00,
Ilfll= = IIfIIL (G) := L vrai lf(t)1 ~ 00.
oo
tEG
Note that if x E Lq(G) for sorne q > 0, then f E Lp(G) for aB p E [0, q], and
in this case Ilfll p ~ Ilfll q and Ilfll p -+ IIfllo with p -+ O. If f E L=(G), then
Ilfll p -+ Ilfll= when p -+ 00.
In the case J-LG = 00, defining the values Ilfll p , we will ornit (J-LG)-l.
For univariate functions we will consider as G: the real axis IR, the half-line Il4,
and finite intervals I. We will also consider the spaces of functions f : ~m -+ ~
that are 2'1r-periodic in each variable, and for such functions we define IIfll p as
IlfIILp(lI''''), where Tm = (-'Ir, 'Ir)m and denote these spaces as L p or Lp(']['m) (L p('1I')
in univariate case).
1991 Mathematics Subject Classification. Primary 26DlO, 41AI0, 41A44, 42A05; Secondary
41A17.
Key words and phrases. Inequalities for norms; Best constant; Markov-Nikolskii type inequality;
Kolmogorov type inequality; Multivariate function; 27r-periodic function; Derivatives of halfinteger order; Additive inequalities for derivatives; Difference operators; Differential operators.
77
G.v. Milovanovic (ed.), Recent Progress in lnequalities, 77-96.
© 1998 Kluwer Academic Publishers.
V.F. BABENKO
78
If Gis R, 1l4, I or T, and n E N, 1 ~ r ~ 00, then we denote by L~(G) the space
of all functions f such that their derivatives f(n-l) (n E N) are locally absolutely
continuous and f(n) E Lr(G). For given 1 ~ p ~ 00 let L;,r(G) = Lp(G) n L~(G).
Note that in the cases G = I or G = T, we have L~(G) C Lp(G) for any p.
It is known that for G = I and any given 1 ~ p,q,r ~ 00, k,n E Z, 0 ~ k < n,
there exist constants A, B such that for f E L;,r(G) the inequality
(1)
holds. If Gis IR. or 1l4, then the inequality (1) holds for all functions jE L;,r(G)
if and only if (see [30])
(2)
n-k
k
n
p
r
q
--+-~-,
and in this case, (1) is equivalent to the multiplicative inequality
(3)
where
a =
n - k - r- 1 + q-l
1
n-r- +p-l
and
k _ q-l + p-l
ß = n-r- 1 +p- 1 .
In this paper, we discuss the problems of finding the exact constants in the inequalities of the form (1) and (3). Note that these inequalities, especially with
exact constants, are dosely connected to many extremal problems of approximation theory (see, for example, [40-42] and [61]).
The problem of finding the best possible constant in (3) can be formulated as
folIows: Given p, q, r, n, k, find the value
(4)
K(Gin,kip,q,r) =
IIf(k) IILq(G)
sup
ß
{!EL;.r(G), !(n)#o} IIfIl1 p (G) IIf(n) IILr(G)
We will write K(Gin,kiP) instead of K(Gin,kiP,P,P). If K = K(Gin,kip,q,r)
we will say that the inequality (1) is exact.
Observe that in the case when condition (2) is satisfied, inf B is equal to zero,
where the infimum is taken over all constants B such that for some constant A the
inequality (1) holds for any f E L~(I) (see, for example [21]). At the same time
A*(n,kip,q,r) := infA is strictly positive, where the infimum is taken over all
constants A such that for some constant B inequality (1) holds for any f E L~(I).
Thus,
A*(n,kjp,q,r) ~ Mn,k(P,q),
where
INEQUALITIES FOR NORMS OF INTERMEDIATE DERIVATIVES
79
is the exact constant in Markov-Nikolskii type inequality for algebraic polynomials
of degree at most n - 1. Therefore, the problem of finding the exact constants in
(1) can be formulated as follows:
(a) For given n,k,p,q,r find A*(n,kjp,q,r)j
(b) Find B*(Aj n, kjp, q, r) = inf B, where the infimum is taken over all B such
that (1) holds with a given A ~ A*(n,kjp,q,r).
We will discuss analogous problems for some multivariate Kolmogorov type inequalities.
2. Some Previous Results
The first results on the investigation of these problems were obtained by Landau
[45] and Hadamard [33]. They proved the following inequalities:
(6)
(J E L;',oo(ll4)) ,
(7)
(J E L;"oo (IR)) ,
and
(J E L;,(I), 1= [0,1]).
They also proved that the constants in (6)-(8) are best possible.
One of the first complete results in this direction is due to Kolmogorov [36-37],
whom such inequalities are named after. Kolmogorov proved that if f E L~,oo
then for any k E N, k < n, the inequality
(9)
Ilflll-k/nllf(n)llk/n
Ilf (k)11 Loo(lR) -< IICPn-kIlLoo(lR)
11 11 1 - k/n
Loo(R)
L (lR)
CPn L oo (lR)
oo
holds, where CPn is the nth periodic integral having zero mean value on aperiod
for the function CPo(x) = sgn(sinx), and
(Note that in all cases when 2 < n < 5 and for n = 5, k = 2, this result has been
proved by Shilov [18]).
Afterwards, there was a great number of papers dealing with exact inequalities of
such type on the line, half-line or finite interval. But there are only a few cases
when the constants K(Gj n, kjp, q, r) are known for all pairs k, n E N, k < n.
Besides Kolmogorov's result mentioned above, the cases for G = IR are:
10 p = q = r = 2 (Hardy, Littlewood, Polya [34]);
V.F. BABENKO
80
2° p = q = r = 1 (Stein [62));
3° q = 00, p = r = 2 (Taikov [63)).
For G = 114 these cases are:
1° p = q = r = 00 (Landau [45], Matorin [51], Schoenberg and Cavaretta [58-59));
2° p = q = r = 2 (Ljubich [48], Kuptsov [43));
3° q = 00, p = r = 2 (Gabushin [31)).
For G = 11' these cases are:
1° 1 ~ q ~ 00, p = r = 00 (Ligun [46));
2° q = r = 1, 1 ~ p ~ 00 (Ligun [47));
3° q = 00, p = r = 2 (Shadrin [55)).
Some new results presented in this paper have been obtained with the help of the
Ligun's inequality [46]:
(10)
lI<Pn-k IILp("ll') Ilflll-klnllf(n) Il kln
Ilf (k) 11 Lp("ll') <
11
II I - kin
L",,("ll')
L",,("ll')'
<Pn L",,("ll')
where p E [1,00], n, k E N, k < n and f E L~(1l').
Not so complete results for G = lR or G = 114 and various special n, k,p, q, r have
been obtained by Arestov, Berdyshev, Ditzian, Gabushin, Magarill-Il'jaev, Nagy,
Soljar and many others (see for references [44] and [66)). Soljar's latest result [60]
result is: If p' = p/(p - 1) and 2k = n, then for any f E Lp(lR) n L;, (lR)
(11)
Exact inequalities for derivatives of functions defined on a finite interval also have
been investigated by many authors, among them Chui and Smith, Karlin, Pinkus,
Sato, Burenkov, Zvyagintsev, Shadrin, and others (see for references [20-21] and
[57)). Note that Burenkov ([20-21)) proved that for any n,k,p,q,r,
(12)
A*(n,n -I;p,q,r) = Mn.n-I(p,q).
The case p = q = r = 00 has been investigated in detail by Shadrin [56-57].
Multiplicative inequalities of the form (3) for multivariate functions played an
important role in the theory of partial differential equations and imbedding theorems. Questions on the existence of such type inequalities have been investigated
in [16-17]. As for the exact constants we refer the reader to papers of Konovalov
[38], Buslaev and Tihomirov [23], Timoshin [65], Timofeev [64], and Ditzian [25].
For known results on additive inequalities of type (1) for multivariate functions
we refer the reader to [22].
Konovalov's inequality [38] for sufficiently smooth functions defined on lR is
INEQUALITIES FOR NORMS OF INTERMEDIATE DERIVATIVES
81
8k+lf
where f<k,l) stands for 8 k8 l'
Xl X 2
There are many ways to generalise inequalities of the type (1) and (3). One of
them was proposed by Hörmander [35] who proved the next result. Let Eo(f)oo
be the best uniform approximation of a function f by constants. Let 'Pn( . ; a, ß)
be the nth periodic integral having zero mean value on aperiod for a 211'-periodic
function 'Po (x; a, ß), which is equal to a for t E [0,211'ß/ (a + ß)) and - ß for
tE [211'ß/(a + ß), 211').
We can now formulate Hörmander's result as the next best possible inequality
which holds for any function f E L~,oo(lR.):
(14)
where /± = max(f(t),O). Another way for generalisations of Kolmogorov type
inequalities is to substitute the operator d/dx in these inequalities by operators of
another nature. For example, in [3] it was proved that for any n, k E N, k < n,
and f E L~(,][,), the following exact inequality
(15)
holds. (j is conjugate function for f [14, p. 519].)
Some special results for fractional derivatives are presented in [2], [32], and [49].
In Section 3 below, we will present general schemes for obtaining the inequalities
of the form (11) and additive inequalities. Wide generalisation of (11) will be
given in Section 4. Using the first general scheme and the univariate inequality
(10), we obtain a rather general inequality of type (13). Using inequality (15),
instead of inequality (10), we obtain, in Section 5, some inequalities for derivatives
of half-integer order and give their applications in approximation theory. Section
6 is devoted to inequalities of Hörmander type on the half-line. In Sections 7 and 8
we are dealing with additive inequalities for derivatives. For details of the results
to be presented in this paper we refer the reader to [3-13].
3. General Schemes
At first, we give a general scheme for obtaining inequalities of type (11). Let X,
Y be real normed spaces, 11'lIx - norm in X, X* - dual space of X, and H - real
Hilbert space. For a linear (in general, unbounded) operator A : X --t Y having a
dense domain of definition V(Ad C X in X, let A* be its adjoint operator.
Theorem 1. Let the operators Al : X --t Hand A 2 : X --t X with dense domains
01 definitions in X be such that lor any
82
V.F. BABENKO
the equality Ai A I A 2 x = AzAi Alx is valid. Then for alt x E Q the inequality
holds. In particular, if A 2 = Ix, then for alt x E V(Ai Ad we have the inequality
(16)
If there exists a set M c X n H such that Aix = ±Alx, then for alt x E M the
following inequality holds:
(17)
Inequality (16) will be called exact if
Note that inequality (16) is exact in this sense if the operator Al is bounded.
Inequalities (16) and (17) give us a wide generalisation of Soljar's inequality presented above.
We now give a general method for obtaining additive inequalities of the form (1)
with the best possible constant A.
Let X, Y be additive groups, II . Ilx and II . Ily be semi-additive real functionals
such that Ilxllx ~ 0 for any x E X and IIYlly ~ 0 for any y E Y. Further, let the
mapping T, X :::> V(T) ~ Y, be such that for any x E V(T),
Ilxllx = 0
===}
IITxlly = 0,
and the functional F, X :::> V(F) ~ lR, be given such that V(F) C V(T).
Conditions providing the existence of constants A and B such that for any x E
V(F) the inequality
(18)
IITxlly :::; Alixlix + BF(x),
holds. In addition, some information on exact constants in this inequality are
given in the next theorem. Set
(19)
A* = A*(X, Y, T, F) = inf A,
where the infimum is taken over all constants A such that for some constant B
inequality (18) holds for any x E V(F).
INEQUALITIES FOR NORMS OF INTERMEDlATE DERIVATIVES
83
Theorem 2. Suppose that there exists H c V(T) such that
(a) M* = M*(X,Y,T,H):=
sup
{YEH,IIYllxi'O}
IITYlly/IIYllx ~ 00;
(b) there exists B > 0 such that for any xE V(F)
inf {M*lly - xlix + IITx - TylIy} ~ B· F(x).
yEH
Then for any xE V(F)
(20)
IITxlly ~ M*llxllx + BF(x),
and consequently
(21)
A*(X,Y,T,F) ~ M*(X,Y,T,F).
1f in addition we suppose that H C V(F) and F(y) = 0 for any y E H, then the
constant at Ilxllx in (20) is best possible, i.e., (21) becomes an equality.
Note that condition (a) means that for elements from H an inequality of MarkovNikolskii type holds true and condition (b) means that for the values of simultaneous approximation of elements x and Tx by elements of Hone can obtain an
estimation of the form BF(x), with some constant B not depending on x.
4. Inequalities for Difference and Differential Operators
For functions ! : IRm -+ IR, hj E IR, j = 1, ... , m, set
6. h;!(t) = !(tI, ... , tj-l, tj + hj /2, tj+1, ... , t m )
- !(~, ... , tj-I, tj - hj /2, tj+1, ... , t m ).
Given h = (h l , ... ,hm), let 6. h := (6. hl'6. h2 , ... ,6.h,..). Given a multiindex
a = (al, ... ,am) E Z+ and a vector t = (t1, ... ,tm) E IRm , set tO!. = tfi ···t~m
m
(we write 1 instead of 0°). For vectors h, t E IRm such that TI h i :f; 0, set t/h =
i=1
P(t) =
L aO!.tO!.
10001=k
be a homogeneous algebraic polynomial of degree k in m variables. We will consider
operators P(6.h), P(6. h /h) (if h i :f; 0 for all i) and P(D).
Theorem 3. Let P(t) be a homogenous polynomial of degree k in m variables.
m
Then for all h = (h l , . .. , hm) such that TI hi :f; 0, q E [0,2], p, r E [1,00] such
i=1
that p-l + q-I ~ 1, and for any junction ! E Lp('lr m ) n Lr('lr m) the inequalities
84
v. F. BABENKO
and
(23)
hold. 1fn = (nI, ... ,nm) E N'" is such that P(n) "10, then for hj = 7rjenj,
e E N, j = 1, ... , m, inequality (22) is exact and becomes an equality for functions
of the form f(t) = sgnsin(e j~I njtj). 1f there exists € E {O, 1, _1}m such that
P(€) "I 0, then for some h inequality (23) is also exact.
Now we consider analogous inequalities for functions f E Lp(lRm).
Theorem 4. Let P(t) be a homogeneous polynomial of degree k in m variables.
Then, for all h = (h I , ... ,hm) P E [1,00], and for any function I E Lp(lRm) n
Lpl (jRm) the inequalities
and
(25)
hold. These inequalities are exact under same assumptions on P as the inequalities
in the previous theorem.
Co
Let G be a domain in jRm. Denote by
(G) the set of compactly supported
infinitely differentiable functions defined in G. By
(']l'm), we denote the set
of all infinitely differentiable functions I: jRm -+ jR that are 27r-periodic in each
variable.
Now let G be a domain in jRm or ']l'm and P(t) be a fixed homogeneous polynomial
of degree k in m variables. We will say that P(D)x E Lr(G) for x E Lp(G) is
defined if there exists a function z E Lr(G) such that for any y E Co(G) the
following equality
Co
1a z(t)y(t) dt = (_1)k 1a x(t)P(D)y(t) dt
holds. Then, by definition, P(D)x = z. We define the dass Wp,r(P; G) as the
set of all functions x E Lp(G) for which P(D)x E Lr(G) is defined and that there
exists a sequence {Yv} ::"=1 offunctions from
(G) such that
Co
Theorem 5. Let G be a domain in IRm or ']l'm, P E [1,00], and let P(t) be a
fixed homogeneous polynomial 01 degree k in m variables. Then for lunctions
I E Wp,pl (P 2 ; G) the exact inequality
(26)
INEQUALlTIES FOR NORMS OF INTERMEDIATE DERIVATIVES
85
holds.
N ow we consider the following problem (see [38]): Let o! = (al, (2), ß = (ßl, ß2) E
Z+, and G be ]R2 or 'll'2. Set
The problem is to obtain the exact estimation IIDI' IIILq(G) in terms of II/IIL=(G)'
liDO: IIIL=(G)' and IIDß IIIL=(G)' when I E W~ß(G) and for a given vector "( E Z2
such that "( = Aa + JLß, A, JL :::: 0, A + JL < 1.
Konovalov [38] solved this problem (see inequality (13)) for functions I E W~,ß (l~.2)
in the case a = (3,0), ß = (0,3), "( = (1,1) = (a/3) + (ß/3), G =]R2 and q = 00.
Timoshin [65] considered the case a = (2,0), ß = (0,3) or a = (3,0), ß = (0,2),
"(= (1,1), G=]R2 andq=oo.
We consider this problem in the case G = 'll'2. The following theorem holds.
Theorem 6. Let rEN, a = (r,O), and ß = (ßl,ß2) E Z~ be such that A -:f ß,
IßI = r, and ß2 be even. Illor a vector"( = ("(1, "(2) E Z~ one of the conditions
(2)
1
1
"2 ßl < "(1 < "2 (ßl + r),
(3)
"(1 =
1
"2(ßl + r),
"(2
1
= "2 ß2,
1
o ~ "(2 < "2 ß2,
is satisfied, then for junctions I E W~,ß('ll'2) the exact inequality
l'
IID III L2 (1l'2) ~
II<Pr-II'III L2 (1l'l)
1->'-1'
0:
>.
ß I'
11 111->.-1' ·II/II L =(1l'2) . IID fIl L =(1l'2) . IID fIl L =(1l'2)
<Pr L=(1l'l)
holds, where the numbers A and JL are defined by conditions "( = Aa+JLß, A,JL:::: 0,
A+JL<l.
Setting r = 2k, k E N, a = (2k,0), ß = (0,2k), and "( = (k, i) or "( = (i, k), we
obtain:
Corollary 1. For junctions I E W~k,O),(O,2k) ('ll'2) and all i = 0,1, ... , k - 1, the
exact inequalities
-(k+l)/(2k) .IID(2k,O)/ll l /(2k) . IID(O,2k)/ll k /(2k)
II D(l,k)/11 L2(y2) -< A .11/11 1L=(1l'2)
L=(1l'2)
L=(1l'2)
and
A ·11/11 1-(k+l)/(2k) IID(2k,O)/ll k /(2k) ·IID(O,2k)fll l /(2k)
II D(k,l)/11 L2(1l'2) <
L=(1l'2)·
L=(1l'2)
L=(1l'2)
-
hold, where
v. F. BABENKO
86
5. Inequalities for Derivatives of Half-integer Order and
Some of Their Applications
In this section we take LI = LI (1l') and L'1 = L1(1l'). For I E LI, we set ao(f) =
(21f)-1 J:1I" I(t) dt. Let r E R, r > 0, and
1 00
L v- r cos(vt - 1fr/2).
Br(t) = -
1f
v=1
A function gELl such that ao(g) = 0 will be called rth derivative in the Weil
sense for I E LI (f(r) = g) if
(211"
I(t) = ao(f) + (Br * g)(t) = ao(f) + 10
Br(t - u)g(u) du.
Denote by L; the set of functions I E LI such that I(r) E L p , and by W; the
set of functions I E L; such that II/(r) IIp :::; 1. Note that, for rEN, L; coincides
with the space previously defined, and
is a standard Sobolev dass of periodic
functions.
W;
For n E N and r E 114, we set
(211"
'Pn,r(t) = 10 Br(t - u)'Pn,O(u) du,
where 'Pn,O(u) = sgn(sinnu). Instead of 'Pl,r(U) we will write 'Pr(u).
In addition to inequality (10), the following theorem holds.
Theorem 7. Let k, rEN, r /2 :::; k < r. Then lor any function I E V:XO,
(27)
11/(k+l/2) 112 < II'Pr-k-l/21I L 2('f) 11 111 1 -(k+l/2)/r 11/(r) 11 (k+ 1 / 2 )/r.
- 11 11 1 -(k+l/2)/r
Loo('f)
Loo('f)
'Pr L oo ('f)
This inequality is exact and becomes an equality for functions I = 'Pn,r, n E N.
Note that Theorem 7 is proved with the help of inequality (15).
The best approximations of function I E Lp or a dass M c Lp by subset H c L p
(1 :::; p :::; (0) in the space L p are defined as
E(f, H)p := inf 11I - hilL ('f)
hEH
and
E(M, H)p = sup E(f, H)p,
P
fEM
respectively.
Theorem 8. Let k,r E N, k:::; r/2. Then lor any N > 0,
E(Wk-1/2 NWr) < r - k + 1/2 11
11 r-k'"-I!2 ( k - 1/2
2
,
1 1 'Pk-l/2 L2('f)
r
r Nil 'Pr 11 L oo ('f)
k-l/2
) r-k-l/2
INEQUALITIES FOR NORMS OF INTERMEDIATE DERIVATIVES
87
Let 72n-l be the set of aH trigonometrie polynomials of the order ~ n - 1 and
S2n,/l- (J.L E Z, J.L ~ 0) be the set of 27r-periodie polynomial splines of order J.L, defect
1, having nodes at the points v7r/n, v E Z.
It is weH known (see [40, Ch. 4, 5]) that for T, n E N,
(28)
where H is 72n-l or S2n,/l-' J.L E N, J.L ~ T - 1, p' = p/(P - 1). For the fractional T,
relation (28) is known only for H = 72n-l and p = 1 (see [40]).
Using Theorem 8, relation (28), and the method of intermediate approximation,
we obtain the next theorem.
Theorem 10. Let k, n E N, H be 72n-l OT S2n,/l-' J.L ~ 2k - 2. Then
6. The Hörmander Type Inequality on the Half-line
After Landau (n = 2) and Matorin (n = 3) for T ~ 4, the problem on exact
constant K(~; T, k; 00) has been solved by Schoenberg and Cavaretta [58-59].
In order to formulate the result of Schoenberg and Cavaretta we need the next
definitions and notations.
Denote by an,r the perfect spline of order T defined on [0,1], having n nodes and
minimal Loo[O, 1]-norm among aH such perfect splines. As it is weH known (see for
example [54]), such a spline an,r exists, is unique (up to multiplier ±1), and has
(n + T + 1)-alternance (ends of the interval [0,1] are points of alternance).
For every n E N, we choose the number An from the condition A~rlan,r(O)1
lao,r(O) land then set
Sn,r(X) = A~r an,r(AnX).
It was proved in [58-59] that there exists the limit
(29)
lim Sn r(X) =: Sr(X),
n--+-oo
'
XE ~
(convergence in (29) is uniform on any interval [0, a] C ~), and the limit function
Sr(X) is a perfect spline of the order T on ~ having an infinite number of nodes.
Now we can formulate the result of Schoenberg and Cavaretta as the foHowing
exact inequality: For f E L~(~)
(30)
We present now a result analogous to Hörmander's inequality (12) for functions
f E L~,oo(~)'
V. F. BABENKO
88
Given rEN, nE Z+ and a, ß E lI4 denote by S;;,r( .; a, ß) and S;;,r( .; a, ß) the
sets of splines of the forms
and
n
r-1
a(t) = -~ t r + (a + ß) I)-1)i-1(t - ~i)+ + 2:avtV,
r.
respectively, where
i=l
0 ~ t ~ 1,
v=O
o
0 < 6 < .,. < ~n ~ 1 (we set i~ = 0). Note that for
E S~r('; a, ß) the derivative a Cr ) takes two values, a and -ß. Moreover, in
the interval (0,6), aCr)(x) = a if 0' E S;;r(-;a,ß) and aCr)(x) = -ß if a E
S;; r( . ; a, ß)· In the case when a = ß = 1 the set S;; r( . ; a, ß) coincides with the
set' of usual perfect splines of the order r having n n~des.
0'
Theorem 10. There exist a~~) ( . ) = a~r( .; a, ß) E S~r( .; a, ß) for which
(i) exist n + r + 1 points 0 = t± < t~ < ... < t;+r+1 = 1, such that
a~~j(ti) = ±(-l)i+ rH lla;,rll oo ,
i = 1, ... ,n + r + 1;
(ii) for k = 0, 1, ... ,r
sgn(a;;,r)Ck) (0) = (_1)r-k,
(iii)
inf
<TES;!',rC' ;a,ß)
sgn(a~,r)Ck)(O)
= (_ly-k+1;
110'1100 = 110';: rlloo.
,
Given n E Z+ choose .\~: satisfying the condition
( i.e., ('x;y =
1a~r(O; a, ß) I) ,
ao,r(O; a, ß)
and set
s;,r(t; a, ß) = (,X;)-r a;,r('xnt; a, ß) ,
tE (0, (,X;)-1) .
Theorem 11. Let r, k E N, nE Z+, 1 ~ k ~ r -1. For functions f E L~,oo(lI4)
the following inequalities hold:
INEQUALITIES FOR NORMS OF INTERMEDIATE DERIVATIVES
89
Theorem 12. Given rEN, a, ß > 0, there exist splines s~(·; a, ß) of the order
r dejined on [0,00), having an injinite number of nodes Yk, k = 1,2, ... ,
°
= Yo < Yl < Y2 < ... < Yk < "',
Yk ~ 00 (k ~ 00),
and such that
(i) for every k E Z+, we have (s~)(rl (t;a,ß) = a, when t E [Yk,Yk+t) or
(s~tl (t;a,ß) = ß, when t E [Yk,YkH)i
(ii) for each c > 0, the sequence {s~,r( . ; a, ß)}~=l (all elements of this sequence are dejined on [0, c] for sufficiently large n) converges to s~( . ; a, ß)
together with all derivatives of the order ~ r - 1 uniformlyon [0, cl.
Letting n ~ 00 in (31) and taking into account Theorem 12 we obtain:
Theorem 13. For any r, k E N, 1 ~ k ~ r - 1, and f E L~,oo(ll4)
(32)
11 ((
-lr- k . f(kl)±IIoo
<
(( -lt- k . s~)~l (0; IIfyllloo, Ilf~llloo)
1
k/
((-I)r-k.s;)±(O;llfYllloo,llf~llloo) - r
l-klr
Eo(f)oo
.
Both inequalities in (32) are exact. The inequality for ((_I)r-kf(k l )+ becomes
an equality if f(t) = A-rst(At;a,ß), A > O. The corresponding inequality for
((-1)r-kf(k l
becomes also an equality if f(t) = A-rs:;(At;a,ß), A > O.
L
7. Additive Inequalities for U nivariate Functions
On the base of Theorem 2 we give a few new additive inequalities. In particular,
we state that for aB n, k,p, q, r
(33)
A*(n, k;p,q,r) = Mn,k(P,q).
Take X = L p ( -1,1), Y = L q ( -1,1), F(f) = IIf(nlll r , T = d k /dt k , n, k E N,
in Theorem 2, and choose P n - l as H, where Pn - l is the set of all algebraic
polynomials of degree less or equal to n - 1. Markov-Nikolskii's inequality for
elements from Pn-l, where the functional F(f) vanishes, gives us condition (a)
in Theorem 2. Condition (b) can be verified by means of Taylor's formula. Thus,
the next theorem holds.
Theorem 14. Let 1 ~ p, q, r ~ 00 and k, n E N, k ~ n - 1. For all functions
f E L~( -1,1), the next inequality holds
(34)
where Mn,k = Mn,k(p, q) is best possible constant in Markov-Nikolskii inequality,
given by (5), and B n is some constant such that B n ~ as n ~ 00. The constant
Mn,k in (34) is best possible.
°
Note that, as it is weB known (see, for example [50], [28], [56]) in the case q =
p = 00, Mn,k = IIT~~llloo, where Tn-l(t) = cos(n -1)arccost is Chebyshev
V.F. BABENKO
90
polynomial of the first kind. Using results from [28], [56], in the case p = q = 00
we can replace the norm IIxll oo on the right in (34) by the value max If(tk)l, where
k
tk = cos(k1r /(n - 1)), k = 0,1, ... ,n - 1. Observe also ([19]) that for arbitrary
nE N, q E [1,00], and p = 00, Mn,l = IIT~_lllq. We refer the reader to [41] and
[52] for other known results on the best possible constants in the inequalities of
Markov-Nikolskii type.
Now we present some inequalities which have been obtained with the help of more
delicate results on simultaneous approximation of functions and their derivatives
by algebraic polynomials.
Let f E C[-I, 1] and
!11f(x) = f(x +~)
- f(x - ~),
As usual we suppose that !1fJ(x) = 0, if at least one of the points x ± ih/2 do
not belong to [-1,1]. Let also cp(t) = Vf=t2 and Pn(t) = n- 1 (n- 1 + cp(t)) =
n- 2 + n- 1 Vf=t2. For 0 ~ A ~ 1 set ([26])
w~>.(f, t) = sup
.
sup
O~h9 zE[-l,l]
1!1~<p>'(z/(x)l.
It is obvious that for A = 0 we obtain the usual modulus of continuity of the order
i for a function f (see, for example, [29, p. 160]).
We now set in Theorem 2, X = Y = C[-I,I], Tf = f(k), H = P n and F(f) =
w~>.(f, I/n). Note that if
(35)
M(n,s,k):=
sup
P n E'Pn ,Pn 1'O
IIp;:;-s+k PAk) 1100
IIp;:;-sPnll oo
then in view of Dziadyk's inequality ([29, p. 262])
(36)
M(s, k) := sup M(n, s, k) < 00 .
nEN
Using (36) and results from [27] we obtain the next theorem.
Theorem 15. For any i, s, k, nE N, k < s, n ~ s + i-I, any A E [0,1], and any
function f, which is s times differentiable on [-1, 1], the inequality
holds, where the constants M(n, s, k) are dejined in (34) and, in view of (35), they
are uniformly bounded in n by M(s, k), and B = B(k, s, i) is sorne constant not
depending on fand n. For jixed s, i and n (= s + i-I), the constant M(n, s, k)
in (37) is best possible, i.e., for considered X, Y, T, F,
A*(X, Y, T, F) = M(n, s, k).
Inequalities (37) make more delicate Besov inequalities [15].
INEQUALITIES FOR NORMS OF INTERMEDIATE DERIVATIVES
91
8. Additive Inequalities for Differentiable Mappings
of Banach Spaces
Let X and Y be Banach spaces over the field IR of real numbers. Given n E N
we denote by Ln(X, Y) the space of an n-linear (Le., linear in each variable when
other variables are fixed) and bounded operators F : xn -+ Y, where xn is the
Descartes product of n copies of the space X. In particular, L(X, Y) = LI (X, Y)
is the space of an linear bounded operators F : X -+ Y. The norm of an operator
FE Ln(X, Y) is defined by
1IFIIcn(x,y) := sup{IIF(XI, ... ,xn)lIy : Xi EX, IIxllx ~ 1, i = 1, ... ,n}.
Given a non-empty open bounded set U C X and mapping f : U -+ Y of the
dass C n = Cn(U; Y), let f(k) be kth derivative mapping for f (we refer the reader
to [24, Ch. 1] for the concept of the theory of difIerentiable mappings of Banach
spaces). As it is known for any n ~ 2 the natural isometry
L(X; Ln-l (X, Y)) ~ Ln (X, Y)
takes place. In view of this isometry, we can understand the kth derivative mapping
for f as a mapping
For f E Cn(U; y) we set
IIlflllu = sup IIf(x)lIy
xEU
and
(1 ~ k ~ n).
(38)
We will discuss a question concerning the existence of the constants A and B such
that the inequality
(38)
holds for any function f E Cn(U; Y), 1 ~ k < n, and fixed n. Also, we will deal
with the quest ion on exact constants in this inequality.
We will show that the question concerning the principal possibility of inequality
(38) is dosely connected to the possibility of the Markov type inequality for polynomial mappings from X to Y and the exact constant in such a Markov type
inequality gives us the best possible constant A in (38).
Denote by P n (X, Y) the space of polynomial mappings from X to Y (polynomials )
of degree at most n, i.e., the space of an mappings Pn : X -+ Y of the form (see
[24, Ch. 1])
n
Pn(x) =
L Uj(X, ... ,X),
j=O
92
V. F. BABENKO
where Uj E .cj(X, Y) for j ~ 1, and Uo is the constant mapping X in Y (homogeneous polynomial of degree 0). Note that if Pn E Pn(X, Y) and 0 ::; k ::; n, then
p~k) E Pn-k(X, .ck(X, Y)).
For fixed n, k E N, k < n, and any bounded open set U C X let
Mn,k(U, Y) := sup{IIIP~k) IIlu : Pn E Pn(X, Y), IIIPnlllu::; 1}.
H X = Y = IR, and U = (a,b) C IR, then Mn,k(U,Y) is an exact constant in
Markov inequality. In the general case, finiteness of Mn,k(U, Y) means that for
polynomials from Pn(X, Y) an inequality of Markov type
(39)
IIIP~k)IIlu ::; MIIIPnlllu
holds, and in this case Mn,k(U, Y) is an exact constant in inequality (39).
It is obvious that if there exist numbers A and B such that inequality (38) holds
true for any f E Cn(U, Y), then Mn-1,k(U, Y) ::; A < 00, and consequently, for
polynomials from P n - 1 (X, Y), inequality (39) holds. H the set U is star, then the
converse statement is also true.
Theorem 16. Let U C X be an open, bounded set which is star with respect to
some 01 its points, n, k E N, k < n. 11 Mn-1,k(U, Y) < 00, then there exists B > 0
such that lor any 1 E Cn(U, Y)
(40)
IIlf(k)IIlu ::; Mn-1,k(U, Y)IIlflllu + Blllf(n)IIlu.
The constant Mn-1,k(U, Y) in (40) is best possible in the sense that
Mn-1,k(U, Y) = inf A,
where the infimum is taken over all A such that there exists a constant B such
that inequality (38) holds true lor all f E Cn(U, Y).
Note that if U is a non-empty, open, bounded convex set, and
R(U) = inf Ra(U) = inf sup \Ix - a\l
aEU
aEU xEU
is the Chebyshev radius of the set U, then, for the constant B in inequality (40),
the following estimate
R(u)n R(u)n-k
(41)
B ::; Mn-1,k(U, Y) - n.
, - + ( _ k)'
n
.
is true.
From the results of [1], it follows that if U is a non-empty, open, bounded convex
set, and
r(U) = sup{r ~ 0 : 3x EU, B(x, r) C U} ,
where B(x, r) is open ball of radius r with the center x, then for the exact constant
Mn,l (U, Y) the following estimates are valid
n2
4n 2
(42)
r(U) ::; Mn,l (U, Y) ::; r(U) ,
where the left inequality is obtained under the additional assumption that a set U
is centrally symmetrie.
Comparing Theorem 16, relations (42), and inequality (41), we obtain the next
result:
INEQUALITIES FOR NORMS OF INTERMEDIATE DERIVATIVES
93
Theorem 17. If U C X is a non-empty, open, bounded convex set, then for every
fE Cn(U, Y),
111f'lllu ::; 4(n - 1)2 IIlfillu + (4(n _1)2 R(u)n + R(u)n-l) IIlf(nlill u .
r(U)
r(U)
n!
(n - 1)!
In [1, Theorem 3) an inequality was proved for the norm of the derivative of a
polynomial mapping IIP~(x)II.cl(X,Yl' that takes into aeeount a loeation of the
point x. Namely, given a non-empty, open, bounded eonvex set U C X, a point
Xo E U, and a number w E (0,1), denote by Uw(xo) the w-eontraetion of U with
respeet to the point xo, Le., the set
Xw
- Xo- EU.
}
Uw = { x EX: Xo + Let also, for Xo, xE U,
r(xo) = sup{r > 0 : B(xo, r) CU},
p(x, xo) = inf{w ~ 0 : xE Uw(xo)}.
Then (see [1 J)
(43)
IIP~(x)II.cl(X,Yl ::; 3b(x)nlllPnlllu,
where
b(x) = inf
1
xoEU r(xoh,!I- p(x,xo)
Now, from Theorem 2 and inequality (43), we obtain the estimate for the norm of
the mapping f(x) that takes into aeeount the loeation of the argument x.
Theorem 18. Let U C X be a non-empty, open, bounded convex set. For given
n E N there exists B > 0 such that for every f E Cn(U, Y) and any x E U
IIf'(x)ll.ctCx,Yl ::; min { 4(~(;~)2 ,3b(x)(n - 1)} IIlfillu + Blllf(nlill u .
For given m, nE N, we denote by W~(( -1, l)m) the dass of real-valued functions
fE L oo (-l, l)m that have, for any multiindex a = (al, ... ,am) E Z+ such that
lai = n, a Sobolev generalised derivative DOt f belonging to L oo ( -1, l)m.
Set for k E N, k ::; n,
and let for n, k E N, k < n,
where Pn,m = Pn(IRm ,lR) is the set of all algebraic polynomials in m variables and
of degree at most n in the sense that surn of degrees in every rnonornial is less or
equal to n.
Using Theorem 16, we ean obtain the following result:
V.F. BABENKO
94
Theorem 19. Let k, m, nE N, k < n. Then tor each function / E W~(( -1, l)m)
the inequality
(44)
holds, with some constant B, independent 0/ /. Moreover, the constant M~-l,k in
(44) is the best possible.
Finally, observe that
and
*
-M n-l,n-l
IIT(n-I)11
-- 2n - 2 ( n - 1)'..
n-l
Loo(-l,l) - T(n-l)
n-l
References
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Banaeh spaces, Mat. Zametki 52 (1992), 15-20. (Russian) [Engl. Trans.: Math. Notes 52
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(1987), 115-119].
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11. ___ , On exaet inequalities of Landau-Kolmogorov-Hörmander type on the half-line, Dokl.
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15. O. V. Besov, Extension of functions to the frontier, with preservation of diJJerential-difference properties in L" Mat. Sb. (N.S.) 66 (108) (1965), 80-96. (Russian)
INEQUALITIES FOR NORMS OF INTERMEDIATE DERIVATIVES
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16. ___ , Multiplicative estimates for integml norms of differentiable functions of seveml
variables, Trudy Mat. Inst. Steklov 131 (1974), 3-15. (Russian)
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18. Yu. G. Bosse (G.E. Shilov), On inequalities between derivatives, Sb. Rabot Stud. Nauch.
Kruzh. Mosk. Univ., 1937, pp. 17-27. (Russian)
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20. V. I. Burenkov, Exaet eonstants in inequalities for norms of intermediate derivatives on a
finite interval, Trudy Mat. Inst. Steklov 156 (1980), 22-29. (Russian)
21. ___ , Exaet eonstants in inequalities for norms of intermediate derivatives on a finite
interval. II, Trudy Mat. Inst. Steklov 173 (1986), 38-49. (Russian)
22. V. I. Burenkov and V. A. Gusakov, On exact constants in Sobolev embedding theorems,
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Math. 204 (1994), 57-67).
23. A. P. Buslaev and V. M. Tihomirov, Inequalities for derivatives in the multidimensional
case, Mat. Zametki 25 (1979), 59-73. (Russian)
24. H. Cartan, Caleul differentiei. Formes differentielles, Hermann, Paris, 1967.
25. Z. Ditzian, Multivariate Landau-Kolmogorov type inequality, Math. Proc. Cambridge Philos.
Soc. 105 (1989), 335-350.
26. Z. Ditzian and V. Totik, Moduli of Smoothness, Springer Verlag, Berlin - Heidelberg - New
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27. Z. Ditzian, D. Jiang and D. Leviatan, Simultaneous polynomial approximation, SIAM J.
Math. Anal. vol 24 (1993), 1652-1664.
28. R. J. Duffin and A. C. Schaeffer, A refinement of an inequality of brothers Markoff, Trans.
Amer. Math. Soc. 50 (1941), 517-528.
29. V. K. Dzyadyk, Introduetion to the Theory of Uniform Approximation Functions by Polynomials, Nauka, Moscow, 1977. (Russian)
30. V. N. Gabushin, Inequalities for norms of a function and its derivatives in Lp-metries, Mat.
Zametki 1 (1967), 291-298. (Russian)
31. ___ , The best approximation of the differentiation operator on the half line, Mat. Zametki
6 (1969), 573-582. (Russian)
32. S. P. Geisberg, A genemlization of Hadamard's inequality, Leningrad. Meh. Inst. Sb. Nauchn.
Trudov 50 (1965), 42-54. (Russian)
33. J. Hadamard, Sur le module maximum d'une fonction et de ses derivees, C.R. Soc. Math.
France 41 (1914), 68-72.
34. G. H. Hardy, J. E. Littlewood and G. P6lya, Inequalities, University Press, Cambridge, 1934.
35. L. Hörmander, New proof and genemlization of inequality of Bohr, Math. Scand. 2 (1954),
33-45.
36. A. Kolmogoroff, Une genemlisation de l'inegaliU de M. J. Hadamard entre les bornes
superieurs des derivees sueeessives d'une fonetion, C.R. Acad. Sei. Paris 207 (1938), 764765.
37. A. N. Kolmogorov, On inequalities between upper bounds of the suecessive derivatives of an
arbitmry function defined on an infinite interval, Uchen. Zap. Moskov. Gos. Univ. Mat. 30
(1939), 3-13. (Russian)
38. V. N. Konovalov, Sharp inequalities for the norms of funetions and their third partial and
second mixed or directional derivatives, Mat. Zametki 23 (1978), 67-78. (Russian)
39. ___ , Supplement to A. N. Kolmogorov's inequalities, Mat. Zametki 27 (1980), 209-215.
(Russian)
40. N. P. Korneichuk, Exaet Constants in Approximation Theory, Nauka, Moscow, 1987 (Russian) [Eng!. Trans.: Cambridge Univ. Press, Cambridge, 1991).
41. N. P. Korneichuk, V. F. Babenko and A. A. Ligun, Extremal Properties of Polynomials and
Splines, Naukova Dumka, Kiev, 1992. (Russian)
42. N. P. Korneichuk, A. A. Ligun and V. G. Doronin, Approximation With Constrains, Naukova
Dumka, Kiev, 1982. (Russian)
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V.F. BABENKO
43. N. P. Kuptsov, Kolomogorov estimates for derivatives in L2[0, 00), Trudy Mat. lnst. Steklov
138 (1975), 94-117. (Russian)
44. M. K. Kwong and A. Zettl, Norm inequalities for derivatives and dijJerences, lnequalities
(Birmingham, 1987), Lect. Notes Pure Appl. Math. 129, Dekker, New York, 1991.
45. E. Landau, Einige Ungleichungen für zweimal dijJerenzierbare Funktion, Proc. London Math.
Soc. 13 (1913), 43-49.
46. A. A. Ligun, Inequalities for upper bounds of funetions, Anal. Math. 2 (1976), 11-40.
47. ___ , Inequalities between norms of derivatives of periodie funetions, Mat. Zametki 33
(1983), 385-391. (Russian)
48. Yu. I. Ljubich, On inequalities between powers of a linear operator, lzv. Akad. Nauk SSSR
Sero Mat. 24 (1960), 825-864. (Russian)
49. G. G. Magaril-Il'jaev and V. M. Tihomirov, On the Kolmogorov inequality for fraetional
derivatives on the half-Une, Anal. Math. 7 (1981), 37-47.
50. V. A. Markov, On functions deviating least from zero in given interval, lzdat. lmp. Akad.
Nauk, St. Petersburg, 1892 (Russian) [German Trans.: Math. Ann. 77 (1916), 218-258].
51. A. P. Matorin, On inequalities between the maxima of the absolute values of a funetion and
its derivatives on a half-line., Ukrain. Mat. Zh. 7 (1955), 262-266. (Russian)
52. G. V. Milovanovic, Extremal problems for polynomials: Old and new results, Open Problems
in Approximation Theory (B. Bojanov, ed.), SCT Publishing, Singapore, 1994, pp. 138-155.
53. D. S. Mitrinovic, Analytie Inequalities, Springer Verlag, Berlin, 1970.
54. A. Pinkus, n-widths in Approximation Theory, Springer Verlag, Berlin, 1985.
55. A. Yu. Shadrin, Kolmogorov-type inequalities, and estimates for spline-interpolation for periodie classes Wr', Mat. Zametki 48 (1990),132-139 (Russian) [Engl. Trans.: Math. Notes
48 (1990), 1058-1063].
56. ___ , On exaet eonstants in inequalities between the L oo -norms of derivatives in a finite
interval, Dokl. Akad. Nauk 326 (1992), 50-53. (Russian)
57. ___ , To the Landau-Kolmogorov problem on a finite interval, Open Problems in Approximation Theory (B. Bojanov, ed.), SCT Publishing, Singapore, 1994, pp. 192-204.
58. I. J. Schoenberg and A. Cavaretta, Solution of Landau 's problem, conceming higher derivatives on halfline, M.R.C. Technical Summary Report, 1970.
59. ___ , Solution of Landau 's problem coneeming higher derivatives on the halfline, Constructive Theory of Functions (Proe. Internat. Conf., Varna, 1970), Izdat. Bulgar. Akad.
Nauk, Sofia, 1972, pp. 297-308.
60. V. G. Soljar, On an inequality between the norms of functions and its derivatives, Izv. Vyssh.
Uchebn. Zaved. Mat. 1976, no. 2 (165), 64-68. (Russian)
61. S. B. Stechkin, Best approximation of linear operators, Mat. Zametki 1 (1967), 137-148
(Russian) [Engl. Trans.: Math. Notes 1 (1967),91-100].
62. E. M. Stein, Functions of exponential type, Ann. Math. 65 (1957), 582-592.
63. L. V. Taikov, Inequalities of Kolmogorov type and formulae of numerical dijJerentiation,
Mat. Zametki 4 (1967), 233-238. (Russian)
64. V. G. Timofeev, lnequality of Landau type for multivariate functions, Mat. Zametki 37
(1985), 676-689. (Russian)
65. O. A. Timoshin, Sharp inequalities between norms of partial derivatives of second and third
order, Dokl. Akad. Nauk 344 (1995), 20-22. (Russian)
66. V. M. Tihomirov and G. G. Magaril-Il'jaev, Inequalities for derivatives, Commentary to
Selected Papers of A. N. Kolmogorov, Nauka, Moscow, 1985, pp. 387-390. (Russian)
TABLE OF INEQUALITIES IN ELIIPTIC BOUNDARY
VALUE PROBLEMS
C. BANDLE and M. FLUCHER
Universität Basel, Mathematisches Institut, Rheinsprung 21, CH-4051 Basel,
Switzerland
Abstract. This contribution contains a compiled list of inequalities that are frequently
used in the calculus of variations and elliptic boundary value problems. The selection
reflects the authors personal taste and experience. Purely one dimensional results are
omitted. No proofs are given. Frequently we refer to textbooks rather than original
sourees. General references are P6lya and Szegö [73], Morrey [59], Giaquinta [33-34],
Gilbarg and Trudinger [35], Kufner, John and Fucik [49], Ziemer [94]. We hope that this
table will be useful to other mathematicians working in these fields and a stimulus to
study some of the subjects more deeply.
1. Introduction
1.1. NOTATIONS
Unless otherwise stated 0 is a bounded, connected domain in Rn with Lipschitz
boundary. The exterior unit normal is denoted by 1/, the distance of a point from
the boundary by
d(x) := inf{lx - Yl : Y i. !1}.
The letter c stands for a generic constants which is independent of the functions involved, c stands for a positive constant that may be arbitrarily small and e E (0,1)
an interpolation parameter. The positive part of a function is u+ := max(u,O).
For a set A c Rn we denote by lAI and laAI its volume and surface area in the
sense of Hausdorff measure. B~ is a ball in Rn of radius p centered at x. The
symmetrized domain 0* is a ball centered at the origin having the same volume
as O. The volume and surface of the unit ball are
Let u
0 --* R be a measurable function. The function
u*(a) := sup{t : 1{lul ~ t}1 ~ a}
1991 Mathematics Subject Classijication. Primary 35Jxxj Secondary 49N60, 35K85.
Key words and phrases. Elliptic partial differential equationsj Calculus of variationsj Isoperimetrie inequalities.
97
G.v. Milovanovic (ed.), Recent Progress in Inequalities, 97-125.
© 1998 Kluwer Academic Publishers.
C. BANDLE AND M. FLUCHER
98
is ealled decreasing rearrangement of u. The funetion
defined on n* is ealled Schwarz symmetrization of the positive funetion u. It is
radially symmetrie and lu* > tl = Ilul > tl for every t ~ O. The relative capacity
of a set A c n is defined as
eapn (A) := min
{k l\7ul
u ~ 1 on B, Ac B, B open}
2 : u E HJ,
and eap (A) := eaplRn (A). The minimum is attained by the capacity potential of A.
1.2. FUNCTION SPACES
All functions (with a few exeeptions) are sealar functions defined on O. Sequenees
are denoted by (Ui). Integrals are taken with respeet to Lebesgue measure. The
mean value of a function is denoted by
Convergenee almost everywhere with respect to Lebesgue measure is abbreviated
as a.e. The convolution ot two functions given on all of jRn is defined as
In
The space LP is endowed with norm lIull~ :=
lul P , where 1 ~ p < 00. A
sequence (Ui) of LI functions is said to be equi-integrable if
lim sup { IUil = 0
i JA
IAI--+o
or
Moreover one says that Ui ~ u in measure if I {x : IUi - ul ~ c} I ~ 0 for every
c > O. The dual exponent p' of pE [1,00] is defined by the relation I/p+ I/p' = 1.
The Sobolev space Hk,p is given by the norm
Ilult,p:= L
i IDauI
P•
lal9 n
If p = 2 we write Hk := Hk,2. in the ease of Orlicz spaces the power functions is
replaced by a more general N-function
A(t):=
1
t
a
with a positive, strictly increasing, upper semi-continuous function a with a(O) = O.
The dual N-function is defined as
Ä(t):=
1
t
a- I .
TABLE OF INEQUALITIES IN ELLIPTIC BVP
99
If A(2t) ~ cA(t) for large t, then
IluliA := inf { c >
0:kA ~ ~ I} ,
0
Ilullk,A:= L IIDoullA
1019
defines a norm on the Orlicz space LA and the Sobolev-Orlicz space Hk,A. In
particular u E LA if and only if
k
Aou < 00.
Another important generalization of LP are the Lorentz spaces L(p, q) on !Rn given
by
1
lIull q
(p,q) := 0
00
(t1/P-1/q-1
1
t
0
u*) q dt,
._ supt 1/p-1
lIull(p,oo) .t>o
l
0
t U*,
with u* as in Section 1.1. If 1 < p, q < 00 this norm is equivalent to
1
00
t q/ p- 1u.(t)qdt.
In particular L(P,p) = LP(!Rn ) [94]. Campanato spaces are given by the norm
The John-Nirenberg space of functions of bounded mean oscillation can be defined
as BMO := LP,n. On !Rn
defines a norm if we identify functions whose difference is a constant. The Hardy
space 11. 1 on !Rn can be defined as folIows. For f E L 1 (!Rn) define
r
IIfllw:= JR dx sup
I [ d! cP(~)
f(y)1 '
ö>o JRn e
e
where
cP E Co(B~) is a mollifying kernel with
A Different 4> lead to equivalent norms. References on Hardy spaces are [26], [81],
[77]. We follow [63]. A local Hardy space was introduced by Goldberg [36]. An
embedding of normed spaces, denoted by X c Y, is a bounded linear injection
j E .c(X, Y). If j is a compact map we write X ce Y.
C. BANDLE AND M. FLUCHER
100
1.3. BOUNDARY VALUE PROBLEMS
In the most general case we consider uniformly elliptic operators of the form
n
Lu:= - L
i,j=l
a2
n
a
i=l
X.
aij{x) a .; . + Lbö(x) au. +c{x)u
X.
xJ
defined for u E H I . Several estimates deal with the Dirichlet problem
(I)
Lu = f
in 0,
u = 0 on ao,
which is the prototype of an elliptic boundary value problem. The natural space for
its solutions is HJ where the subscript refers to homogeneous Dirichlet boundary
values. The corresponding principal Dirichlet eigenvalue is denoted by >'1. For
simplicity most results are stated for the Laplacian although they carry over to
more general elliptic operators. The Dirichlet Green's function Gy is the solution of
-!J.G y = 8y
in 0,
Gy = 0 on ao,
where 8y is the Dirac distribution with singularity at y.
2. LP-spaces
Most inequalities of this section are proved in standard books on functional analysis
(see e.g. [1), [3]).
2-1 Cauchy-Schwarz's inequality:
2-2 Hölder's inequality: If 1 ~ P ~ 00 then
with the Orlicz norm as defined in Section 1.2 [1, p. 237]. If l/p = I/PI + l/p2
and l/q = l/ql + l/q2 then
II U lU211(p,q) ~ lI u dl(Pl,Ql)ll u 211(P2,q2)·
In fact the dual space of the Lorentz space L(p, q) is L(p', q') [94].
2-3 Calderon's lemma: If PI ~ P2 then
hence LP2 C LP1. If ql ~ q2 then
TABLE OF INEQUALITIES IN ELLIPTIC BVP
hence L(p, ql) C L(p, q2) [94, p. 37].
2-4 Young's inequality: [46]. If 1 < P < 00 then
,
inr uv ::; p~lIull: + ~lIvll::
p
10 uv ::; cllull: + p;,1 c1/(1-P)llvll:: ,
10 uv ::; 10 A u + In Ä v ,
0
0
where A is an N-function with dual Ä as defined in Section 1.2 and u, v 2: o.
2-5 Bank's inequality: [11, p. 69]. If Ul, U2, 4> E L 2 with
2-6 Jensen's inequality: If 4> 2: 0 is convex then
2-7 Minkowski's inequality:
2-8 Clarkson's inequalities: [4, p. 89].
Ilu + vII: + Ilu - vii: ::; 2P-l (Ilull: + Ilvll:) , 2::; p < 00 ,
Ilu + vII:' + Ilu - vii:' 2: 2(llull: + Ilvll:)P'-l, 2::; p < 00,
lIu + vii:' + Ilu - vii:' ::; 2(llull: + Ilvll:)P'-l,
lIu + vii: + Ilu - vII: 2: 2P-l (Ilull: + Ilvll:) ,
1< p::; 2,
1 < p ::; 2.
2-9 Interpolation inequality: [35, p. 146]. If p ::; r ::; q and
1
()
1-(}
r
p
q
-2:-+--,
then
101
C. BANDLE AND M. FLUCHER
102
2-10 Riesz-Thorin theorem: [45). If a linear operator T satisfies
with
1
1- (I
(I
P
Po
PI
- = - - + -,
then
0 ~ (I ~ 1,
IITullq ~ c~-8 cf Ilulip .
2-11 Convolution inequality: [4, p. 89), [94, p. 96). If
1
1
P
1
q-l
- = - + --,
r
1 ~ P,
q ~ 00,
then
with u * v as in Section 1.2. If
11111
1
- = - + - - 1 and - = - + - ,
P
PI
P2
q
ql
q2
then
IIUI *U211(p,q) ~ 3pllulll(Pl,qtlllu211(P2,q2)'
If one of the factors is the Riesz kernel K),,{x) := Ixl-)" then K)" E L{n/>.., 00) and
2-12 Hardy-Littlewood-Sobolev inequality: If 0< >.. < n, 1 < P < n/{n - >")
and l/p + >../n = l/q + 1, then
2-13 Hardy-Littlewood maximal function theorem: [82, pp. 55-58). The
maximal function
Mu{x) := sup IB1 I [ lul
p>O
p JB~
of u E LI satisfies
2-14 Hardy inequality: [45), [94, p. 35). If P > 1, r > 0,
11:1: u
U{x):= -
x
0
for
x< 0
TABLE OF INEQUALITIES IN ELLIPTIC BVP
l
and
eu
O(x) := sup -c-1
e>x .. - x x
103
x E IR,
for
then
and
2-15 Hardy inequalities in one dimension: If 1(0) = 0 then
11 ~ ~ 41111'1
1
2
1
2
•
More generally, if a > 2k - 1 then
1
00
o
X",-2k 111 2 <
4k
- (1 - a)2 ... (2k - 1 - a)2
If a < 1 and l(i) (0) = 0 for i = 0,1, ... ,k - 1, then
1
1
00
0
x'" II(k) 12 •
1
00
X",-2k 111 2 <
4k
x'" II(k) 12 •
- (1 - a)2 ... (2k - 1 - a)2 0
2-16 Hardy inequalities in higher dimensions: [65]. If p > 1 then
00
o
kl ~ I ~
P
C
kl'Vu lP
for all u E H~'P(O), d = distance from boundary, C ~ {P/{P - 1))p. For convex
domains c = (p/{P - I))P. In three dimensions
[ lul 2 < 4 [ l'Vul 2
in 1 + Ixl 2 - in
for every u E HJ(O). If P"l- n and 0"1- n then
[lul P (p)P (
in Ixlp ~ In - pi in l'Vul P
for u E H~'P(O), whereas
klxln1~:l~lxl/r) ~
(n: 1) kl'Vuln
n
for u E H~,n(Rn \ Bfir).
2-17 Hardy-Littlewood-Sobolev inequality: [45]. If 1 < P < n/2 then the
solution u = K n - 2 * 1 of -~u = 1 satisfies
lIullnp/(n-2p) ~ cn,pll~ullp'
2-18 Monotonicity of p-Laplacian: If p ~ 2 then
(lV'ulp-2V'u -1V'vIP-2'Vv) . (V'u - 'Vv) ~ cpl'Vu - 'Vv1 2 (I'VuI P- 2+ l'VvI P- 2)
with cp ~ C2 = 1/2 and cp = 1 for p ~ 3.
104
C. BANDLE AND M. FLUCHER
3. Convergence Theorems in LP
3-1 Lebesgue's differentiation theorem: [43]. If U E LI then
JB~ = u(x)
lim IB\I [
p-to
x
U
for a.e. x E n.
3-2 Absolute continuity theorem: If U E LI and c > 0 then
for all A c n with lAI< <5(c).
3-3 Lusin's continuity theorem: If U E LI and c > 0 then U is uniformly
continuous on n \ E with lEI< c.
3-4 Egoroff's theorem: If Ui --t U a.e. (all measurable) and c > 0 then
Ui --t U
uniformlyon
n\E
with lEI< c.
3-5 Lebesgue's convergence theorem: If Ui --t U a.e. and IUil ~ Vi --t V in LI
then
3-6 Vitali's convergence theorem: [3]. If (Ui) is equi-integrable and Ui --t U
in measure then Ui --t U in LI. If Ui E LP, Ui --t U a.e., and (uD is equi-integrable
then
Ui --t U in LP.
3-7 Fatou's lemma: [3]. If Ui ~ 0 then
In
liminfui ~ liminf
If Ui ~ Vi --t V in LI then also
limsup
In ~ In
Ui
In
Ui·
limsupui'
4. Sobolev Spaces
Most inequalities of this section can be found in [59], [1], [49], (35), [53), [94].
4-1 Poincare's inequalities:
(a) For every u E HJ
105
TABLE OF INEQUALITIES IN ELLIPTIC BVP
(b) For every u E H~'P
lIullp ~ C IIVulip .
(c) For every u E H I and B~ C 0
(d) [59]. If 0< 0< 1 then
for every u E H I with I{u = O}I 2:: 0101.
(e) [31, p. 15]. If u vanishes on a set of non-vanishing capacity then
klIu 1~
2
cap({: = O})
kIVuI
2
.
(f) [20]. If u E HJ(O x IRm ) then
(g) If 0 bounded in one direction then
for every u E H I .
(h) A one dimensional version is Wirtinger's inequality: If u E H I (O,27f) is
periodic with vanishing mean value then
Equality holds if and only if u(t) = a cos t + bsin t [87], [15].
4-2 Gärding's inequality: [34, p. 7-9]. If A is a uniformly positive definite
matrix and A E Loo, bELn and d E Ln/2 then there is a constant Cl > 0 such
that
k
Vu· A(x)Vu + 2u b(x) . Vu + d(x)uv 2:: Cl
kIVul + klul
2
C2
2
for every u E H I . The same is true for systems with continuous A satisfying the
Legendre Hadamard condition.
C. BANDLE AND M. FLUCHER
106
4-3 Korn's inequality: [93]. In terms of the symmetrie gradient
k +
lul 2
1
E: = - (Du + DuT ) ,
2
k
Tr (ETE)
for u E HJ(n,JRn ),
IIDulip ~ c(llull p + IIEll p)
for u E HP(n, jRn) .
IDul 2 ~ c
4-4 Poincare's inequality for capacity potentials: [28]. Let (Ui) be a sequence of capacity potentials with cap (Ai) -+ 0 and p < 2n/(n - 2). Then
IIUilip
IIV'Uill p -+ O.
4-5 Gagliardo-Nirenberg's inequality: [32], [64], [4, p. 38].
1
lIulln/(n-l) ~ 211 V' u 1l 1
for every u E H~,l. This implies: If ()/p + (1 - ())/q = () then
lIulln/9(n-l) ~ (2()) -911V'ull!lIull!-9
for every u E H~'p.
4-6 Ehrling-Browder's inequality: [1]. If kik' ~ () ~ 1 and
! = ~ + ()
n
p
(.!. _ +
p'
kl )
n
1 - () ,
q
then
If n = 2 then
lIullp ~ c lIV'ull~-l/Pllull~/p .
For n ~ 3 Sobolev's inequality follows.
4-7 Sobolev's inequalities:
(a) [65], [3]. If 1 ~ p < n/k then
Ilulinp/(n-kp) ~ c(IIDkuli p + Ilullp)
for all u E H1,p(jRn).
(b) [4, p. 39]. If 1 ~ p < n then
p - 1 ( n _ p ) l/p
lIullnp/(n_p) ~ n _ p n(p _ 1)
x
X
(
r(n + 1)
) l/n
r(n/p)r(n + 1 _ n/p)18B1 1
lIV'ull p
107
TABLE OF INEQUALITIES IN ELLIPTIC BVP
for u E H~'p. Extremal functions are of the form
u(x) = ( C + Ix - xo IP/(P-l) )
l-n/p
.
(c) [29]. If 0> 0 then
Il uI1 2n/(n-2) ::; collV' ul1 2
for every u E HJ with I{u = O}I ;::: 0101·
(d) If 0< 0: < k - n/p then
(e) If I/p' = l/p - (k - k')/n, k;::: k' ;::: 0 and 1 ::; p ::; p' then
Ilullk"p' ::; C Ilullk,p .
4-8 Weighted Sobolev inequality: [53]. If p ;::: 2, n + 0: > 0, ß/2 ;::: o:/p,
0: + n/p = ß + (n - 2)/2 (special homogeneity) and u = 0 on ao then
If
ao is Lipschitz and 1 < P < 00 then
10 luIPd(x)"'-P ::; 10 lV'uIPd(x)'" if > P 10 luI Pd(x)-1+ 10 lV'uIPd(x)'" if > P - 1
0:
C
e ::; Ce
1,
0:
for every E > 0 [48].
4-9 Generalized Sobolev inequality: [27]. If 0::; f(7) ::; CI71 2n /(n-2) then
where
Si := sup
{ln f(v) :
V E C,:"'(IRn ),
IIV'v11 2 ::; I} .
This statement can be localized. For every f, > 0 there is an optimal ratio k(f,)
such that
C. BANDLE AND M. FLUCHER
108
for every p/R ~ k(8), x ERn, and u E D 1 ,2(Rn).
4-10 Traces: [3, p. 168]. If 1 ~ p < 00 then
{
ian
~ e (IVuI P , i.e. H1,p c LP(an).
in
The embedding is compact for p < 00 and continuous for p = 00. [1, p. 114],
[46, p. 328 and 337]. If an is Ck then
Hk,p C L(n-l)P/n-kp(an)
Hk,p C U(an)
if kp< n,
for every q if kp ~ n,
H1,p c H1-1/p,p(an).
If M is am-dimensional submanifold of 0 and j5 < mp/(n - (k - k)p) then
Hk,p C Hk,P(M n 0) .
5. Critical Sobolev Embeddings
In this section we consider the spaces Hk,p with kp = n. In this case the measure
IID kull pdx which contributes to the leading term of the norm is conformally invariant. By Sobolev's theorem Hk,p ce U for every q < 00 but Hk,p ~ LOO. See
Section 1.2 for definitions of spaces and norms.
5-1 Poincare-Sobolev inequality for BMO functions: If p < 00 then
If kp = n then
lIull BMo ~ e lIullk,p'
5-2 Orlicz' inequality: [4, p. 63].
In exp(u) ~ e exp (a llu ln + ß llvuln)
5-3 Strichartz's inequality: [1, p. 242]. If n = kp and
A(t) := exp (tP/(P-l)) - 1,
then
lI u llA ~ ellullk,p'
5-4 Trudinger-Moser's inequality: [4, p. 65].
l exp (nlaBlll/(n-l)luln/(n-l)) ~ elnl,
In exp (41l'u ~ elnl (n = 2)
2)
TABLE OF INEQUALITIES IN ELLIPTIC BVP
109
for every u E H~,n with IIVulin ~ 1.
5-5 Orlicz-Sobolev embedding: [1, p. 252]. If
1
00
I
A(t)
t(n+1)/n dt < 00,
then
for every u E HI,A. n fact u is continuous.
5-6 Wente's inequality: [91], [41]. For /,g, hE H I (JR2 ) one has
5-7 Higher integrability of J acobians: [24], [56], [62]. If u E HI,n(JRn, JRn)
then
IIdet Dull1il ~ c IIDull~n .
5-8 Fefferman-Stein duality: [26], [77), [88).
J/g ~ 1I/llwllgII
C
BMO ·
In fact BMO is the dual space of 11 1 .
6. Maximum and Comparison Principles
In this section all functions are supposed to be C 2 (except for the weak maximum
principle) .
6-1 Maximum principle: [74), [35). If ßu+g(·,u) ~ ßv+g(·,v) in 0 and
u ~ von ao.
(a) If 9 (x,·) is non-increasing for every x, then
u~v
in O.
(b) If in addition 9 (x,·) is Lipschitz and u -:j:. v then
u>v
in O.
(c) If 0 satisfies an interior ball condition and if u(x) = v(x) for some xE ao
and u -:j:. v, then
au
av
av (x) < av (x) .
(d) If (-ß - A)u ~ 0 with u = 0 on ao, A < Al and u -:j:. 0 then
u > 0 in O.
C. BANDLE AND M. FLUCHER
110
6-2 Weak maximum principle: [35, p. 179]. H u E Hl is subharmonic
(10 "Vu"VljJ::; 0 for all
ljJ E HJ,ljJ
~ 0),
then
sup u ::; sup u+ .
o
ao
6-3 Giraud's maximum principle: [58]. H an is Hölder continuous -Au::; 0
and u assurnes its maximum at a point x E an, then
lu(x) - u(y)1 ::; clx - Yl
for every yEn.
6-4 Bernstein type inequalities:
(a) HAu = 0, then AI"VuI 2 ~ 0 and the maximum of l"Vul 2 is attained on an.
(b) [71]. H Lu = 0 then for some constant c the maximum of l"Vul 2 + clul 2 is
attained on an.
(c) [53], [75]. Let u be the solution of the torsion problem
- Au = 1 in
n u = 0 on an.
Then the maximum of l"Vul 2 + 21ul 2 is attained on an.
6-5 Payne-Philippin maximum principle: [70]. Let u be a solution of the
elliptic problem
"V. (g (I"VuI 2 ) "Vu) + P (l"VuI 2 ) f (u) = 0 in n,
with g (t) + 2tg' (t) > O. Define
P (x):=
r'Vu(x W g (~)
10
+ 2~g' (~)
p (~)
r(x)
d~ + 2 10
f ('TJ) d'TJ.
Then the maximum of P is attained on an or at a critical point of u.
6-6 Miranda's biharmonic maximum principle: [57]. H n is sufficiently
smooth and A 2 u = 0 then max l"Vul 2 - uAu is attained at the boundary.
6-7 Boundary blow up: [50], [13]. H Au ~ u P then
u(x) ::; cljJ(d(x))
u(x) -ljJ(d(x)) ::; c
where
~(t) ,~ (t;~:
ifp>l,
ifp>3,
1))
-2/1.-1)
6-8 Whitney's inequality: [76]. Given a domain n there exists a function
d E COO(n) such that l"Vdl is bounded and
1
-
- d ::; d ::; cd,
c
TABLE OF INEQUALITIES IN ELLIPTIC BVP
111
7. Elliptic Regularity Theory
We start with the weakest hypothesis on Lu.
7-1 Weinberger's inequalities: [90]. Let Lu := V . A (x) Vu be an elliptic
operator in divergence form and
A := inf Al (A).
(1
Then the Dirichlet Green's function Gy of L satisfies:
p
2
cp,n ..!.lnI
II G y 11 pi <
A /n-l/ '
where
and ß is the beta function [18]. For p > n
II VGY 11 pi <_ Cp,n Alnl l / n - l / p ,
where
Cp,n := IBII- I / n
(;
=~
r-
l p
/ nl/n-':'IP.
For the Laplacian equality holds when n is a ball centered at x. As a consequence
the solution of (1) satisfies
If f = V . v, then
7-2 Grisvard's inequality: [37-38], [54]. If an is smooth, -ßu E L 2 and
avu E H I / 2 then
7-3 Regularity in Lorentz spaces: If f E L(p.q) with 1 < p < n/2 and ßu = f
in !Rn then u = Uo + h with -ßh = 0 and
Il u OIl(np/(n-2p),q) ~ cllfll(p,q)'
This follows by convolution with K n - 2 (x) = Ixl-(n-2) and the Hardy-LittlewoodSobolev inequality.
C. BANDLE AND M. FLUCHER
112
7-4 Regularity in Hardy spaces: [26]. If f E 11 1 and -Au = f in IRn then
u = Uo + h with -Ah = 0 and
In two dimensions also
7-5 Riesz operators: The operators
-A-1a·a·
. LP -t,
LP
L oo -t BMO ,
• J.
BMO -t BMO,
are bounded.
7-6 Calder6n-Zygmund's inequality: [35, Lemma 9. 17]. If an is
00, and u E H 2 ,p n HJ, then
e2 , 1 < P <
7-7 Meyers' inequality: [55]. If Ais bounded and Ip - 21 small enough, then
lIulll,p ~ C 1IV'(AV'u)lI-l,p.
7-8 Regularity theorem for smooth operators: [4, p. 85]. If L has e oo
coefficients, k ~ 0, 1 < P < 00 and 0 < a < 1, then
Ilullk+2,p ~ C IILullk,p ,
lIullck+2+o< ~ C IILullck+o< .
7-9 Schauder estimates: [3], [34, p. 48-53]. If an is e2+ a then
lI u ll c loc
2+O< ~ c{lI u llLoo + 11 Au 11 Co< ) ,
lIV'ullcO< ~ c{lI u llp + IIAullcO<) ,
Il ull c2+o< ~ c{lIullco + IIAullcO< + lI ullc2+O«öo)) .
If L has trivial kernel then
7-10 Cordes-Nirenberg's inequality: [4].
7-11 De Giorgi-Nash-Moser regularity theorem: [60], [33, p. 53]. If u is a
weak solution of V' . A(x)V'u = 0 with uniformly positive A E Loo and n' ce n
then
TABLE OF INEQUALITIES IN ELLIPTIC BVP
113
for some a > O.
1-12 Campanato's theorem: [49], [33, pp. 70-72], [34, p. 41]. If 0< a < 1 and
r lu - uB~IP ~ cpn+o p,
JB~
then
If n < A ~ n + p then
Ilul!c>,-n/p ~ IluIlLp,A'
In fact LP'>" = c>..-n/p are isomorphie.
1-13 Morrey's Dirichlet growth theorem: [33-34]. If a > 0, u E Hl~: and
r lV'ul ~
P
JB~
cpn-p+op
for every ball, then
u E Cl~c'
1-14 John-Nirenberg's inequality: [81], [35]. For p < 00 and 7 > 0 one has
1;~ll~ lu - uB~IP ~ cpllull~MO'
I {y E B~ : 1u(y) - uB~ I> 7} I ~ c11B~1 exp (-llu~:~o) .
If n is eonvex, u E H 1 ,1,
for all balls then
10 exp (blu - unI) ~ clnl·
1-15 Estimates for the Green's function near the boundary: The Diriehlet Green's function of a seeond order uniformly elliptie operator L with C 1 ,0_
eoeffieients on a C 1 ,0 domain satisfies
with positive eonstants Cl, C2, and
Ix -
g(x, y) =
2-n'
y1
(1 d(X)d(Y))
mm, Ix-y 12
(n ;::: 3),
Iog (1 + d(X)d(Y))
IX-Y 12
(n = 2),
y'd(x)d(y) min( 1, y'~(~~~y) )
(n = 1).
(see: G. Sweers, Positivity jor a strongly coupled elliptic system by Green's junction
estimates, J. Geometrie Analysis 4 (1994), 121-142).
114
C. BANDLE AND M. FLUCHER
8. Further Integral Inequalities for Solutions of Elliptic
Differential Equations
8-1 Mean value properties: If -ßu :::; 0 then
whenever Bg C n. If -ßu = 0 then
1
ID"'u(x)1 :::; c'" pn/2+I"'lllullL2(B~) .
If n = 2 and -ßu:::; Ke u then
für p small enüugh. Best constants are known [8].
8-2 Harnack's inequality: [35]. If -ßu = 0, u > 0 in n and K ce n then
sup u < c inf u .
K
-
K
8-3 Weak Harnaek inequality: [62], [35, p. 194]. If u ~ 0, 1 < P < n/(n - 2)
and q > n then
8-4 Caecioppoli's inequality: [33, p. 77]. If -ßu = 0 then
8-5 Reverse Hölder inequality: [33, p. 119, 136]. If -ßu = 0 then
für p > 2.
8-6 Monotonicity formula for harmonie maps: [78]. If u : n -+ jRn is a
harmonie map and R ~ p then
TABLE OF INEQUALITIES IN ELLIPTIC BVP
115
8-7 Kato's inequality: [46, Lemma 9]. If U E C 2 , <P E Co, <P ~ 0, then
l ~<plul ~ l sign(u)<p~u.
8-8 Inequality for sub- and supersolutions: [52]. A pointwise maximum
(minimum) of subsolutions (supersolutions) of Lu = f is a subsolution (supersolution). The same is true for H 1 solutions.
8-9 Pohozaev identity: [83]. If
in n,
-~u j'(u)
then
n;
2l 12 - l
lV'U
n
If n is starshaped, then
u = 0 on an,
f 0 u+~
hn
lV'ul 2x . v = O.
9. Calculus of Variations
9-1 Direct method: [83, p. 4]. A weakly lower semicontinuous coercive functional
on a reflexive Banach space attains its minimum. Le. if F( Ui) -t inf Fand Ui --'" U
weakly then
F(u) ::5 liminf F(Ui) = inf F.
9-2 Weak lower semi-continuity of norm: If (Ui) is a bounded sequence in a
reflexive Banach space then
Ui --'" U weakly
for a subsequence. If Ui --'" U weakly in a Banach space then
lIulI ::5 liminf IIUill·
If Ui --'" U weakly and IIUili -t Ilull in a uniformly convex Banach space then
Ui -t u.
9-3 Brezis-Lieb's lemma: [18]. If a bounded sequence (Ui) in LP converges
pointwise a.e. to a function U then
liminf Ilui - ulI: = liminf Iluill: -lIulI:·
9-4 Maximal distance to weak LP-limits: [27]. If Ui --'" U weakly in LP then
liminf IIUi - ulI: :::; c liminf Iluill:
with
c = max (aV- 1 + (1 - a)V-l) (a1/(V-l) + (1 _ a)l/(V-l»)V-l
O~o:9
9-5 Semicontinuity theorem: [33, pp. 23-25], [34, p. 13]. If fE C(n, IR.m ,lR.mn)
is bounded below and convex in the last argument, Ui --'" U weakly in Hl~'~ or
Ui -t U in Lfoc then
C. BANDLE AND M. FLUCHER
116
10. Compactness Theorems
10-1 Ascoli's compactness theorem: [3]. If (Ui) is a bounded sequence of
equi-continuous functions in C(K) with compact K then
Ui -t U in C(K)
for a subsequence.
10-2 Dunford-Pettis compactness criterion: [3, p. 176]. If the sequence (Ui)
is bounded and equi-integrable in Li then
for a subsequence.
10-3 Frechet-Kolmogorov compactness theorem: Suppose (Ui) is a bounded
sequence in LP with p < 00. If for every ca compact set K ce n exists such that
s~p IIUiIlLP(fl\K) < c
•
and
s~p
then
•
Ilui(· - h) - uill p -t 0
as
h -t 0,
for a subsequence.
10-4 Rellich-Kondrachov compactness theorem: [3].
np
Hk,p
ce Lq
for
q< --k-'
n- p
Hk,p
ce COI.
for
a<k--.
n
p
10-5 Weak compactness in non-reflexive Sobolev spaces: [33, p. 29]. If
(Ui) is bounded in Hi,i with (V'Ui) uniformly absolutely continuous then
Ui ->. U weakly in Hi,i
for a subsequence.
10-6 Murat's compactness theorem: [83, p. 30]. If Ui ->. U weakly in HJ and
(ßUi) is bounded in Li then
for every q < 2 and a.e.
10-7 Ehrling lemma: [3]. For every tripIe of nested Banach spaces X ce Y c Z
one has
117
TABLE OF INEQUALITIES IN ELLIPTIC BVP
11. Geometrical Isoperimetrie Inequalities
The perimeter of a set A c ]Rn is defined as
while the relative perimeter of A c 0 is given by
IBAlo := sup
{i
\7. v : v E Cgo (O,]Rn), lvi ~
1} .
For smooth sets IBAlo = IBA \ BOI.
11-1 Isoperimetrie inequality for perimeter: [66], [21], [39].
with equality for balls. The Fourier analysis proof of Hurwitz and Lebesgue in two
dimensions can be found in [87] as well as a variational approach to the general
case. A similar inequality holds in spaces of constant curvature [21].
11-2 Bonnesen's inequality: For every set A C ]R2 one has the following quantitative stability estimate involving yhe deviation from a disk
where h denotes the minimal width of an annulus containing BA. Similar results
for higher dimensional convex sets can be found in [67].
11-3 Relative isoperimetrie inequality: [21], [68]. If 0 satisfies an interior
cone condition, then
min{IAI, 10 \ AI} ~ cIBAI~/(n-l) .
If 0 is a ball equality holds for half balls.
11-4 Relative isoperimetrie inequality for planar sets: [7], [9]. Suppose
A C ]R2 is simply connected with BA = BAI U BA 2 (disjoint). Denote by K, the
curvature of BA with respect to the exterior normal. Then
Equality holds for sectors.
11-5 Isoperimetrie inequality for two-dimensional manifolds: [2], [10],
[17], [44]. Let 0 C ]R2 be a simply connected domain endowed with the conformal
metric pldxl of Gaussian curvature K, i.e. -ßlog(p) = K p2. Then
C. BANDLE AND M. FLUCHER
118
where
Equality holds for balls in the limit as K tends to a Dirac measure at the center.
Moreover
L ~ 47rIOl
2
p -
(S~PK) 101~,
Equality holds e.g. if 0 is a ball, K a constant and p(x) = 1/(1 + Klxl 2 /4).
11-6 Gromov's isoperimetrie inequality: [16]. Let M be a compact Riemannian manifold of dimension n and A c M. If the Ricci curvature of M satisfies
Ric (M) ~ Ric (sn) = n - 1 then
18AI > (IMI) (n-l)/n
18A*1 - ISnl
where A* is a gedodesic ball on sn = 8Bl c ]Rn+l with IA*I = lAI.
11-7 Isoperimetrie inequality of Reilly and Chavel: If Ac ]Rn has smooth
boundary then
where /-l2 denotes the first nonzero eigenvalue of Laplace-Beltrami operator on 8A
[87].
12. Symmetrization
The decreasing rearrangement of a function u : 0 -+ ]R+ has been defined in
Section 1.1.
12-1 Cavalieri's principle: [73]. The decreasing rearrangement of a positive
function satisfies
{
{In l
Jn fou = Jo fou*,
.
12-2 Rearrangement inequalities: [40].
if cf> is non-decreasing and convex and
for every a ~ 101.
TABLE OF INEQUALITIES IN ELLIPTIC BVP
119
12-3 Schwarz symmetrization: [73], [11], [47], [85-86], [6]. The symmetrized
function u* defined in Section 1.1 satisfies
for every convex, non-decreasing positive function rjJ and every u E HJ. In partieular
rf
10.
ou*
=
rf
10
0
u
for 1 ~ p < 00. Equality in first relation with p > 1 implies that u = u* a.e. up
to translation provided that no level set below the top level has positive measure
[20].
12-4 Sehmidt's inequality: [39]. For every A c B c ]Rn one has
dist (8A, 8B) ~ dist (8A*, 8B*) .
12-5 Brunn-Minkowski's inequality: [39], [21]. For A, B C ]Rn one has
where (JA + (1- (J)B := {(Ja + (1 - (J)b : a E A, bEB}. The same is true for the
exterior Lebesgue measure. If A and Bare convex and 0 < (J < 1 then equality
holds if and only if A and B are homothetie.
12-6 Riesz' rearrangement inequality: [40].
Ln dx Ln dy f(y)g(x - y)h(x) ::::; Ln dx Ln dy J*(y)g*(x - y)h*(x).
12-7 Weinberger-Talenti's inequality: [85]. If
-~u =
f in 0,
u = 0 on 80,
J* in 0*,
u = 0 on 80*,
and
-~u =
then
u* ~ U in 0*.
12-8 Harmonie transplantation: [42], [12]. Let r(x) denote the harmonie
radius of 0 at x. For radially symmetrie u = J.toG o : B~(x) -t ]R define U x := J.toG x .
Then
for every f : ]R -t ]R+. This fact allows to derive upper bounds for eigenvalues and
related quantities while symmetrization gives lower bounds.
120
C. BANDLE AND M. FLUCHER
12-9 Isoperimetrie inequality for capacity: [73], [30].
capo(A)
IAI(n-2)/n
>
= n(n _ 2)IB l I2 / n
cap(B l )
IB l l(n-2)/n
Inl
(P)
IBJI
capo ()
A log jAf 2: capBJ B o log IBgl = 471"
(n 2: 3),
(n= 2).
Equality holds if and only if A is a ball and n = lRn (in two dimensions if n and
A are concentric balls).
12-10 Subadditivity of modulus: [69], [42], [12]. If Ac B c ethen
1
1
1
capc(A) - capB(A) + capc(B) .
>
Equality holds if and only if B is a level set of the capacity potential of A with
respect to C.
13. Inequalities for Eigenvalues
Let Al < A2 ~ A3 ~ ... be the Dirichlet eigenvalues of n with corresponding
L 2 orthogonal eigenfunctions (<Pi) and Ei := span( <PI, ... ,<Pi). The Neumann
eigenvalues are denoted by 0 = J.tl < J.t2 ~ J.t3 ~ .• '. A survey on this subject can
be found in [65].
13-1 Rayleigh-Ritz characterization of eigenvalues: [25], [7].
13-2 Poincare principle: [72].
l'nf
sup
ECHö uEE\{O}
dimE=i
Iof lV'ul2 2
Jo U
and similarly for J.ti with HJ replaced by H l • This implies:
13-3 Barta's inequalities: [14], [75]. For every u E C 2 , U > 0,
\
Al
. f -~u(x)
> In
-
zEn
u(x)
.
If in addition u = 0 on an then
In both cases equality holds for the principal eigenfunction.
TABLE OF INEQUALITIES IN ELLIPTIC BVP
121
13-4 Rayleigh-Faber-Krahn's inequality:
where itn-2)/2 is the first zero of the Bessel function J(n-2)/2' Equality holds for
balls.
13-5 Cheeger-Yau's inequality: [23], [5], [92], [51].
Al > ~ inf (IÖAI)2 = ~ inf
- 4 Acn lAI
4 UEH6,1
(In IVU2I)2
In luI
13-6 Szegö-Weinberger's inequality: [84], [89].
J-t2 ~ J-t2 (0*) .
13-7 Payne-Weinberger's inequality: [71]. If 0 is convex then
13-8 Lichnerowicz-Obata's inequality: [92]. The first nontrivial eigenvalue of
a compact Riemannian manifold M is
J-t2(M) ~ _n_ infRic(M).
n-l
Acknowledgment. Several mathematicians have contributed one or several inequalities
to this list. In particular we thank S. Müller and W. Dörfler for a number of valuable
suggestions.
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A CATALOGUE OF HELP AND HELP-TYPE
INTEGRAL AND SERIES INEQUALITIES
M. BENAMMAR
Science Wing, Air College (Dafra), P.O. Box 45373, Abu Dhabi,
United Arab Emirates
C. BENNEWITZ
Mathematical Institute, University 0/ Lund, Box 118, S-22100 Lund, Sweden
M. J. BEYNON
Cartliff Business School, Aberconway Building, Column Drive, CartliJJ CF13EU,
Wales, UK
B.M.BROWN
Department 0/ Computer Seien ce, University 0/ Wales CardijJ, CardiJJ CF23XF,
Wales, UK
N. G. J. DIAS
Department 0/ Mathematics, University 0/ Kelaniya, Kelaniya, Sri Lanka
W. D. EVANS
School 0/ Mathematics, University 0/ Wales CardiJJ, Mathematical Institute,
Senghennydd Road, Cardiff CF24AG, Wales, UK
W. N. EVERITT
School 0/ Mathematics and Statistics, University 0/ Birmingham, Edgbaston,
Birmingham B152TT, England, UK
V. G. KIRBY
Department 0/ Mathematics, Dublin City University, Dublin 9, Ireland
L. L. LITTLEJOHN
Department 0/ Mathematics and Statistics, Utah State University, Logan,
UT 84322-3900, U.8.A.
Abstract. This catalogue of the HELP and HELP-type integral and series inequalities records the contributions made to this area of analytic inequalties from the years
1971-1996. The original HELP integral inequality came from the results of Hardy and
Littlewood in one of their seminal papers, in this case written in 1932. The main analytic
tools for the study of these inequalities are the properties of linear, ordinary, self-adjoint
differential operators, and the properties of the Titchmarsh-Weyl / Hellinger-Nevanlinna
m-coefficient and its ramifications. It is appropriate then, that this catalogue records
some of the many distinguished contributions made to mathematical analysis in the first
half of this century, by these named mathematicians. Likewise it is appropriate that this
catalogue is dedicated to D.S. Mitrinovic whose contributions to the study and recording
of analytic inequalities in the second half of this century, are now legendary.
1991 Mathematics Subject Classijication. Primary 26DI0, 26D15j Secondary 34B20, 34L05.
Key words and phrases. Integral inequalitiesj Titchmarsh-Weyl m-coeflicientj Ordinary differential and difference operators.
127
G. V. Milovanovic (ed.), Recent Progress in Inequalities, 127-160.
@ 1998 Kluwer Academic Publishers.
128
M. BENAMMAR ET AL.
1. Introduction
The remarkable and lasting contributions to the subject of analytie inequalities
from the long and dedicated labours of D.S. Mitrinovic in the vineyard of mathematies, are to be seen in the two books Analytic Inequalities [47] and Inequalities
Involving Functions and Their Integrals and Derivatives [48]. The HELP and
HELP-type integral and series inequalities fall within this category of inequalities.
As a tribute to the memory of Mitrinovic we have gathered together this catalogue
of HELP and HELP-type inequalities. We are convinced that this collection is put
together within the spirit of the first book [47], and the book [48] that followed in
the same style and form.
In writing this paper we have excluded names of contributors from the text in
order to simplify the presentation. The names of those mathematicians involved
in the programme of HELP inequalities, extending now over aperiod of more
than twenty five years, can be readily seen on looking through the collection of
references at the end of this paper.
However here in this Introduction we recall the three names now legendary in
mathematics for the original creation of the subject inequalities as aseparate
discipline, those of Hardy, Littlewood and P6lya; the book Inequalities [42) is one
of the great classie texts in mathematies, and still in print after more than sixty
years since publication in 1934. In partieular the first HELP integral inequality is
due to Hardy and Littlewood and appeared in 1932 in the seminal paper [41].
The HELP and HELP-type inequalities may form one of the few outstanding
examples of a structured family of inequalities; the common theme running
through these inequalities is a dependence upon the original Titchmarsh-Weyl
m-coefficient, see [54), and its extensions and ramifications.
The HELP inequalities are quadratie in terms of the basie function elements but
all of them stern from consideration of linear, ordinary, symmetrie (formally selfadjoint) differential and difference expressions defined on intervals of the realline.
The analysis of these inequalities depends, in a subtle way, on the spectral properties of the self-adjoint operators generated by these expressions in an integrable
or summable-square Hilbert function space, say H.
All the inequalities are of the form
(1.1)
F(f) ~ KG(f)
(f E ~),
where the linear manifold ~ is, in a well defined sense, a maximal linear manifold
of H determined by the linear differential or difference expression; here F, G : ~ -7
where
is the set of non-negative real numbers. The inequality is said to
be valid if there exists a positive number K such that (1.1) holds for all f E ~.
The inequality is said to be not valid if (1.1) fails to hold for any positive number
K; in these cases we write, symbolically, K = +00.
Ifthe inequality (1.1) is valid then our notation assurnes that the symbol K is the
best possible number, Le., the smallest positive, real number for whieh validity
holds. With K so determined cases 0/ equality consist of those elements f E ~ for
which equality holds in (1.1).
JRt,
JRt
HELP AND HELP-TYPE INTEGRAL AND SERIES INEQUALITIES
129
For all the HELP and HELP-type inequalities we have, for the null element 0 of ß,
(1.2)
F(O) = G(O) = 0,
so that this element 0 is always a case of equality. For a valid inequality there may
or may not be non-null cases of equality.
The study of the inequality (1.1) is, in general, in three stages:
1° To determine if the inequality is valid or not valid.
2° If the inequality is valid to determine or characterise in some sense, the best
possible number K.
3° If the inequality is valid to determine or characterise all possible cases of equality.
For some inequalities the best possible number K can be found as a "known" or
"familiar" number; examples are K = 4 or K = (cOS(O))-2 for some 0 E [0, ~).
In other cases K may be determined as the root of some transcendental equation;
in many of these cases a numerical approximation for K can be obtained.
In a similar manner the non-null cases of equality may be expressed in terms of
"known" special functions, but in other cases less explicit information only may
be available.
The analytical problems for the validity and determination of cases of equality can
be very demanding. For this reason numerical techniques have been established
to seek out validity or non-validity of HELP and HELP-type inequalities, and the
existence of non-null cases of equality. These numerical techniques are now so well
tried and established as to inspire confidence in their findings. For details of these
numerical methods see [15], [18], [19] and [45].
An overall view of the analytical and numerical techniques required for the study of
the HELP second-order integral and series inequalities may be found in the survey
paper [11]. There is a survey of the higher-order HELP integral inequalities,
together with a valuable list of examples, in the recent Ph.D. thesis [23]; the
HELP-type integral inequalities are studied in the thesis [6]; the HELP series
inequalities are studied and surveyed in the thesis [2].
This catalogue lists known special cases that have been studied since the HELP
inequality was first considered in 1971. For each inequality we report brieflyon
the results of analytical and numerical techniques. In each case reference is made
to the original publications listed at the end of the paper.
2. Notations
Z, Nt and N+ denote the sets of all integers, non-negative and positive integers
respectively; IR. and C denote the real and complex number fields; JRt denotes the
set of all non-negative real numbers. Open and compact intervals of IR. are denoted
by (a, b) and [0:, ß] respectively.
The symbols LP and AC denote p-integration and absolute continuity with respect
to Lebesgue measure; Lfoc(a, b) and ACloc(a, b) denote sets of complex-valued
functions on (a, b) that are LP and AC on all compact sub-intervals of (a, b).
130
M. BENAMMAR ET AL.
The symbol '( x E E)' is to read as 'for aB the elements x of the set E'.
If w : (a, b) -* ~ then L2 (( a, b) : w) denotes the weighted integration spaee
{I : (a,b) -* C : (i) I is Lebesgue measureable on (a,b)
(ii)
l
b
w(x)l/(x)1 2 dx < oo} .
With due regard to equivalenee classes L 2 (( a, b) : w) also represents a Hilbert
function spaee H with norm and inner-produet respeetively
II/II! :=
l
b
w(x)l/(xW dx
and
(f, g)w :=
l
b
w(x)/(x)g(x) dx.
In all eases eonsidered in this paper it is assumed that w(x) > 0 for almost aB
x E (a, b).
Notations for the HELP series inequalities are given in the appropriate section
below.
3. Real and Complex Inequality Domains
The HELP inequalities involve either I : (a, b) -* IR and then P, or I : (a, b) -* C
and then 1/1 2 • This quadratie dependenee of the inequalities on the element I
enables the eomplex ease of the inequality to be dedueed from the real ease; for
ease of presentation all the particular inequalities in this paper are presented in
the real ease.
The real and eomplex forms of the classieal HELP integral inequality are given in
the next seetion in order to illustrate this point.
4. The Classical HELP Integral Inequality
This inequality is determined by
(4.1)
(i)
(ii)
(iii)
(iv)
(v)
an interval [a, b) of IR,
two eoeffieients p, q : [a, b) -* IR,
a weight w : [a, b) -* ~ with w(x) > 0
for almost aB x E [a,b),
a loeal integrability eondition p-l, q, w E L\oc[a, b),
a real parameter T E IR.
For eonvenienee the quasi-differential expression M r : D(Mr ) x IR x [a, b) -* C is
defined by
(4.2)
(i)
(ii)
D(Mr ) := {I : [a, b) -* C : I, PI' E ACloc[a, b) }, }
Mr[J](x):= -(p(x)l'(x))' + (q(x) - Tw(x))/(x)
(x E [a, b), I E D(Mr ), TE IR).
HELP AND HELP-TYPE INTEGRAL AND SERIES INEQUALITIES
131
The linear manifold that gives the domain Ll C L 2 ((a, b) : w) of the HELP inequality is defined by
Ll:= {f E D(MT ) : (i) f: [a,b) --t IR and
(4.3)
(ii) f, w- 1 MT[fl E L 2([a, b) : w) } .
It is readily seen that Ll is independent of the parameter 7" and this independence
is reflected in the notation.
With these definitions made the HELP integral inequality takes the form, for all
fELl,
(4.4)
(l
a
-tb
{p(X)f'(X)2 + (q(x) -7"w(X))f(X)2} dX)
~K
l
b
w(X)f(X)2 dx
l
b
2
w(x) {W(X)-l MT [J](x) }2 dx.
Remarks:
1. The inequality (4.4) is a typical example of the general inequality given in (1.1) with
the mappings Fand G determined by the above given left- and right-hand sidesj note
the quadratic dependence on fj also that (1.2) is satisfied, Le., F(O) = G(O) = O.
2. The notation Ja-tb on the left-hand side of (4.4) indicates that this integral is, in
general, only conditionally convergentj this integral is often called the Dirichlet integral
of the differential expression MT' In the cases when it is known that this integral is
absolutely convergent the --t will be omitted.
3. The spectral theory background of this inequality is derived from the spectral properties of the Sturm-Liouville quasi-differential equation
(4.5)
MT[y]=>.wy
on
[a,b),
where >. = I-' + iv E C is the spectral parameterj the spectral analysis takes place in the
Hilbert space L 2 ([a,b) : w). It is for this reason that the parameter 7" is introduced into
the inequalitYj this is a shift parameter that allows any point on the real line IR to be
regarded as the origin of the spectral plane.
4. The validity of the inequality (4.4) depends upon all the 'variables', Le., the coefficients
p, q, w and the shifting parameter Tj these quantities also influence the value of the best
possible number K in the case when there is a valid inequality. The best possible value
is often shown as K(7")j if other parameters are involved in the coefficients then this may
be shown also in the expression for K.
5. The end-point a for the differential expression MT is regular (see [24, Section 3])j it is
essential for the application of the Titchmarsh-Weyl m-coefficient to the study of (4.4),
that at least one end-point of the interval (a, b) be in this classificationj however it should
be noted that no boundary condition at this regular end-point is required.
6. HELP integral inequalities on the open interval (a, b) for which neither end-point is
regular, are also of interestj however in these cases it is necessary to appeal directly to
the spectral properties of the underlying self-adjoint differential operatorsj a number of
examples are given below to illustrate these techniques.
132
M. BENAMMAR ET AL.
1. The domain ß C L 2 ([a,b) : w) is determined by the classification of the differential
expression M r at the end-point b in this L 2 space; in the three subsections given in this
section the classifications considered are strang limit-point, limit-circle and regular at b.
8. The complex form of (4.4) is, to be considered on the domain ß + iß,
(4.6)
(l--tb {p(x) 1!,(x)1 + (q(x) - rw(x))lf(x)1
2
::; K
l
b
w(x)lf(x)1 2 dx
l
b
2}
dx
f
w(x)lw(x)-l M r [f](x) 12 dx.
This inequality is valid if and only if (4.4) is valid on ß, with the same best possible
number K; likewise all the cases of equality are determined from the equalising functions
for the real case; see [24].
9. The general theory of the HELP integral inequality is developed in the series of papers;
[4], [24], [26], [30-33], [36] and [49].
4.1. THE REGULAR/STRaNG LIMIT-POINT CASE
In this subsection we consider the general case when the HELP integral inequality
on the interval (a, b) has one regular end-point, say a, and one singular end-point,
say b, in the function space L 2 ((a, b) : w).
Examples.
1. a = 0, b = +00; p(x) = w(x) = 1, q(x) = 0 (x E [0,00))
(4.7)
ß = {f : [0,00) -+ ~ : f, l' E ACloc[O,OO) and f, 1" E L 2[0, 00) };
K = 4 is best possible; an cases of equality are described by, for an A E ~ and an
p> 0,
(4.9)
f(x) = Aex p ( -~ px) sinG px -
i)
(xE [0,00)).
This is the original HELP integral inequality; see [41] and [42, Section 7.8], and
the papers [1], [20-21], [44].
2. a = 0, b = 00; p(x) = w(x) = 1, q(x) = 0 (x E [0,00)) with shift parameter
rE~
(4.10)
(1 {J'(X)2 _ rf(x)2} dX) 2
00
~ K(r)
1
00
f(x)2 dx
1
00
{fl/(x) + rf(x)}2 dx.
HELP AND HELP-TYPE INTEGRAL AND SERIES INEQUALITIES
133
In this case the domain a is given by (4.7) and the best possible number K(7) is
K(7) = { 4
+00
°
(7 E [0,00)),
(7 E (-00,0)).
For 7 = this inequality reduces to (4.8) and all cases of equality are given by
(4.9); for 7 E (0,00) the only case of equality is given by the null function.
This example is considered in [9], [24, Section 9], [26, Section 4] and [11, Section
4.4.1].
3. a = 0, b = 00; p(x) = 1, q(x) = 0, w(x) = x Cl (x E (0,00)).
In this example it is necessary to take 0 > -1 in order to make the end-point
regular; the corresponding differential equation
-y"(x) = AXCly(X)
(4.11)
°
(x E (0,00))
has explicit solutions in terms of Bessel functions and this allows a detailed analytical analysis of the corresponding HELP inequality, Le.,
In this case the domain is given by
a = {! : (0, 00) ~ lR: !, !' E AC\oc[O, oo) and
(4.13)
!,x- Cl !" E L 2( [0, 00) : x Cl )}.
For the analysis of this example see [39] and [24, Section 9] to give the best-possible
result
(0 E (-1,00))
(4.14)
with all cases of equality determined by, for all A E lR, p>
°and E [0,00),
x
here H~l) is the Hankel function of type 1 and order 11, and
11 =
(0 + 2)-1 ,
(0 E [-1,00)).
°
For 0 = this example reduces to (4.8).
The importance of this example is not only that it can be analysed in such explicit
terms, but that it shows the full range of the best possible number K in the HELP
integral inequality, Le., in general
1< K < +00,
M. BENAMMAR ET AL.
134
sinee K (.) is monotonie inereasing on (-1, 00) and
lim K(o:) = 1,
",-+-1 +
lim K(o:) = +00.
"'-++00
This example has not yet been analysed either analytically nor numerieally for
the effeet of the shift parameter T, but the elose link with Example 1 given above
suggests that for all 0: E (-1,00) the inequality is valid for all T E [0,00) and not
valid for all T E (-00,0).
4. a = 1, b = OOj p(x) = x"', q(x) = 0, w(x) = 1 (x E [1,00)).
For this example it is possible to ehoose 0: E lR sinee the end-point 1 is regular for
all 0: in this range. As in the previous example the associated differential equation
ean be solved in terms of Bessel functionsj the equation is
(x E [1,00))
- (x"'y'(x))' = AY(X)
and the HELP inequality
in this ease the domain is given by
ß ={J: [1,00) ---+ lR: J, x"'!, E AC1oc [I,00)
(4.16)
and
J, (x"'!,)' E L 2 [1, 00) } .
For this example the analysis ean be found in [38], but see also [24, Seetion 9] j the
result is
(0: E (-00,0]),
K(o:) = {
~ eos [(3 - 0:)-1 71"]}-2
(0: E [0,1)),
(o:E[I,oo)).
+00
°
For 0: = this example reduees, essentially, to the inequality (4.8) with the eorresponding eases of equalitYj for 0: E (-00,1) \ {o} the only ease of equality is the
null function.
5. a = 0, b = OOj p(x) = w(x) = 1, q(x) = x 2 (x E [0,00)) with shift parameter
E lR. The inequality is
T
(4.17)
(1
00
{J'(X)2 + (x 2 _ T) J(X)2} dX) 2
:s K(T)
1
00
J(X)2 dx
1
00
{J"(x) - (x 2 - T) J(x)} 2 dx
with domain
ß = {J : [0,00) ---+ lR : J,!' E ACloc[O, 00) andJ, !" - x 2 JE L 2 [0, oo)}.
HELP AND HELP-TYPE INTEGRAL AND SERIES INEQUALITIES
135
The differential equation in this case is the linear harmonie oscillator equation
_y"(X) + (x 2 - r) y(x) = .xy(x)
(4.18)
(x E [0,00)).
To see the effect in this example of the shift parameter r define the two sets of
integers
NN := { 4n + 1 : n E No}
and ND:= { 4n + 3 : n E No } .
For the analysis of this example see the results in [29]; the inequality is valid if
and only if rENN U ND. For rENN
K(4n + 1) = 4
(n E No)
with equality in (4.17) if and only if, for some A E IR,
J(x) = Aexp (_x 2/2) H 2n (x)
(XE [0,00))
where H m represents the Hermite polynomial of degree m; in this case both sides
of (4.17) are zero.
For r E ND
K(4n + 3) > K (4(n + 1) + 3) > 4
(n E No)
and
lim K(4n + 3) = 4;
n-too
there is equality in (4.17) if and only if either
(x E [0,00))
in whieh case both sides of (4.17) are zero, or J is a transcendental function derived
from the differential equation (4.18) in whieh case both sides of (4.17) are not zero.
For details of the analysis for this example see [29]; see also [26].
6. a = 0, b = 00; p(x) = w(x) = 1, q(x) = _x2 (x E [0,00)) with shift parameter
r E III The inequality is
(4.19)
(1-t00 {J'(X)2 _ (x 2 + r) J(x)2} dX)
~ K(r) 1 00 J(X)2 dx 00 {J"(X) + (x 2 + r) J(x)} dx
2
1
2
with domain
L\ = {J : [0,00) -t IR: J, f' E AC\oc[O,oo) and J, !" + x 2JE L 2[0, 00) } .
In this example the Diriehlet integral on the left of (4.19) is, in general, only
conditionally convergent. The differential equation in this case is
_y"(X) - (x 2 + r) y(x) = .xy(x)
(x E [0,00))
M. BENAMMAR ET AL.
136
and has explicit solutions in terms of Weber functions.
The analysis of this example is considered in [26, Section 4], but see in particular
[37, Sections 7 and 8, Example 3] with analytical details in the forthcoming paper
[28]; these results show that the inequality (4.19) is valid for r = 0 with the best
possible result
K(O) = 4 + 2V2.
There are no cases of equality other that the null function.
The numerical consideration of this example shows that the inequality is valid for
all r E IR and that K (.) is monotonically decreasing on IR with
lim K(r) = +00;
T--+-OO
also that for some positive number ro
K(r) > 4 (r E (-oo,ro))
K(r) = 4 (r E [ro, 00).
and
The numerical results show that the approximate value of ro is 0.35, and that
for all these valid inequalities there are no cases of equality other than the null
function.
7. a = 0, b = 00; p(x) = w(x) = 1, q(x) = x
parameter r E IR. The inequality is
(4.20)
(1 {J'(X)2 + (x - r)/(x)2} dx
00
~ K(r)
1
00
r
1
00
I(X)2 dx
(x E [0,00)) with the shift
{f"(X) - (x - r)/(x)} 2 dx
with domain
ß = {J : [0, 00) ~ IR : I, !' E ACloc[O,OO) and I, f" - xl E L 2 [0, 00) } .
The differential equation in this case
(4.21)
_y"(X) + (x - r) y(x) = AY(X)
(x E [0,00))
has explicit solutions in terms of the Airy functions.
The solution of the analytical problem presented by this inequality was presented
in the notes [43] and [37, Sections 7 and 8, Example 5]; further details are to be
given in the forthcoming paper [28]. There exist two sets of positive numbers
IRN = {A~ : n E No}
with
and
IRD = {A~ : n E No }
(n E No)
HELP AND HELP-TYPE INTEGRAL AND SERIES INEQUALITIES
137
and lim >..~ = lim >..;; = +00 such that
n~oo
n~oo
K(T) = { 4
+00
(TE {>"~}U{>..;;}),
(T E R \ ({ >..~ } u { >..;; } )).
There is equality for the valid cases in the form, for all A E Rand using the above
notation again,
f(x) = Acp~(x) or Acp~(x)
(x E [0,00) and nE No)
where the {cp~} and {cp;;} are non-null functions depending on the Airy solutions
of (4.21); however in all these cases of equality both sides of the inequality (4.20)
are zero.
8. a = 0, b = 00; p(x) = w(x) = 1, q(x) = -x (x E [0,00)) with shift parameter
E R. The inequality is
T
(4.22)
(1--+ {J'(X)2 - (x + T)f(x)2} dX) 2
00
:S K(T)
1
00
f(x)2 dx
1
00
{f"(x) + (x + T)f(x)}2 dx
with domain
~ = {f : [0,00) -+ R : f, l' E ACloc[O,OO) andf, 1" + xf E L 2[0, oo)}.
Again in this example the Dirichlet integral is, in general, only conditional convergent; for analytical details of this convergence see the specific results for this
example in [24, Section 3] and [25].
In this case the differential equation
-y"(x) - (x + T) y(x) = >..y(x)
(x E [0,00))
has explicit solutions in terms of Bessel functions of order 1/3.
The analysis of this example is considered in [25], see also [37, Sections 7 and 8,
Example 4]; the results given show that (4.22) is a valid inequality for T = 0 with
K(O) = 4.
There is a continuum of cases of equality, similar to Examples 1, 3 and 6, given
explicitly by, with A E Rand p E ~ ,
(x E [0,00))
and >.. = pexp(i7r/3).
M. BENAMMAR ET AL.
138
The numerieal consideration of this example shows the inequality (4.22) is valid
for all r E IR and that K (.) is monotonieally decreasing on IR with
K(r) > 4
K(r) = 4
and
(r E (-00,0))
(r E [0,00)) ;
also that there are there are no cases of equality for r E IR \ {O} other than the
null function.
9. a = 1, b = 00; p(x) = w(x) = 1, q(x) = x- 2/2 (x E [1,00)) with the shift
parameter r E IR. The inequality is
(4.23)
(1 {f'(X)2 + (2!2 - r) J(X)2} dX) 2
00
~ K(r)
1
00
J(x)2dx
1
00
{!"(x) - (2!2 -r) f(X)f dx
with domain
ß = {J: [1,00) -+ IR: J, J' E AC1oc [l,00) andJ,!" - 2!2 E L 2[l, 00) } .
The differential equation in this case
-y"(x) + (2!2 - r) y(x) = '\y(x)
(x E [1,00))
has explicit solutions in terms of Bessel functions of order ±V3/2.
Consideration of the analytieal and numerical solutions is given in [13]; these
results show that the inequality (4.23) is not valid for all r E (-00,0], Le.,
K(r) = 00
(r E (-00,0])
and that K(·) is monotonie decreasing on (0,00), and that there exists a number
ro E IR(j such that
lim K(r) = +00
T-tO+
and
K(r) = 4
(r E [rO' 00));
the numerieal value of ro is approximately 0.13. The cases of equality are as
follows:
(i)
(ii)
(iii)
for r E (ro, 00) there is a one-dimensional case of equality, that is, the set
of equalising functions is one-dimensional in the space L 2 [1, 00),
for ro there is a continuum of cases of equality,
for r E (0, ro) only the null function.
HELP AND HELP-TYPE INTEGRAL AND SERIES INEQUALITIES
139
10. a = 0, b = 00; p(x) = w(x) = 1, q(x) = xO< (x E [0,00)) with the shift
parameter r E IR, and with the parameter a E [0,00). The inequality is
(4.24)
(1 {J'(X)2 + (xO< _ r) j(x)2} dX)
00
~ K(r,a)
1
00
2
j(x)2 dx
1
00
{f"(X) - (xO< - r) j(x)}2 dx
thereby indicating that the best-possible number K depends upon both the parameters rand a. The domain is given by
6. = {j : [0,00) -t IR : j, !' E ACloc[O,OO) and j, !" - xO< jE L 2 [0, 00) } .
The differential equation in this case
_y"(X) + (xO< - r) y(x) = >.y(x)
(x E [0,00))
has solutions in terms of known transcendental functions only when a = 0, 1
and 2; these special cases are covered by examples 2, 8 and 5 respectively. For
general values of a E (0,00) the analytic consideration has not received detailed
analysis; however certain general operator theoretic results are known and then
the established numerical methods give reliable additional information. These
results are reported on in [11, Section 1, page 276 and Section 4.4.2] and [14,
Section 4.1]. These works show the existence of two sets of non-negative numbers
{ >.~ (a) : n E No } and {>.~ (a) : n E No } (these are the Neumann and Dirichlet
eigenvalues and depend upon the value of the parameter a) with the properties,
for each a E (0,00),
(n E No)
and lim >.~(a) = lim >.~(a) = +00, such that
n--+oo
(4.25)
n--+oo
(i)
(ii)
(iii)
(iv)
K(>.~(a),a) = 4 (n E No and a E (0,00)),
for a = 1 K(>'~(l),l) = 4 (n E No),
for a E (0,00) \ {I} 4< K(>.:?(a),a) < +00
for a E (0,00)
lim K(>.:?(a),a) = 4.
}
(n E No),
n--+oo
The inequality (4.24) is not valid for all a E (0,00) when
r E IR \ ({ >.~ (a)} U {>.:? (a)} ) .
For all a E (0,00) there is equality in (4.24), with r E {>.~ (a)} U {>.:?(a)}, with j
taken to be the corresponding eigenfunction from the set {cp~} U {cp:?} but with
both sides of (4.24) equal to zero. For a E (0,00) \ {I} and r E {>.:?(a)} there is
a one-dimensional case of equality with both sides not zero.
140
M. BENAMMAR ET AL.
It is interesting to look at the form of the best-possible value of K for the first
Diriehlet eigenvalue translate A:?(a) as a function of the parameter a, Le., to
consider the mapping 1I:+(a) := K(A{?(a), a) for all a E (0,00). This mapping is
considered in [11, Section 4.4.2, Figure 12] and in [14, Section 4.1, Figure 10]. The
outcome is still not complete but the firm indieations are
(i)
11:+0 is continuous on (0, 00),
}
(ii) 11:+0 is monotonie decreasing on (0,1],
(4.26)
(iii) 11:+0 is monotonie increasing on [1,00),
(iv) 11:+0 has an absolute minimum at a = 1 with 11:(1) = 4.
With the definitions
u--tO+
L o := lim 1I:+(a) ,
u--too
L oo := lim 1I:+(a)
then it would follow from (4.26) that L o and L oo exist; the numerieal evidence is
that both these numbers are finite but these results have not been established.
See also the account in [37].
11. a = 0, b = 00; p(x) = w(x) = 1, q(x) = -x u (x E [0,00)) with the shift
parameter r E IR, and with the parameter a E (0,2]; this upper restrietion on a is
necessary to place the problem in the strong limit-point case. The inequality is
(4.27)
(1--t00 {J'(X)2 _ (X + r) J(X)2} dX)
00
00
$ K(r,a) 1 J(x)2dx 1 {J"(x)2+(x +r)J(x)}2 dx
U
2
U
with domain
ß = {J : [0,00) -t R : J, !' E AC\oc[O, 00) and J,!" + (X U + r) JE L 2[0, 00) } .
The differential equation in this case
-y"(x) - (X U + r) y(x) = AY(X)
(x E [0,00)).
With the restrietion placed on the parameter a this equation has solutions in terms
of known transcendental functions only when a = 1 and 2; these special cases are
covered by examples 7 and 6 above respectively.
As with example 10 above the general case has been considered with the use of
both analytical and numerical techniques. The most comprehensive report on
this general case is to be found in [14, Section 4.4]; these results show that the
inequality (4.27) is valid for all a E (0,2] and for all r E III Apart from the special
case a = 1 considered above in Example 7, there are no cases of equality except
for the null function.
It is ofinterest to consider the mapping 11:_ (a) := K(a,O); (a E (0,2]). The analytieal and numerieal evidence is that this mapping has the properties
(i)
(4.28)
(ii)
(iii)
(iv)
11:-0 is continuous on (0, 2],
}
1I:_(a) = 4 (a E (0,1]),
11:-0 is strietly monotonie increasing on [1,2],
11:_(2) = 4 + 2V2.
HELP AND HELP-TYPE INTEGRAL AND SERIES INEQUALITIES
141
4.2. THE STRONG LIMIT-POINT/STRONG LIMIT-POINT CASE
In this subsection we consider the HELP integral inequality when both end-points a
and b are in the strong limit-point classification in the corresponding L2( (a, b) : w)
space.
The general references for this section are [35] and [50].
12. a=O, b=oo; p(x) =1, q(x) =0, w(x)=xO!
(XE (0,00)).
°
This is areturn to Example 3 but now with the parameter CI! ~ -1; this places
the differential expression in the strong limit-point case at both the end-points
and +00. As before, see (4.12), the HELP integral inequality takes the form
but this time with the domain, since now both end-points are singular,
~ := {J : (0,00) ---+ lR. : J, !' E ACloc(O, 00) and J, x-O! f" E L 2( (0, 00) : xO!) } .
For the analysis of this inequality, again based on the solutions of the differential
equation
(x E (0,00)),
see the account in [34]; in this case with two singular end-points, the method
cannot be based on the Titchmarsh-Weyl m-coefficient. The result is
K(CI!) = { 1
+00
(CI! E [-3, -1]),
(CI! E (-00, -3)).
For CI! E [-3, -1] there are no non-null cases of equality.
13. a = -00, b = +00; p(x) = w(x) = 1, q(x) =
parameter r E IR. The inequality is
(4.29)
(1:
°
(x E (-00,00)) with shift
r
1: 1:
{J'(X)2 - rJ(x)2} dx
~ K(r)
J(X)2 dx
{f"(X) + rJ(x)}2 dx
with domain
~:={J: (-00,00) ---+ lR.: J,!' E ACloc(-oo,oo) andJ, f" E L 2 (-00,00)}.
°
For r = this inequality is one of the original HELP inequalities considered in
[41] and [42, Section 7.9]; there is an alternative, operator theoretic analysis in [35,
Section 6, (1)]. The result is that the inequality (4.29) is valid for all r E lR. with
K (r) = 1 (r E lR.) and that there no cases of equality other than the null function.
142
M. BENAMMAR ET AL.
14. a = -00, b = +00; p(x) = w(x) = 1, q(x) = x 2 (x E (-00,00)). In this case
the inequality is considered in [35, Section 6, (2)] and takes the form
(4.30)
(1:
1: 1:
{f'(X)2+ x2f(X)2}dX)2
~K
f(X)2 dx
{f"(x) - x 2f(x)} 2 dx
with domain
ß:= {J: (-00,00) -+ IR: f, f' E AC1oc(-00, 00) and f, f"-x 2f E L 2(-00,00)}.
It is shown that the best possible number K in (4.30) is K = 1. For A E IR and
n E No all the cases of equality can be given explicitly in the form
f(x) = Aexp(x 2/2) Hn(x)
(x E (-00,00)),
where {Hn : n E No } is the set of all Hermite polynomials.
15. a = -00, b = +00; p(x) = w(x) = 1, q(x) = _x 2 (x E (-00,00)). This
example is considered in [35, Section 6, (3)]; the inequality is
(4.31)
(L~: {J'(X)2 - x 2f(x)2} dX)
~K
1: 1:
2
f(x)2 dx
{f"(x) + x 2f(x)} 2 dx
with domain
ß := {J : (-00,00) --t IR : /, /' E AC1oc (-00,00) and /, /" + x 2/ E L 2(-00, oo)}.
It is shown that the best possible number K for this inequality is given by K = 1,
and that the only case of equality is the null function.
16. In this last example in this subsection a case is given for which the coefficient
p changes sign on the interval (a, b).
a
= -00, b = +00; w(x) = 1, q(x) = 1 (x E (-00,00)) and p is defined by
p(x) := {
-I
+1
The HELP in equality takes the form
(x E (-00,0)),
(x E [0,00)).
HELP AND HELP-TYPE INTEGRAL AND SERIES INEQUALITIES
143
with domain
~ := {J : (-00, 00) ~ IR : I, PI' E ACloc ( -00,00) and I, (Pl')' E L 2(-00, oo)}.
In this example the functions in the domain ~ warrant careful examination:
10 Pi' E AC1oc(-00,00) implies that l' E AC1oc(-oo,0] and I' E AC1oc[0,00)
and so lim I(x) and lim I'(x) both exist finitely in IR; then again using PI' E
:1:-+0-
:1:-+0+
AC1oc ( -00,00) it follows that these two limits are equal in magnitude but opposite
in sign.
2° (Pl')' E L 2 (-00, 00) is equivalent to requiring 1" E L 2 (-00, 00).
This example is considered in [35, Seetion 6, (6)]; it is shown that the best-possible
number K is given by K = 1, and that only the null function gives a case of
equality.
4.3. THE REGULAR/REGULAR OR LIMIT-CIRCLE CASE
In this subsection we consider the case when, for the interval (a, b), one end-point
is regular and the other end-point is either regular or singular. The main source
of reference is the paper [27].
17. a = 0, b = 1; p(x) = 1 - x 2, q(x) = 1/4, w(x) = 1 (x E [0,1)) with the shift
parameter r E IR. The inequality is
(4.33)
(11 {(1- x 2) j'(X)2 + (~_ r) I(X)2} dX)
~K
1
1
I(X)2 dx
1{((11
2
x 2) I'(x))' -
(~- r) I(x)
r
dx.
The differential equation in this case is the Legendre equation
(4.34)
- ((1 - x 2) y'(x))' + (~ - r) y(x) = '\y(x)
(x E [0,1))
°
with a regular end-point at and a singular limit-circle end-point at 1 in the space
L2[0,1).
In order to apply the HELP procedure it is necessary in this case to place a restriction on the maximal domain of the differential equation in order to determine a
Titchmarsh-Weyl m-coefficient for the singular end-point 1. Full analytical details
of this restriction are given in [27, Sections 2 and 3]. In this case the domain ~1
for the inequality is determined by
~ := { I: [0, 1) ~ IR: I, l' E ACloc [0, 1) and I, ((1- x 2)1')' E L 2[0, 1)}
and then
~l:={IE~: :1:-++1lim (1-x 2)I'(x)=0};
144
M. BENAMMAR ET AL.
for technical details see [27, Section 5, (5.2)].
It is shown that the inequality (4.33) is valid if and only if
{(n +
n
No }. The analytical problems of determining the best possible values of Kare
formidable but the numerical methods are very successful; these methods yield the
following given table for values of K against the integers in No to determine the
translate.
rE
1/2)2 : E
n
0
1
n = 2,3,4, ...
3
5
K
4.25
4.98
4
4.21
4.09
These numerical results also indicate that für all n there is a one-dimensional case
of equality, involving the Legendre polynomials, such that both sides of (4.33)
are zero. For those n for which K > 4 there is an additional one-dimensional
case for which both sides of (4.33) are not zero; here the equalising function is
not a solution of the equation (4.34) but is functionally dependent on Legendre
functions.
18. a = 0, b = 1; p(x) = w(x) = 1, q(x) = x- 2 /2 (x E (0,1]) with the shift
parameter r E IR. The inequality is
(4.35)
(11 {/'(x)2 + (2!2 - r) l(x)2} dX) 2
~ K(r)
1
1
l(x)2 dx
1
1
{j"(x) -
(2!2 - r) f(x)
r
dx.
°
This example is dearly related to Example 9; here end-point 1 is regular, as
before, but end-point is limit-circle in the space L 2 (0, 1]. As with the previüus
Example 17 it is necessary to restrict the maximal domain; here ß is defined by
ß := {I: (0,1] -+ IR : I, !'
EAC (0,1] and I, j" - 2!2 EL (0, 1] } ,
2
loc
and the inequality domain ßo by
ßo :=
{I ß: lim [f,u](x) = o},
E
x-tO+
where u(x) := x,,-/S+1)/2 and [I, u](x) := (f . u' - f' . u)(x) both for all x E (0,1].
This inequality is considered in [13, Section 3]; there exist two sets of non-negative
numbers {>.~ : n E No } and {>.~ : n E No } with the properties
(nENo)
HELP AND HELP-TYPE INTEGRAL AND SERIES INEQUALITIES
145
and lim >.t: = lim >.~ = +00. The results for the inequality are:
n-tOC>
n-+OC>
1° The inequality is not valid for all translates T E IR \ ({>.t:} u {>.~}).
2° For T E {>.t: : n E No} the inequality is valid with K(>.t:) = 4; there a
one-dimensional case of equality for whieh both sides of (4.35) are zero.
3° For T E {>.~ : n E No} the inequality is valid with K(>'~) > 4 and the
sequence {K (>.~) : n E No} of real numbers is monotonie decreasing with limit 4
at +00; in addition to the one-dimensional case of equality as for 2° above, there
is an additional one-dimensional case for whieh both si des of (4.35) are not zero.
19. For an inequality that is regular at both end-points but which requires a
boundary condition at one end-point consider:
a = 0, b = 11"; p(x) = w(x) = 1, q(x) = (x E [0,11"]) with shift parameter T E IR.
The inequality is
°
As with Examples 17 and 18 we have to restrict the maximal domain
ß := {J : [0,7f] -+ IR : J, j' E AC[O,7f] and J,
r L [0, 7f]}
E
2
by means of a boundary condition at one end-point, say 7f; to give either
or
ß v := {f E ß : j'(7f) = O}.
For both of these domains there is a countable set of real translation numbers for
which the inequality (4.36) is valid; the corresponding K numbers have properties
similar to those given for Example 18. For all other translation numbers the
inequality is not valid. The details are given, together with reports on the outcome
of the numerical techniques, in [27, Section 5.2].
20. For this last example in this Section we consider a case that is regular at both
end-points and for whieh the inequality is valid on the maximal domain:
a = 0, b = 11"; p(x) = w(x) = 1, q(x) = -1 (x E [0,7f]).
The inequality is
(4.37)
with maximal domain
ßo := {J: [0,11"] -+ IR : J, j' E AC[0, 1I"] and J,
r L [0, 7f] } .
E
2
It is rare for regular integral inequalities to be valid on the maximal domain but
this is such an example. The inequality (4.37) was first studied in [5] with methods
developed in [4]; additional details are given in [12].
146
M. BENAMMAR ET AL.
Define two sub-domains of do by
dl := {f E do : f(7r) = O}
and
d2 := {f E do : f(O) = f(7r) = O}.
Then the following results hold:
1° The inequality (4.37) is valid on do with K = K o ~ 6.15 obtained as the root
of a transcendental equation; there are two distinct cases of non-trivial equality:
(a) f(x) = Asin(x) + Bcos(x) (x E [0,7r]) with A,B E IR when both sides ofthe
inequality (4.37) reduce to zero.
(b) A two-dimensional set of trigonometrically based functions, dependent upon
the transcendental equation which determines K o , for which both sides of the
inequality (4.37) are not zero.
2° The inequality (4.37) is valid on dl with K = K 1 ~ 4.64 obtained by the analysis given in [27, Section 5.2]; there are two distinct cases of non-trivial equality:
(a) f(x) = Asin(x) (x E [0,7r]) with A E IR when both sides of the inequality
reduce to zero.
(b) A one-dimensional set of trigonometrically based functions, dependent upon
the analysis in [27], for which both sides of the inequality are not zero.
3° The inequality (4.37) is valid on d2 with K = K 2 = 1 obtained by operator
theoretical analysis; all cases ofequality are given by f(x) = Asin(nx) (x E [0,7r]
and n E N) with A E IR; when n = 1 both sides of the inequality are zero but
when n E N \ {1} both sides are not zero.
5. The Classical HELP Series Inequality
5.1. NOTATIONS
This series inequality is determined by
(5.1)
(i)
a sequence p ={ Pn : n E No } of real numbers with
Pn:f:0 (nENo),
(ii)
(iii)
a sequence q ={ qn : n E No} of real numbers,
a sequence W ={ W n : n E No } of real numbers with
W n > 0 (n E No).
We make the following definitions:
1° The sequence space l2 (No : w) of all sequences of real numbers x = {x n : n E
No} such that
L wnx; is convergent in IR.
00
n=O
It is well known that l2 (No : w) may be regarded as a Hilbert sequence space with
norm and inner-product defined by
Ilxll~ :=
L wnx; and (x, Y)w := L WnXnYn .
00
00
n=O
n=O
HELP AND HELP-TYPE INTEGRAL AND SERIES INEQUALITIES
147
We write l;" as a shortened form of l2(No : w).
2° Let So and Sl denote the two collections of real sequences
So:= {x ={x n : n E No}: X n E ~ (n E No)},
Sl := {x ={x n : n E N} : X n E ~ (n E N)} ;
then the forward difference operator ß : So --+ So is defined by
ßx := {ßx n := Xn+1 - Xn : n E No} .
00
3° Note that if x E Sl then x E l;" is taken to imply that I: wnx;' < +00.
n=l
4° The difference expression M : So --+ Sl, given p and q as above, is defined by
The difference expression M can be used to define difference operators in the space
l;" in a form similar to the differential operators generated by the differential
expression M in the function space L 2 ((a, b) : w), as outlined in the opening
remarks of Section 4 above. There are essential differences between the differential
and difference operator theories; for an account of the difference theory see the
papers [3], [10], [16-17]; see also the account in the survey paper [11, Section 3].
The Titchmarsh-Weyl m-coefficient for differential expressions is replaced by the
m-coefficient of Hellinger-Nevanlinna; see the account in [17, Section 2]. This
theory leads to consideration of the HELP series inequality to compare with the
integral inequality of the previous section.
Given the definitions and notations in (5.1) and (5.2) the HELP series inequality
can be expressed as
(2: (Pn(ßx n )2 + qn X;') + Po ( ßXO)2)
00
(5.3)
2
n=l
00
00
n=O
n=l
< K 2: WnX~ 2: Wn {W;;,-l MXn )2
with domain
D:={XES:XEl;' and
here
w- 1 MxEl;'};
w- 1 Mx:= {W;;,-l MX n : n E N}.
Note that for x E D the right-hand side of (5.3) is finite.
The comments made in Sections 1 and 2, and the opening remarks of Section 4 are
equally appropriate to the analysis of the series inequality (5.3), but now working
with the Hellinger-Nevanlinna m-coefficient. The differential equation M[y] = AWY
on (a, b) is now replaced by the symmetric difference equation
(5.4)
Mx =AWX
on
N
148
M. BENAMMAR ET AL.
or equivalently M x n = AWnX n (n E 1'1) .
We give four examples of the series inequality (5.3) in the subsections that follow.
The last three examples are based on aremarkable connection between the difference equation (5.4) and the classical orthogonal polynomials of Legendre, Hermite
and Laguerre as developed in [17].
5.2. THE COPSON SERIES INEQUALITY
Let the sequences p, q and w be defined by Pn = Wn = 1, qn = 0 (n E No); then
the inequality (5.3) takes the form
CL (~Xn)2 + (~xO)2) ~ K L:>~ L (~2Xn)2
00
(5.5)
n=l
2
00
00
n=O
n=O
with domain
here ~2xn := ~(~Xn) = Xn+2 - 2x n+1 + x n (n E No). With this in place we may
rewrite (5.5) in the more convenient form
(5.6)
In the original paper [22] it is shown that in this fundamental HELP series inequality the following results hold:
(i)
(ii)
the best possible number K = 4,
the only case of equality is the null sequence {x n = 0: (n E No ) } .
Also in [22] the corresponding series inequality on the set Z of all integers is
considered; this is the series inequality equivalent to the integral inequality (4.8)
of Section 4 above. The result is
(5.7)
for all sequences {x n E IR : n E Z} such that the right-hand side of (5.7) is finite,
with equality if and only {x n : n E Z} is the null sequence.
In [17] there is also a discussion of the inequality (5.6) when the shift parameter is
introduced, Le., when the sequence {qn : n E No} is replaced by {qn - T : n E No}
with T E IR. In spite of the seeming simplicity of this problem the analysis turns
out to be very difficult; the numerical results are reported on in [17, Section 5.2].
HELP AND HELP-TYPE INTEGRAL AND SERIES INEQUALITIES
149
5.3. THE LEGENDRE SERIES INEQUALITY
This example illustrates the remarkable connection between the HELP series inequality and the classieal orthogonal polynomials. The connection sterns from the
property that such orthogonal polynomials are generated by a three term recurrence relation; such a relation, since it is of the second order, can be rewritten as a
symmetrie difference equation of the form given by (5.4). The spectral parameter
in this difference equation becomes the independent variable of the polynomials,
whilst the integer variable, i.e., n E No, becomes the degree or order of the individual orthogonal polynomials. For accounts of this theory see [16-17].
Here we quote the application of this theory to the Legendre polynomials and
thereby gives the corresponding HELP series inequality.
Let the sequences p, q and w be defined by
Pn
= - (n + 1),
qn
= 2n + 1,
Wn
= 2n + 1
(n E No).
After some simplification of the various terms the series inequality is
(2 [~)n + l)x
00
(5.8)
x~])
n X n +1 -
2
n=O
00
00
n=O
n=1
< K L(2n + l)x; ~)2n + 1)-1 [(n + l)x n +1 + nx n ]2
with domain
D:=
{X E S
xE l;',
00
i.e. L(2n + l)x; < +00
n=O
It is shown in [17, Section 5.3], by analytieal means, that
(i)
(ii)
the best possible number K = 2·/Jr 2 + 4 (V7r2 + 4 - 7rr1 ,
the only case of equality is the null sequence.
The form of this best possible number K makes it unlikely that there is an "elementary" proof of this series inequality.
5.4. THE HERMITE SERIES INEQUALITY
Let the sequences p, q and w be defined by
M. BENAMMAR ET AL.
150
After some simplification of the various terms the series inequality is
(~Xnxn+1
(59)
.
~
2n n!
1
2)2
-"2 Xo
<K~ x; ~(xn+1
-
~2nn!~
Xn _ 1
2 n+ln!+2 n (n-1)!
)2
with domain
D:=
{X E S xE
and
00
[2
w'
2
n
.
""' X
l.e.
~ 2n n! < +00
n=O
~ [2~:i~! + 2n~:=-11)!] < +00 } .
2
It is conjectured in [16, Section 5], supported by the numerical methods, that
(i)
the best possible number K = 2y'7r + 4 (y'7r + 4 _ y1r) -1 ,
(ii)
the only case of equality is the null sequence.
As in the case of the Legendre series inequality the form of this best possible
number K makes it unlikely that an "elementary" proof of these results can be
found.
5.5. THE LAGUERRE SERIES INEQUALITY
Let the sequences p, q and w be defined by
(n + I)!
Pn
= r( a + n + 1) ,
qn = 0,
Wn
n!
= -.,--------,-
f(a+n+1)
(nE No),
where a E (-1, (0) is the real parameter that appears in the Laguerre orthogonal
polynomials.
The Laguerre series inequality can be obtained by substitution of these sequences
p, q and w in the general form of the series inequality (5.3).
This inequality is considered both analytieally and numerically in [16, Section 6].
The analytic results show that
(i)
(ii)
the inequality is not valid for a E (-1,0],
the inequality is valid for a E (0,00).
Further these results indieate that there may not be an explicit formula for the
best number K(a), for the range a E (0, (0), in terms of the known transcendental
special functions. The proof of validity is an existence proof only.
There are interesting numerieal results given in [16, Seetion 6] that lead to the
conjecture
(i)
(ii)
the best-possible function K(·) is monotonie decreasing on (0, (0),
lim K(a) = +00,
lim K(a) > 0.
<>-+-1 +
<>-++00
HELP AND HELP-TYPE INTEGRAL AND SE RIES INEQUALITIES
151
6. The Higher-Order HELP Integral Inequalities
6.1. EXTENSIONS OF THE ORIGINAL HELP INEQUALITY
The original HELP integral inequality, see (4.8),
(6.1)
with domain
D := {I: [0, (0) ---+ lR : I, f' E AC1oc[0, (0) and
I, f" E L 2 [0, oo)}
has been extended in a number of forms to include derivatives of higher order. For
a survey of such inequalities see [46], and the recent thesis [23). There is also a
connection with the results given in [6-7); these results are considered in the next
Section below.
There is a direct extension of (6.1) to the fourth-order case in [8), using methods
similar to the original proof in [41) and in the first proof given in [42, Section 7.8).
This inequality takes the form
with domain
D:= {J: [0,(0) ---+ lR: I(r) E ACloc[O,OO) (r = 0,1,2,3) and 1,/(4) E L 2 [0,oo)}.
It is shown in [8) that the inequality (6.2) is valid on D, that the best possible
number K can be characterised as one of the roots of a fourth-order real polynomial
and that K ~ 78.82. It is also shown in [8) that there is a continuum of non-null
cases of equality similar in form to those equalising functions for the inequality
(4.8) as given in (4.9).
The best-possible number K in the inequality (6.2) plays a significant röle in the
fourth-order examples considered below; for this reason we give a special symbol
for this number
(6.3)
and make reference to this definition as required below. It is worth recording
that in the second-order case the corresponding number is 4 as determined for the
original HELP integral inequality (4.8).
As with Example 13, Section 4.2 the inequality (6.2) mayaiso be considered on
the realline (-00, (0) with the result that
152
M. BENAMMAR ET AL.
is valid on the domain
D := {I: (-00,00) --t ~ : I(r) E ACloc (-00,00) (r = 0,1,2,3)
and 1,/(4) E L 2(-00,00)}.
The only case of equality is the null function.
The inequality (6.2) can be extended to all even-order integers n = 2m (m E N)
to give
with a domain of the same form as for (6.2). The best possible number K(m)
for m E N can be characterised in a number of ways, and some of them lead to
numerical approximations. For additional details and references see [48, Chapter I,
Section 38].
6.2. THE FOURTH-ORDER CASE
The fourth-order HELP integral inequality case is considered in [51-53] and in [23,
Chapters 8 and 9]. Some examples are now given from these references to illustrate
the results that can be obtained by using an extension of the Titchmarsh-Weyl
m-coefficient for higher-order equations.
There is a fourth-order extension of the second-order theory developed in Section 4
above; this has been developed for the quasi-differential expression
with coefficients PO,PI,P2 : [a, b) --t IR, and satisfying certain local integrability
conditions on [a, b). The associated differential equation is
M[y] = >..wy
(6.6)
on
[a, b)
for a weight function w. The HELP integral inequality generated by this differential equation has the form, for a real-valued domain,
(6.7)
{-tb
(Ja
{p2(x)/(2) (X)2 + PI (x)/(1) (x)2 + Po/(x)2} dX)
~K
l
b
w(X)/(X)2 dx
l
2
b
w(x) {W(X)-I M[J](x)} 2 dx.
This inequality has been studied in [51] and more recently in [23]; the two methods
are equivalent as is shown in [23, Section 8.2].
There are few fourth-order differential equations for which there are explicit solutions in terms of known transcendental functions. For this reason examples of
HELP AND HELP-TYPE INTEGRAL AND SERIES INEQUALITIES
153
the inequality (6.7) have been studied analytieally only for the cases when the
differential equation (6.6)
(i)
(ii)
has constant coefficients on the realline ~
is the formal square of a second-order differential equation.
6.2.1. Gonstant Goefficient Gase
a = 0, b = +00; P2(X) = 1, Pl(X) = Po(x) = 0, w(x) = 1 (x E [0,00)), with shift
parameter r E lR.. The inequality is
with domain
~ := {I : [0,00) -t lR. : I(r) E AC1oc[0, 00) (r = 0, 1, 2, 3) and I, 1(4) E L 2 [0, 00) }.
The differential equation is
(x E [0, 00) ) .
The results in [52] and [23, Theorem 8.2] show that the inequality (6.8) is valid
only for r = 0 with K(O) = OC (see (6.3)); thus K(r) = +00 (r E lR. \ {O}). This
result for r = 0 is in confirrnation with the results for the inequality (6.1), although
the method used in [8] is quite different from the HELP analysis in [52] and [23].
There is confirrnation also for the form of the continuum of equalising functions.
It should be noted that the validity ofinequality (6.8), as dependent upon the shift
parameter rE lR., is in marked contrast to the second-order case (4.10), Example 2
of Section 4.l.
6.2.2. The Formal Square Gase
There is a discussion in [23, Section 9.2] of the method of constructing a fourthorder symmetrie differential equation from the formal square of the second-order
symmetrie differential equation -(py')' + qy = >..wy on [a, b). It is shown that
the Titchmarsh-Weyl m-coefficient matrix for the fourth-order equation can be
calculated from the m-coefficient of the second-order equation; the analytie and
numerieal methods for the general fourth-order case are then applied to this special
case in order to obtain information ab out the corresponding fourth-order HELP
integral inequality.
The examples that follow give the interval [a, b), the coefficients P, q and w, and
the resulting fourth-order integral inequality, together with the analytieal and
numerical results available.
1. a = 0, b = 00; p(x) = w(x) = 1, q(x) = -p. (x E [0,00)), where p. E lR. is areal
parameter. The fourth-order differential expression is
(x E [0,00))
M. BENAMMAR ET AL.
154
and the corresponding HELP inequality
(1
(6.9)
00
{J(2) (X)2 - 2,."f(1) (X)2 +,.,,2 f(X)2} dx
::::; K(,.,,)
1
00
f(X)2 dx
1
00
{
f
f(4) (x) + 2,."f(2) (x) + ,.,,2 f(x)} 2 dx.
It is shown in [52-53] and [23, Section 9.3.1], both analytieally and numerically,
that the inequality (6.9) is valid if and only if ,." = 0, in which case it reduces to
inequality in (6.2) together with the continuum of cases of equality.
2. a = 0, b = +00; p(x) = w(x) = 1, q(x) = _x 2 (x E [0,00)) with the shift
parameter T E IR. The fourth-order differential expression in this case is
M[f](x) = f{4>(x) + (2x 2!,(x))' + (x 4 + 2) f(x)
(x E [0,00))
and the associated HELP inequality, including the parameter T,
(6.10)
(1
00
::::; K(T)
{f(2)(x)2 - 2x 2!,(x)2 + (x 4 + 2 - T)f(x)2 } dx
1
00
f(x)2 dx
1
00
f
{f(4)(X) + (2x 2!,(x))' + (x 4 + 2 - T)f(x)} 2 dx.
The numerieal analysis for this inequality is given in [23, Section 9.3.3] and leads
to the conjecture; there exists Tl ::::: 5.5 such that
(i)
(ii)
(iii)
(iv)
K(T) = +00 for all TE (-00,0),
K(·) is continuous and monotonie increasing on [0,00),
K(T) = lK for all T E [0,T1],
K(·) is strictly monotonie increasing on (T1,+00),
and lim K(T) = +00.
r--++oo
Some of these result are confirmed analytieally in [23, Section 9.3.3, Theorem 9.2],
hut there is no discussion of the cases of equality.
3. a = 0, b = +00; p(x) = w(x) = 1, q(x) = x 2 (x E [0,00)) with the shift
parameter T E IR. The fourth-order differential expression in this case is
M[!](x) = f(4)(x) - (2x 2!,(x))' + (x 4 - 2) f(x)
(x E [0,00))
and the associated HELP inequality, including the parameter T,
(6.11)
(1
00
::::; K(T)
{f(2)(X)2 + 2x 2!,(x)2 + (x 4 - 2 - T)f(x)2 } dx
1
00
f(X)2 dx
1
00
f
{f(4)(x) - (2x 2!,(x))' + (x 4 - 2 - T)f(x)} 2 dx.
HELP AND HELP-TYPE INTEGRAL AND SERIES INEQUALITIES
155
There is a numerieal assessment of this inequality in [23, Seetion 9.3.7]; it is shown
numerieally that the inequality is valid if and only if T E {(4n)2 : n E N}. The
numerieal results indieate that
the sequenee {K((4n)2): n E N} is monotonie inereasing,
K((4n)2) = lK for n = 1,2,3,4,
{ K (( 4n) 2 )} is strietly inereasing for n ~ 5
and lim K ((4nn = +00.
(i)
(ii)
(iii)
n-++oo
For all these valid eases there is a two-dimensional eigenspaee of equality for whieh
both sides of the inequality are zero; there may be additional eases of equality.
4. a = 1, b = +00; p(x) = 1, q(x) = 0, w(x) = x a (x E [1,00)), where the
parameter 0: is restrieted to the range 0: E (-1,00). The fourth-order differential
expression in this ease is
M[J](x) = (x-ay" (x))"
(x E [1,00))
and the associated HELP inequality
(6.12)
(1
00
x-a j"(x)2 dx
r
:::; K(o:)
1
00
X
a f(X)2dx
1
00
xa{x-a(x-aj"(x))"}2dx.
This inequality is eonsidered numerieally in [23, Seetion 9.3.6, pp. 162-163]; it is
shown that this inequality is valid for all 0: E (-1,00) and the results lead to the
eonjecture
(i)
(ii)
(iii)
K (.) is eontinuous and monotonie deereasing on (-1, 00 ),
K(o:) = lK for all 0: E [0,00),
K(·) is strietly deereasing on (-1,0) and lim K(o:) = +00.
a-+O+
There is no definite information eoneerning the eases of equality.
7. HELP-type Integral Inequalities
The general Landau-Kolmogorov normed inequality is diseussed in detail in [48,
Chapter I]. Here we eonsider one partieular integral inequality, from this general family, that ean be studied, both analytieally and numerieally, through the
Titehmarsh-Weyl m-eoefficient.
Let the notations and definitions of Seetions 2 and 4 above, hold for the eoefficients
p, q and w on the interval [a, b); thus
M[J](x) = - (p(x)!'(x))' + q(x)f(x)
(x E [a,b))
with the linear manifold ß C L 2 ([a, b) : w) defined by
(7.1)
ß := {J: [a, b) -t IR : fand w- 1 M[J] E L 2 ([a, b) : w)} .
M. BENAMMAR ET AL.
156
For the HELP-type inequality it is necessary to consider the "product"
on a suitable domain of real-valued functions. It is possible to effect this product
without imposing additional conditions on the coefficients p, q and w, but using the
properties of quasi-derivatives; for details see [40]. However it is not necessary here
to enter into these details as in the examples considered below these coefficients
have all the smoothness properties required for the product to be written out in
full. It is convenient to write the original differential expression and this product
in the form w- 1 M[j] and (w- 1 M)2 [j] respectively.
The HELP-type inequality takes the form
(7.2)
(!ab w(x)(w- 1M[j])(x)2 dx )2 $ K
l
b w(x)j(x)2
dx
l
1
b w(x)(w- M[j])2(x)dx
with domain in the space L 2 ([a, b) : w) as given by (7.1).
To give an example to illustrate the form of this inequality and to link it with earlier
results in this paper, consider the case a = 0, b = +00; p(x) = w(x) = 1, q(x) =
(x E [0,00)) for which the inequality (7.2) takes the form
°
this is the inequality (6.4) with K = 1K as the best possible number. It is clear
that this example should emerge from the general theory for the inequality (7.2).
The general theory is extensively developed in [6] and reported on with examples
in [7]. The validity of the general inequality can be made dependent upon the
Titchmarsh-Weyl m-coefficient for the second-order differential equation M[y] =
>..wy on [a, b); the criterion also includes a full description of the cases of equality. There are two technical conditions that have to be placed on the differential
expression M in the space L 2 ([a, b) : w) for this general theory to be applied;
we mention these conditions here without additional comment; M has to be in
the limit-point condition at end-point bin L 2([a, b) : w), and (w- 1 M)2 has to be
partially separated in L2([a, b) : w); for details see [7, Section 2]. All the examples
quoted below satisfy these two technical conditions.
7.1. EXAMPLES
1. a = 0, b = +00; p(x) = 1, q(x) = 0, w(x) = xO! (x E [0,00)) with the shift
parameter TE IR and with the parameter Q; E (-1,00). Then
(x E [0,00))
and a calculation shows that
HELP AND HELP-TYPE INTEGRAL AND SERIES INEQUALITIES
157
The inequality in this case thus becomes
(7.3)
(1
00
~ K(T,a)
r
X-o.(J"(X) + TXo. f(X))2 dx
1
00
xo. f(X)2 dx
1
00
x-o.((x-o. !,,(X))" + 2T!,,(X) + T2xo. f(x))2 dx
with domain
D := {I: [0,00) --+ lR: 1(3) E ACloc[O, 00) and I, (w- 1MT )2[/1 E L 2([O, 00) : xo.)}.
It can be shown that the differential equation, for all T E lR,
MT[y](x) = AXo.y(x)
(x E [0,00))
has explicit solutions in terms of Bessel functions; see [39].
It is shown in [7, Sections 2 and 3] that
(iv)
K(T,a) = +00 for all T E (-00,0) and for all a E (-1,00),
K(T, a) < +00 for all T E [0, +00) and for all a E (-1,00),
K (T, .) is continuous and monotonie decreasing on (-1, 00),
for all T E [0,00),
lim K(O, a) = +00,
(v)
(vi)
K(O,O) = JI{ (for JI{ see (6.3)),
lim K(O, a) > o.
(i)
(ii)
(iii)
0.--+-1 +
0.--++00
These results follow from use of both analytieal and numerieal methods.
The result (v) in the above table is the required consistency with the separate
analysis of the inequality (6.12); in this case the results in [7] show that there is a
continuum of cases of equality.
2. It is tempting to prediet that there are interesting examples of the inequality
(7.2) arising from a choiee of the p, q and w coefficients such as a = 0, b = +00;
p(x) = w(x) = 1, q(x) = x or x 2 (x E [0,00)); in these cases the classieal HELP
integral inequality is considered in Section 4.1 above. The reason for making this
prediction, at an earlier stage, was that the spectra of the second-order Neumann
and Diriehlet problems are discrete and this has a marked effect on the classical HELP integral inequality. However predietion in mathematies is not always
successful; the remarkable result proved in [6] is that for all such choiees of the
coefficients the discrete spectrum leads to the HELP-type inequalities being not
valid for all values of the shift parameter T E lR. It seems that for the HELP-type
inequality to be valid it is necessary for the shift parameter to be in the essential
spectrum of the second-order Neumann, or equivalently Dirichlet, operator. The
next two examples illustrate this interesting observation.
3. a = 0, b = +00; p(x) = w(x) = 1, q(x) = -x (x E [0,00)).
M. BENAMMAR ET AL.
158
In this case (w- 1 M)[f](x) = - f"(x) - xf(x) (x E [0,00)) and
(w- 1 M[J])2 (x) = f(4) (x) + (2xj'(x))' + x 2f(x)
(XE [0,00)).
The inequality (7.2) takes the form
(7.4)
(1
00
{f"(x) +Xf(X)}2 dxf
::; K
1
00
f(X)2dx
1
00
{f(4)(X) + (2xj'(x))' +x2f(x)r dx
with domain
The numerical study of this inequality in [7, Section 3) leads to the results (for lK
see (6.3))
(i) the best possible number for the inequality (7.4) is K = lK,
(ii) the only case of equality is the null function.
4. a = 0, b = +00; p(x) = w(x) = 1, q(x) = _x 2 (x E [0,00)). In this case
(w- 1 M[f]) (x) = - f"(x) - x 2f(x) (x E [0,00)) and
(XE [0,00)).
The inequality (7.2) takes the form
(7.5)
(1 {J" +
00
(x)
::; K
x 2f(x)} 2 dx
1
00
f
f(X)2dx
1
00
{f(4)(X) + (2x 2j'(x))' +x4f(x)r dx
with domain
The numerical study of this inequality in [7, Section 3) leads to the results (for lK
see (6.3))
(i) the best possible number for the inequality (7.5) is K = lK,
(ii) the only case of equality is the null function.
Acknowledgement. The co-ordinating author (WNE) thanks Professor Gradimir Milovanovic for his help and patience during the preparation of this catalogue.
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REMARKS ON THE JACKSON AND WHITNEY
CONSTANTS
BORlSLAV BOJANOV
Department 01 Mathematics, University 01 Sofia, Blvd. James Boucher 5,
1126 Sofia, Bulgaria
Abstract. The paper is devoted to the constants in the Jackson theorem about approximation of continuous functions by polynomials on [a, b] and the Whitney type estimation
of the interpolation error. The J ackson theorem is derived here on the basis of the
Tchebycheff alternation theorem. This approach leads to an algorithm for computation
of the exact Jackson constant. In the second part we give a new representation of the
remainder in the Lagrange interpolation formula and then use it to get estimates of
Whitney type for certain classical approximation schemes.
Introduction
For any function 1 defined on [a, b] the n-th modulus of 1 is defined by
In particular,
Wl (f; 8)
== w(f; 8) := sup{l/(x) - l(y)1 : x, y E [a, b], Ix - Yl ~ 8}
is the modulus 01 continuity of 1 on [a, b]. Many estimations of the rate of convergence of approximation processes are expressed in terms of these modulL For
example, according to one of the central results in approximation theory, the Jackson theorem (see, for example, Natanson [6]),
(1)
En(f) ~ Cw(f; 1/n)
(C is an absolute constant),
where En(f) is the best approximation of Ion [a, b] by algebraic polynomials of
degree n, Le.,
En(f) := inf{ max I/(x) - p(x)1 : pE 7rn }.
zE[a,b]
1991 Mathematics Subject Classification. Primary 41A25j Secondary 41A10.
Key woms and phrases. Best approximationj Degree of approximationj Jackson theoremj Whitney constant.
The research was partially supported by the Bulgarian Ministry of Science under Contract No.
MM-414.
161
G.v. Milovanovic (ed.), Recent Progress in lnequalities, 161-174.
© 1998 Kluwer Academic Publishers.
162
B. BOJANOV
(As usual, lI"n denotes the set of all algebraic polynomials of degree less than or
equal to n).
In Section 2 we study the exact constant C in the Jackson inequality (1) and
discuss a quest ion of Frank Deutsch: could one derive the Weierstrass theorem
from the Tchebycheff alternation theorem.
Another result due to Whitney [10] asserts that
(2)
where W is a constant which does not depend on 1. Sendov has conjectured that
the Whetney constant W is bounded independent of n and moreover W ::; 1 in
case [a, b] = [0,1]. He proved in [8] that W ::; 6. Later Kryakin [4] showed that
W ::; 3.
We consider here the corresponding problem for interpolating polynomials. For
any given set x of points Xl < ... < X n we denote by P,(x) the polynomial of
degree n - 1 that interpolates 1 at Xl, ... , X n and present estimates of the form
(3)
111 - P,II := max 11(x) - P,(x)1 ::; const. w(x,1; I/n),
xE[a,b]
where w(x, 1; I/n) is a certain quantity that reduces to the ordinary n-th modulus
in the case of equidistant nodes. The problem have been studied before in the particular case of equally spaced points, mainly for Xk = kl(n+ 1), k = 1, ... , n. The
corresponding constant is usually denoted by W' for this choice of the interpolation
nodes. The well-known conjecture of Sendov [8] says that W' ::; 2.
Dur efforts to understand the nature of this problem resulted in obtaining certain
curious relations involving divided differences and Lagrange interpolation. They
are presented in Section 3. Using this general representation of the error in the
particular case of equally spaced nodes we derive Whitney type estimates for the
classical differentiation rules and interpolatory quadrat ure formulas. The usefulness and the exactness of this techniques is demonstrated by the fact that one of
the remainder representations obtained here reduces to the expression that yields
the famous Sendov bound W' ::; 6 for the interpolatory Whitney constant.
The J ackson Inequality
Denote by g[h, ... ,tm] the divided difference of 9 at the points h, ... ,tm. For
given 1 E C[-I,I] and points Xo < ... < xn+l in [-1,1], consider the best approximation En(x; j) oft he function 1 on the set x = (xo, ... ,XnH) byalgebraic
polynomials of degree n. By the Tchebycheff alternation theorem the polynomial
P of the best approximation satisfies the relations
with e = +1 or e = -1. Taking the divided difference on the both sides one get
REMARKS OF THE JACKSON AND WHITNEY CONSTANTS
163
where sex) is any function on [-1,1] satisfying the conditions S(Xi)
i = 0, 1, ... , n + 1. Therefore
En(Zj f) =
Ij[xo,
... ,Xn+1] I,
s[xo, ... , Xn+l]
which is the well-known representation of the best approximation. In the next
lemma we give an estimate of En(zj f) and then using the relation
En(f) =
(4)
sup
En(zj f)
"'0<"'<"',,+1
we derive the Jackson inequality.
The best uniform approximation of any given continuous function ! is completely
defined by (4). It follows from the Weierstrass theorem that En(f) tends to zero as
n goes to infinity. All of the known proofs of this classical result are constructive
and present certain sequence of algebraic polynomials approaching ! uniformly
on the interval. Frank Deutsch posed the problem of showing directly that the
quantity (4) tends to as n -+ 00. Studying this question we came to the approach presented below. It was applied in [3] to a more general setting treating
approximation by arbitrary Tchebycheff system of functions.
°
Lelllllla 1. For any natural number n and points Xo < ... < Xn+1 in [0,1], there
exist constants Co, ... , Cn such that
(5)
I
n
En(zj f) = :L>k[!(Xk+1) - !(xk)]I·
k-O
M oreover, the constants {Ck} satisfy the conditions
Proof. It follows from the recurrence relation
g[h, ... , t
]-
g[to, ... , tm-I]
m
9 [to, . .. , t m ] = =-=-,;:.;.,._..:........;;.:-=-.--='-!....:-'-----'--.:.:c'--=
t m - to
that there exist constants {dd such that the divided difference may be presented
in the form
(6)
j[xo, ... ,Xn+1] =
n
L dkj[Xk,Xk+1]'
k=O
In order to find dk we apply (6) to the function
'Pi(X) :=
{°Xk+1 - xk forfor
x - Xk
for
xE [O,Xk],
xE [Xk+I, 1],
xE (Xk, Xk+d.
164
B. BOJANOV
We shall show that dk "I- 0 and Sgndk = (_I)nH-k. Indeed, let qk be the polynomial from 7rn +l which interpolates <{)k at Xo, ... ,XnH. By Rolle's theorem, q~(x)
vanishes at each subinterval (Xi, xHd for i = 0, 1, ... ,k -1, k + 1, ... ,n. Thus q~
should be of degree exactly n. Since qHx) has no more zeros, qk(X) - (XHl - Xk)
changes sign at XkH, ... ,Xn+l. Finally taking into account that q~(x) is positive
at XkH we find the sign of qk(X) for large X and consequently, the sign ofthe leading coefficient of qk (x) which is actually the divided difference <{)k [xo, ... ,XnH]·
Thus
sgn dk -- sgn<{)k [Xo,··· ,Xn+l ] -- ( - l)n+l-k .
Applying (6) to the divided differences of 1 and sand taking into account that
sex) takes alternatively the values +1 and -1 at {xd, we rewrite En(x; f) in the
form
n
E dk[I(XHd - I(Xk)]
En(x; f) = k:O
E dk[s(XHd - S(Xk)]
n
= IL Ck[J(XkH) - I(Xk)]1
k=O
k=O
with
Ck := (-ltH-kdk (2
L Idkl) -, k = 0,1, ... ,no
n
1
k=O
All stated properties of {Ck} follow now from the corresponding properties of dk ·
The proof is completed.
An immediate consequence of (5) is the fact that: En(f) tends to zero il the set 01
alternation points 0/1 are dense in [-1,1], that is, provided the maximal distance
Hn(f) between two consecutive points 0/ TchebycheJJ alternation tor 1 approaches
zero as n -+ 00. The latter holds indeed, according to the well-known Kadec
theorem. Moreover, Tashev [9], improving the result of Kadec, has shown that
liminf Hn(f)n(logn)-l ~ 8(b - a).
So, this could be accepted as an answer to Frank Deutsch question (even with
an estimate En(f) ~ const. w(f; n-1logn). Looking for a more direct way of
establishing the convergence of E n (f) to 0 and for a more precise estimation, we
transform further the quantity (5) and estimate it.
Making use of some elementary properties of the modulus of continuity we derive
from (5) the estimation
En(x; f) ~ tiCk Iw(f; IXkH - Xkl) ~ t ICkl
k=O
k=O
CXkH8- xkl + 1) w(f; 8)
REMARKS OF THE JACKSON AND WHITNEY CONSTANTS
165
which holds for every 8 > 0. In particular, choosing
n
8 = 8(x) :=
L ICkllxk+l - xkl,
k=O
we arrive at the estimation
Therefore
where
n
an := sup8(x) = sup
'"
L ICkllxk+l - xkl·
'" k=O
Here the supremum is extended over all sets of points Xo < ... < xn+l in [-1,1).
Clearly we may assume without loss of generality that Xo = -1 and xn+l = 1.
If in addition the modulus of continuity of f is convex, that is,
for each O! E [0,1) and 81 ~ 0, 82 ~ 0, then we get
Thus the exact value of an would yield the exact constant in the Jackson inequality
at least in the dass of functions defined by a given convex modulus of continuity
w(8).
Observe on the basis of Lemma 1 that the quantity 8(x) is the best approximation
by polynomials of degree n of a particular function sn(x; x) on the set x. The
function Sn is continuous, piece-wise linear with knots at {Xi} and changes its slope
alternatively from 1 to -Ion the consecutive subintervals (Xi, XHt). Precisely,
1 n
sn(x; x) = Ax + B + 2
-l)kl x - xkl,
L(
k=l
with some coefficients A and B.
The problem of estimating E n (I) for arbitrary continuous function f is reduced to
the estimation of an. Our remarks above show two ways of doing it. The first one
is to find some good constructive approximation of Sn and the other is to find the
explicit expression of Ck in terms of Xl, .•. ,Xn and then maximize the function
8(x) over x. We shall go further both ways.
B. BOJANOV
166
First, following an idea of EH Passow [7], we construct a polynomial approximation
to S(x) := sn(x) - Ax - B of order 27f"/n. Let P be a polynomial of degree n and
n ~ 1. Set G(t) := Ix - tl- p(x - t) and assume that G(xo) = G(xn+1) = O. Then
n
1 n
12S(x) - ~)-I)kp(x - xk)1 < 1"2 ~)-I)k[lx - xkl- p(x - Xk)] 1
k=l
k=l
n+1
1 I:(-I)k[G(xk) -
k=l
G(Xk- 1 )]1
= [11 IG'(t)1 dt
[11 I sgn(x - t) - p'(x - t)1 dt.
For any xE [-1,1] the last quantity is clearly dominated by the L 1-approximation
of sgn t over [-2,2] by p' (t). Now we choose p' to be the polynomial of best L 1approximation to sgnt in [-2,2]. As follows from the classical result of Markov
(see Akhiezer [1, p. 98]), p' should interpolate sgn tat the zeros of the Tchebycheff
polynomial ofthe second kind Un (t/2) (here n is an even number) and the deviation
is given by
[22 (sgn t) . sgn U (t/2) dt.
n
This integral equals 4tan(1I"/(2n + 2)) ~ 411"/n (see Lemma 2 in Passow [7]). Thus,
using the described choice of p we get an approximation of sn(x) of order 211"/n
and therefore ß n ~ 211" In. Finally
3
En(f) ~ "2 w(f; 211"/n).
The exact evaluation of ß n seems to be very difficult problem. The extremal
points to the problem
n
I:
sup
ickllxH1 - xkl
.. k=O
are not known. The coefficients Ck are uniquely defined by the nodes {Xk}. They
can be computed by the formula
Pk(XHd -1
( ) Ck, k = 0,1, ... n - 1,
Pk Xk+1
where Pk is the polynomial from 1I"n which takes value 1 at Xo, ... ,Xk and vanishes
at Xk+2, ... ,Xn+l. Indeed, since
CH1 =
n
(7)
I: Ck[!(Xk+1) - !(Xk)] = 0
k=O
REMARKS OF THE JACKSON AND WHITNEY CONSTANTS
167
for each f E 7rn , we just set f = Pk and get the above recurrence relation.
Let us mention two necessary conditions for the optimality of the no des {xd:
There exists a polynomial q E 7rn and a number>. such that
Xj+l -
Xj = >. + (-l)j[q(xj+l - q(Xj)].
Summing the both sides of the last equality multiplied by Cj and using the orthogonal property (7), we get
The extremal nodes can be computed for small n. We give below these nodes and
the corresponding coefficients {Ck} for n = 10 and the interval [-1, 1].
0.926
-X2 = X9 = 0.771
-X3 = X8 = 0.575
0.010
Cl =C9 = 0.019
C2 =C8 = 0.037
C3 =C7 = 0.059
C4 =C6 = 0.079
C5 = 0.087
-Xl = Xl =
-X4
-X5
Co =ClO =
= X7 = 0.353
= X6 = 0.119
The results of computer experiments suggest that the extremal points are unique.
Whitney Type Estimates
Let Xo < ... < Xn be given points in [0,1]. Denote by j[xo, ... ,Xk] the divided
difference of f at Xo, . .. ,X k. Let
n
Pp(x) :=
L F(Xk)lk(X)
k=O
be the Lagrange interpolation polynomial for F with no des Xo, ... ,Xn . Set w(x) :=
(x - xo) ... (x - x n ).
By the Newton interpolation formula
(8)
F(x) - Pp (x) = F[xo, ... ,xn,x]w(x),
The divided difference representation of the remainder is a basic tool in the error
estimation of various approximation schemes based on interpolation. The next
lemma gives one more transformation of this expression.
B. BOJANOV
168
Lemma 2. Let F'(x) = f(x) and ~ be any point from [0,1]. Then
(9)
where Xk(t) := ~ + (Xk - ~)t, k = 0,1, ... ,n.
Proof. This relation was mentioned in [2, p. 11]. It could be derived also from
a recurrence relation for multivariate B-splines discovered earlier by Micchelli [5].
The relation (9) can be proved directly in a very simple way: Denote the integral
by I(F). Clearly I(x k ) = tSk,nH' Besides,
and thus I(F) is a linear combination of the values of F at xo, ... ,xn,~' This
two properties defined the divided difference F[xo, ... ,xn,~] uniquely. The proof
is completed.
An immediate application of Lemma 2 yields the following Whitney type estimate
for the interpolation error.
Theorem 1. Let P E 11'n interpolates the function F at the equidistant points
Xk = kh, k = 1, ... ,n + 1, h = l/(n + 2). Then
1
t
1
IIF - PIIC[-l,l]::; 0 wn(F'; n + 2) dt.
Proof. According to Lemma 2,
Since Iw(x)1 ::; Iw(l)1 = (n + l)!h nH < n!h n we get
IIF - PIIC[-l,l] ::;
1
1
wn(F'; th) dt
and the proof is completed.
Note that
Similarly we can obtain a Whitney type estimate for an approximate differentiation
formula based on Lagrange interpolation. Indeed, differentiating (8) one get
REMARKS OF THE JACKSON AND WHITNEY CONSTANTS
169
This is a differentiation formula of interpolatory type. It is widely used in numerical analysis in the particular case of equidistant nodes. Let Xk = kh, k =
0,1, ... ,n. Consider the remainder
Applying Lemma 2 with ~ = Xk and recalling that F'(x) = /(x), we get
Finally,
One important conclusion from the last example is that the quantities
8k(t) := F[xo + t, ... ,Xn + t,xk + t]W'(Xk),
k = 0,1, ... ,n,
are bounded by Wn (J i h) in case {x j} are equally spaced with a step size h. Thus
any estimation ofthe error expressed in terms of 8j (t) would yield a Whitney type
estimates. The next relation (10) is the key to obtaining such estimates in terms
of 8j (t).
Because of the fundamental role of 8j (t) it seems reasonable to introduce the
following generalized n-th modulus w(x, /i 8) associated with a fixed skeleton set
x of points 0 = Xo < ... < Xn = 1.
W(x, /i 8) := sup {
I:t
f(~(T I : -1 ~ ~ ~ 1, xo(t), xn(t) E [-1,1], t ~ 8}.
k=O W Xk
Clearly w(x, /i 8) reduces to the ordinary Wn(Ji 8) in case the skelation set is
Xk = kin, k = 0,1, ... ,n.
Set rk(t) := /(Xk) + 8k (t) - 8k (0).
Theorem 2. Let P(ti z) be the polynomial of degree n which takes values rj(t) at
the points Xj + t respectively, j = 0,1, ... ,n. Then
(10)
l
T
P(ti Xk + T) dt =
l
T
f(Xk + t) dt
B. BOJANOV
170
fOT each T E [-Xo, 1 - x n ] and k = 0, 1, ... ,n.
Proof. Denote by {lj(t; x)}8 the Lagrange fundamental polynomials corresponding
to the nodes Xo + t, ... ,Xn + t. In case t = 0 we shall use the abbreviation
lj(x) = Ij(O; x). Thus the polynomial
n
Pg(t;x):= Lg(Xj +t)lj(t;x)
j=O
interpolates the function g at {Xj + t}8. Differentiating the Newton identity
n
F(x) - PF(t;x) = F[xo + t, ... ,Xn + t,x] II(x - Xi - t)
i=O
at x = Xk + t, we get
n
f(Xk + t) = L F(xj + t)lj(t; Xk + t) + 6k(t).
j=O
Observe that lj(t; Xk + t) = lj(xk). Thus
n
f(Xk + t) - f(Xk) = L[F(Xj + t) - F(Xj)]lj(Xk) + 6k(t) - 6k (0),
j=O
and consequently
n
t
f(Xk + t) = Tk(t) + L i f(Xj + t) dt lj(xk).
j=O 0
Now let us multiply the both sides of this equality by lk(t; x) and sum for k =
0,1, ... ,n. Using the fact that
I·)(t·, x) - I)
.(0'
- )
I .(x - t) ,
' x - t) we get
t
k=O
f(Xk + t)lk(t; x) =
t {t
j=O k=O
n
=- L
)=0
d
lj(xk)lk(t; x)}
l
0
t
t
!(Xj + s) ds +
t
k=O
dt lj(x - t) in !(Xj + s) ds + p(t; x).
0
rk(t)lk(t; x)
REMARKS OF THE JACKSON AND WHITNEY CONSTANTS
Therefore
d lj(x - t)
p(tj X) = Pj(tj X) + Ln -d
j=O
t
l
t
171
f(Xj + S) ds
0
and hence
l
T t
T
p(tj X) dt = lT Pj(tj X) dt + Ln l l f(Xj + S) ds dlj(x - t).
o
0
j=O 0
0
An integration by parts in the last integral yields
1
T
o
1
T
p(tjx)dt=
n
T
Pj (t j x)dt+L{l f(Xj+s)dslj(x-T)
0
n
L
j=O
0
f(Xj + t)lj(Xk + T - t) = Pj(tj Xk + T)
j=O
and the relation (10) follows. The proof is completed.
Let us give an application of Theorem 2 in the case Xk = kh, k = 0,1, ... ,n,
h = 1/(n + 1), T = h. Then, by virtue of (10),
But lj(tj Xk +T) = lj(Oj Xk +T- t) =: lj(Xk +T- t). Therefore, denoting by Pj(t)
the polynomial from 1I'n that interpolates f at {Xk}ö, we get
Let Q be the polynomial from 11'n-1 which interpolates f at Xl, ... ,X n . Then the
error en (J) of the interplatory quadrat ure formula
1
1
f(x) dx
~
1
1
Q(t) dt
172
B. BOJANOV
is bound by
sup
11 f(x) dx 1
1
over all functions that vanishes at Xl, .•• ,Xn and have a preassigned n-th modulus
Since for such functions
W n (J ; 8).
and clearly If(xo)1 ::; w(J; l/(n + 1), the integration error en(J) can be estimated
in the following way
Now using the obvious inequalities
where
'Y:= max
L ~J )-1 Ilj(t)l,
n
(
O~t~Xl . 0
J=
and the estimates for 'Y and Jl.k given in Lemma 1 and Lemma 2 of Sendov [8], we
get
Next we derive sorne consequences from (10).
Theorem 3. For every continuous function f in [0,1] and any k = 0,1, ... ,n,
we have
Proof. Differentiating (10) with respect to T we get
(12)
REMARKS OF THE JACKSON AND WHITNEY CONSTANTS
173
Next we transform further this expression. Recall first that
Making use of this observation we obtain
n
= - L f(xj )[lj (Tj Xk + T) - lj (Oj Xk + T)]
j=O
n
= - L f(xj)[lj(Xk) -lj(Xk + T)]
j=O
Thus
Inserting this expression in (12) we get the desired presentation of the interpolation
error. The proof is completed.
Similarly we derive the following.
Theorem 4. For every continuous function f there exists a polynomial p* E 7rn
such that
(13)
To prove the assertion, we show as in the proof of Theorem 3 that
where q interpolates {dj(O)}~ at {Xj(O)}~. Then the theorem follows with p*(x) =
Pf(x) + q(x).
In the particular case of equidistant points {x k} an estimation of the interpolation
error based on Theorem 4 leads to Sendov's result [8]. Let us sketch the proof of
this important application.
B. BOJANOV
174
Let Xk = k/(n + 1), k = 0,1, .. . n. We are going to estimate the error 111 - Pli
where PE 1rn -l interpolates 1 at Xl, ... ,X n . Since the quantities t5j (t) are the
same for every function of the form 1 - Q, Q E 1rn -l, we may assume that 1
vanishes at Xl, ... ,Xn . Then we get from (13)
I(Xk + T) - I(xo)lo(xk + T) -
n
L t5j (O)lj(xk + T)
j=o
and therefore
I/(Xk + T)I < I/(xo)lo(xk + T)I +
n
L lt5j (O)lllj(xk + T)I
j=o
But I/(xo)1 $ w(f; l/(n + 1), 11o(xk + t)1 $ 1 for k = 0, 11o(xk + t)1 $ H~) -1 for
1 $ k $ n, and
Estimating further these bounds on the basis of Sendov's Lemma 1 and Lemma 2
from [8] one get I/(Xk + T)I $ 6wn (f; l/(n + 1)).
References
1. N. I. Akhiezer, Lectures on Approximation Theory, Nauka, Moscow, 1965. (Russian)
2. B. Bojanov, H. Hakopian and A. Sahakian, Spline Flmctions and Multivariate Interpolations,
Kluwer, Dordrecht, 1993.
3. B. Bojanov, A Jackson type theorem /or TchebychelJ systems, Math. Balkanica (to appear).
4. Yu. V. Kryakin, On the theorem 0/ H. Whitney in spaces L p , 1 ~ P ~ 00, Math. Balkanica
4 (3) (1990), 258-271.
5. C. A. Micchelli, On a numerically efficient method /or computing multivariate B-splines,
Multivariate Approximation (W. Schempp and K. Zeller, eds.), Birkhäuser Verlag, Basel,
1979, pp. 211-248.
6. I. P. Natanson, Constructive Function Theory, Gizdat, Moscow - Leningrad, 1948. (Russian)
7. E. Passow, Another proo/ 0/ Jackson's theorem, J. Approx. Theory 3 (1970), 146-148.
8. BI. Sendov, On the theorem and constants 0/ H. Whitney, Constr. Approx. 3 (1987), 1-11.
9. S. Tashev, On the distribution 0/ points 0/ maximal deviation, Approximation and Function
Spaces (Z. Ciesielski, ed.), North-Holland, New York, 1981, pp. 791-799.
10. H. Whitney, On functions with bounded n-th diIJerences, J. Math. Pures Appl. 36 (1957),
67-95.
ON THE APPLICATION OF THE PEANO
REPRESENTATION OF LINEAR FUNCTIONALS
IN NUMERICAL ANALYSIS
HELMUT BRASS
Institut für Angewandte Mathematik, Technische Universität Bmunschweig,
Pockelsstr. 14, D-38106 Bmunschweig, Germany.
KLAUS-JÜRGEN FÖRSTER
Institut für Mathematik, Universität Hildesheim, Marienburger Platz 22,
D-31141 Hildesheim, Germany.
Abstract. For more than 80 years, Peano kernel theory has proven to be an important
tool in numerical analysis. It is one aim of this paper to elucidate the wide range of
possible applications of Peano's representation of linear functionals. In the literature,
Peano kernel theory is mostly considered for restricted classes of linear functionals. In this
paper, it is also our objective to give an elementary but general approach for continuous
linear functionals on G[a, b].
1. Introduction
Let R be a continuous linear functional defined on C[a, b) with the property
R[:1's-d = 0; here, :1'8-1 denotes the space of all polynomials of at most degree
8 - 1. Then, for every / E C8-1 [a, b) for which /(8-1) is absolutely continuous in
[a, b), we have
(1.1)
where
(1.2)
(. - X)S-l]
Ks(x) := R [ (8
-I)! '
is the sth Peano kernel of R. The representation given by (1.1) is the Peano
representation of R. It is the aim of this paper to elucidate the wide range of
possible applications of (1.1) in numerical analysis which does not seem to be
sufficiently weH known.
1991 Mathematics Subject Classification. Primary 65D30; Secondary 41A55, 65D32.
Key woms and phrases. Peano kernel theory; Inequalities for linear functionals; Error estimates;
Quadrature; Interpolation; Optimal formulas.
175
G. V. Milovanovic (ed.), Recent Progress in Inequalities, 175-202.
© 1998 Kluwer Academic Publishers.
176
H. BRASS AND K.-J. FÖRSTER
Far this purpose, we give an elementary description of the basic facts and most
important results of Peano kernel theory in Section 2. In particular, we have to
discuss here how to interpret (1.2) in the case s = 1 where we have to take into
consideration that (. - x)~ is not a continuous function.
In the third section, we illustrate the theory by systematically applying it to
a particular functional, namely the remainder functional of Mehler's quadrature
formula. Furthermore, we report about the results obtained when the theory is
applied to the classical Gaussian quadrature formula.
In the fourth section, we look at some examples for various applications of Peano
kernel theory.
The most sophisticated applications of (1.1) are those in the theory of numerical
quadrature. Recent papers on this topic are (apart from those mentioned below)
e.g. those by Fiedler [14], Brass [5], Petras [29] and Brass [6]; see also Davis and
Rabinowitz [10] and Brass [4]. However, we will not elaborate on this in a systematic way because it is our intention to initiate work concerning the application
of (1.1) outside of quadrature theory.
The representation (1.1) can be generalised in many different ways. For example,
G[a, b] may be replaced by Gr[a, b], this can be lead back to (1.1) in an obvious
way by introducing an integral operator. Another variation is the substitution
of G[a, b] by aspace of 1-periodic functions. Here, R[P 8-1] = 0 is not a useful
condition, but large parts of the theory can be carried over to this case (under
the assumption R[Po] = 0) if the function (. - X)~-1 is replaced by B s (' - x)
(for the definition, see (3.21) below). Finally, the expression j(s) can be replaced
by applying a linear differential operator to j, see Radon [35] or Ghizzetti and
Ossicini [19]. The representation theorems of Sard [37] are even more general.
Owing to the lack of space, we cannot' go into details about this here. It seems to
us that these general representations have hardly ever been applied in a concrete
way to problems in numerical analysis.
2. Fundamentals of Peano Kernel Theory
In the following let R be a continuous linear functional on G[a, b] with R[P s - 1 ] = 0
and R[P 8 ] t- O. The norms under consideration are those of the space G[a, b].
2.1. For fixed ~ E 1R and fixed v E Niet the function sf"v be defined by
(2.1)
0,
{
Sf"v(x) :=
v(x - ~),
1,
for x <~,
for x E [~,~ + v-I],
for x > ~ + v-I.
Using (2.1) we can state the following
Lemma 1. Let ~ E [a, b]. Then, the limit
(2.2)
lim R[sf"v]
V-H)()
PEANO REPRESENTATION OF LINEAR FUNCTIONALS
177
exists.
Proof. The following method of proof was first given by Riesz (see, e.g., Riesz and
Sz.-Nagy [36]). We have that
11-1
R[S(,lIl = R[s(,tJ + 2:)R[s(,I<Hl- R[s(,I<])
(2.3)
1<=1
and Lemma 1 is proved if
00
(2.4)
L
IR[s(,1<+11- R[S(,l<ll
1<=1
converges.
The latter follows from the boundedness of
A
(2.5)
I
L
R[s(,I<Hl- R[se,l<ll
K=1
A
= R
[~[Se'KH - se,l<l sgn(R[s(,I<Hl- R[S('I<])] .
Note that the argument of R is bounded by
A
(2.6)
L
A
IS(,KH(X) - S(,I< (x) I = L(Se,K+1(X) - S(,K(X»
which proves the result.
0
We are now able to state the following
Definition 1. The nmction K 1 defined by
(2.7)
K 1 (e) := {
0,
for e= a,
r R[Se,lI,1 for a < e~ b,
II~~
is called the first Peano kernel 0/ R.
In particular, we see that
(2.8)
Furthermore, we will use the following lemma.
178
H. BRASS AND K.-J. FÖRSTER
Lemma 2. Let a = Xo < Xl < ... < Xn = band let 'Y := sup Ix,,+! - x"l. Then,
"
it follows that
IR[f]- ~[J(X,,+d - f(x,,)]Kl (x,,) I:::; IIRII w(J; 'Y),
(2.9)
where w(J; .) denotes the modulus of continuity of f.
Proof. Let
n-l
j := f(Xl) + L[f(x,,+d - f(x")]sx K,,,·
(2.10)
,,=1
For sufficiently large v, satisfying x" + v-I< X,,+l, a short calculation shows that
(2.11)
Therefore, using R[Po] = 0,
I R[J]- ~[f(X"+!) - f(x,,)]Kl(x,,) I
= lim
(2.12)
,,~oo
IR[J]- ~ [f(x,,+!) - f(x,,)] R[sx K,,,] I
,,=1
= v-+oo
lim IR[J] - R[I] I :::; v-+CX)
lim IIRllllf - 1II
Theorem 1. K l is of bounded variation and we have
(2.13)
Var K l = IIRII.
Furthermore, for every ~ E (a, b), K l (~) is between K l (~ - 0) and K l (~ + 0).
Proof. Let a = Xo < Xl < X2 < ... < Xn = b. Then,
n-l
I
L
K l (x,,+d - K l (X,,)
,,=0
=
(2.14)
I
)~ (IR[sxl,,, - 1]1 + ~ IR[sxK+I,"]- R[SXK,,,]I)
= v
lim
R [(SXl " - 1) sgn(R[sXl , " - 1])
-+oo'
PEANO REPRESENTATION OF LINEAR FUNCTIONALS
179
In (2.14), for sufficiently large v, the argument of R is bounded by
(2.15)
11 -
SX1.V(X) 1 +
n-1
I: ISX,,+loV(X) - sx".v(x)1 = 1.
1<=1
Therefore, the inequality
VarKl ~ IIRII
(2.16)
follows. For the proof of the reversed inequality we will use Lemma 2. By partial
summation we have
R[J] =
(2.17)
n
I: !(XI<) [K (xl<-d - K (XI<)] + p,
1
1
1<=1
where
Ipl ~ IIRII w(f; 7)·
(2.18)
Now, we directly obtain
IR[!l1 ~ II!II Var K 1 + P
(2.19)
and since (2.19) is valid for every decomposition of the interval [a, b] it follows that
IR[J] I ~ li/li Var K 1 ,
which yields the inequality Var K 1 ~ IIRII.
(2.20)
Finally, for the proof of the second part of Theorem 1, we assume that K 1 (~) is not
between K 1 (~ - 0) and K 1 (~+ 0). We consider the function [(1 which is identical
with K 1 outside the jumps of K 1 and which is continuous from the left-hand side
at the jumps (from the right-hand side, resp., if x = a). Then, Var [(1 < Var K 1 •
Note that the set of jumps of K 1 is at most countable. Considering in (2.19) only
such XI< with K 1 (XI<) = [(1 (XI<) we obtain by the same method as above Var K 1 ~
IIRII. Since IIRII = Var K 1 this is a contradiction to Var [(1 < Var K 1 • 0
The first main result is the following theorem.
Theorem 2. Let / be absolutely continuous on [a, b], then
R[J] =
(2.21)
l
b
!,(u)K1 (u) du.
Proof. For a = Xo < Xl < ... < Xn = b we obtain
I: [J(XI<+1) - /(XI<)] K (XI<) = I:K (xl<) 1. ,,+1 !,(u) du
n-1
(2.22)
n-1
1
1<=1
x
1
1<=1
x"
180
H. BRASS AND K.-J. FÖRSTER
1:
Furthermore, estimating the second term in (2.22), we have
I~
K
1
+ !,(u)[K1(xl() -
1:
L
1:
~ (s~p
(2.23)
K
+
1
K 1(u)] du
1!,(u)1
I
dU) x
n-1
X
sup{IK1(xl() - K 1(u)1 : XI( ~ u ~ XI(+!}
~ (s~p
K
+
1
1!,(u)1
dU) Var K
1•
The first factor in (2.23) tends to zero if sup IXI(+! - xI(I tends to zero. Therefore,
I(
the result follows from Lemma 1 and (2.22). 0
Now, the fundamental definition is the following:
Definition 2. The function K v defined by
(2.24)
Kv(x):=
l
b
for v = 2,3, ... ,8
K v- 1(u)du
is called the v-th Peano kernel of R.
Note that, for v > 1,
(2.25)
while it is possible that K 1 is not continuous in [a, b].
We are now able to prove the main theorem.
Theorem 3 (Peano representation). Let f(v-1) be absolutely continuous on [a, b].
Then,
(2.26)
for
v = 1,2, ... ,8.
Proof. The result can be proved by induction. For v = 1, (2.26) holds by Theorem 2. Let (2.26) be proved for v = 0' and let now R[Pu] = o. Then, using the
notation Pu (u) := uU,
(2.27)
Ku+!(a) = [ab Ku(u) du = (0'!)-1 [ab p~)(u)Ku(u) du
Ja
Jn
= (O'!)-lR[pu] = O.
Since Ku+! (b) = 0 it follows by partial integration that
(2.28)
R[/] =
l
b
/(u) (u)Ku(u) du =
l
b
/(u+1) (u)Ku+1(u) du.
0
Note that, by the above proof, K v has a zero in both end-points of the interval
[a,b],
(2.29)
Kv(a) = Kv(b) = 0
for v = 1,2, ... ,8.
PEANO REPRESENTATION OF LINEAR FUNCTIONALS
Theorem 4.
(2.30)
Kv(x) = R
[(.(v_X)~-l]
_ I)!
181
lor v = 2,3, ... ,s.
Proof. The proof follows directly from Theorem 3,
(2.31)
(. - x t - 1]
fb
R [ (v _
= Ja (u - X)~KV-1(U) du
i)!
=
l
b
K v- 1(u) du = Kv(x).
0
Now, we have completed the proof of the Peano representation (1.1) stated in the
introduction. It should be emphasised that the proof would be much simpler if we
restriet consideration to special classes of functionals or if we use Stieltjes integrals
and the Riesz' representation theorem for functionals on G[a, b].
2.2. In this section we consider bounds for R[/] whieh are based on Peano kernel
theory. An immediate consequence of Theorem 3 is the following estimate.
Theorem 5. Let 1 E G8-1 [a, b] and let 1(8) be bounded in [a, b], then
IR[/]I ~ 11/(8)11
(2.32)
l
b
IK8(x)1 dx.
(2.32) is sharp, namely we have
(2.33)
By Theorem 5 error estimates for many functionals in numerieal analysis can be
obtained in a systematic and uniform manner. Therefore, this theorem is one of
the most important (and most well-known) results in Peano kernel theory. The
following result is less well-known but also of wide practieal applicability.
Theorem 6. Let v E {O, 1, ... ,s - I}. 111(v) is 01 bounded variation (and il
1 E G[a, b] when v = 0), then the lollowing sharp bound holds,
(2.34)
Proof. If v = 0 then the result is an immediate consequence of Lemma 2. Therefore, let v > O. Since there exist two monotonie functions li v) and IJv) such that
I(v) = li v) + IJv) and Var I(v) = Var li v) + Var IJv) in the following we may
assurne that I(v) is a monotonie function on [a, b]. Using the second mean value
theorem for integrals, there exists a ~ E [a, b] such that
R[/] =
(2.35)
l
b
I(v) (u)Kv(u) du
l
= I(v)(a) e Kv(u) du + I(v)(b)
lb
Kv(u) du.
H. BRASS AND K.-J. FÖRSTER
182
Since v < s, we have
J:
Kv(u) du = 0 and therefore
which implies
IR[!ll ::; Var !(v) IIKv +1l1.
(2.37)
The sharpness of this estimate follows directly from Theorem 4.
0
The next theorem is a useful supplement to Theorem 5 and Theorem 6, since it
applies to all ! E G[a, bl. At first we require the following definition
Definition 3. If either K 1 (y) = 0 or sgn(K1 (y + 0)K1 (y - 0)) ::; 0, y is called a
generalised zero of K 1 .
Theorem 7 (Köhler 1994). Let a = ~o < 6 < '"
o! K 1 and let
(2.38)
< ~r+1 = b be generalised zeros
'Y:=sup{I~V+1-~vl: v=O,l,oo. ,r}.
Then,
(2.39)
Proof. Let c > 0 such that ~v + 2c < ~v+1 - 2c for every v E {O, 1, ... ,r}. We
define Iv and Cv for v E {O, 1, ... ,r} by
(2.40)
1
(2.41)
Cv := -2 [max!(x)
xElv
+ xEl
min !(x)].
v
Then it follows that
(2.42)
Now, we define, for sufficient small c,
(2.43)
Se,E,V := {
K1(€+c)Se-E,v-K1(€-c)sHE,v
K1 (€+c) - K1 (€ -c)
far K1(€+0)#K1(€ - 0),
se,v
far K1(€+0)=K1(€-0),
r
(2.44)
He,v := Co + L(cl< - cl<-dSe",e,v,
1<=1
PEANO REPRESENTATION OF LINEAR FUNCTIONALS
183
where 8(,,, is defined in (2.1). Using (2.42) we can prove, for sufficient large v,
1
I/(x) - He,,,(x)1 ~ 2 w(fj'Y+ 4c)
(2.45)
and therefore
1
IR[/]- R[He,,,] I ~ IIRIIIII - He,,,11 ~ 2 11RII w(fj 'Y + 4c).
(2.46)
Since l~ R[S(",e,,,] = 0, (2.46) implies for v -+ 00
,,~OO
1
IRU]I ~ 2I1Rllw(fj'Y+4c)
(2.47)
and (2.47) holds for every c > O.
0
We now state a further estimate using the modulus of continuity, which in case of
applicability often gives a sharper estimate. We will use the following notation,
(2.48)
Theorem 8 (Ligun 1976). Let the modulus 0/ continuity w(fj .) 0/1 be concave.
Then,
(2.49)
IRU]I
~ ~ IIRII w (fj 211K11I 1 / IIRID·
Proof. We require the following theorem from approximation theory (see, e.g.,
Lorentz [25, pp. 122-123]: For every h > 0 there exists
(2.50)
IIg'lI ~ M
and
1
111 - gll ~ 2 (w(fj h) - Mh).
Applying Theorem 5, we have
(2.51)
1
IRU]I ~ IRU - gll + IR[gli ~ 2 11RII (w(fj h) - Mh) + IIK1IiIM.
Now, substituting h = 211K1lldliRII the result follows.
0
Köhler [21] has given a proof of Theorem 8 based on completely different ideas.
In his paper further estimates using the modulus of continuity are stated.
In the next estimate for RU] we assurne the convexity of I. This is of interest
since convexity is a type of smoothness which, e.g., does not imply the boundness
of the first derivative of I.
H. BRASS AND K.-J. FÖRSTER
184
Theorem 9 (Förster and Petras 1990). Let f be a convex junction on [a, b]. Then,
the following sharp bound holds,
IR[fli ~ 8 (f(a) - 2f((a + b)/2) + f(b)),
(2.52)
where
_
{ x- a
K 2 (x) :=
b- x
(2.53)
for x E [a, ~] ,
b
for x E [at ,b] .
Proof. Using a suitable approximation of f (e.g., a polygon whose vertices are
smoothed) we may restrict to functions having a nonnegative second derivative.
We have
(2.54)
R[f] =
~
l
b
81
j"(x)K2 (x) dx =
b
l
b
j"(x)
~:~:~ K (x) dx
2
j"(x)K2 (x) dx = 8 [f(a) - 2f ((a + b)/2) + f(b)].
The lower bound in (2.52) can be proved analogously. For the proof of the sharpness of the result, consider f(x) = (x - xo)+, where Xo is a maximum point of
(2.53). 0
Petras [29] has given a generalisation of Theorem 9 for convexity of higher order.
Further bounds for R[J] can be obtained by applying Hölder's inequality to (2.26),
but more important is a closer look to the structure of Peano kerneis.
2.3. For several applications the changes of sign of Peano kerneis are of interest.
To be more precise, we will use the following definition.
Definition 4. The function f defined on [a, b] has at least r changes of sign if
there exist numbers a ~ 771 < 772 < ... < 77rH ~ b such that
(2.55)
for
K,
= 1,2, ... ,r.
We will use the notation SC(f) = r, if f has at least r changes of sign but not at
least r + 1 changes of sign. In this situation there exist numbers a = ~o ~ 6 ~
..• ~ ~rH = b such that, for v E {O, 1, ... ,r}
(2.56)
(-ltfJf(x) ~ 0 for x E [~v,~vH]
and
sup
(-ltfJf(x) > 0,
~v~"'~~v+l
where either fJ = 1 or fJ = -1 is fixed. If f = K>.., then we always have ~v < ~vH'
This is trivial for >. > 1 and follows from Theorem 1 for >. = 1. Now we can prove
the following result.
PEANO REPRESENTATION OF LINEAR FUNCTIONALS
185
Theorem 10. K>. has at least s - A changes 01 sign.
Prool. We assume that K>. has r < s - A changes of sign. Using a decomposition
as in (2.56), then
(2.57)
since K>. is of bounded variation and therefore in each interval [~/I' ~/I+ d there is
a subinterval in whieh K>. has a fixed sign. The term on the left hand side of
(2.57) is equal to R[P] with a pE Pr+A C :1's-l and this means R[P] = O. By this
contradiction the result is proved. 0
Theorem 11. SG(K>.) = r implies SG(K>.+d ~ r - 1.
Proof. We use a decomposition as in (2.56). K>'H is a primitive of -K>. and
therefore monotonie in [~/I' ~/I+1]. This implies that KoHl has at most one change
of sign in [~/I' ~/I+1]. Since K>'+1 (a) = K.>'+1 (b) = 0 and since KoHl E G[a, b] there
are no changes of sign in [a,6] and [~r, b]. 0
The following result is an immediate consequence of Theorem 11.
Theorem 12. SG(Kd = s - 1 implies SG(Ks ) = O.
J:
The situation that K s has no change of sign is of practieal interest for the use of
Theorem 5, since the (often laborious) calculation of IKs(x)1 dx can be simplified
drastically. Then we have
(2.58)
where Ps(x) := x S. If 1 E GS[a, b], applying the mean value theorem to (2.26) we
obtain
(2.59)
RU] = j(s)(~) R[ps]
s!
with ~ E [a, b].
Generally, a functional R is called definite 01 order A if there exists a fixed constant
c such that
(2.60)
~ E [a,b],
for every j E G>'[a, b].
Theorem 13. R is definite il and only il K s has no change 01 sign. In particular,
il R is definite, then R is definite 01 order s.
Proof. Let R be definite of order A. Then R[/] has the same sign for all 1 with
j(>') > O. Under the assumption that K>. has a change of sign, it is easy to
construct functions j with 1(>') > 0 such that
(2.61)
H. BRASS AND K.-J. FÖRSTER
186
has different signs for two different functions f. By an application of Theorem 10
the result folIows. 0
2.4. More information on Peano kerneis can be obtained using the expansion of
K.\+1 in Chebyshev polynomials. For this technique see Brass and Förster [7]. In
the following let A ? 1. In the following let A ? 1 and let [a, b] = [-1,1]. Our
starting point is the following expansion,
;! x)~ f a~.\)Tv(t),
(2.62)
=
(t -
~
a(A) :=
v
'Ir
(2.63)
1
1
-1
v=O
(t - x)~ Tv(t) dt.
,x!
v'f=t2
Note that the series in (2.62) converges uniformly in [-1,1]. Using the Rodrigues
formula (for the theory of orthogonal polynomials and the notations used here, see
Szegö [40]) we have
Tv(t) = (-I)Vv! 2V (~) v [(1 _ t2t-1/2]
viI - t 2
(2v)!
dt
(2.64)
Using partial integration we obtain for v > ,x
(2.65)
a(A)
v
= ~ (-lt+ A v! 2
V
(2v)!
'Ir
1
1
-1
(t _ x)O (~) V-A [(1 _ t2t-1/2] dt
+ dt
= ~ (-lt+.\+1 v! 2 (.!!...) v-A-1 [(1 _ x2t-1/2] .
V
dx
(2v)!
'Ir
A comparison with the Rodrigues formula for ultraspherical polynomials shows
that
(2.66)
(A)
av
= ~ 2A(v(- ' A)-1I)! ,x! (1 _ X 2)A+1/2 p(A+1)
( )
v-A-1 X •
v+".
'Ir
By a result of Durand [12] we have, for every x E [-1,1],
(2.67)
(1-
x
)1 <
2)A+1/2Ip(A+1)(
n
X
-
r(n/2 +,x + 1)
r(,x + 1) r(n/2 + 1)"
(2.66) and (2.67) imply the following estimate,
(2.68)
1a(.\)
I<- ~ (v - ,x) (v - ,x + 2)(v1 - ,x + 4) ... (v + ,x) .
v
'Ir
Furthermore by (2.61) we have
00
(2.69)
K A+1(x) = La~A) R[Tv ].
V=S
PEANO REPRESENTATION OF LINEAR FUNCTIONALS
187
Using (2.68) we obtain
~ I (A)I
IKA+1(x)1 :5 IIRII ~ a",
~
2
1
:5 -; IIRII t:-; (v-.\)(v-.\+2) ... (v+.\)
=~IIRII..!...~[
1
7r
2.\ t:-; (v - .\)(v - .\ + 2) ... (v + .\ - 2)
(2.70)
1 ]
- (v - .\ + 2)(v - .\ + 4)· .. (v + .\)
= ~7r IIRII..!...
[
1
2.\ (s - .\)(s - .\ + 2) ... (s + .\ - 2)
~
+ (8 - .\ + l)(s - .\ 3)· .. (s + .\ - 1)] .
Therefore we have proved the following result:
Theorem 14. Let.\ E {2, 3, ... , s}. Then,
(2.71)
IIKAII :5 7r(.\ 2_ 1) (8-.\+1)(S-.\+3)(~-.\+5) ... (s+.\-3) IIRII·
Using the method deseribed above it is also possible to obtain estimates of IKA(x)1
which depend on x. Instead of (2.67) we have to use loeal bounds, for more details
see Förster [16]. Finally we state the following conjecture
(2.72)
(1 _ x 2)"'-1
IK",(x) I :5 2"'(v _ I)! IIRII for xE [-1,1],
v = 1,2, ... ,s.
3. An Example: Mehler's Quadrature Rule
3.1. For the numerical approximation of J~l /(x)(l - X)-1/2 dx mostly the following well-known quadrat ure formula of Mehler's is used,
(3.1)
M
7r
~ .(
Qn eu] := ;, ~ /
2v - 1 )
eos - n - 7r •
As an example, we eonsider here
(3.2)
Q:!e is the Gaussian formula for the Chebyshev weight function (l- x 2 ) -1/2 and
we have R[P2n-1] = O. For estimates of the remainder term R we now apply the
Theorems 5-9. We obtain estimates of the type
(3.3)
(3.4)
(3.5)
(3.6)
IRU]I :5 Cl", 11/("') 11,
v = 1,2, ... ,2n,
IRU]I :5 ß", Var /(",-1),
v = 1,2, ... ,2n,
IR[/]I :5 7rw(f; 1'),
IRU]I :5 81/(-1) - 2/(0) + /(1)1,
188
H. BRASS AND K.-J. FÖRSTER
where we have to investigate the numbers a", ß", 7, 8. Let us emphasise that
a large number of bounds is important in order to utilise fully the properties of
f. E.g., though (3.5) generally can be applied for f E G[a, b], in most cases the
obtained bound leads to a large overestimation of the error. Furthermore, even if
all derivatives of fexist, the use of the highest derivative in (3.3) or (3.4) often
does not result in the best estimate.
For R defined in (3.2) we easily obtain
(3.7)
K 1 (x) = arccosx - - (n - A)
n
7r
where
(3.8)
2A-1
XA := -COS~7r
1, ... ,n),
(A =
Xo := -1, X n +1 := 1.
K 1 has zeros (of order 1) in - COS(A7r In) (A = 0,1, ... ,n) and generalised zeros
in X A (A = 1,2, ... ,n). Theorem 7 therefore yields
(3.9)
Furthermore, a simple calculation gives
(3.10)
and doing some more work we obtain
1
1
(3.11)
7r
IK1 (x)1 dx = 2tan -4 .
n
-1
So we have obtained a1 and ß1 (Ponomarenko [33]) and we also can apply Theorem 8. Furthermore,
(3.12)
r.;---;;
7r
7r
sin(A7rln)
K 2 (x) = V 1- x 2 - x arccos x + x(n - A) - - - -..,.:-,...,.:,..-:-:n
2n sin(7r/(2n))
for x E [x A , XAH] and this implies
7r
(3.13)
IIK2 11 = {
odd n,
;n [1 - 2~'- cot - ] forfor even
n,
7r
COS -
2n
and
7r
1--cot-
2n
2n'
PEANO REPRESENTATION OF LINEAR FUNCTIONALS
189
On the other hand, for J~1 IK2(x)1 dx an explicit simple expression seems to be
hardly obtainable. A lengthy but straightforward calculation gives the following
estimate for every n > 1,
(3.14)
1
1
4V2 ( rr )2 (rr
1
_1IK2(x)ldx=-3-n cos 4n
2nsin(rr/(2n»
)3/2
1
(1+e: n ),
-1(1 - tan 2rr)
Ie: n I<
- 2n
4n '
(3.15)
Using Cauchy's inequality, from Theorem 3 it follows that
(3.16)
where the first factor on the right hand side of (3.16) is sharp. An explicit lengthy
computation gives for s = 1 and s = 2 the following results,
(3.17)
1
1
-1 (K 2(x»
2
32 ( rr)2 cos(rr/n)
rr
dx = 27 - 2n sin(3rr/2n) cot 2n
rr
+-
1
2n sin(3rr /2n)
(3.18)
(4--3cos
2 -rr)
9
2n
4 (rr)4
-5
= 135
2n + 0 (n ).
An application of Theorem 9 yields after some calculation
(3.19)
IR(fll:::; (cot
;J
(1- 2: cot 2:) (/(-1) - 2f(0) + f(I»,
where the constant
(1- ~ cot~) = ~ + 0 (n- 3 )
( cot~)
4n
2n
2n
3n
is sharp.
The explicit calculation of 0:2, 0:3, ... , ß3, ß4, ... fails because of the complexity of
the formulas for K 2 , K 3 , ••.• This, of course, is true only if we want to express
H. BRASS AND K.-J. FÖRSTER
190
a,. and ß,. as funetions of n. For every single fixed n, the evaluation of a,. and ß,.
does not eause any problems. All praetical needs will be satisfied with a table for
seleeted values of n that may easily be established. (Stroud and Seerest [39] have
calculated such a table).
The best known error bound for Mehler's quadrature rule is (3.3) with v = 2n. It
can be proven easily because R is definite of 2n-th order. The definiteness follows
immediately from Theorem 12, the changes of sign have been stated above. We
obtain
(3.20)
7r
R[J] = 22n-1
j(2n)(e)
(2n)! '
where we have used the relation R[P2n] = R[T~]/22n-2 for the ealculation of R[P2n]'
Simple expressions for the Peano kerneis of higher order are available, however, if
we aecept representations that hold asymptotically (as n -t 00). For this purpose,
we define the Bernoulli function B). by
(3.21)
B ( ) = -2 ~ eos(2v7rx - 7r >'/2)
). x
~
(2v7r).
,
>. = 1,2, ....
FIG. 1: First Peano kernel K1 of Mehler's formula Q!fe for n = 8 (scaled
10- 1 )
FIG. 2: Second Peano kernel K2 of Mehler's formula Q!fe for n = 8 (scaled
10- 3 )
PEANO REPRESENTATION OF LINEAR FUNCTIONALS
191
FIG. 3: Third Peano kernel K3 of Mehler's formula Q~e for n = 8 (scaled
10- 4 )
FIG. 4: 4th Peano kernel K4 of Mehler's formula Q~e for n = 8 (scaled
10-5 )
FIG. 5: 8th Peano kernel Ks of Mehler's formula Q~e for n = 8 (scaled
10- 9 )
BA is a 1-periodic function whose restrietion to the interval (0,1) is a polynomial,
the so-called Bernoulli polynomial. In particular, we have
(3.22)
1
Bt{x)=x-LxJ- 2,
H. BRASS AND K.-J. FÖRSTER
192
0.5
FIG. 6: 16th Peano kernel K16 of Mehler's formula Qffe for n = 8 (scaled
10- 17 )
and therefore (3.7) leads to
(3.23)
1)
_ 7r B (narccosx
K 1 (X ) -1
+-2'
n
7r
Now, we integrate successively and take into consideration that B~+1 = BA' In
the first step, we obtain
(3.24)
K 2 (x) = [1 '!!..B 1 (narccosu +~) du
}z n
7r
2
= [1 '!!..B 1 (narccosu + ~)
}z n
2
7r
1
VI - u 2
\11 _ u2 du.
A partial integration yields
K 2 (x) = 7r 2 B 2 (narccosx +~) ~
(3.25)
n2
2
7r
7r211 B 2 (narccosu +-1)
-n2
z
7r
2
u
d
VI - u 2 u '
and, by another partial integration, we obtain
K 2(x) = 7r 2 B 2 (narccosx +~) \11 _ x2
n2
7r
2
(3.26)
7r
3 B (narccosx
-- 3
n3
7r
+-1)
x
2
7r311 B 3 (narccosu +-1) du
7r
2
'
n3 z
--
and thus
PEANO REPRESENTATION OF LINEAR FUNCTIONALS
193
More generally, this method leads to
(3.28)
Using well-known analytical results, from (3.28) we can easily obtain the asymptotic behaviour of a s and ßs.
For K 2n , this method does not lead to any results, hut we may use (2.68). We
have
(3.29)
[ v ] -_{ (-1)k+17r,
RT
0,
if v=2kn, k=1,2, ... ,
else,
and thus, taking into consideration (2.65) and (2.67),
(3.30)
K 2n (x) =
2 2n +1(2n)!
(4n)!
[(1 _x 2)2n-1/2 + O((247)n)]
folIows.
3.2. In practice, the classical Gaussian quadrature rule
n
(3.31 )
Q~[j] =
L avf(xv)
v=l
is much more important than Mehler's rule. In (3.31), the weights a v and the
nodes x v E [-1, 1] are determined in such a way that Q n [P] = f~ 1 p( x) dx holds
for every p E P2n-1. Here, we thus have to investigate
(3.32)
R~[f] := [11 f(x) dx - Q~[f].
Again, we can ask for the values of a K, ßK' ,,(, eS from (3.3)-(3.6). The technical
problems are much more difficult than in Mehler's case because now, no simple
expressions for a v and Xv are available. The only simple case here is a2n because
R~ is definite [Proof: By definition of K 1, we have that K 11 (a=v.:t v +l ) (x) = -x +
const v ; therefore, K 1 cannot have more than 2n - 1 changes of sign. Now, apply
Theorems 10-13].
ß1 has been determined explicitly by Förster and Petras [18] who have shown in
the case n = 2m - 1 that
(3.33)
= KG(O 0) = ~ m = [((n - 1)/2)!]4 22n - 2
II KGII
1
1
+
2a
[n!J2
From the structure of K 1 described above, this directly implies "( = IIKrll, and
hence
(3.34)
194
H. BRASS AND K.-J. FÖRSTER
as conjectured by Baker [2, p. 789].
In all other cases, only bounds are known: Asymptotically sharp bounds only for 6
from Förster and Petras [17], for al from Petras [27], and for a2 from Petras [30].
In particular in the latter case, a very large amount of analytic work is necessary
in order to obtain the results.
In an important paper, Petras [26] has shown that the application of Peano kernel
theory need not necessarily fail due to analytic difficulties. He has shown for a
large dass of quadrat ure rules (induding Q~) how the asymptotic behaviour of K >(for an increasing number of no des) can be expressed by simple formulas similar
to (3.27). This allowed hirn also to find out the asymptotic behaviour of a). and
ß).. For asymptotic behaviour of an for increasing n, see Section 4.7.
Bounds for
have first been stated by DeVore and Scott [11], they have been
irnproved by Petras [29].
Kr
4. Selected Applications
4.1. Proofs of Definiteness. Definiteness often can be proved by Theorem 12.
A first example was given in Section 2 for the Gaussian formulas. In the following
we give three further examples.
Example 1. Let R = dvd (Xl, X2, ... , X n ), where dvd (Xl, ... , X n ) is the divided
diJJerence for the nodes Xl < X2 < ... < X n uniquely defined by
n
(4.1)
dvd(Xl,X2,'" ,xn)[f] = Lc"f(x,,),
(4.2)
0 for v = 0,1, ... ,n - 2,
dvd(Xl,X2,'" ,xn)[P,,] = { 1
for v = n -1,
,,=1
p,,(X) := x".
Therefore R[:Pn - 2 ] = 0, Le. s = n -1. The first Peano kernel K l is a step function
having jumps at the points x". Therefore, SC(Kd ~ n - 2 and by Theorem 10
we obtain SC(K l ) = n - 2. Theorem 12 shows that R is definite of order n - 1.
Example 2. Let
where u E [0,1] is fixed. This is the remainder term of the Bernstein operator
which is well known in approximation theory. We have R[:P l ] = 0, Le., s = 2.
We directly obtain Kl(x) ~ 0 for X < u and Kl(x) ~ 0 for X > u which gives
K 2(x) ~ O. The calculation of R[P2] is simple, we obtain (see Stancu [38])
(4.3)
- u) f"(I:),
R[f] = - u(12n
~
I:
[0 1]
~E,.
PEANO REPRESENTATION OF LINEAR FUNCTIONALS
Example 3. R[f] =
l
b
195
f(u)qn(u)w(u) du, where qn is the orthogonal polynomial
of degree n for the fixed nonnegative weight function W. We have R[Pn-1] = 0,
Le., s = n and
(4.4)
We assume that there exist numbers 111 < 112< ... < 11n+1 such that sgnK1(11I1) =
(_1)11. Then we have
(4.5)
["V+l
sgn J..
'Iv
qn(u)w(u) du = (-lt
for v = 0,1, ... ,n + 1,
where 110 := a, 11n+1 := b. Therefore, qn has n + 1 changes of sign which is a
contradiction to the algebraic degree n of qn. We have shown that SC(K1 ) :::;
n - 1 and the proof of definiteness of R order n follows again by Theorem 10 and
Theorem 12.
For more sophisticated methods to prove definiteness, see Brass and Schmeisser
[9].
4.2. Proof of Non-Definiteness. Often the investigation of definiteness may be
hard. In the following we give an example for a simple proof of non-definiteness.
We consider the remainder term of the interpolatory quadrat ure formula of Clenshaw and Curtis,
(4.6)
where the numbers all are uniquely determined by R~C[Pn_1] = O. It is wellknown that all a ll are positive. Akrivis and Förster [1] have solved the problem of
non-definiteness of R~c as follows. Let n > 2 be an even number. We have
(4.7)
Kn(x) =
(1 - x)n
(1 - x)n-1
,
an
(n
_ I)'
n.
.
n-2
for X > - cos --1 7r.
n-
Therefore, we have Kn(x) < 0 in a nonempty interval (1 - c, 1). The definiteness
of R~c of order n would imply K n :::; 0 everywhere in [-1,1]. Since it is easy to
calculate that
(4.8)
we obtain a contradiction. Rabinowitz [34] has applied this method to prove nondefiniteness of Gauss-Kronrod quadrature formulas.
4.3. Error Estimates in Lagrange Interpolation. Let
196
H. BRASS AND K.-J. FÖRSTER
be the Lagrange interpolatory polynomial to f for the nodes Xl < X2 < ... < X n
and let R be the associated remainder term,
(4.9)
R(f] = f(u) - intpol (Xl, X2,· •• ,xn)[f](u).
Let u f/. {XI,X2, ... ,xn }, then s = n. K I has jumps at U,XI,.·· ,Xn . Since
min( u, Xl) and max( u, x n ) can not be points of changing sign by Theorem 10 all
other points of jump are generalised zeros of K I . The application of Theorem 7
gives
(4.10)
IR(f] I ~ -2111RII w(fj sup
O~v~n
lxv+! - xvi),
Xo:= a,
Xn+l:= b,
a result of Brass and Günttner [8]. For generalisations see Köhler [22].
4.4. Error Estimates for the Bernstein Operator. We consider as in 4.1
(Example 2)
(4.11)
It is easy to show that
(4.12)
and by some algebraic transformation we have that
(4.13)
K, :=
LnuJ.
Therefore, for a concave modulus of continuity w(fj . ) we obtain by Theorem 8
the estimate
(4.14)
a result of Gonska and Meier [20].
4.5. Numerical Integration of Functions having Singularities. Let Rn be
the remainder term of a positive interpolatory quadrat ure rule,
(4.15)
with
(4.15)
Rn[Pn-d = 0,
Xv E [-1,1],
av ~ 0 for v = 1,2, ... ,n.
PEANO REPRESENTATION OF LINEAR FUNCTIONALS
197
Here, as an example, we consider the function f given by
f(x) = (1- x) log(1 - x).
(4.16)
The following method can be applied easily to a larger dass of singularities at the
endpoints of [-1,1] (for singularities inside [-1,1] see Petras [31]), but the basic
idea can be demonstrated more dearly by the example (4.16). Theorem 5 cannot
be applied, Theorem 6 gives at most Rn(f] = O(n- 1 ). Theorems 7-8 do not even
yield this asymptotic result. We will prove by Peano kernel theory
(4.17)
We have
(4.18)
For n > 5 and for 0 < c < 2 we obtain
(4.19)
Using the trivial bound IKv(x)1 ~ (1 - xy-l IIRII/(v - I)! we see that the limit
for c -t 0 exists. Therefore,
(4.21)
j (1 _6
l
(4.20)
Rn[f] =
Rn(f] =
j
l-l/n 2
-1
(1
-1
6
-
X
X)4 K 5 (x) dx,
)4 K 5 (x) dx +
1
1
1-I/n2
For the first integral in (4.21) we use the estimate
while for the second integral in (4.21) we use the bound
6
( ) 4 K 5 (x) dx.
1- x
198
H. BRASS AND K.-J. FÖRSTER
Now, (4.17) follows from (4.21). Instead of (4.22), more generally we have
for all quadrature rules considered here (see Petras [26], Brass [6]). For the proof
of (4.23), from the definition of K>. we obtain
IK5 (x)1 :::; (1 - x) sup IK4(U)I:::; (1 - X)2 sup IK3 (u)1
(4.25)
x~u~l
x~u~l
:::; (1- x)3 sup IK2(u)1
x~u9
and now we apply (4.24) with r = 2.
The technique used here was first introduced by DeVore and Scott [11] in a special
situation.
4.6. Asymptotic Behaviour for Smooth Functions. By the zeros of multiplicity (8 -1) in a and b of K s , for large 8 the kernel K s is small near the endpoints
ofthe interval [a,b]. Therefore, for smooth f, the value of f(s)«a+b)/2) will have
a strong influence on R[J]. For [a, b] = [-1,1] we have
(4.26)
Therefore, we have the following result. If R is definite of order 8 and if T is even,
then, for every f E CS+T[-I, 1],
(4.27)
where TJ E [-1,1] and, as above, Pu(x) := x u . (4.27) can be applied in several situations. As an example, let intpol (Xl, X2, ••. ,xn)[f] be the Lagrange interpolatory
polynomial to f for the nodes XII = - cos(211 - l)rr/(2n) and let
(4.28)
R[f] = f(x) - intpol (Xl, X2,· .• ,xn)[f](x).
Using (4.27) with T = 2 we have
(4.29)
_ Tn(x) [f(n)(o)
f(n+1) (0)
f(n+2)(TJ)
R[f] - 2n-l
n! + (n + I)! + (n + 2)!
(2
X
~)]
+4
'
PEANO REPRESENTATION OF LINEAR FUNCTIONALS
199
which is of interest in approximation theory (see Brass [3]).
4.7. Construction of Optimal Quadrature Formulae. We consider the
functional R defined by
1
n
1
(4.30)
R[fl =
f(x)w(x) dx -
-1
L avf(x v )
v=1
with w ~ 0 and R[Pol = O. By Theorem 6 we have
c=IIKdl·
IR[Jll":;' cVarf,
(4.31)
J:
How should we choose the numbers av and Xv to minimise c? The structure of
K 1 shows that this problem is equivalent to the approximation of w(u) du by a
step function. The solution of this problem is not hard. We obtain for the minimal
c = Cmin,
(4.32)
Cmin
1
1
= -2
1
w(x)dx
n -1
and the associated quadrat ure formula can be given easily.
Obviously, the investigation of minimal constants for estimates of the type
IR[fli ..:;, ßv Var f(v-l)
(4.33)
leads to the investigation of special approximations by spline functions. Concerning this subject, many results in the literature can be found (see, e.g. Levin and
Girshovich [23]). The calculation of explicit values for the minimal constants a v
and ßv seems to be hard.
4.8. The Superiority of Gaussian Quadrature. Considering (4.30), if w == 1
and R[Pn - 1 l = 0 the problem of an asymptotically sharp estimate for the minimal
an in (4.33) is open. For the (dassical) Gaussian rule Q~ we obtain by Theorem 14
the following simple bound
(4.34)
aG
< 211K G II < 16
n -
-
n
'Ir
1
(n - 1)(n + l)(n + 3) ... (3n - 3)'
Using Stirling's formula we have
(4.35)
lim sup (n! a~) I / n..:;,
n--+oo
l"" = 0.192 ....
3v3
(For sharper results see Förster [16]) and up to now there is no quadrature rule
known which gives a better result. For the Clenshaw-Curtis rule we have
(4.36)
.
( n!acc)l / n = -1 = 0.5,
hm
n
2
n--+oo
200
H. BRASS AND K.-J. FÖRSTER
while for the Newton-Cotes rule we obtain
(4.37)
(n.a
' NC)
Iln -_ -2 - 0
·
11m
. 735 . ...
n
e
n-too
4.9. Exit Criteria Based on Peano Kernel Methods. In numerical software
packages (see, e.g. Piessens et al. [32]) is often used a functional S of the form
m
(4.38)
S[/]:= Lbvl(Yv)
v=1
to estimate (not to bound!) the value R[J]:
R[/] ~ S[J].
(4.39)
If we want to get rid of the vagueness of ~ , we may ask for the validity of
IR[/li ::::; IS[/lI·
(4.40)
The computation of S[/] is mostly much simpler than the computation of bounds
of the types (3.3) or (3.4), therefore the discussion of (4.40) is of high practical
interest. Evidently an inequality like (4.40) can only hold for a restricted dass of
functions. Generalising the idea of Theorem 9 we define the dass G+ by
(4.41)
G+ := {I
I 1 E G[a, b] and I(v) has no change of sign} .
We obtain as an immediate consequence of the Peano representation, that (4.40)
is valid for 1 E G+ if S has a v-th Peano kernel Kv(S) and
(4.42)
holds on [a, b], where Kv(R) denotes the v-th Peano kernel of R. We are mostly
interested in functionals S satisfying (4.42) with small Kv(S). To be more precise
we call sopt an optimal (v, m) Peano stopping functional lor R if sopt minimises
J~1 Kv(S)(x) dx among all functionals oftype (4.38) (with fixed m and fixed nodes
Yl, ... ,Ym) satisfying (4.42). This theory is developed in Förster [15] and in Ehrich
and Förster [13], we can cite here only one example:
Example 4. We consider the remainder term R;;,c of the interpolatory quadrature formula Q;;,c of Clenshaw and Curtis, see (4.6). Let
v-I
{ Yv = - cos n _ 1 11',
v= 1, ... ,n}
be the set of the nodes of Q;;'c. Applying a definiteness criterion of Brass and
Schmeisser [9] we easily obtain for odd n = 2m + 1 and v = 2m the optimal
(2m, 2m + 1) Peano stopping functional S~:::+1 for Rf';;'+1'
(4.43)
2
S~:::+1[/] = 4m 2 -1 L
2m+1"
v=1
-
1
1(- cos v2m 11').
PEANO REPRESENTATION OF LINEAR FUNCTIONALS
201
In (4.43) the double prime indicates summation where the first and last terms are
halved.
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type, Computing 33 (1984), 363-366.
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Numer. Anal. 5 (1968), 783-804.
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8. H. Brass and R. Günttner, Eine Fehlerabschätzung zur Interpolation stetiger Funktionen,
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1975, pp. 353-374.
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Z. Angew. Math. Mech. 75 (1995), 625-628.
14. H. Fiedler, Das asymptotische Verhalten der Peanokerne einiger interpolatorischer Quadraturver/ahren, Numer. Math. 51 (1987), 571-581.
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Suppl. Rend. Circ. Mat. Palermo, Serie 11 33 (1993), 311-330.
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term in quadrature tor convex functions, Numer. Math. 57 (1990), 763-777.
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Studia Sci. Math. Hung 22 (1987), 287-297.
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Akademie Verlag, Berlin, 1995, pp. 141-150.
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26. K. Petras, Asymptotie behaviour 01 Peano kernels 01 fixed order, Numerical Integration III
(H. Brass and G. HäInmerlin, eds.), Birkhäuser Verlag, Basel, 1988, pp. 186-198.
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(1986), 225-227.
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ii
INEQUALITIES DUE TO T. S. NANJUNDIAH
P. S. BULLEN
Department 0/ Mathematics, University 0/ British Columbia, Vanccuver BC,
Canada V6T lZ2
Abstract. In this note we give Nanjundiah's proofs of his mixed geometric-arithmetic
mean inequalitiesj in particular his use of inverse means is explained.
1. Introduction
Recently two proofs have been given of a mixed geometrie-arithmetie mean inequality (see [2-3]). The authors seem to be unaware that this result, and even
more, was proved over forty years aga by Nanjundiah [4-5]. His basie result is
stated in [1, p. 121]. However, the inductive proof given there contains a glaring
error, as was pointed out to the present author by H. Alzer in a private communieation.
Professor Nanjundiah has never published his proof for reasons that will be explained later. His announcement of the result [5] that forms part of his Ph.D.
thesis, only states that the inequality is derived from a simple but brilliant idea
he had used in [4] to give elegant proofs of the classieal inequality between the
geometrie and arithmetie means. This simple idea is given in [1, pp. 67-96], but
without the emphasis it deserves.
The object of this paper is to present these results of Professor Nanjundiah, with
perhaps some slight generalisations here and there.
2. Notation
In this note we use the following notations and conventions:
a = (al,"')' b = (bi, ... ), W = (Wl,"') are sequences of positive numbers;
n
W n = L Wi, n = 1, ... , and W = (Wl , ... );
i=l
When necessary we will let ao = 1 and Wo = 0;
If c, d are two sequences and if A, J.L E IR, then
AC + J.Ld = (ACl + J.Ldl , ... ), cAdIJ. = (crdi,···),
C ,...., d
when 3 A :I 0 such that C = Ad,
d1
Cd -- (c1 , ... ).,
To say that c and d are similarly ordered is to say that for some simultaneous
permutation they are both increasing or decreasing.
1991 Mathematics Subject Classification. Primary 26D15, 26D20.
Key words and phrases. Geometric-arithmetic mean inequalitiesj Inverse meanSj Mixed mean
inequalitYi Carleman's inequalitYi Rado's inequalitYj Popoviciu's inequalitYj Hölder's inequality,
Cebisev's inequality, Sequence of the power means.
203
G. V. Milovanovic (ed.), Recent Progress in lnequalities, 203-211.
© 1998 Kluwer Academic Publishers.
P. S. BULLEN
204
3. Nanjundiah's Inverse Means
Using the above conventions,
n = 1, ... ,
are, respectively, the sequenees 0/ arithmetie and geometrie means 0/ a with weight
w. We will write
A(aj w) = A = (Ai(aj w), ... )
and
G(aj w) = G = (Gi(aj w), ... )
for these two sequences.
Regarding a ~ A, a ~ G as two maps of positive sequences into positive sequences Nanjundiah's ingenious idea was to define inverse mappings as follows.
-1
Wn
Wn - 1
(1) An (ajw) = -an - - - a n-1,
Wn
Wn
n = 1, ....
These will be called, respectively, the sequenees 0/ inverse arithmetie and geometrie
means 0/ a with weight w, and we will write A -1 (aj W) = A -1 = (All (aj w), ... )
and G-1(aj w) = G- 1 = (G l 1 (aj w), ... ).
These inverse means have same of the elementary properties of the original means.
(2)
A;;l(,xa + J.tbj w) = ,xA;;l(aj w) + J.tA;;l(bj w),
G;;l(a A bl'jw) = (G;;l(ajw))A (G;;l(ajw))I'.
In case of equal weights, W1
omittedj thus An (a), etc.
= W2 = ... , all reference to the weights will be
The following simple lemma is easily deduced and justifies the names of these
means of Nanjundiah:
Lemma 1. With the above notations
(3) A n (A- 1jw) = A;;l(Ajw) = Gn (G- 1jw) = G;;l(GjW) = an, n = 1 ....
Nanjundiah's idea then was to obtain inequalities between the classical means by
first proving simpler but analogous inequalities for these inverse means. Before
going to the main result of the paper, Nanjundiah's mixed mean inequality, we
give a ßavour of his method by looking at some more classical results.
4. Nanjundiah's Proofs of Some Classical Results
4.1. THE GEOMETRIC-ARITHMETIC MEAN INEQUALITY
This result is
(GA)
with equality only if a1 = ... = an.
To prove (GA) we first consider the analogous inequality for the inverse means
which is very easy to prove.
INEQUALITIES DUE TO T. S. NANJUNDIAH
205
Lemma 2. We have
(4)
with equality only if an-l
= an.
Proof. If we put a = Wn/w n then a > 1, a - 1 = Wn-t/w n , and if further we
put a = an, b = an-l (4) becomes
(5)
a Ot
aa - (a - l)b ,
bOt --1 >
with equality only if a = b. There are several ways to see this.
For instance if we put x = loga, y = 10gb, (5) becomes
exp(ax + (1 - a)y) > ax + (1 - a)y,
x#- y,
which is immediate from the strict convexity of the exponential function; the last
inequality just says that the extension of a chord to the graph lies below the graph.
Alternatively we can recognise (5) as a form of Bernouilli's inequality [1, p. 6]. D
Theorem 3. 1f n ~ 1, then (a)
with equality only if an = G n - 1 (a; w); and (b)
(P)
with equality only if an = A n- 1 (a; w).
Proof. The following follows by simple applications of Lemmas 1 and 2.
by (4) applied to the sequence A,
=an ,
-- G-1(G·w)
n
,
,
by (3),
by (3),
by (4) applied to the sequence G.
Prom the first and third lines we get that
1
1
Gn (A·, w) -> Gn (G·, w)
which is, on rewriting using (2), the inequality (R); from the first and fourth lines
we get that
which is the inequality (P).
The cases of equality are easy to deduce from those in Lemma 2.
D
206
P. S. BULLEN
The inequality (R) is known as Rado's inequality, while (P) is Popoviciu's inequality; both imply (G-A). For instance from the first,
~ W1 (A1(a;w) - C1(a;w»
= o.
In general an inequality can be interpreted as saying that a sequence is positive;
thus (G-A) says that A - G ~ O. The "Rado" extension sharpens this to saying
that the sequence W(A - G) is increasing, and so in particular A - Gis positive.
On the other hand (G-A) can be written AIG ~ 1 and the "Popoviciu" extension
sharpens this to saying that (AIG) W is increasing, which implies that AIG ~ 1.
It is a feature of Nanjundiah's method that for each inequality that he considers
it is a "Rado" or "Popoviciu" extension, or both, that is obtained. The inequality
is then obtained by an iteration similar to the one above.
4.2. HÖLDER'S INEQUALITY
The following inequality is sometimes referred to as Hölder's inequality
(H)
with equality only if a '" b (see [1, p. 171]).
Nanjundiah obtains this from the "Popoviciu" extension proved below; Theorem 5.
Lemma 4. 1/ n > 1 then
(6)
Praof. By (2)
G;:;-l(a; w) + G;:;-l(b; w) = C-1(al(a + b). w) + C- 1(bl(a + b)· w)
C;;l(a+b;w)
n
'n
,
~ A;:;-l(al(a + b); w) + A;:;-l (bl(a + b); w),
by (4),
= A;:;-l(al(a + b) + bl(a + b); w) = 1,
by (2).
The case of equality follows from that of Lemma 2.
Theorem 5. 1/ n > 1 then
0
INEQUALITIES DUE TO T. S. NANJUNDIAH
207
with equality only if anGn-1(bj W) = bnGn-1(aj W).
Proof. By (6) and (3)
G;;:l(G(ajw) + G(bjw)jw):::; G;;:l(G(ajw)jw) +G;;:l(G(bjw)jw)
=an+bn
= G;;:l(G(a + bjw)jw),
which gives the above inequality.
The case of equality follows from that of Lemma 4.
0
4.3. CEBISEV'S INEQUALITY
A companion inequality to (H) is that of Cebisev
(T)
provided that a and b are similarly ordered . Equality occurs only if al = ... = an
or b1 = ... = bn .
Nanjundiah proves (T) from its "Rado" sharpeningj Theorem 7 below.
Lemma 6. Ifn> 1 then if (an-I, an), (bn-1,b n) are similarly ordered
with equality only if an = an-l or bn = bn- 1.
Proof. This is an immediate consequence of the elementary computation
Theorem 7. If n > 1 and if a and b are monotone in the same sense then
(7)
W n (An(ajw)An(bjw) - An(abjw))
:::; W n- 1 (An-1(ajw)An-1(bjw) - An_1(abjw)),
with equality only if an = A n- 1(aj w) or bn = A n- 1(bj w).
Proof. Since a and b are monotone in the same sense so are A(aj w) and A(bj w).
So we can apply Lemma 6 using these sequences to get
A;;:l(A(aj w)A(bj w)j w) :::; A;;:l(A(aj w)jW)A;;:l (A(bj w); w)
= A;;:l(A(ab;w);w)
by (3);
which is just (7). The case of equality follows from that of Lemma 6.
0
208
P. S. BULLEN
5. Nanjundiah's Mixed Mean Inequality
As the basis of Nanjundiah's method of proof was to apply one mean to the
sequence of other means it was natural that he should try now to see what would
happen if he applied the arithmetie mean to the sequence of geometrie means, and
viee versa. To do this he again followed his technique of first doing this to the
inverse means.
To obtain the results of this section some restrietions have to be placed on the
sequences a, w.
Lemma 8. 11 n > 1 and il Wna n and Wn/w n are strictly increasing sequences
then
(8)
with equality only if a n-2 = an-l = an.
Proof. On writing out (8) we have to prove that
Wn-l
) w.. /w ..
Wn
( -an---an-l
(9)
Wn
Wn
Rewriting (9) with a simpler notation what we have to show is that
(10)
_('--r_a_-_('--r_----.:1)'--c.:....t. > r_a_r_ _ (r _ l)_c_q_
(qc - (q - 1)bt- 1 - cr - 1
bq-I'
subject to ra > (r - l)c, qc > (q - l)b. In addition it is dear that r > 1, q > 1,
and r > q.
Let us put
ß = (r - q)c + (q - l)b
r-1
when the left hand side of (10) becomes
(11)
(ra - (r - l)ct
(rc - (r - 1)ßt- 1
whieh by (6) is greater than or equal to
(12)
(rar
(rc)r-l
((r - l)ct
((r - 1)ßt- 1 '
INEQUALITIES DUE TO T. S. NANJUNDIAH
209
Now by (GA), ß ~ (cr-qb q- 1 )l/(r-l), or, equivalently
~ < (c)q.
ßr-l - bq-l
(13)
Collecting (13), (12) and (11) we have proved (10).
For equality in the use of (GA) we need that b = c, when ß = c. For equality
in the application of (6) we then need a = c. This completes the proof of the
lemma. 0
The reasons for the restrictions on a and w are clear from the proof: (i) the left
hand side of (8) needs Wna n to be strictly increasing; (ii) for ß to be an arithmetic
mean the weights must be positive; and the condition Wn/w n strictly increasing
ensures that r - q > O.
When Lemma 8 is applied to different sequences a we must check that the first
condition holds for that sequence. In particular, in the case of equal weights the
first condition reduces to na n being strictly increasing and the second is satisfied.
Theorem 9. If n > 1, and if Wn/w n is strictly increasing then
( Gn(A;W))Wn ~ (Gn_l(A;W))Wn_l
An(G;w)
An-1(G;w)
with equality only if an = Gn-da; w) = An-1(G; w).
Proof. Since WnAn(a; w) is strictly increasing we can apply Lemma 8 to A to get
using Lemma 1
A;;1(G-1(A;w);w) ::; G;;1(A-1(A;w);w) = G;;l(a;w) = A;;1(A(G-1;w);w).
In other words, as in the deduction of (P) in Theorem 1, Wn(An(G-l;w) G~l(A;w) is an increasing sequence. This implies that
(14)
In (14) replace a by G to get
G;;l(A(G; w); w) ::; An(a; w) = G;;l(G(A; w); w),
which completes the proof since the cases of equality follow from those of Lemma8. 0
Corollary 10. If n ~ 1 and if Wn/w n is strictly increasing then
(N)
with equality only if al = ... = an.
The case of equal weights, proved in [2-3] is a particular case of this corollary.
Theorem 9 is a "Popoviciu" extension of (N) and we can also prove a "Rado"
extension by a similar argument. All we have to do is to start the proof of Theorem
9 by applying Lemma 8 to the sequence G. For this however we need an extra
lemma since it is not immediate that WnGn(a; w) is strictly increasing.
P. S. BULLEN
210
Lemma 12. II Wna n is strictly increasing then so is WnGn(a;w) provided
Wn/w n is also strictly increasing.
Proof. Note that WnGn(a; w) = Gn(o:; w), where
if n> 1,
if n = 1.
(
ßn -- {
1 + ~) Wn-l/W ..
Wn - 1
if n> 1,
ifn=1.
1
By hypothesis Wna n is strictly increasing and so is ßn by the weIl known inequality
l)l/q
( 1+< ( 1+-l)l/P ,
P
q
O<q<p;
see [1, p. 8]; [incidentally the inequality is quoted the wrong way round in that
reference.] Hence an is strietly increasing and so therefore is WnGn(a;w). 0
Theorem 13. Iln> 1, and Wna n , Wn/w n are strictly increasing sequences, then
with equality only if n = A n- 1 (a; w) = G n - 1 (A; w).
In the case of equal weights Nanjundiah made a important deduction from (N).
For in that case we can write (N) as
n
~ Gn(a) ~ ~Gn(S),
where Gn(s) is the geometrie mean, with equal weights, of the sequence s =
(al. al + a2, al + a2 + a3, ... ). However, n/ V'nf < e and so
n
n
LGn(a) < eGn(s) < e Lai.
i=l
i=l
an inequality called Carleman's inequality; see [1, p. 116 and p. 273].
Now Carleman's inequality has been subject to various generalisations and it was
the hope of Nanjundiah that his methods would lead to proofs of these further
inequalities, and his delay in publishing has been attributed to his desire to bring
this idea to a successful conclusion.
INEQUALITIES DUE TO T. S. NANJUNDIAH
211
6. Further Results
As might be expected these ideas can be extended to other power means The
arguments being similar they will not be given here. If -00 < r < 00, r "# 0, then
is the rth power mean 0/ a with weight w, and the corresponding inverse Nanjundiah mean is
n 1 r ) l/r
(Mnlr])-l(.
a,w ) = (Wn anr _ W - an-l
Wn
Wn
Calling the sequence of rth power means MIr] we can extend (N) to:
Theorem 10. I/ n ~ 1 and i/ Wna n , Wn/w n are strictly increasing, then
MB (MIr]. w) > MIr] (MIBj. w)
n
,
-
n
"
r < s.
References
1. P. S. Bullen, D. S. Mitrinovic and P. M. Vasic, Means and Their Inequalities, Reidel Pub-
lishing Co., Dordrecht - Boston, 1988.
2. K. Kedlaya, Prool 01 a mixed arithmetie-mean geometrie-mean inequality, Amer. Math.
Monthly 101 (1954), 355-357.
3. T. Matsuda, An induetive prool 01 a mixed arithmetie-geometrie mean inequality, Amer.
Math. Monthly 102 (1955), 634-637.
4. T. S. Nanjundiah, Inequalities relating to arithmetie and geometrie means I, II, J. Mysore
Univ. Sect. B6 (1946), 63-77 and 107-113.
5. ___ , Sharpening some classieal inequalities, Math. Student 20 (1952), 24-25.
MARCINKIEWICZ-ZYGMUND INEQUALITIES:
METHODS AND RESULTS
D. S. LUBINSKY
Mathematics Department, Witwatersrand University, Wits 2050, South Africa
Abstract. The Gauss quadrature formula for a weight W 2 on the realline has the form
tAjnP(Xjn) =
j=l
!
PW 2
for polynomials P of degree :5 2n - 1. In studying eonvergenee of Lagrange interpolation
in L p norms, p =1= 2, one needs forward and eonverse quadrature sum estimates such as
AjnW - 2 (Xjn)IPWI P (xjn)
t
:5~ C
j=l
!
IPWI P
with C independent of n and P. These are often ealled Marcinkiewicz-Zygmund inequalities after their founders. We survey methods to prove these and the results that ean he
achieved using them. Our foeus is on weights on the whole real line, hut we also refer to
results for (-1,1) and the plane. In partieular, we present four methods to prove forward
estimates and two to prove eonverse ones.
1. Introduction
There is an intimate connection between Gauss quadrature sums and mean convergence of Lagrange interpolation - hardly surprising, when both involve zeros of
orthogonal polynomials.
Let da be a non-negative measure on IR, and {Pn(x)}~o be its orthonormal polynomials, so that
f
PnPm da
= tSmn ·
=
If da(x)
w(x)dx, and we need to indicate the dependence on w, we write
Pn(w, x), etc. Let
-00
< Xnn < Xn-l,n < ... < Xl n < 00
1991 Mathematics Subject Classification. Primary 41A55, 42C05j Secondary 65D04.
Key words and phrases. Quadrature sumSj Gauss quadraturej Marcinkiewicz-Zygmund inequalities.
213
G. V. Milovanovic (ed.), Recent Progress in lnequalities, 213-240.
@ 1998 Kluwer Academic Publishers.
D. S. LUBINSKY
214
denote the zeros of Pn (x) and let {Ajn} denote the Christoffel numbers. The Gauss
quadrat ure formula is
Here Pm denotes the polynomials of degree ~ m. Let Ln[f] E P n- 1 denote the
Lagrange interpolation polynomial to f at the zeros of Pn, so that
1 ~ j ~ n.
The connection between convergence of Lagrange interpolation and convergence
of Gauss quadrature is nowhere dearer than in the following result: Let f be in
the L 2 (da) dosure of the polynomials. Then
(1.1)
lim
n-too
JU - [J])2
Ln
da = 0
if and only if for every polynomial P
These relations are of course part of Shohat's extension to the infinite interval of
the dassic result of Erdös-Tunin on L 2 convergence of Lagrange interpolation.
For L p , P :f. 2, things are far more complicated and we need forward and converse
quadrature sum estimates, often called Marcinkiewiez-Zygmund inequalities. Zygmund's dassie treatise contains a particularly elegant proof of both forward and
converse estimates in the case of trigonometrie polynomials [48, Ch. X, pp. 28-29].
Let us illustrate the use of these in the context of weights of the form
where W(x) is a non-negative function, the archetype being
W(X) = Wß(x) := exp( -lxI ß ) ,
ß > 1.
At first sight, the use of W2 for the weight, rather than W, seems strange, but is
standard for weights on the whole realline: it simplifies formulation of results.
1.1. FORWARD QUADRATURE SUMS IN LAGRANGE INTERPOLATION
Let us assume that we have a forward quadrature sum estimate of the form
(1.2)
tAjnW - 2 (Xjn)</J(Xjn)IPWIP(Xjn) ~ c
j=1
J
IPWIP</J,
MARCINKIEWICZ-ZYGMUND INEQUALITIES: METHODS AND RESULTS
215
P E P n- 1 , where C i- C(n,P). Here ljJ is a slowly changing function. For our
purposes, we can take ljJ(x) := (1 + X2)-I/p .
The main part of proving mean convergence of Lagrange interpolation in a weighted
L p norm is showing uniform boundedness in n of the operator Ln. Let 1 < P < 00
and q := p/(P - 1). Then by duality
IILn[f]WIILp(IR) = s~p
1
L n[f]gW 2
where the sup is taken over all 9 with IIgWIILq(IR) = 1. To proceed, we use the
partial sums
Sn[g](x) =
n-l
2: Cjpj(x); Cj= Igpj W 2 (j=O,1, ... ,n-1).
j=O
of the orthonormal expansion of g. Since 9 - Sn[g] is orthogonal to Pn-l, we have
1
Ln [!]gW 2 =
1
L n[f]Sn[g]W 2 =
t
Ajnf(Xjn)Sn[g](Xjn)
3=1
by the Gauss quadrature formula. Let us now assume that
IfWI(x) ~ ljJ(x) = (1 + x2) -1/p ,
(1.3)
Then we obtain
11 Ln[!] gW2 1~
t,
~C
xE lR..
AjnW- 2(Xjn)ljJ(xjn)I Sn[g]WI(Xjn)
1
ISn[gllW ljJ,
if we use (1.2) with P = 1. Setting an := sign (Sn[g]) and then using the symmetry
property of the operator Sn, we can continue this as
=C
1
Sn[g](anljJW- 1 )W 2 = C
1
gSn[anljJW- 1 ]W 2
~ ClIgWIILq(IR)IISn[anljJW-I]WIILp(IR).
Assuming a suitable mean boundedness of the operator Sn from L p to L p with
suitable weights, we can continue this as
Thus we have shown that 'Vf satisfying (1.3),
D. S. LUBINSKY
216
Here C2 "I- C 2 (n, f). This and the reproducing property
Ln[P] =P,
PE 'Pn -1
and the density of polynomials give convergence of {Ln[J]}~l in weighted L p
norms.
We emphasise that this is just an illustration. The complete proofs are more
complicated and require breaking up the L p norm of Ln[J] into several different
piecesj the quadrature sum often includes Xjn only for those j satisfying IXjnl ~
(1 - c)X1n with fixed 0 < c < 1j and suitable factors are often inserted into the
weighted L p norms.
1.2. CONVERSE QUADRATURE SUMS IN LAGRANGE INTERPOLATION
Assurne that we have a converse quadrature sum inequality of the form
(1.4)
for P E 'Pn - 1 with C"I- C(n, P). Then
IILn[J]WIILp(R)
~ c{t, >-jn W-2(Xjn)lfWIP(xjn)} l/p
~ c{t >-jn W- (Xjn)(1 + X]n)-l }l/P
2
3=1
provided (1.3) holds. This last quadrature sum converges as n -+ 00 to
So again we have uniform boundedness in n for functions f satisfying (1.3), and
hence convergence.
Clearly converse quadrature sum estimates yield boundedness of {Ln} in a far
simpler way than forward ones. However as we shall see, they are usually more
difficult to prove and more restrictive in scope. There is also an almost incestuous
duality between forward and converse estimates, as we shall see.
Historieally, forward and converse quadrat ure sum estimates were first considered
by Marcinkiewiez and Zygmund in the 1930's [19-20]. As we have remarked, Zygmund's treatise contains a concise elegant treatment of both forward and converse
estimates for trigonometrie polynomials. Askey seems the first to have applied
these estimates in studying Lagrange interpolation for Jacobi weights in the 1970's
[1], and subsequently Nevai studied and applied these for the Hermite weight and
Jacobi weights [34], [36-37].
MARCINKIEWICZ-ZYGMUND INEQUALITIES: METHODS AND RESULTS
217
Indeed, it seems Nevai and his coIlaborators have been responsible for intensively
studying and developing these inequalities [33-37], [15]. The author and his students have concentrated on the case of weights on the whole real line [3-4], [8],
[16-18] while Y. Xu has considered generalised Jacobi weights [42-45]. A particularly interesting method has been developed by König [9-10] in the context of
Banach spaces, but yields new results even in the scalar case. Complex methods
such as Carleson measures and Hp space techniques have been developed by Zhong
and Zhu [47], see also PeIler [39].
In spirit, estimates for Lebesgue functions of LagrangejHermitejHermite-Fejer
interpolation are related to the quadrature sum estimates we consider here, but
we shaIl not discuss them. See for example [23-26], [38], [41].
This paper is organised as foIlows: In Section 2, we outline four methods to prove
forward quadrature sum estimates and discuss some of the results that can be
proved using them. In Section 3, we outline two methods to prove converse quadrature sum estimates and results that they yield. In Section 4, we present some
condusions, and some open problems.
As apreparation for subsequent sections, we present more notation, and we also
define dasses of weights on the realline. Throughout, C, Cl, C2 , •.. denote positive
constants independent of n, x and P E P n . The same symbol does not necessarily
denote the same constant in different occurrences. Gi yen real sequences {b n }, { cn }
we write
if there exist Cl, C2 such that
for the relevant range of n. Similar notation will be used for functions and sequences of functions.
Our weights on IR always have the form W 2 (x) = e- 2Q (x) where Q is even and
convex. Much as one distinguishes between entire functions of finite and infinite
order, one distinguishes between Q of polynomial growth at 00 (the so-caIled Freud
weights) and of faster than polynomial growth at 00 (the so-caIled Erdös weights).
We define first a suitable dass of the former:
°
Definition 1.1. Let W := e- Q , where Q : IR -+ IR is even, continuous in IR, Q" is
continuous in (0,00), Q' > in (0,00), and for some A,B > 1,
A< 1 + XQ"(X) < B
Q'(x) - .'
x
E (0
)
,00.
Then we write W E F.
The most important examples are W(x) = Wß(x) = exp( -lxI ß ), ß> 1.
D. S. LUBINSKY
218
Definition 1.2. Let W := e- Q , where Q : IR --+ IR is even, continuous in IR, Q"
is continuous in (0,00), Q", Q' > 0 in (0,00) and T(x) := 1 +xQ"(x)/Q'(x) is
increasing in (0,00) with
lim T(x) > 1;
x-+O+
lim T(x) = 00.
x-+oo
Moreover, assume that for some Cj > 0, j = 1,2,3,
and for every c > 0,
T(x) = O(Q(x)c),
x --+ 00.
Then we write W E c.
The most important examples are W(x) = exp( - eXPk(lxI ß )), where ß > 1, k ~ 1
and eXPk = exp(exp(· .. exp())) denotes the kth iterated exponential.
For both Freud and Erdös weights, the Mhaskar-Rahmanov-SafJ number an plays
an important role. It is the positive root of the equation
21
n =-
(1.5)
7r
0
1
dt
antQ'(ant) v'f=t2.
1 - t2
One of its important properties is
(1.6)
PEPn
and for 0 < P < 00,
(1.7)
PEPn
where C '" C(n, P) [27-28], [11), [13). Concerning its growth, we note that an is
increasing in n, and grows roughly like Q[-11(n), where Q[-ll is the inverse of Q
on (0,00). For those to whom it is new, a good example to think of is W = Wß'
Q(x) = IxI ß , for which an = Cn 1 / ß , n ~ 1.
In presenting the various methods, we shall use the following estimates that hold
for the dass :F of Freud weights (all of which can be found in [11), [2)). Define
(1.8)
and
(1.9)
Then
(1.10)
XO
n := X1n
(1 + n-
2/ 3 );
x n+1,n:= Xnn
(1 + n-
2/ 3) •
MARCINKIEWICZ-ZYGMUND INEQUALITIES: METHODS AND RESULTS
219
and uniformly in j, n
(1.11)
The Christoffel numbers {Ajn} are special cases of the Christoffel functions
(1.12)
which admit the estimate
(1.13)
The orthogonal polynomials {Pn (x)} ~=o for W 2 satisfy
(1.14)
2. Forward Quadrature Sum Estimates
In illustrating the four methods to prove forward quadrature sum estimates, we
shall assurne that W E :F, and that our weight is W 2 . We shall also often use the
estimates (1.10)-(1.14).
2.A NEVAI'S METHOD
This simple method requires an estimate like (1.11) and a suitable Markov-Bernstein inequality. The most influential papers (and possibly the first) papers in
which it was used were those of P. Nevai for Jacobi and Hermite weights [34], [3637]. Given u E [Xjn, Xj-l,n], we have from the fundamental theorem of calculus,
We can assurne that u is the point in [Xjn,Xj-l,n], where IPWIP attains its minimum. H we now use our estimate (1.11) for the Christoffel numbers, we obtain
AjnW- 2(Xjn)IPWIP(Xjn) :::; C 1~~-1'" IPWIP(u) du
+ C an 1/J;;1/2(Xjn)
n
l
x
;-I,n IPWIP-l(S)I(PW)'(s)1 ds.
x; ..
Summing over j, and using the fact that 1/Jn does not change much in [Xjn,Xj-l,n]
(see [11] if you want a proof), we obtain
(2.1)
t
j=l
Ajn W- 2(Xjn)IPWIP(Xjn) :::; C lIPWIP(u) du
R
+ Cl an f IPWIP-1(s)I(PW)'(s)I1/J;;1/2(S) ds.
n JIR
D. S. LUBINSKY
220
At this stage, we need a quite sophisticated Markov-Bernstein inequality of the
form
(2.2)
PEPn.
This was proved in [12] for the dass:F, using Carleson measures. Applying Hölder's
inequality with parameters q := p/(P - 1) and p to the second term in (2.1) and
then this Markov-Bernstein inequality give
~
LIPWIP-I(S)I(PW)'(s)I1/!~1/2(S)
ds
~ ~ IIPWII~:tlR)II(PW)'1/!~1/2I1Lp(lR) ~ ClIPWII~p(lR)·
So we have shown
The real bugbear of this method, at least for fuU quadrature sums, is the sophisticated Markov-Bernstein inequality (2.2). In his treatment of the Hermite weight,
Nevai used a somewhat weaker inequality, namely
(2.3)
PEPn·
Later authors [8-9] did likewise. Since for fixed 0 < c < 1, and IXjnl ~ (1 - clan,
the same arguments as above yield (assuming (2.3))
This last inequality is typicaUy enough for mean convergence of Lagrange interpolation. In fact, in it one may allow for fixed k ~ 1, P E Pkn, rather than just
PEPn.
The foUowing result is what D. Matjila and the author [17] could prove using this
method:
Theorem 2.1. Let W := e- Q E:F. (a) Let 1 ~ p < 00, r > 0 and -00 < b ~ 2.
Then
(2.4)
t
j=1
Ajn W-b(Xjn)IPWIP(Xjn) ~ C lIPWIP(t)W2-b(t) dt,
lR
MARCINKIEWICZ-ZYGMUND INEQUALITIES: METHODS AND RESULTS
221
for all P of degree at most n + rn 1/ 3.
(b) More generally, let ljJ : IR -t (0,00) be even, continuous with ljJ" continuous for
large x, and
.
xlljJ'(x) I
h~s~p ljJ(x)Q(x)1/3 < 00;
lim _1_~ (xep'(X))
x-too Q'(x) dx
ep(x)
= o.
Then for every P of degree at most n + rn 1 / 3
(2.5)
:t
AjnW-b(Xjn)ep(Xjn)IPWIP(Xjn) ~ C lIPWIP(t)ep(t)W2-b(t) dt.
R
j=l
For example, we could choose ep(x) := (1 + Ixl)b where b E IR, or ep(x) := exp(lxl a )
where a > 0 is small enough. The upper bound of n + rn 1 / 3 on the degree of P
is curious, but essential. If m = n + n 1 / 3 ,
-+ 00, one can choose P E Pm for
which (2.4) faHs with b = p = 2 as n -+ 00 (see [17]).
This method has also been used by König [9-10] in the context of Banach spaces
with the Hermite weight and Jacobi weights, where instead of scalar polynomials
P, one has polynomials P with vector values or values in a Banach space. The
inequalities take the form
en
en
where 11 . 11 is the norm of the Banach space in which P takes values.
2.B THE LARGE SIEVE METHOD
This method is in spirit dosely related to the large sieve of number theory, and
was already used by Marcinkiewicz and Zygmund [48, Ch. X, pp. 28-29]. Let us
illustrate this for Freud weights. Our starting point is the estimate (1.13) for the
Christoffel function, which gives
x E IR.
The definition of Am+! and the infinite-finite range inequality (1.7) lead to
PE Pm, xE 1Il
In order to deal with L p norms other than p = 2, we fix a large positive integer l,
and we replace P by pi and W by W'. Since the Mhaskar-Rahmanov-Saffnumber
of order ml for W ' is just the Mhaskar-Rahmanov-Saff number of order m for W,
we obtain
D. S. LUBINSKY
222
Hence if 0 < P < 2l,
II pw llt,(IR) :::; C : .
I::
IPWIP(t)IIPWII~'::(IR) dt
so for x E lR,
IPWIP(x) :::; IIPWlli oo (lR) :::; C : .
(2.6)
I::
IPWIP(t) dt.
It is now that the idea of the large sieve enters: It is largely Nevai and his coIlaborators that have been responsible for developing the method in this form [15], [22],
[35], [37]; Askey's [1] variant of this depends on having a suitable non-negative
kernel for the Jacobi weight to replace Kn(x, t). We need the reproducing kernel
Kn(x, t) for the Chebyshev weight
n-l
Kn(x, t) = ~ (1 + 2?: Tj (x)Tj (t) )
1=1
(as usual Tj (cos B) = cos(jB)). It is weIl known [35, p. 108] that
Kn(x,x) '" n,
xE [-1,1];
x, tE [-1,1].
IKn(x, t)1 :::; nmin { 1, nix1_ tl } ,
We now apply (2.6) to the polynomial P(t)Kn (x/a2kn,t/a2kn)k, where k and x
are fixed and P E Pkn' This polynomial has degree :::; 2kn in t, so we can apply
(2.6) with m = 2kn:
IPWIP(x) JKn
(_x_,~) JkP :::; 2knC j a 2kn IPWIP(t) JKn (~, _t_) JkPdt
a2kn a2kn
a2kn
a2kn a2kn
-a2kn
and hence if kp > 2, oUf bounds for K n give
IPWIP(x):::; Cl j a 2kn IPWIP(t)K~ (~, _t_) dt.
nan -a2kn
a2kn a2kn
This holds uniformly for lxi:::; a2kn with Cl 1= Cl (n, P, x). Choosing x = Xjn in
the above inequality gives
t
AjnW - 2(Xjn)1/!;/2(Xjn)IPWIP(xjn) :::; C j a
2kn
-a2kn
j=l
where
:::;C2: 1,-J---J _1}2
n
j=l
.
mm
{
1 Xjn
t
n a2kn
a2kn
IPWjP(t) Z)t) dt,
MARCINKIEWICZ-ZYGMUND INEQUALITIES: METHODS AND RESULTS
223
by our bounds (1.11) on the Christoffel numbers Ajn and the bounds on K n. Since
uniformly in j and n, the zeros of Pn(W 2 , x) satisfy
an
Xj-1,n - Xjn ~ C n
it is not difficult to estimate the sum by an integral, so that
1
~)t) ~ Cl ~
[ min{l).l~ _ _
t 1- }2 dx
an JR
n a2kn
a2kn
Thus we have for P E Pkn,
(2.7)
t, AjnW-2(Xjn)tP~/2(Xjn)IPWIP(Xjn) ~ L
C
IPWIP(t) dt
with C =F C(n, P). As tPn(t) ~ C(c), Itl ~ (1 - c)a n, we also deduce that
(2.8)
L
I"'jn 1~(l-e)an
AjnW - 2(Xjn)IPWIP(xjn) ~ C
L
IPWIP(t) dt.
One of the powerful features of the estimates (2.7) and (2.8) is the freedom to
allow polynomials of degree kn and not just n. Thus if ljJ : IR -+ (0,00) is such
that one can find polynomials Rn of degree O(n) such that
(2.9)
and ljJ±l is "smali" relative to W- 1, one can insert ljJ in (2.7) to obtain
with C =F C(P,n). In particular ljJ(x) := (1 + Ixl)b, bE IR will do.
The first time the full quadrature sums (2.7) or (2.10) have been considered, with
the factor tP~j2 in the left, is in the recent Ph.D thesis of Haewon Joung [7],
a student of Nevai. There not just ordinary polynomials, but generalised nonnegative polynomials P were considered. These have the form
P(x) = lei
where c, Zj E C, and
m
II Ix - zjlr
j=l
m
d
= deg(P) = L Tj.
j=l
j
D. S. LUBINSKY
224
Of course in the polynomial case, all Tj are positive integers. Joung's estimate
depends on first finding estimates for Christoffel nmctions that involve generalised
non-negative polynomials.
The "large sieve" method has many advantages over the method that we called
Nevai's method. It works for all p > 0, not just p :2: 1; it requires only estimates
on spacing of Christoffel functions and spacing of zeros, not the deeper Bernstein
inequality. Nevertheless, it does not seem to be able to yield the full quadrature
suro estimates (2.4), (2.5) in Theorem 2.1; the latter do not involve the factor
'IjJ~,f2 .
We note that in both the large sieve method and Nevai's method, we are not really
using intrinsic properties ofthe zeros {Xjn}, only estimates on their spacing. Thus
if
(Vj, n) tj+l,n - tj,n :2: C an;
n
then the same methods yield
We remark that both (2.7), (2.8) and (2.10) hold for W E :F and P E 'Pkn and
more generally probably for generalised non-negative polynomials of degree ~ kn,
via Joung's method of proof. For W E E, the function 'ljJn has to be replaced by
another more complicated function in (2.7) and (2.10); (2.8) is still true, but is
not sufficient for mean convergence of Lagrange interpolation. Damelin and the
author [3] found it necessary to prove (using the large sieve method) that given
0< TI < 1,
L
AjnW - 2 (Xjn)(p(xjn)IPWIP(xjn) ~ C
I"'in 1~(l-e)an
L
IPWIP(t)I/>(t) dt,
where I/> is any function for which (2.9) is possible. This is sharper than (2.8),
since for Erdös weights,
a1jn/an --t 1,' n --t 00.
The reader may find further applications and developments of this method in [15],
[30-31], [37].
2.C THE DUALITY METHOD
This method is based on applying duality to a suitable converse quadrat ure sum
estimate, and is in a way indicative of the almost incestuous relationship between
forward and converse estimates. It was apparently first used by König [9-10].
Let n be fixed and let f.Ln be the discrete (pure jump) measure having mass
AjnW- 2 (Xjn) at Xjn. Then
MARCINKIEWICZ-ZYGMUND INEQUALITIES: METHODS AND RESULTS
225
where the SUp is taken over all 9 with
Here of course q = p / (p - 1). Since 9 needs to be defined only at the n points
{Xjn}j=l' we can assurne that 9 E Pn-l· So
Now we make our major assumption: There is a converse quadrature sum estimate
of the form
SE P n - 1 •
(2.11)
Then
Thus we obtain
The attraction of this method is that it comes "for free". After spending a lot
of effort proving a converse quadrature sum inequality involving the L q norm,
we immediately obtain a forward quadrature sum estimate for the dual L p norm,
and one that holds for full quadrature sums. The disadvantage of this method is
that usually the range of q for which we can prove (2.11) is quite restricted. For
example, in König's work, he showed that (2.11) is true for the Hermite weight
only for 1 < q < 4, and so one deduces the forward estimate for 4/3 < P < 00,
whereas it should hold in some form for all 1 ~ p < 00. Another disadvantage is
that it works only for P E Pn-l.
2.D COMPLEX METHODS AND CARLESON MEASURES
Complex methods have been used primarily by Zhong and Zhu [47] for forward
and converse quadrature sum estimates in the plane. A principal ingredient are
Carleson measures. The latter also underlie the Markov-Bernstein inequality (2.2).
Recall that a Carleson measure is a positive measure da on the upper half plane,
that satisfies
(2.12)
a
([a- ~h,a+ ~h] x [O,h]) ~ eh
D. S. LUBINSKY
226
for all a E IR, h> O. Thus the a-measure of any square S in the upper-half plane
with base on the real line should be bounded by a constant times the side of S.
The smallest C in (2.12) is called the Carleson norm N(a) of a.
The point ab out Carleson measures is the following: Let 0 < P < 00, and HP be
the Hardy space of the upper-half plane, that is, the set of all functions f analytic
in the upper-half plane with boundary values f(x) satisfying
Then
(2.13)
Thus Carleson measures can be used to pass from the upper-half plane back to the
realline. To illustrate how this idea can be used in the context of Freud weights;
we follow dosely the proofs given in [12] for (2.2).
Our first step is to pass from an estimate for IPWIP(x), x E IR, to one over an arc
in the upper-half plane, via Cauchy's integral formula. The problem is that W is
not analytic! So for a given x, define
Hz(z) := e-[Q(z)+Q'(z)(z-z)].
Let us assume P has real coefficients. Cauchy's integral formula and the reflection
principle give
1111" IPHzl(x + ceie) dO.
IPWI(x) ~ -
7r
0
If we set W(z) := W(lzl) and choose
c := cn(x) := ~ 'IjJ;;,1/2(x),
it can be shown [12, Lemma 2.1, p. 234] that for lxi ~ X1n,
with C =I C(n, x, P). Hölder's inequality gives for p ~ 1,
We deduce that
n
L AjnW - 2(Xjn)IPWIP(Xjn)
j=1
MARCINKIEWICZ-ZYGMUND INEQUALITIES: METHODS AND RESULTS
227
We see that the measure an is supported on the union of semicireular ares, eentred
on the points {Xjn}. If we ean show that the Carleson norms N(an ) of an satisfy
(2.14)
and if PW belongs to the Hardy spaee of the upper half plane, we eould use (2.13)
to deduee that
As W is not in general analytic, we have to use a function Gn(z) that is, in
essenee, derived from solving the Diriehlet problem for the domain C\[-a n , an],
with suitable boundary values on [-an, an]. It was used by Mhaskar and Saff in
proving (1.6) and in a different form by Rahmanov [27-28], [40]. The properties
of G n that we need are that G n is analytic in C \ [-an, an] with a simple zero at
00 and for x E IR, [12, pp. 234-235],
Then
f
IPWIP dan:::; C
f
IPGnlP dan
:::; CN(an )
L
IPGnIP(x) dx
= CN(an )
{jan
:::; CN(an)
l:nn IPWIP dx.
-an
IPWIP dx +
r
IPGnlP dX}
JJR\[-an,an]
In this last step, one uses a representation of PGn as a Hilbert transform of a
function supported on [-an, an], and boundedness of the Hilbert transform from
L p to L p , p > 1.
What about (2.14)? Our estimate (1.11) for the Christoffel numbers gives
f
IPWI P dan:::; C t(Xj-l,n - Xjn)
=: C
1r
IPWIP (Xjn + en(xjn)e ilJ ) d8
r l IPWIP (x + en(x)e
JXnn
j=l
:::; C
l
0
1r
1n
f
0
IPWIPdUn.
ilJ )
d8 dx
228
D. S. LUBINSKY
Of course the second last step requires proof, but is intuitively reasonable. In [12,
Lemma 2.4], it is shown that
and the same proof shows that (2.14) holds.
This method of proof is attractive, but as already remarked, it involves essentially
the same tools as to prove the Markov-Bernstein inequality (2.2).
Perhaps the only published paper where this method has been used to prove quadrature sum estimates is that of Zhong and Zhu [47]. They proved:
Theorem 2.2. Let r be a C2+ö smooth simple are in C, that is r = {-y(t) :
t E [a, b]} where ,../' satisfies a Lipschitz condition 01 order /) > o. There exist
{Zk,n}~~J er, n ~ 1, such that lor 1 < P < 00 and PE Pn- 1 ,
Here zn,n := ZO,n.
Essentially the authors use a conformal map W of the exterior of the unit ball onto
C\r, and form the Fejer points
O~k~n-l.
As some ofthese may be too elose, they modify these to obtain {Zk,n}~~J. Instead
of estimating P(Zk,n) in terms of values of P on a semi-cirele centre Zk,n, the
authors estimate P(Zk,n) in terms of values of P on the "level curve"
r n := { W( (1 + ~) eit ) : t E [0, 27r] }
which encireles r. A suitable Carleson measure is formed, and moreover it is shown
that for all 1 in a suitable Smirnov space of functions analytic inside r n,
C =f. C(n, f).
Other ingredients are Lagrange interpolation and careful estimation of the spacing
IZk+1,n - zk,nl, and of
rr
n-l
nn(z) :=
(z - Zk,n).
k=O
A related paper of Zhong and Shen is [46]. Unfortunately this paper is not available
in South Africa, and the British Lending Library could not provide a copy to the
author. So the reader should please take note that [46] is exeluded from this
survey.
MARCINKIEWICZ-ZYGMUND INEQUALITIES: METHODS AND RESULTS
229
3. Converse Quadrature Sum Estimates
We shall present two methods for these, illustrated in the case of Freud weights.
3.A THE DUALITY METHOD
This method already appears, in the setting of trigonometrie polynomials, in the
treatise of Zygmund [48, Ch. X, pp. 28-29]. It is based on duality and "deep"
results on mean boundedness of orthogonal expansions. Let 1 ~ P < 00 and
q = p/(P - 1). Let P E Pn-1. We have
IIPWIILp(lR) = s~p
J
gPW 2,
where the sup is taken over all 9 with IIgWIILq(lR) = 1. By orthogonality of g-Sn[g]
to P n -1, and then by the Gauss quadrature formula,
!
gPW 2 =
J
Sn [g]PW 2 = tAjn(PSn[g])(Xjn)
{t,
{t,
3=1
~
Ajn W- 2(Xjn)IPWIP(Xjn)
x
f/P
x
AjnW-2(Xjn)ISn[g]WIQ(xjn)} l/Q
=: Tl x T2 •
Let us suppose that we have a suitable forward quadrature sum estimate like (2.4)
and that the partial sum operators {Sn} are bounded uniformly in n in a suitable
weighted setting. Then
T2 ~ CtllSn[g]WIILq(R) ~ C2I1gW IILq(R) = C2.
So we have shown that
IIPWIILp(lR)
~ c{~ AjnW-2(Xjn)IPWIP(Xjn)} l/P.
This duality method is elegant but it depends on having a forward quadrature sum
estimate, and, much deeper, results on mean boundedness of orthogonal expansions. It is the difficulty of proving the latter that severely restriets this method.
Chiefly it is a tool to pass from results on mean convergence of orthogonal expansions to corresponding results for Lagrange interpolation.
Typieally, the mean boundedness required above is valid only for 4/3 < q < 4;
to ensure its validity for other values of q, one needs to insert suitable powers of
1 + lxi as weights on Sn[g] and/or g. Moreover, in proving even these, one needs
bounds on functions of the second kind or on PnH - Pn-1. For Jacobi weights,
the requisite bounds are classieal, but these bounds are not generally available in
the setting of Freud weights. This explains the severe restrietions of the following
result [14]:
D. S. LUBINSKY
230
Theorem 3.1. Let W(x) := exp(-lxIß), ß = 2,4,6, .... Let 4 < P < 00 and
r, R E IR. satisfy
1
R> - - j
(3.1)
P
and
(3.2)
r - min
1 - l/p,
{R, 1_!} + ~6 (1 _~) {<~ 0,0, ifif RR '" 1-I/p.
P
P
=
Then for PE P n - 1,
II(PW)(x)(1 + IxlrIlLp(R)
(3.3)
n
~ C { ~AjnW-2(xjn)IPWIP(xjn)(1 + IXjnl)Rp
}l/P .
For p = 4, (3.3) holds if (3.1) holds and
r - min {R, 1- l/p} < 0.
(3.4)
The conditions on r, R are disconcerting, but it was shown in [14] that (3.2) is
necessary for (3.3). (It is not clear if (3.1) is also necessary). ln particular, (3.2)
requires r < R, so that for p > 4, we can never have r = R in (3.3). However,
there are always r,R that satisfy (3.2), (3.1), one just needs to choose r small
enough. More generally, we proved:
Theorem 3.2. Let W E F, with the additional condition that the orthonormal
polynomials {Pn} for W 2 satisfy
Then if (3.1) holds and
(3.6)
ar - min{R,1-1/p}n(1-4/p)/6 = { 0(1),
n
O((1ogn)-R) ,
R", 1 - l/p,
R = 1 -1/p,
we have (3.3). For p = 4, if (3.1) and (3.4) hold, then we have (3.3).
It was also shown in [14] that a slightly weaker form of (3.6) is necessary for the
converse estimate. In both the above results, we restricted ourselves to p ~ 4j For
p < 4, the next method will give better results.
Since the theory of mean convergence of orthogonal expansions for weights on
( -1, 1) is far more developed than that for weights on IR., it is hardly surprising
that more impressive results can be achieved by this duality method for weights
on (-1,1). Here is a result of Yuan Xu [45, p. 82] for generalised Jacobi weights
MARCINKIEWICZ-ZYGMUND INEQUALITIES: METHODS AND RESULTS
231
that extends earlier results of Nevai. Recall that a generalised Jacobi weight has
the form
M
w(x) =
II Ix - tjlßi,
xE (-1,1),
j=l
where
-1 = tl < t2 < ... < tM = 1;
ßj E lR,
1:::; j :::; M.
(It is possible that some ßj < -1). We call w a generalised Jacobi distribution
if it is integrable and has the form 'l/Jw, where w is a generalised Jacobi weight,
and'l/J is a continuous function on [-1,1] with 'l/J-1 bounded on [-1,1], and whose
modulus of continuity w satisfies
r w(t) dt <
1
Jo
Corresponding to
00.
t
w, we define
Theorem 3.3. Let u, v be generalised Jacobi distributions on (-1,1) and w be a
generalised Jacobi weight on (-1,1). Let 1 < P < 00 and q := p/(p - 1). Then iJ
{Xjn}, {Ajn} are the Gauss points and weights Jor u,
PE Pn-1
provided the Jollowing Jour conditions hold:
w 1- q u E L 1[-1, 1];
w 1- Q(x)u(x) (u(xh!L~) -Q/2 E L 1[-1, 1];
wu ~ Cv in (-1,1);
v(x) (u(x)~) -p/2 E Ltl-1, 1].
Xu's paper also contains a converse quadrature sum inequality that involves not
just the values of P, but also of its derivatives [45, p. 83]. These are useful in
studying mean convergence of Hermite interpolation.
3.B KÖNIG'S METHOD
König's method is based on Lagrange interpolation, a clever estimate for Hilbert
transforms of characteristic functions of intervals, and bounds on the norms of
linear operators derived via Hölder's inequality. It is technically the most difficult
D. S. LUBINSKY
232
amongst those we have presented, but is extremely powerful, and relatively direct
- it does not depend on deep results such as mean convergence of orthogonal
expansions or Markov-Bernstein inequalities.
We shall need some extra notation. Let
1 ~ j ~ n;
(recall XO n = Xln (1 + n- 2/ 3 )) and the characteristic function of I jn is denoted by
Xjn(X) := Xljn (x).
The fundamental polynomials of Lagrange interpolation are
Pn(x)
l . ( ) ._
Jn X . - PIn(Xjn
)(
X -Xjn ) ,
1 ~ j ~ n;
The Hilbert transform H[J] of J E Li (IR) is
H[J](x):= lim
",-+0+
1
J(t) dt
a.e. x E IR.
lx-tl:::::'" t - x
We write, for fixed PE Pn-l,
.
._
-1/2 P(Xjn)
YJn .- an
I (
Pn Xjn
)
so that
P(x) = Ln[P](x) = a;;2 pn (x)
(3.7)
-_ an1/2Pn(X) ~
~Yjn
j=l
n
L
Yjn
j=l X - Xjn
{I
1Jn IH [Xjn ](x) }
_ . - -11.
X XJn
n
+ a;;2pn (x) ~ I~;:I H[Xjn](X) =: J 1 (x) + J 2(x).
We first deal with the easier term J 2 . We shall use the bound (1.14) in the form
x E IR,
and also the following bound on the Hilbert transform: For all functions 9 with
support in [-2a n , 2a n ],
(3.8)
I 11 -1/411
< C 11 g(x) 11 - ~
I 11-1/411
11 H[g] (x) 11- ~
an
L (lxl9 an) an
L (lxl9 an)
p
p
MARCINKIEWICZ-ZYGMUND INEQUALITIES: METHODS AND RESULTS
233
with C t- C(n, g) provided 1 < P < 4. This bound is a variant ofM. Riesz' theorem
that the Hilbert transform is a bounded operator from L p to L p , 1 < P < 00. For
the case an = V2ri" this lemma already appears in a 1970 paper of Muckenhoupt
[32]. It was in this paper that the modern study of weighted inequalities for the
Hilbert transform began, leading ultimately to Muckenhoupt's A p condition. Then
It can be shown that uniformly in j, n,
(3.9)
and hence
(3.10)
Hence we deduce that
The estimation of J 1 (defined by (3.7)) is more difficult. If we set
-1/4() {
f Jn. (x )..- 'l/Jn
x x
-lII '
Ix - xjnl ~ 2l I jnl,
I jn
11· I
Ix ~~jnl {Ix-~;nl + HI;;nl}' Ix - xjnl > 2l I jnl,
then it can be shown that uniformly in j, n and x E [x nn , X1n],
1x
1 I H[Xjn ](x) 1 an1 / 2 IPn W I(x) ~ CfJn(x)
_1x . - -11.
Jn
Jn
[14, p. 542, Lemma 5.2] so that
n
IJ1WI(x) ~ CL IYjnlfJn(x),
j=l
D. S. LUBINSKY
234
Then as each !in does not change much in each Ikn,
Taking account of the form of !in, we see that
(3.12)
where
Exactly as for h, we deduce that
If we use
t/J;;1/4(Xjn) '"
[~ IIjnl] 1/2
and our estimate (3.10) for Yjn, then we obtain
SI ::;
c{~ [~bkjIIjnll/PIPWI(Xjn)] } I/P,
where bkk = blk = 0 for all k, and otherwise
bkj := IIknll/P+1/2IIjnI3/2-1/P(Xkn - Xjn)-2.
Defining the n x n matrix
B := (bkj );:,j=1
(note the reversed order of our indices), we obtain
MARCINKIEWICZ-ZYGMUND INEQUALITIES: METHODS AND RESULTS
235
If we can show that
(3.13)
then we obtain the desired estimate for S1. Similarly
where D := (dkj)~,j=1 and
(Vj,k)
If we can show that
(3.14)
we obtain the desired estimate for S1. Then (3.12) and (3.11) yield
We proceed to prove (3.13) and (3.14). König's method to bound these depends
on the following:
Proposition. Let (0, J-L) be a measure space, and s, r : 0 x 0 -t IR. Define the
operator
Tk[J](X) :=
10 s(x, y)f(y) dp,(y).
Let M > 0, p, q > 1 with l/p + l/q = 1. Assume that
sup ( Is(x,y)llr(x,y)lq dJ-L(Y) ~ M
x
10
and
sup ( Is(x,y)llr(x,y)I-P dp,(x) ~ M.
Y
10
This proposition is easily proved using Hölder's inequality - see [4], [9]. To prove
(3.13) one chooses 0 := {I, 2, ... ,n}, J-L( {j}) = 1, 1 ~ j ~ n, and
s(k,j) := bkj;
r(k,j) := (IIjnl 1 + IX k nl) 1/pq
IIknl 1 + IXjnl
236
D. S. LUBINSKY
One can show that [14]
n
sup
k
L Is(k,j)llr(k,jW ~ M
j=l
n
and
s~p
3
L Is(k,j)llr(k,j)I-P ~ M,
k=l
with M =I M (n). The actual proof of these involves re-expressing certain sums in
terms of integrals and then careful estimation of the integrals. Similarly to prove
(3.14), one chooses the same n, f..t and chooses
s(k,j) := dkj;
r(k J.) .= (IIjn I 1 + IXknl) l/pq (1 + Ix. I)l/q nJ.l/4q (x. )nJ.l/4 q (x )
, . IIknl 1 + IXjnl
3n
'f/n
3n 'f/n
kn·
This is what we could prove using König's method:
Theorem 3.4. Let 1 < P < 4 and W E:F. Let
1
R> --.
(3.16)
p
Then
In [14] it is also shown that the first two conditions in (3.16) are necessary. For
Erdös weights, we proved [4, Thm. 3.1]:
Theorem 3.5. Let 1< p < 4 and W E c. Then (3.17) holds with r = R = O.
For Jacobi weights, König and Nielsen [10] proved the following elegant theorem
using this method. In fact they worked in the more general setting of Banach
spaces that admit a Hilbert transform bound. For simplicity, we quote this result
for the case of polynomials:
Theorem 3.6. Let u(x) = (1- x)Q(l + x)ß, a,ß > -1 be a Jacobi weight and
{Ajn}, {Xjn} be the corresponding Gauss weights and points. Let
4(a + 1) 4(ß + 1) }
f..t(a,ß) := max { 1, 2a + 5 ' 2ß + 5
;
4(a + 1) 4(ß + 1) }
m(a,ß) := max { 1, 2a + 3 ' 2ß + 3
;
and
m(a,ß)
M(a,ß) := m(a,ß) -1 .
MARCINKIEWICZ-ZYGMUND INEQUALITIES: METHODS AND RESULTS
The /ollowing are equivalent:
(i) Forn ~ 1 and P E P n ,
(l,
IP( t) I'u(t) dt) 1/, ,; C
[t ~;n
!PI'(X;n)
237
r'·
(ii) JL(a,ß) < p < M(a,ß)·
In comparing to Xu's Theorem 3.3, we note that in the "unweighted case" of that
theorem, for which there v = u; w = 1, and u is a Jacobi weight, König's result is
more extensive.
3.C COMPLEX METHODS
In Section 2D, we discussed the results of Zhong and Zhu for forward quadrature
sum estimates. The same framework of ideas of Carleson measures and Smirnov
spaces, contour integral error formulas for Lagrange interpolation, and conformal
maps, enabled them to prove:
Theorem 3.7. Let 1< p < 00. Under the hypotheses 0/ Theorem 2.2, the points
{Zk,n}~';;;;5 there satisfy
PE Pn-l.
It is notable that all the methods we have presented for converse quadrature sum
estimates work only for p > 1. Using operator theoretic methods, and complex
ones, Peller [39, p. 480] proved a converse quadrature sum for (4n - l)st roots of
unity that works even for p < 1, but involving polynomials of degree at most n-l:
Theorem 3.8. Let 0< p < 00. For polynomials P 0/ degree :::; n -1,
It seems likely that the same methods should allow one to replace 4 by 1 + c.
4. Conclusions
We have seen four methods for proving forward quadrat ure sum estimates. For
purposes of weighted approximation, I believe that the "Iarge sieve" method is
the most versatile, and the most generally applicable, yielding adequate results
most generally. However when full quadrature sums need to be estimated, without
damping factors, Nevai's method is the most appropriate. The duality and complex
methods seem to yield less in weighted approximation, though are powerful in some
circumstances.
For converse quadrature sum estimates, I believe that König's method is the most
direct and powerful, though at present it works only for p < 4. The method based
238
D. S. LUBINSKY
on duality and mean convergence of orthogonal expansions is elegant but the mean
convergence results required are very deep. Perhaps chiefl.y it can be used to pass
from mean convergence of orthogonal expansions to mean convergence of Lagrange
interpolation.
There are several worthwhile open problems:
1° Make König's method for converse estimates work for all 1 :::; p < 00. One of
the main sticking points seems to be to extend the Hilbert transform inequality
(3.8) in some form to p ~ 4, by inserting suitable damping factors on both sides.
2° There seems to be little on converse quadrature sum estimates in L p for p :::; 1.
As far as the author could determine, Peller's Theorem 3.7 is about the only one.
Surely 4n there can be replaced by n? And what about weights on (-1,1) or IR?
3° There are gaps between the necessary and sufficient conditions for converse
quadrature sum estimates in [14]. The gaps arise because the necessary conditions
are derived from results on mean convergence of Lagrange interpolation, while the
sufficient ones are derived via König's and the duality method. Close these gaps!
4° Yuan Xu's extensive result for generalised Jacobi weights Theorem 3.3 involves
sufficient conditions. Find the necessary and sufficient ones, thereby extending the
scope of König's Theorem 3.6. Most probably, König's methods will have to be
used.
5° Explore the implications of Peller's methods for converse quadrature sum estimates.
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MARCINKIEWICZ-ZYGMUND INEQUALITIES: METHODS AND RESULTS
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240
D. S. LUBINSKY
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Rassias et al., eds.) (to appear).
SHAPIRO'S INEQUALITY
A. M. FINK
Iowa State University, Ames, IA 50011, U.S.A.
Abstract. Shapiro proposed an inequality in the 1954 American Mathematical Monthly,
which now goes by his name. The inequality is now settled but work on the subject
continues. The discussion of the history of this inequality does not always give a clear
picture of the chronology and the results. In addition, the work still to be done is not
always made dear. Here, we try to separate the various aspects of the problem: give the
chronology and priorities; give a hint on the role of computer proofs; give the shortest
route to the results; indicate the methods used in the shortest route; conjecture about
the continued interest in the problem; and indicate where furt her work can be done.
1. Introduction
Harold Shapiro raised the following question in the 1954 American Mathematical
Monthly (Problem 4603.): Given Xi 2: 0, i = 1,2, ... ,n. Establish
__
X_I_
X2
+ X3
+
X2
X3
+ X4
+ ... + _X.:..:.n_-_l_ + X n > ~
X n + Xl
Xl + X2 - 2 '
equality occurring only if all denominators are equal.
The case with n = 3 was earlier proposed by A. M. Nesbitt in 1903. Earlier
vers ions may exist but none have been confirmed.
The problem is settled. We exclude the cases n = 1 and n = 2 which are identities.
The inequality is correct for odd integers less than or equal to 23 and for even
integers less than or equal to 12. For all other n the proposed inequality is false.
It is nearly true in apreeise sense to be given later.
Then why write about the inequality? We claim that the history is interesting
and that eertain problems remain. The plan of the paper is to eomment on the
methods of proof in the next seetion and make eonjectures about the allure of the
problem. Then sueeeeding seetions will give a ehronological history for eaeh n. We
then give achart which will give the shortest route to a solution to the problem
for all n. More comments on ideas follow and then achallenge or two for the
interested reader.
1991 Mathematics Subject Classification. Primary 26-03, 26D20i Secondary 01A80.
Key words and phrases. Cyclic inequalitYi History.
241
G.v. Milovanovic (ed.), Recent Progress in Inequalities, 241-248.
© 1998 Kluwer Academic Publishers.
242
A. M. FINK
2. The Allure of the Problem
The problem seems to be a very special inequality and it is surprising that many
mathematicians have spent time on it, some apparently a very great deal of time.
No known connection with any other problem has ever been given. I conjecture
that part of the allure is the simple lower bound n/2. If it had been some other
complicated number, it probably would not have attracted the same amount of
interest. Secondly, it was shown soon after the problem was posed, that the
inequality was not true for all n. It seems to me that this adds to the intrigue.
Why should such a lower bound be true for some n and not for others? The
counterexamples showed that even in the cases where the inequality is not true,
it is barely false. When a Mathematician of the stature of Mordell is interested
in the problem, it also adds to the intrigue of the problem. This is especially so
since Mordell conjectured that the inequality is false for n ~ 7.
Finally, a lot of the work was a result of the use of modern high speed computers.
The interplay between "computer proofs" and analytic (other authors use 'algebraic' here) proofs adds to the mystique of the problem. Using the computer to
find counterexamples is one level of computer use. To reduce the problem to a
numerical minimization of a function of one variable is a slightly deeper use of the
computer and lends itself to a high degree of confidence in the result. A numerical
minimization of a function of more than one variable is sometimes viewed as a
different use of the computer because convergence is rarely capable of proof.
3. Chronology
In order to have a precise way to talk about the problem, we formulate the problem
in a slightly different way. Let
where {Xi} is extended to the positive integers by making it periodic of period n.
Let
J.L(n) = inf.! f(XI, ... ,Xn ),
n
where the inf is taken over Xi ~ 0 and no denominator = O. Note that since
f(l, ... ,1) = n/2,J.L(n) ~ 1/2.
What we call the conjecture is that J.L(n) = 1/2.
We will give two charts. First achart with the priorities for each n ~ 3. The second
one gives only those events which changed the status of the conjecture. This chart
also gives the running status of the conjecture. Outside these chronologies there
are several papers that must be separately cited as especially important in the
development of proofs. First is Lighthill's counterexample (1956) (n = 20) cited
in [31] which first showed that the inequality did not always hold.
The second is Zulauf's result of 1958, ([41]), who showed that if the conjecture is
false for some n, then it is also false for n + 2, n + 4, . .. so that he showed the
SHAPIRO'S INEQUALITY
n
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
> 25
243
Contributions in chronological order
Nesbitt, 1903j MordelI, 1958j Diananda published three proofs in
1977 which are reproduced in Mitrinovic, Pecaric, and Fink [20]
Shapiro, 1954 (unpublished)j proof by MordelI, 1958.
Phelps, 1956 (not published)j MordelI, 1958.
MordelI, 1958j Diananda, 1959j direct proof Freidkin, 1990**.
K. Goldberg numerical evidence, 1960j Djokovic and Diananda, 1963*.
Djokovic, 1963.
Nowosad, 1968 and Diananda, 1963*.
Nowosad, 1968j Bushell, 1992 analytic proof.
Godunova and Levin, 1976 and Diananda, 1963*.
Stuart, 1974, in an unpublished dissertation improves a proof by
G.K. Kristiansen (unpublished)j Bushell and Craven, 1975,
numerical evidence for yeSj Godunova and Levin, 1976.
Troesch, 1985.
Zulauf, 1958.
Troesch, 1989 and Diananda, 1963*.
Zulauf, 1958~.
Troesch, 1989 and Diananda, 1963*.
Zulauf, 1958~.
Troesch, 1989 and Diananda, 1963*.
Lighthill, 1956.
Troesch, 1989 and Diananda, 1963*.
Zulauf, 1958~.
Troesch, 1989.
Zulauf, 1958~.
Malcolm, 1971j Daykin, 1971. (independently)
Zulauf, 1958, for even nj Rankin, 1958, large nj
Zulauf, 1959, n > 52j Diananda, 1963, n 2: 27.
* Author X and Diananda 1963 means that the result follows by combining author X's
result and Diananda's theorem of 1963.
'" The counterexample is constructed by induction starting from n = 14.
** Freidkin and Freidkin give a direct proof for all n < 7 by using Fourier analysis.
conjecture failed for all even n ~ 14. Diananda [10], (1963), extended this result
to show that if the conjecture is true for some even n, then it is true for all smaller
n and if it is false for some odd n, then it is false for alilarger numbers. The next
far reaching result is that of Nowosad [24], who in 1968 showed that interior local
minima of f are always ~ n/2 and showed that the only other points that need be
considered are on what he called the regular boundary. More of this later. This
allowed the reduction in the number of cases that needed to be considered and led
to the numerical proofs that are used for all cases where n 2: 12.
An independent event was the discovery of Rankin [25] in 1958 that the conjecture
failed for all large n and his introduction of the asymptotic number A. The defin-
A. M. FINK
244
itive determination of this number by Drinfel'd [13], in 1971, ended the results in
this direction.
Finally, Troesch [39] (1989) solved the case for n = 23 and hence for all lesser
odd n. Meanwhile the proof for n = 12 by Godunova and Levin [17], (1976), had
settled the even cases and all n ~ 12.
CHRONOLOGY OF THE STATE OF THE CONJECTURE
1903 Nesbitt [23]
1954 Shapiro [31]
1956 Phelps [31], Lighthill [31]-
conjecture
holds for n
=3
::;4
::;5
1958 Mordell [21], Zulauf [41], Rankin [25]-
::;6
1959 Zulauf [42]
::;6
1963 Djokovic [12], Diananda [10]-
::;8
1968 Nowosad [24]
::;10
1971 Maleolm [19], and
1971 Daykin [5]
::;10
1976 Godunova and Levin [17]
::; 12
1985 Troesch [38]
n ::; 13
1989 Troesch [39]
even n::; 12
odd n::; 23
date and contributor
conjecture
fails for n
n= 20
even n ~ 14
alilarge n
even n ~ 14
odd n ~ 53
even n ~ 14
odd n ~ 27
even n ~ 14
odd n ~ 27
even n ~ 14
odd n ~ 25
even n ~ 14
odd n ~ 25
even n ~ 14
odd n ~ 25
even n ~ 14
odd n ~ 25
* The changes in this row from the previous row are in the same order as the order of
the listed contributors.
4. What is p.(n)?
Since we know that J.L(n) is sometimes 1/2 and sometimes less, what is it? Rankin
[25] considered the number
A = lim J.L(n)
n-oo
and showed that A = inf J.L(n).
n>2
Various estimates for A have been given. Rankin showed that it was less than
.49999999 thereby showing that Shapiro's conjecture is false for alllarge n. Other
bounds were obtained in [42], [26], [8] and [5] by computing bounds for J.L(n) < 1/2.
The problem is completely solved by Drinfel'd [13] who showed that A = .49456682
and further digits may be obtained by the following. Let S be the convex hull of
the regions above the graphs ofy = exp( -x) and y = 2/(exp(x)+exp(x/2)). Then
SHAPIRO'S INEQUALITY
245
A is the y coordinate of the interseetion of S with the y axis. To get a feel for the
values of JL(n) for small we give some estimates:
upper bounds for p,(n)
Daykin, .4999898*
n
14
16
18
20
22
24
25
27
111
452
Zulauf, .49953; Diananda, .49919
Malcolm, .49994; Daykin, .49995
Diananda, .4999646
Daykin, .49656
Troesch, .49484
numerical estimates"
.499975, .4999978
.499875
.499955
.499495
.499275
.49904
.499940, .499944
.499464
* Maleolm also gives a lower bound of .4999388 .
** These estimates are all from Bushell and Craven. They claim to be the numerical
minimum, Le. the exact value of p,. The second numbers in n = 14 and n = 25 are
numbers that are offered elsewhere as the minimum.
5. Outline of Proof and Methods
First, to get the most direct proof of the solution of the conjecture one now reads
the theorems of Diananda [10] and the proofs for n = 12 by Godunova and Levin
[17] and for n = 23 by Troesch [39]. Both ofthese papers use the results ofNowosad
[24] and the notions of regular boundary. Then one needs counterexamples for
n = 14 and 25. The cleanest (and integral) examples are from Troesch [39] and
are
X14 = (0,42,2,42,4,41,5,39,4,38,2,38,0,40)
and
X25=(25,0,29,0,34,5,35, 13,30, 17,24,18, 18,17,13,16,9, 16,5,16,2,18,0,20,0).
We should note that only a few direct proofs exist. For n = 3, the result follows
from use of the geometric-arithmetic-harmonic mean inequalities, see [6] or [20].
The cases with n :::; 6 were done by MordeIl [21] in the following way. He finds
that taking by B r = X r + X r +1, that
and that
(f: r
X
I
r: :
min(n/2, 3)
f: xrBr , from which the result folIows.
I
Diananda's results [10] follow from two lemmas.
Lemma 1. fn+2(XI, ... ,Xn , Xl, X2) = fn(XI, . .. ,Xn ) + 1.
A. M. FINK
246
Lemma 2. 11 n is odd and In(XI, ... ,xn ) < n/2, then there is a Y so that
In+1(YI,". ,Yn+l) < (n+1)/2.
The proof of the latter is to write
In+I(XI, ... ,Xr,Xr,Xr+I,'" ,xn) - In(XI, ... ,xn) -1/2
= (xr-Xr+1)(xr-xr-t)/2xr(xr+xr+t).
Remembering that the x's are periodie and that n is odd, it is not possible for all
of these numerators to be positive, we get a Y this way.
Djokovic [12] gives a direct analytie proof for n = 8 by defining ai = Xd(Xi+1 +
Xi+2), from whieh it follows that these numbers are nonnegative and the sum of
n
two consecutive ones is positive. The conjecture then is equivalent to E ai 2: n/2
I
with the ai 's satisfying the identity
1
0
-al
1
-al
-a2
0
-a2
0
0
0
0
1
=0.
0
-an-l
-an
0
0
-an
0
1
0
0
-an-2
1
0
-an-2
-an-l
1
Finally, Bushell [3] gives a direct algebraie proof for n = 10. He actually does
more. He gives an easy proof of Nowosad's lemma that at interior minima the
value of 1 is n/2. See also his paper [2]. He also gives symmetry properties of
the global minimizing points whieh allows one to halve the number of cases to
consider. This is useful for n = 10 and in the case of n = 12 where the reduction
is not sufficient to give an analytie proof at this writing.
The proofs of the two pivotal cases, n = 12 and 23 are partly based on careful
numerics and the results of Nowosad. Briefly, Nowosad shows that non-interior
local minima must be on the regular boundary. The regular boundary are those
points where some of the variables are zero but no term in 1 is indeterminate. The
cases to consider are then reduced to those with a specified number of nonzero
entries between the zero entries of x. Such a set of consecutive nonzero entries is
called a segment. Nowosad then makes a change of variable, essentially using the
partial sums of a segment as the new variables. In this way, Shapiro's conjecture
becomes a matter of minimizing a quadratic form with quadratie constraints. Some
of the cases can be done with known inequalities, for example with the arithmetiegeometrie mean inequality and the rest are done on an ad hoc basis. For n = 10
his results lead to only 5 cases than are "hard" and need to be dealt with on an ad
hoc basis and use some numeries. For the proofs of n = 12, Godunova and Levin
use Nowosad's ideas and carry out the various cases, again dealing with most of
them easily and reducing the few remaining to ad hoc arguments, some of which
require numerieal estimates. As noted above, Bushell has reduced the number of
SHAPIRO'S INEQUALITY
247
cases but still there are too many to complete the proof. Troesch, in doing the
n = 23 case also proceeds as Godunova and Levin, finding many cases that can be
done with known inequalities.
6. Praspects far Further Research
The remaining quest ion is now obvious. Can the pivotal cases of n = 12 and 23 be
done analytieally? There are three threads of research which are the beginnings of
an approach. Nowosad's work has been carried on by Bushell by considerations of
proving new symmetries and reducing the problems to an estimation of eigenvalues. Perhaps more can be done in this direction. 1t would be of interest to include
a quote from Troesch [40, page 664]. "... but this did not constitute a proof, but
rather an example of [13], because a definite distribution of the zero-components
of x was assumed. The assumption appeared reasonable, based on previous experience. 1t would be desirable to prove that for any n this partieular distribution
of nonzero components always gives the lowest sum (for 1), except of course for
the case with all components equal to 1. A result 01 this kind would make the
investigations reporled here (the proollor n = 23) essentially trivial."
Other recent work includes that of R.E. Scraton [27], who also numerically (apparently independently) has come to the correct conclusions. He notes that for n
even, (a, b, a, b, ... ) is a stationery point for which the eigenvalues of the Hessian
can be explicitly computed. These are all positive for n = 4,6,8, 10, 12 but for
n ~ 14 there are at least one pair of negative eigenvalues. He also offers data for
the conjecture that the minimum occurs at points where X n , X2, X4, ... ,X2k are
zero and Xn-l, Xl, X3, . .• ,X2k+1 are in a geometrie progression.
A second direction is to look at Mordell's work. For n ~ 6 he interpolated a
quadratic form between fand the number n/2. Are there other interpolants for
larger n?
Finally, it seems to me that most of the proofs heavily use the classical arithmetiegeometrie-harmonie mean inequalities to settle some of the cases. 1s there a set of
related inequalities whieh are strengthenings or interpolants of these inequalities,
that would be helpful in the pivotal cases?
I give the challenge to the interested reader to follow up one of these lines of study,
or a more appropriate one of their own choosing.
References
1. B. Bajsanski, Aremark conceming the lower bound 0/ Xl/(X2 + X3) + X2/(X3 + X4) + ... +
Xn /(Xl + X2), Univ. Beograd. Pub!. Elektrotehn. Fak. Sero Math. Fiz. No. 70-76 (1962),
19-20.
2. P. J. Bushell, Shapiro 's cyc/ic sum, manuscript.
3. _ _ , Analytic proo/s 0/ Shapiro's cyc/ic inequality /or ellen n, manuscript.
4. P. J. Bushell and A. M. Craven, On Shapiro's cyc/ic inequality, Proc. Roy. Soc. Edinburgh
75A 26 (1975/76), 333-338.
5. D. E. Daykin, Inequalities /or functions 0/ a cyc/ic nature, J. London Math. Soc. (2) 3
(1971), 453-462.
6. P. H. Diananda, Extensions 0/ an inequality 0/ H. S. Shapiro, Amer. Math. Monthly 66
(1959), 489-491.
248
A. M. FINK
7. ___ , A eye/ie inequality and an extension 0/ it, I, Proc. Edinburgh Math. Soc. (2) 113
(1962/63), 79-84.
8. ___ , A eye/ie inequality and an extension 0/ it, II, Proc. Edinburgh Math. Soc. (2) 13
(1962/63), 143-152.
9. ___ , Inequalities tor some eye/ie sums, J. London Math. Soc. 38 (1963), 60-62.
10. ___ , On a eye/ie sum, Proc. Glasgow Math. Assoc. 6 (1963), 11-13.
11. ___ , Inequalities lor some eye/ie sums, Math. Medley 5 (1977), 171-177.
12. D. Z. Djokovic, Sur une inegalite, Proc. Glasgow Math. Assoc.6 (1963), 1-10.
13. V. G. Drinfel'd, A eye/ie inequality, Mat. Zametki 9 (1971), 113-118 (Russian) [English
trans!. Math Notes 9 (1971), 68-71].
14. C. V. DureIl, Query, Math. Gaz. 40 (1956), 266.
15. A. M. Fink, Letter to the editor, Math. Gaz. 79 (1995), 125.
16. E. S. Freidkin and S. A. Freidkin, On a problem by Shapiro, Eiern. Math. 45 (1990),137-139.
17. E. K. Godunova and V. 1. Levin, A eye/ie sum with 12 terms, Mat. Zametki 19 (1976),
873-885 (Russian) [English trans!. Math Notes 19 (1976), 510-517].
18. M. Herschern and J. E. L. Peck, Problem 4603, Amer. Math. Monthly 67 (1960), 87-88.
19. M. A. Malcolm, A note on a eonjecture 0/ L. J. MordelI, Math. Comp. 25 (1971), 375-377.
20. D. S. Mitrinovic, J. E. Pecaric, and A. M. Fink, Classieal and New Inequalities in Analysis,
Kluwer Academic, Dordrecht, 1993.
21. L. J. MordelI, On the inequality L;=l x r /(Xr+l + Xr +2) ~ n/2 and some others, Abh.
Math. Sem. Univ. Hamburg 22 (1958), 229-240.
22. ___ , Note on the inequality Lk=l Xk/(Xk+l + Xk+2) ~ n/2, J. London Math. Soc.37
(1962), 176-178.
23. A. M. Nesbitt, Problem 15114, Educational Times (2) 3 (1903), 37-38.
24. P. Nowosad, Isoperimetrie eigenvalue problems in algebras, Comm. Pure App!. Math. 21
(1968), 401-465.
25. R. A. Rankin, An inequality, Math. Gaz. 42 (1958), 39-40.
26. ___ , A eye/ie inequality, Proc. Edinburgh Math. Soe. (2) 12 (1960/61), 139-147.
27. R. E. Scraton, An unexpeeted minimum value, Math. Gaz. 78 (1994), 60-62.
28. J. L. Searcy and B. A. Troesch, The eye/ie inequality, Notices Amer. Math. Soc. 23 (1976),
A-604-605.
29. ___ , A eye/ie inequality and related eigenvalue problem, Pacific J. Math. 81 (1979),
217-226.
30. H. S. Shapiro, Problem 4603, Amer. Math. Monthly 61 (1954),571.
31. ___ , Problem 4603, Amer. Math. Monthly 63 (1956), 191-192.
32. _ _ , Problem 4603, Amer. Math. Monthly 97 (1990),937.
33. J. Stuart, On Kristiansen's Proo/ 0/ Shapiro 's Inequality tor n = 12, Diss. Univ. of Reading,
1974 (not published).
34. D. G. S. Thomas, On the definiteness 0/ eertain quadratie /orms arising in a eonjeeture 0/
L. J. Morde/I, Amer. Math. Monthly 68 (1961), 472-473.
35. B. A. Troesch, The eye/ie inequality lor a large number 0/ terms, Notices Amer. Math. Soc.
25 (1978), no. 6, A-627.
36. ___ , Shapiro 's eye/ie inequality with eleven terms, Notices Amer. Math. Soc. 26 (1979),
no. 7, A-646.
37. ___ , The shooting method applied to a eye/ie inequality, Math. Comp. 34 (1980), 175184.
38. ___ , On Shapiro's eye/ie inequality tor N = 13, Math. Comp. 45 (1985),199-207.
39. ___ , Full solution 01 Shapiro's eye/ie inequality, Notices Amer. Math. Soc. 39 (1985),
no. 4, 318.
40. ___ , The validity 0/ Shapiro's eye/ie inequality, Math. Comp. 53 (1989), 657-664.
41. A. Zulauf, Note on a conjecture 0/ L. J. MordelI, Abh. Math. Sem. Univ. Hamburg 22
(1958), 240-241.
42. ___ , On a eonjecture 0/ L. J. MordelI, II, Math. Gaz. 43 (1959), 182-184.
43. ___ , Note on an inequality, Math. Gaz. 46 (1962), 41-42.
BERNSTEIN TYPE INEQUALITIES FOR RATIONAL
FUNCTIONS WITH PRESCRIBED POLES
N. K. GOVIL
Department of Mathematics, Auburn University, AL 36849, U.S.A.
R. N. MOHAPATRA
Department of Mathematics, University of Central Florida, Orlando,
FL 32816, U.8.A.
Abstract. The paper surveys polynomial inequalities and their generalisations to rational functions with prescribed poles. We also mention results concerning generalisations
of Bernstein's polynomial inequalities through the use of Functional Analysis. Finally, a
Bernstein type inequality associated with wavelet decomposition is mentioned.
1. Introduction and Notation
Let p~ and p~ be the sets of all algebraie polynomials of degree at most n with
real or eomplex eoefficients, respeetively. The sets of all trigonometrie polynomials
of degree at most n with real or eomplex eoeffieients, respeetively, will be denoted
by r;: and r:;,. For a eontinuous function f defined on a set A, we shall use the
notation
IIfliA = sup If(z)l·
zEA
As usual IR and C denote the fields of real or eomplex numbers, and K .IR(mod.27T). We shall write
D_ := {z E C : Izl < I},
T:= {z E C : Izl = I},
D+:= {z E C : Izl > I}.
In order to show that a smooth eurve ean be approximated by sueeessions of
quadratie ares, Mendeleev [79] eonsidered p(x) = ax 2 + bx + c with a, b, cE IR,
Ip(x)1 ~ 1 for -1 ~ x ~ 1 (see [52] and [11]). He showed that Ip'(x) I ~ 4 for
-1 ~ x ~ 1 and this result is best possible sinee, for p( x) = 1 - 2x 2 , Ip( x) I ~ 1 on
[-1,1] and Ip'(±I)1 = 4.
Motivated by this result, A.A. Markoff [76] investigated the eorresponding problem
in a more general set up and proved the following.
1991 Mathematics Subject Classijication. Primary 41A17, 26D07j Secondary 26D05, 30CI0.
Key words and phrases. Polynomial inequalitiesj Inequalities with rational functionsj Prescribed
polesj Bernstein's inequalitYj Wavelet decomposition.
249
G.Y. Milovanovic (ed.), Recent Progress in Inequalities, 249-270.
© 1998 Kluwer Academic Publishers.
250
N. K. GOVIL AND R. N. MOHAPATRA
n
Theorem 1. If p(x) = E akxk E p~ and Ip(x)1 :::; 1 on [-1,1], then
k=O
(1)
Ip'(x)l:::;n 2
for -l:::;x:::;l.
The equality in the above inequality is possible at only x = ±1 and only when
p(x) = ±Tn(x), where Tn(x) = cos(narccosx), is the n-th Chebyshev polynomial
of the first kind.
Several years later Serge Bernstein needed the analogue of Theorem 1 for the unit
disk in the complex plane. He [7] proved the following result.
n
Theorem 2. If p(z) = E akzk E p~, then
k=O
(2)
max Ip'(z)1 :::; n max Ip(z)l.
Izl9
Izl~l
The result is best possible and the equality holds for p(z) = AZ n (A, z E q.
Theorem 2 has an analogue for trigonometrie polynomials and can be stated as
Theorem 3. If t(8) =
(3)
n
E akeikO E T;,(8), then
k=-n
It'(8)1:::; n,
0:::; 0:::; 27r,
whenever It(O)1 :::; 1 for 0 :::; 0 :::; 27r. In (3) equality holds if and only if t(O)
ei ,,! cos(nO - a), where'Y and aare arbitrary real numbers.
Bernstein proved Theorem 3 with 2n in place of n by using a variational method.
Inequality in the form (3) appeared in print for the first time in a paper of Fekete
[36] who attributes its proof to Fejer [34]. Bernstein [7] attributes the proof to
E. Landau (see [101] and [35]). Alternative proofs of (3) have been given by
F. Riesz [95], M. Riesz [96], de la Vallee Poussin [107] and many others, and each
of these methods has led to the interesting extensions of the inequality (3).
Theorem 1 and Theorem 2 are generally known as Markoff and Bernstein inequalities, respectively. These inequalities play an important role in the proof of inverse
theorems in polynomial approximation (see Dzyadyk [31], Ivanov [63], Pekarskii
[87], Meinardus [78], Telyakovskii [105], Milovanovic, Mitrinovic and Rassias [80],
Borwein and Erdelyi [14] and Petrushev and Popov [85]). Inequalities (1) and (2)
were extended in many direction and turned out to be the center of considerable
research activity, see ([8-10], [30], [33], [60-61], [63], [65], [67-74], [82], [84], [86],
[88-89], [92], [94], [97], [102-103], [106] and [109]).
In what follows we shall mention some refinements and generalisations of Markoff
and Bernstein theorems and their extensions to rational functions and their generalisations. It is not possible to give a cursory look at the vast literat ure developed
over the years in this article. We only mention certain segments of current research.
BERNSTEIN TYPE INEQUALITIES FOR RATIONAL FUNCTIONS
251
2. Some Generalisations and Refinements of Markoff's
Inequality
Theorem 1 shows that if Ip(x)1 ~ M, -1 ~ x ~ 1, then Ip'(x) I ~ Mn 2 on [-1,1].
If we repeatedly use this result to find an upper bound for Ip(k)(x)l, k ~ n, then
we can obtain Ip(k) (x) ~ Mn 2k . A. A. Markoff [76] showed that this result is not
sharp and proved the following:
I
Theorem 4. 1/ p(x) E p~ and Ip(x)1 ~ 1 on [-1,1], then on [-1,1]
(4)
tor every k = 1,2, ... ,n.
The right hand side of the inequality (4) is equal to T~k)(I) where Tn(x) is the
Chebyshev polynomial of the first kind and hence (4) is sharp.
Given k (1 ~ k ~ n), and x" E [-1,1], let p .. be an extremal polynomial in the
sense that
max Ip*(x)1 = 1. V. Markoff
-1<:z:<1
[77] was able to characterise and even identify the extre~al polynomials for differ-
It is easily observed that such a P .. (x) exists and
ent values of x*. Let 6 < 6 < ... < en-k and "11 < "12 < ... < "In-k be the zeros
of (x + I)T~k+l)(x) + kT~k)(x) and (x _1)T~k+1)(x) + kT~k)(x) respectively. Not
only that ei,"Ii E [-1,1], they interlace, i.e., -1< e1 < "11 < 6< "12 < 6<··· <
en-k < "In-k < 1. v. Markoff showed that the polynomial Tn is extremal for x*
belonging to any of the intervals
He also showed that the points
Further, he showed that at a point x" E (eil c5j ], with
'Ir
1 < C ~ 1 + 2 tan2 2n '
the polynomial
T. (2X+I-C) =T. (I+ ej )(X-x*)
n
C
+1
n
1 + x*
co)
+ '>3
252
N. K. GOVIL AND R. N. MOHAPATRA
is extremal. The point
Aj := (sec2 2:) fJj - tan2 2: E (ej,fJj),
for j = 1,2, ... ,n - k and at a point x* E [Aj, fJj) the polynomial
Tn (
(l- fJj )(X-X*)
)
1 _ x*
+ fJj ,
.
J = 1,2, ... ,n - k,
is extremal. However, in the intervals (Oj, Aj) for each j E {I, 2, ... ,n - k}, the
equation
d~k ((x 2 -1) T~_l(X)) = 0,
has a root bj in (Oj, Aj) for j = 1,2, ... ,n - k. The polynomial T n - 1 is extremal
when x* is any of the (n - k) points b1 , b2 , ••• ,bn-k' It can be shown that when
x* E (t5 j ,bj ) and (bj,Aj), the extremal polynomial is a solution of a first order
differential equation (for details see [1]).
Among other things he proved
n
Theorem 5. Ifp(x) = E akxk is as in Theorem 1, thenfor 1::; k::; n, we have
k=O
if (n - k) is even,
(5)
if (n - k) is odd,
and
(6)
Voronovskaja, in a long series of papers developed an alternative approach to the
problems considered by V. Markoff. For discussion and literature related to these
problems, see [108] and [92].
Erdös [32] has shown that it is possible to improve Theorem 1 if the zeros of p(x)
lie on IR. \ (-1,1). He [32] showed
Theorem 6. Let p(x) be as in Theorem 1. Let the zeros of p be all real and lie
on IR. \ (-1,1). Then
(7)
Ip'(x)
1
I ::; ~ ( 1-;;:
)-n+l ,
for
- 1 ::; x ::; 1.
In (7) equality holds only at ±1 for
.
nn
p(x) := e''"( 2n (n _ l)n-l (1 + x)(l- x)n-l,
p(x) := ei '"( 2n (n
respectively.
'Y E IR.,
:n1)n_l (1 + x)n-l(l_ x), 'Y E IR.,
BERNSTEIN TYPE INEQUALITIES FOR RATIONAL FUNCTIONS
253
Theorem 7. Let p(x) be a polynomial of degree at most n, and Ip(x)1 :::; 1 for
-1:::; x :::; 1. If p(x) is real for real x and p(z) =I- 0 for Izl < 1, z E C, then
(8)
Ip'(x)
I : :; 4y'ri,f(1-lxI)2,
for xE [-1,1].
Remarks:
i) fo can not be replaced by a function of n tending to infinity more slowly.
ii) The bound 4fo/(1 -lxl)2 is not best possible for any x.
In [1], Arsenault and Rahman have obtained sharp estimates for Ip'(x*) I for arbitrary x* E [-1,1]. They have also obtained the exact bound for Ip(x*)1 at an
arbitrary point x* E IR \ [-1, 1]. In fact [1] contains many interesting results and
historie al development of the estimates of Markoff type. (Also see [14], especially
pages 234-235).
2.1. WEIGHTED-MARKOFF INEQUALITIES
Let p~. be the subcollection of p~ consisting of the monie polynomials of degree
less or equal to n. Let W denote the collection of real weight functions, w, such
that the following hold:
(a) w(x) > 0 for all x E IR,
(b) w'(x) is continuous on IR,
(c) lim [xnw(x)] = lim [xnw'(x)] = 0, n = 1,2, ....
Ixl-->oo
Ixl-->oo
Let all norms be supremum norm on IR. Let us define
(9)
An = sup
pEP~'
IIWP'II}
{ -11-11
wp
and
Iln = sup
rEP:,
{ 11 (wp)' 11 }
11 wp 11 .
By standard arguments one can show that An and Iln are finite and that there exist
monie polynomials p, q E p~. for which Ilwp'lI/llwpll = An and II(wq)'II/llwqll =
Iln· Such polynomials p or q will be called extremal polynomials for An or Iln,
respectively. In view of these the following inequalities of Markoff type hold for
pE p~:
(10)
Ilwp'lI :::; Anllwpll
and
lI(wp)'II:::; Ilnllwpll·
An and Iln are best possible constants for these inequalities. Let T n be the monic
polynomials of exact degree n whieh are extremal in the sense that
(11)
IIwTn 11 = inf{ 11 w(x) [x n - q(x )]11 : q E p~-d
.
Since {xkw(x) : k = 0,1, ... ,n -1} is a Haar system on IR, T n is uniquely
characterised by the fact that wTn has an alternant of size (n + 1) (see [82] for
details). In [82], Mohapatra et al have proved the following results:
254
N. K. GOVIL AND R. N. MOHAPATRA
Theorem 8. Let w E Wand p E p~., n 2': 2, is any extrem al for f..tn. Then the
following hold:
(a) A maximal alternant for wp is of size n + 1.
(b) If w' / w is decreasing on IR then there is exactly one maximal altemant for
wp. Moreover if this maximal altemant Xl < X2 < ... < Xn-l is of size n (i.e.
n
if p =j:. Tn ) then (wq)'(to) = 0, where g(x) = TI (x - Xi) and to is such that
i=l
l(wp)'(to)1 = IIwp'll·
Theorem 9. Let w E Wand suppose p E p~. is any extrem al for An. Then the
following hold:
(a) If n = 1, then p = Tl.
(b) If n 2': 2, then a maximal altemant for wp is of size n + 1.
(c) If w' / w is decreasing on IR then there is exactly one maximal altemant for
wp. Moreover if this maximal alternant, Xl < X2 < ... < Xn, is of size n (i.e. if
p =j:. Tn ) then g'(to) =
°
n
where g(x) = TI (x - Xi) and to is such that
i=l
I(wp') (to)1 = IIwp'll·
Theorem 10. If w(x) = exp (_x 2 ) and p E p~. is any extremal for f..tn, then
p = Tn - l or p = Tn , where To := 1.
In [69], Li, Mohapatra and Rodriguez have shown that p = Tn - l in Theorem 10
is not a possible solution by using a representation theorem and the analysis of
Voronovskaja [108]. Thus they have characterised the best possible constant in
Markoff's inequality in IR for the Hermite weight in terms of weighted Chebyshev
polynomial.
Remark. Duffin and Schaeffer [29] asked if it is necessary for the inequality (4) in
Theorem 4 to assume that Ip(x)1 :5 1 for aU x E [-1,1]. They [29] found out that (4)
holds whenever Ip(x)1 :5 1 at the (n + 1) points x = cos(k1r/n)j k = 0, 1, ... , n. In fact,
they [29] proved
Theorem 11. If p(z) is a polynomial of degree n with real coefficients and
(12)
Ip(cosk1r/n)1 ~ 1,
(k = 0,1, ... ,n),
then also the inequality (4) holds.
It is natural to ask if there are other (n + 1) points in (-1, 1) such that Ip( x) I ~ 1
at these points will imply the inequality (4). Duffin and Schaeffer [29] showed that
if E is any closed subset of ( -1, 1) which does not contain all the points cos( krr / n),
k = 0,1, ... ,n, then there exists a polynomial of degree n which is bounded by 1
on the set E but for which the inequality (4) does not hold. This refined inequality
known as Duffin-Schaeffer inequality has applications in numerical analysis (see
Berman [6]).
For Markoff type inequalities with constraint, see Borwein and Erdelyi ([14, A5]).
BERNSTEIN TYPE INEQUALITIES FOR RATIONAL FUNCTIONS
255
3. Bernstein's Inequality and Its Generalisations
As mentioned in Section 1, Bernstein proved inequality (3) (see Theorem 3). Note
n
E akxk is a polynomial of degree n in p~ or p~, -1 < x < 1,
k=O
then p( cos 0) E r;, or r:;, respectively, and by Theorem 3 of Bernstein, we have,
that if p(x) =
Ip'(cosO) sinOI ::; n, which is equivalent to Ip'(x) I ::; n(l- X2)-1/2, -1::; x::; l.
Thus we get the following result which is also due to Bernstein [7].
Theorem 12. Ifp(x) E p~ and Ip(x)l::; 1 on [-1,1], then
(13)
Ip'(x)
I ::; ~,
-1< x< 1.
The equality holds at the points x = Xk = cos((2k -l)7r/n), 1::; k ::; n, if and only
if p(x) = eTn(x) where lei = 1 and Tn(x) is the Chebyshev polynomial of the first
kind.
Remar k. Theorem 12, proved by Bernstein in [7] yields a better estimate for
when x is not near ±1, but it does not yield Markoff's theorem, viz. Theorem 1.
Ip' (x) I
Related to Theorem 12 and Theorem 1 is the following inequality known as Schur's
inequality (see [14, p. 233]):
Theorem 13. For every p E P~-l'
(14)
By Theorem 3 and Theorem 13, one can prove Theorem 1, since
(15)
Theorem 3 usually known as Bernstein's inequality can be derived from the Bernstein-Szegö inequality given below (see [104], this inequality was first explicitly
stated by Van der Corput and Schaake [22], also see [23]):
Theorem 14. For tE r:; and 0 E ~,
(16)
Equality in (16) holds if and only if It(O)1 = IItlllR or t is of the form t(O)
ß) with Cl, ß E R
Cl cos(nO -
From (16) it clearly follows that Ilt'lllR < nlltlllR for every t E Tn.
mathematical induction one can obtain
By using
Theorem 15. Let tE r:;. Then,
(17)
Remark. By standard argument one can show that Theorem 15 holds for every t E T,;,
and for a proof, see [14, p. 232].
Since p E P~ implies t(O) := p (e iO ) Er;" we get
Ip'(z)1 = l-ieiOt'(O)1 ::; nlltlllR = nllpIID_,
z = ei () .
Now, by the maximum modulus principle applied to the unit disc, we get
256
N. K. GOVIL AND R. N. MOHAPATRA
Theorem 16. I/ p(z) E p~, then
(18)
maxlp'(z) 1 ~ nmax Ip(z)l.
Izl9
Izl9
The equality in (18) holds tor p(z) = AZ n , A E C.
Writing IIpll = max Ip(z)l, (18) can be written as
zET
(19)
IIp'll ~ nIlplI·
There are several proofs of Theorem 16. The proof of de Bruijn [26] (also see
Rahman [88]) uses the following two propositions to prove Bernstein's inequality
for t E T:f:
Proposition 1. I/ p(z) E p~ with all its zeros in Izl ~ 1, and i/ q(z) = zn p(l/z),
then tor Izl ~ 1,
(20)
Iq'(z) 1 ~ Ip'(z) I·
Proposition 2. I/ p(z) is a polynomial 0/ degree n such that Ip(z)1 ~ 1 tor Izl ~ 1
and i/ q(z) is as in Proposition 1, then tor Izl ~ 1,
(21)
Another proof of Theorem 16 can be seen in Milovanovic, Mitrinovic and Rassias
[80, p. 532]. There is also an interesting proof by O'Hara [83] which depends
upon the use of Lagrange interpolation and an identity. The latter method has
been exploited by Mohapatra, O'Hara and Rodriguez [81] to obtain simple proofs
of some refinements of Bernstein's inequality and later by Li, Mohapatra and
Rodriguez to prove Bernstein-type inequality in rational spaces (see [70]). O'Hara
[83] deduced Theorem 16 from the following identity (also see [52]).
Proposition 3. I/ p(z) is any complex polynomial 0/ degree at most n, and
Zl, Z2, •.• ,Zn are the zeros 0/ zn + 1, then tor every complex number t,
(22)
n
1~
2Zk
tp'(t) = "2 p(t) + ~ ~P(tzk) (Zk _ 1)2 .
3.1. REFINEMENTS OF BERNSTEIN'S INEQUALITY
Szegö [104] proved inequality (18) under a weaker condition, viz.
max I Rep(z) I ~ 1.
zET
Precisely, he [104] proved
BERNSTEIN TYPE INEQUALITIES FOR RATIONAL FUNCTIONS
257
Theorem 17. 1/ p( z) E p~ with I Re p( z) I ~ 1 tor z E T, then Ip' (z) I ~ n tor
Izl ~ 1. Equality holds tor p(z) = AZ n with A E C so that lAI = 1.
Malik [75] has given a proof of Theorem 17 based on a result of de Bruijn (see
Govil [52] and de Bruijn [26]). In the same paper he [75] also proves the following
improvement of Bernstein's inequality (also see Rahman [88]):
Theorem 18. 1/ p E p~ and q be the sel/-inversive polynomial associated with p
as in Proposition 1, then
(23)
max [lq'(z)1 + Ip'(z)ll = nmaxlp(z)l·
zET
zET
Further generalisations can be found in [39-40]. Frappier, Rahman and Ruscheweyh [40, Theorem 8] provided the following generalisation:
Theorem 19. 1/ p E p~ and Zl, Z2, ... , Z2n are any 2n equally spaced points on
Izl = 1, then
Mohapatra, ü'Hara and Rodriguez [81] proved an identity similar to (22) and
deduced Theorems 17, 18 and 19. Their main result [81] was
Theorem 20. 1/ p E p~, A E C with lAI = 1 and Tl, 1'2, . .. ,Tn are n th roots 0/ A,
then tor all z E T,
(24)
np(z) - zp'(z) +
A p'(z) = -1 "'p(Tk)
n
n
A ,
-=-1
~
zn
n L..J
z - Tk
I
2
1
k=l
and
(25)
- AI
-1 n Izn
--n
nLz-Tk - .
k=l
Note that (25) is a special case of (24) when p(z) = zn. Replacing A by -A in (24)
and subtracting the resulting identity from (24) we abtain a companion identity
given by the following:
Let p E p~, A E C with lAI = 1. Let Tl, ... ,Tn be as in Theorem 20 and 0"1, •.• ,O"n
be the n th roots of -A. Then far all z E T,
(26)
2
2
n
I zn - A 1
n
I zn + A 1
-2A
_ p'(z) = -1 "'p(Tk)
- - - -1 "'p(O"k)
-Zn 1
n L..J
z - Tk
n L..J
z - O"k
k=l
k=l
In [81] the following result is proved which is an improvement of Theorem 19.
N. K. GOVIL AND R. N. MOHAPATRA
258
Theorem 21. Let Zl, ..• ,Z2n be 2n equally spaced points on T, say Zk = ue ikrr In,
lul = 1, 1 :::; k :::; 2n. Then for any p E p~, we have
maxlp'(Z)I
:::; ~2 [max
Ip(Zk)1 + max IP(Zk)l]
zET
k odd
k even
(27)
In Theorem 16 and Theorem 17, equality holds if and only if p(z) = AZ n , A E
Co
n
Hence, if we write, p(z) = l:: akzk, then we have equality if and only if
k=O
ao = al = ... = an-l = O. Thus, it is natural to conclude that if any of the
ai, i = 0,1, ... ,n - 1 is non-zero, then it should be possible to improve upon
the bound in Bernstein's inequality. This observation by Frappier, Rahman and
Ruscheweyh [40]led to
Theorem 22. Let p(z) E P~. Then for R > 1,
(28)
IIp(Rz) - p(Z) 11 + 1Pn(R)lp(O)1 :::; (Rn - 1) Ilpll,
where
(29)
1Pn(R) =
(R-1)(Rn-1 + Rn-2){Rn+1 + Rn - (n+1)R + (n-1))
Rn+! + Rn - (n-1)R + (n-3)
if n ? 2 and 1PI (R) = R - 1.
The coefficient of Ip(O)1 is the best possible for each R. If we divide both sides of
(28) by (R - 1) and let R -+ 1, then we get
Corollary 1. If p(z) is a polynomial of degree at most n, then
(30)
2n
where En = --2 if n ? 2, whereas EI = 1. The coefficient of Ip(O) I is the best
n+
possible for each n.
In order to prove the above inequalities, Frappier, Rahman and Ruscheweyh [40]
developed a method based on convolutions of analytic functions (see Ruscheweyh
[98]). This method provides a dependence of IIp'll on the coefficient lall. They [40]
have proved the following result.
Theorem 23. For p E p~,
(31)
where Cl = 0, C2 = /2 - 1, C3 = 1//2, whereas for n ? 4, Cn is the unique root
of the equation fex) := 16n - 8(3n + 2)x 2 - 16x 3 + (n + 4)x 4 = 0, lying in the
interval (0,1). The coefficient of Ip(O)1 is the best possible for each n.
Frappier [37] obtained Cn appearing in Theorem 23 and also proved a result where
IIp'll depended on a2 (see [38]).
BERNSTEIN TYPE INEQUALITIES FOR RATIONAL FUNCTIONS
259
3.2. CONSTRAINED BERNSTEIN INEQUALITIES
In this seetion we mention results where polynomial p(z) satisfies some requirements regarding the loeation of its zeros.
Erdös [32] eonjeetured that if p E p~ has no zeros in Izl < 1, then IIp'lI :5 ~ IIpll·
This eonjeeture was proved in the special case when all the zeros of p(z) lie on
Izl = 1, independently by P6lya and by Szegö. However Lax [66] proved the
conjeeture in full generality and showed the following:
n
Theorem 24. I/ p Ep~ and p( z) t:- 0 tor z ED -, then IIp' 11 :5 '2 IIpll. The result
is sharp and equality holds tor any polynomial which has all its zeros on Izl = 1.
For simpler proofs and generalisations of the above theorem, see ([5], [12], [26],
[41] and [90]).
Professor R.P. Boas proposed to obtain results analogous to Theorem 24, when
p(z) has no zero in Izl < K, K > O. In this connection, Malik [75] provided the
following first partial result:
Theorem 25. I/pE p~ and has no zeros in Izl < K, K ~ 1, then
(32)
IIp'll:5
(1: K) IIpll·
The result is best possible and equality holds tor p(z) = (z + K)n.
For analogous results when 0 < K :5 1 see [45-46]. Govil and Rahman [57]
generalised Theorem 28 for higher order derivatives of p(z). Precisely, they proved
Theorem 26. Let p E p~ and p(z) t:- 0 tor Izl < K, K ~ 1. Then
(33)
Other generalisations and refinements of the above results are obtained among
others in [21], [47], [55] and [58].
A refinement of Lax's result (Theorem 24) is due to Aziz and Dawood [4]. Their
result is
Theorem 27. I/pE p~ has no zeros in Izl < 1, then
(34)
IIp'll :5 ~2 {lIpll- Izl=l
min Ip(z)l} .
The result is best possible and equality holds tor p(z) = az n + ß where IßI ~ lai.
Theorem 30 has been generalised by Govil [51]. His result also sharpens Theorem 4
of Govil and Rahman [57].
Bernstein type inequalities for polynomials when all the zeros lie in a circle have
been investigated in [24], [48], [50], [75] and [106]. Bernstein type inequalities with
restricted zeros are studied in [13] and [15].
'furan [106] proved
N. K. GOVIL AND R. N. MOHAPATRA
260
Theorem 28. If P E p~ and has alt its zeros in Izl ~ 1, then IIp'lI ~ (n/2) IIpll.
The equality holds for p(z) = (z + l)n.
Govil [48] generalised the above result of Turan, when p(z) has all its zeros in
Izl ~ K, K > O. A simpler proof of this result of Govil [48] was given by Datt [24].
Rahman [91] generalised Theorem 28 to entire functions of exponential type and a
generalisation of this result of Rahman [91] was given by Govil [49]. A refinement
and generalisation of Theorem 28 is also due to Giroux, Rahman and Schmeisser
[44] who proved:
n
Theorem 29. Let p(z) = an TI (z - Zk) be of degree n. If IZkl ~ 1, 1 ~ k ~ n,
k=l
then
IIp'll ~ ~
(35)
(1 +llzkl ) IIpll·
If the zeros of p(z) are all positive, then there is equality in (35).
A generalisation of Theorem 29 is due to Aziz [2]. Further results occur in [4] and
[50-51].
When the polynomial p(z) == znp(l/z) or p(z) = znp(l/z), we expect a better
bound in Bernstein's inequality. The initial results are due to O'Hara and Rodriguez [84] and Saff and Sheil-Small [99]. They have shown the following
Theorem 30. Ifp(z) is a polynomial of degree n satisfying p(z) == znp(l/z), then
IIp'll = (n/2) IIpll·
On the other hand Govil, Jain and Labelle [54] proved
Theorem 31. If p(z) is a polynomial of degree n satisfying p(z) == znp(l/z) and
having all its zeros in the lejt half-plane or right half-plane, then
n
IIp'll ~ ..;2llpll.
For further results related to Theorem 31, see ([25], [27], [39-40], [44], [52], [59]
and [64]).
3.3. BERNSTEIN-TYPE INEQUALITIES IN THE L r NORM
The first result in this direction is due to Zygmund [109], who proved:
Theorem 32. If p(z) is a polynomial of degree n, then for r ~ 1,
(36)
( 2~
10211" Ip' (e Ir dO)
i9 )
l/r
(211"
~ n 2~ 10 Ip (e i9 ) Ir dO
) l/r
The result is best possible and equality holds for p(z) = AZ n , A E Co
If r -t 00, then (36) reduces to (19).
The L r analogue of Theorem 24 was proved by de Bruijn [26], Rahman [90] (see
also Rahman and Schmeisser [92]) and Aziz [3]). Govil and Jain [53] proved the
analogue for Theorem 30 (also see Dewan and Govil [28]). For polynomials not
vanishing in IZI < K, K ~ 1, Govil and Rahman [57] proved
BERNSTEIN TYPE INEQUALITIES FOR RATIONAL FUNCTIONS
261
where Er = 27r/ J:1I" IK + ei<>lr da.
Remarks. As M ~ 00, Theorem 33 reduces to Theorem 25, since E;!r ~ 1/(1 + K).
Inequality (37) is not sharp. Gardner and Govil [42-43) have generalised Theorem 33.
Related results are in [17), [68), [71-74) and [93). For further discussion ofthese types of
results, see Govil [52).
3.4. DENSE MARKOV SYSTEMS AND BERNSTEIN INEQUALITIES
Let A be a sub set of G1 [a, b]. Then A is said to have an everywhere Bernstein
inequality if for every [a, ß] C [a, b], a :f- ß,
(38)
IIp'II[<>,ßl .
} _
sup { Ilpll[a,bl . pE A, p:f- 0 - 00.
If for some [a, ß], the supremum in (38) is finite then we say that the Bernstein
inequality is bounded (see [13] and [16]).
Borwein and Erdelyi [16] have proved among other things
Theorem 34. Suppose M := Uo, 11, h, ... } is an infinite Markov system on
[a, b] with each fi E G2[a, b]. Then span M is dense in G[a, b] if and only if span
M has an everywhere unbounded Bernstein inequality.
It may be remarked that the collection of all polynomials of the form
{ x 2 p( x) : p is a polynomial}
has an everywhere unbounded Bernstein inequality. Proof of Theorem 34 requires
careful examination of Chebyshev polynomials associated with a Chebyshev system.
4. Bernstein Type Inequalities for Rational Functions
Recently Borwein, Erdelyi and Zhang [19] have proved Bernstein-Markov inequalities for real rational functions. Their results deal with both algebraic and trigonometrie polynomials on a finite interval. Subsequently Borwein and Erdelyi [18]
have studied extensions of Bernstein inequalities for rational spaces (also see [14,
Chapter 7]). Meanwhile, Li, Mohapatra and Rodriguez [70] have used the method
developed in [81] to obtain Bernstein type inequalities for rational functions.
For aj E C, j = 1,2, ... ,n, let w(z) =
B(z) =
(39)
rr (z - ai) and let
i=l
n
Ir (1- äjZ) ,
j=l
Z - aj
IRn = IRn(a1,a2,'" ,an) := {~~~) : p E p~}.
Then IRn is the set of all rational functions with poles a1, a2, ... ,an, at most, and
with limit at infinity. Clearly B(z) E IRn.
N. K. GOVIL AND R. N. MOHAPATRA
262
Definition 1. (a) For r(z) = p(z)/w(z) E 1Rn, the conjugate transpose r* is
defined by r*(z) := B(z)r(l/z).
(b) The rational function r E 1Rn is called self-inversive if r*(z) = 'xr(z) for some
>. E T.
Note that if r E 1Rn and r = plw, then r* = p* Iw and hence r* E 1Rn. So r = plw
is self-inversive if and only if pis self-inversive.
Let all the poles of r, viz. al, a2, ... , an lie in D + or D _. Then we have the
following result due to Li, Mohapatra and Rodriguez [70, Theorem 1].
Theorem 35. Suppose'x E T. Then the following hold:
(a) The equation B(z) =,X has exactly n simple roots h,t2, ... ,tn , say, and all
ti 's lie on T. If r E 1Rn and z E T, then
(40)
B'(z)r(z) _ r'(z)[B(z) -'x] = B(z) tCkr(tk)IB(Z) - 'x1
Z
k=l
Z - tk
2
,
where Ck is defined by
(41)
for
k= 1,2, ... ,no
(b) M oreover, for z E T
(42)
z B'(z) = t
B(z)
k=l
Ckl B(z) -,X 12
z - tk
Using the above Theorem 35, Li, Mohapatra and Rodriguez [70] obtain
Theorem 36. Let tk be as defined in Theorem 35 and let Sk, k = 1,2, ... , n be
the n roots of B(z) + >. = O. Then for z E T,
(43)
The inequality is sharp with equality for r (z) = uB (z ), u E T.
Theorem 36 implies the following Bernstein-type inequality for r E 1Rn (see [70]).
Theorem 37. If z E T, then
(44)
Ir'(z)1 ~ IB'(z)llIrll·
The inequality is best possible and the equality holds for r (z) = uB (z) with u E T.
The next result which sharpens Theorem 37 is also due to Li, Mohapatra and
Rodriguez [70].
BERNSTEIN TYPE INEQUALITIES FOR RATIONAL FUNCTIONS
263
Theorem 38. If rE IRn and z ET, then
(45)
l(r*(z))'1 + Ir'(z)1 ~ IB'(z)lllrll·
Again the inequality is best possible and the equality holds for r(z) = uB(z) with
uET.
Borwein, Erdelyi and Zhong [19] have proved
Theorem 39. If z E T, and aj E C \ T for j = 1,2, ... ,n, then for rE IRn, we
have
(46)
Moreover the inequality is sharp.
Remark. In Theorem 39, the poles al, ... , an of r can lie any where except on T while
in Theorem 38, they have to lie either in D_ or D+. But if we consider rational functions
whose poles lie in D_ or D+, then Theorem 38 is better than Theorem 39.
Following two results which are rational analogues of of Erdös-Lax and Tunin
Theorems are again due to Li, Mohapatra and Rodriguez [70].
Theorem 40. Let r E IRn and alt zeros of r lie in T U D +. Then, for z E T,
(47)
1
Ir(z)1 ~ 2 IB(z)1 Ilrll·
Equality holds for r(z) = aB(z) + ß with lai = IßI = 1.
Theorem 41. Let r E IRn has exactly n poles at al, ... , an and has exactly n
zeros which lie in T U D _. Then for z E T,
(48)
1
Ir'(z)1 ~ 2 IB'(z)llr(z)l.
Again the equality holds fOT r(z) = aB(z) + ß with lai = IßI = 1.
Recently Govil and Mohapatra [56] have proved the following refinements of Theorems 40 and 41, respectively.
Theorem 42. Let rE IRn be as in Theorem 40. Then, fOT z ET,
(49)
1
Ir'(z)1 ~ 2IB'(z)I(llrll- m),
where m = min Ir( z) I. The inequality is best possible and becomes equality fOT
Izl=l
r(z) = aB(z) + ß with lai = IßI = 1.
264
N. K. GOVIL AND R. N. MOHAPATRA
Theorem 43. Let r E IR"" be as in Theorem 41. Then, for z E T,
(50)
1
Ir'(z)1 :::: 2 IB'(z)I(lr(z)1 + m),
where m = min Ir(z)l. The equality in (50) holds again for r(z) = aB(z) + ß with
Izl=l
10'.1 = IßI = 1.
They also prove
Theorem 44. If rE IR"" has no poles in D_ U T, then for Izl :::: 1,
(51)
Ir(z)1 :::; IIrIIIB(z)l·
The result is best possible with equality holding for r(z) = )..B(z), 1)..1 = 1.
If r(z) has no zeros in D_, one would expect a better bound in (51) and for this,
they prove
Theorem 45. If r E IR"" has no poles in D_ U T and has no zeros in D_, then
for Izl :::: 1,
(52)
Again the result is best possible and the equality in (52) holds for r (z) = aB (z) + ß,
with 10'.1 = IßI = 1.
Remarks. Rational approximations are discussed in detail in [85] and inequalities for
derivatives of rational functions are given by Gonchar and Rusak (see [87] for reference).
Bernstein type inequalities for rational functions and inverse theorems for rational functions are discussed in [86-87]. In [87] there are results on Hardy spaces and in BMOA,
i.e. the space of analytic functions of bounded mean oscillation.
5. Bernstein Polynomial Inequalities in Hilbert Space
Harris [61) has shown how classieal inequalities for the derivative of polynomials
can be proved in real or complex Hilbert spaee. There exists a clear intereonnection
between equality of norms of symmetrie multilinear mappings due to Banaeh (see
[100)) and an inequality for the derivatives of trigonometrie polynomials due to
Van der Corput and Sehaake [22). A result of Hörmander ([62), Lemma 1) plays a
key role in proving polynomial inequalities. Harris [60] eontains functional analytie
approaehes to polynomial inequalities in Hibert spaee. Browder [20) deals with the
relation between Bernstein's inequality and the norms of Hermitian operators.
Let X and Y be real or eomplex normed linear spaees and
F:XxXx···xX--tY
be a eontinuous symmetrie m-linear mapping with respeet to the underlying sealer
field where m = 1,2, .... Define F(x) = F(x, x, ... ,x) for x E X. We say that
BERNSTEIN TYPE INEQUALITIES FOR RATIONAL FUNCTIONS
265
P : X -+ Y is a homogeneous polynomial of degree m if P = P for some eontinuous
symmetrie m-linear mapping F. We say that P : X -+ Y is a polynomial of degree
:S m if
P = Po + PI + ... + Pm ,
where Pk : X -+ Y is a homogeneous polynomial of degree k for k = 1,2, ... ,m
and a eonstant function when k = O. Let .c(X, Y) be the spaee of all bounded linear
mappings L : X -+ Y with the operator norm IILII. Let the Freehet derivative of
P at x be denoted by DP(x).
The following result due to Harris [61] is an analogue of Bernstein's inequality.
Theorem 46. 1f X is a complex Hilbert space and P : X -+ Y is a polynomial of
degree :S m, then liDPlI :S mllPlI·
The following theorem whieh is also due to Harris [61, Theorem 2] yields the results
of de Bruijn [26], Malik [75] and Szegö [104].
Theorem 47. Let X be a complex Hilbert space and let P : X -+ C be a
polynomial of degree :S m. Define S(x) = mP(x) - DP(x) for x E X and let
Xl = {x EX: Ilxll :S I}. Then
DP(x)y + S(x) E mP(X I )
for all x,y E Xl.
From the above theorem follow (see, Harris [61])
Corollary 2. 1f I ReP(x)1 :S 1 for all x E Xl, then
IDP(x)yl + IReS(x)1 :S m
for all x, y E Xl.
Corollary 3. Let r 2: 1. 1f IP(x)1 :S 1 for all x E Xl and if P has no zeros in the
closed ball in X about 0 with radius r, then IIDP(x)11 :S m/(l +r) for all x E Xl.
Applieations to trigonometrie polynomials is also given by Harris in [60-61].
6. Bernstein Type Inequality Associated With Wavelet
Decomposition
Let cp E Lo,)Rd , d = 1,2, ... , with eompaet support. Let Zd be the d-dimensional
lattiee eonsisting of all d tuples of integers. Together with cp we have its dyadie
dilates cp(2 k .), k E Z and their translates cp(2 k . - j), j E Zd. With n := [O,l]d,
let I = j2- k + 2- k n, cp E W~(lRd), r, s E Z and cpI(X) := cp(2 k x - j), x E IRd .
For any f E Lp(lRd ) (0< p :S 00), f = l: aICPI is ealled a wavelet deeomposition
IED
of f, where D = U D k , being the set of diadie eubes 2- k (j + n), j E Zd. Let <I>
kEZ
be a finite eolleetion of eompaetly supported functions cp. Then
S(cp) := {
L cp(. - j)a(j) : ais a sequenee on Zd} ,
jEZ d
N. K. GOVIL AND R. N. MOHAPATRA
266
the space generated by the shifts of 4>, is shift invariant. Moreover, if CI> is a
finite collection of compactly supported functions, then 8(CI» := L 8(4)) is shift
<pEil>
invariant. We say that a shift invariant space is refinable if f E 8 =? f(. \ j) E 8.
The Besov space B~(Lp(E)) is the collection of functions f E Lp(E) for which
0< q < 00,
q = 00,
where wr(f, t)p := sup 1I~~Jllp (E(rh)) , ~h is the rth order forward difference
Ihl:<:;t
operator and h E !Ra, Ihl is the Euclidean length of the vector h and E(rh) is the
set of x such that the line segment [x, x + rh] is contained in E. We conclude this
section by stating the following result due to Jia [65].
Theorem 48. Let CI> be a collection of compactly supporled junctions in W~(Rd),
s = 1,2, .... Let 8(CI» be refinable and the shifts of the junctions in CI> are locally
linearly independent. Then, for each a, 0< a < s, and each p, 0< p ~ 00,
IfIB'" ~ Cn a / d Ilfllp, for every f with a wavelet decomposition,
where BO: = B:;(L q ) with a = (ald + IIp)-l, C being a constant depending on p,
when p is smalI.
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SOME GENERALISATIONS AND REFINEMENTS
OF THE HARDY INEQUALITY
H. HEINIG
Department 01 Mathematics, McMaster University, Hamilton,
Ontario L884K1, Canada
A.KUFNER
Mathematical Institute, Academy 018ciences, Zitna 25, 115 67 Prague,
Czech Republic
L. E. PERSSON
Department 01 Mathematics, Lulea University, 8-971 87 Lulea, 8weden
Abstract. Some recent results connected with the one-dimensional Hardy operator are
given. Namely, some fractional order analogues of the c1assical Hardy inequality are
discussed, and results concerning the two-dimensional Hardy operator are extended to
Hardy operators defined on !RM x lltN . The main tools are the interpolation theory and
some direct approaches for the fractional order case, and a recently derived N -dimensional
Hardy inequality for operators on !RN .
1. Introduction
The one-dimensional Hardy operator H,
(1.1)
(H f)(x) =
l
x
I(t) dt,
a< x< b,
as a bounded operator from one weighted Lebesgue space into another has been
extensively studied during the last decades, and the Hardy inequality
(1.2)
where IIgllr,W denotes the norm in U(W),
(l
a
b
Ig(tWW(t) dt
)l/r
,
1< r < 00,
is investigated in detail in the book [11].
1991 Mathematics Subject Classification. Primary 26D15, 46E30.
Key WOMS and phrases. Hardy inequalities; Fractional order derivatives; Weighted Lebesgue
spaces; More-dimensional inequalities.
271
G.Y. Milovanovic (ed.), Recent Progress in Inequalities, 271-288.
© 1998 Kluwer Academic Publishers.
H. HEINING, A. KUFNER AND L. E. PERSSON
272
Furthermore, recently some results concerning the operator (1.1) have been extended to the more-dimensional case, with the N -dimensional Hardy operator H,
(Hf)(x) = f
(1.3)
f(z) dz,
JBN(X)
involved, where BN(X) , x E RN, denotes the ball with center at origin and with
radius lxi, Le., BN(X) = {z E RN, Izl ~ lxi}.
In particular, in [4] it was proved that the N -dimensional Hardy inequality
(LN Wo (x)[(Hf)(xW dX)
(1.4)
l/q
~ C(LN W(x)fP(x) dX) l/p
holds for every measurable and nonnegative function f
only if one of the following two conditions is satisfied:
(i) 1 < p ~ q < 00 and
(1.5)
A := sup( f
<»0
J1xl?<>
Wo (x) dx f/q ( f
J1x I5:.<>
= fex), x E RN, if and
W1-pf (x) dx f / pf < 00
with p' = p/(P - 1);
(ii) 1 < q < P < 00 and
(1.6)
A:=
(f (r
JRN
J1xl?lyl
Wo(x) dX) r/q X
X
(1
Ixl5:.lyl
qf
w1-p f (x) dx )r/ w1-p f (y) dy
)l/r <
00
with q' = q/(q - 1) and I/r = I/q - I/p.
We note that the conditions (1.5) and (1.6) are exactly some N-dimensional generalisations and counterparts of the usual one-dimensional (necessary and sufficient)
conditions.
In this paper, we will deal with two types of extensions of Hardy's inequality:
A. Obviously, inequality (1.2) can be rewritten in the "differential" form
(1.7)
with u' the derivative of u, for functions u satisfying the condition u(a) = O. This
indicates the possibility to investigate also fractional order Hardy inequalities of
the form
(1.8)
and
(1.9)
GENERALISATIONS AND REFINEMENTS OF HARDY INEQUALITY
where
[b [b Iu(x) - U(yW
J,\,r(U,W)= ( Ja
(1.10)
Ja
Ix-YI1+'\r W(x,y)dxdy
273
)l/r
with 0 < >. < 1, 1 < r < 00, and W is a (two-dimensional) weight function.
In asense, the usual Hardy inequality (1.7) may be regarded as an "endpoint
inequality" in a scale of inequalities (1.8) and (1.9).
In Section 2, we will review and discuss some results obtained recently in [8] by
using interpolation techniques. In particular, we point out that this approach give
us a good understanding of weighted fractional order inequalities, but also that
technique has some "shortcomings". Therefore, we will deal in Section 3 with
some recent results derived by more direct methods which avoid some of these
shortcomings. We will review here only some of the results from [6], where also
proofs and more extensions (e.g., to Orlicz norms) can be found.
B. The two-dimensional Hardy operator H 2 ,
o < x < A,
(1.11)
0 < Y < B,
as a bounded operator between weighted Lebesgue spaces appears only rarely in
the literature. However, in [1] the following mixed norm Hardy-type inequality
(1.12)
( J[A([B
J Wo(x, y)(H2!)q2 (x, y) dy )ql/q2 dx )l/ql
o
o
~
c(l (l
A
B
W(x,y)jP2(x,y)dyYl/P2 dxf/Pl
was investigated for a large scale of parameters Pi > 1, 0 < qi < 00, i = 1,2,
but for the special case when both of the involved weight functions Wo and Ware
products of weight functions depending on x and on y separately. Moreover, in [12]
necessary and sufficient conditions for the validity of (1.12) were given for general
weights but for the following special choice of parameters Pi, qi: Pl = P2 = p,
ql = q2 = q, 1 < P ~ q < 00, and for A = B = 00.
In this paper we will prove some assertions which in a sense unify and extend
some of the results mentioned above. More exactly, we will study the double-sized
multidimensional Hardy operator H 2 ,
(1.13)
(H2!)(x,y) = [
[
JBM(X) JBN(Y)
f(~,17) d17d~,
cf. (1.11) and (1.3), and prove inequalities of the type
(1.14)
(LM (LN Wo(X, y)[(H f) (x, y)]Q2 dy) Ql/Q2 dX) l/Ql
~ c(LM(LN W(x,y)jP2(x,y)dyyl/P2 dxf/Pl.
2
In Section 4 we state and prove an extension of a result of Muckenhoupt [10] and
Sawyer [12] and in Section 5 we derive an inequality of the type (1.8) for special
weights thereby generalising a result from [1].
274
H. HEINING, A. KUFNER AND L. E. PERSSON
2. Hardy Inequalities of Fractional Order Via
Interpolation
First, let us consider the following inequality:
(2.1)
1o
00
x-Aplf(x)IP dx :S CP
1 1 Ix - yl{(~)IP
00
0
00
0
If(x) -
+ P
dx dy.
This inequality was derived by Grisvard [5) provided 1 < P < 00, A i- l/p, u E
Cü(O,oo), but in fact, he rediscovered an earlier result of Jakovlev [7).
Inequality (2.1) is of the type (1.8) with p = r, wo(x) = x- Ap , W(x,y) == 1,
(a,b) = (0,00); a more general case - again with p = q but with more general
weight functions - was investigated by Kufner and '!riebel [9).
In this section we will present a special interpretation of the inequalities (1.8) and
(1.9). In particular, we will show that (2.1) can be interpreted as an "intermediate"
inequality between the classical Hardy inequality - Le., inequality (1.7) for p = q,
(a,b) = (0,00), wo(x) = x- P, w(x) == 1 - and the trivial imbedding LP C LP Le., IIull p :S Cllull p. Therefore, we can call such inequalities Hardy inequalities of
fractional order.
Without going into details (we refer, e.g., to the book [2)) we will use the following
result from the theory of interpolation of Banach spaces: If (A o , At), (B o, Bt) are
two compatible Banach couples and A = (A o, At}A,r, B = (B o, Blh,r (0 < A :S
1, 1 :S r :S 00) the corresponding interpolation spaces (constructed by the real
method of Lions and Peetre), then for any bounded linear operator T such that
T : A o -+ B o (with norm Mo) and T : Al -+ BI (with norm Md it holds that
T : A -+ B with norm M = M~-A Mt. This can be written in terms of inequalities
in the following way:
If
(2.2)
IITfliBO :S MollfllA o,
then
(2.3)
If we choose for A o the homogeneous Sobolev space WI,P(w) of all function f for
which the expression IIf'IIp,w is a norm and for B o the weighted Lebesgue space
Lq(wo) (with norm 11 . IIq,wo) then the Hardy inequality (1.7) is the first inequality
in (2.2) (with T equal to the identity operator). The choice Al = BI = LP(w)
leads trivially to the second inequality in (2.2), and consequently, the resulting
inequality (2.3) can be regarded as the desired fr action al order Hardy inequality
(of the type (1.8)).
The main problem in applying this approach is that the corresponding interpolation spaces
B = (U(wo), LP(w)h,n
GENERALISATIONS AND REFINEMENTS OF HARDY INEQUALITY
275
can be described in a suitable (simple) form only for some special cases, e.g.
(i) for the unweighted case w = 1 we have
· l,p , LP) .x,r -- iJ)..p,T'
(w
(2.4)
the Besov space;
(ii) for r = p(,X) where l/p('x) = 'x/p + (1 - 'x)/q, we have
(U(wo), LP(w)h,r = LP()..) (w)..)
with
_
p()")(l-)..)/q p()..) ../p
w).. - Wo
w
.
For more details we refer to [8] and here, we will mention only some special cases
for illustration. Let us emphasise that this interpolation approach can be shortly
described as folIows: Using the Hardy inequality (1.7) and a trivial imbedding
(= the two inequalities in (2.2)) we obtain the fractional order Hardy inequality
(= (2.3)).
Proposition 2.1. Let 1 < p ~ q < 00, l/p('x) = (1 - 'x)/p + 'x/q, 0 < ,X < 1,
-00 < a < b ~ 00, and let f be a differentiable function on (a, b) such that
f(a) = o. If
(2.5)
sup
a::;x::;b
(l
b
wo(t) dt
)l/q
x
(x - a)l/ pl = Cl < 00
then for any 8 > 0
(2.6)
l If(x)IP()..)w~p()..)/q(x)
b
dx
~ C [6 r)..p()..)-l ([b If(x + t) - f(x)IP dx Y()..)/P dt.
Jo
Ja- t
Example 2.1. If p = q, then p('x) = p and (2.6) reads
(2.7)
l If(x)IPw~(x) ~ 1(l~t
b
dx
C
6
If(x
+t~P~ f(x)IP dX) dt.
This is a "fractional" version of the Hardy inequality
In particular, if a = 0, b = 00, 8 = 00 and wo(x) = x- P, then (2.7) essentially
coincides with the Jakovlev-Grisvard inequality (2.1).
H. HEINING, A. KUFNER AND L. E. PERSSON
276
Remark 2.1. Proposition 2.1 deals with the special case w(x) == 1 where we are able
to describe the corresponding interpolation space - see (2.4). Condition (2.5) then guarantees that the Hardy inequality (1.7) holds for our special choice of weights, i.e., (2.5)
makes sure that the first inequality in (2.2) holds.
The approach just described allows to derive fractional order Hardy inequalities
of type (1.8).
Choosing now Al = BI = wI,q(w) with the trivial inequality lIu'lIq,w ~ Cllu'lIq,w
for the second inequality in (2.2), we can obtain as a result a fractional order Hardy
inequality (2.3) which corresponds to (1.9). The next proposition deals again with
the special case Wo (x) == 1.
Proposition 2.2. Let 1 < p ~ q < 00, l/p(>..) = (1 - >")/p + >../q, 0 < >.. < 1,
-00 ~ a < b < 00, and let f be a diJJerentiable junction on (a, b) such that
f(a) = O. If
then for any 8 > 0
(2.8)
1
6
cÄp(Ä)-1 (l~t If(x + t) - f(xW dxY(Ä)/q dt
~C
l
b
1!'(x)IP(Ä)W(I-Ä)P(Ä)/P(x) dx.
Example 2.2. If p = q, then p(>..) = p and (2.8) reads
1(l~t
6
If(x +t~P;1 f(x)IP dX) dt ~ C
l
b
1!,(x)IPwl-Ä(x) dx.
This is again a "fractional" version - of the type (1.9) - of the Hardy inequality
l
b
If(x)IP dx
~C
l
b
1!'(x)IPw(x) dx.
Remark 2.2. The double integral which appears at the right hand side of (2.6) or at
the left hand side of (2.8) is different from the expression h,r(u, W) (defined in (1.10))
which appears at the corresponding place in (1.8) or (1.9), respectively: Namely, we
have a mixed norm in this integral. This is closely connected with the problem of the
description of the corresponding interpolation spaces. In the next section we will avoid
this difficulty by using another more direct approach. But before, let us give one more
example showing that the approach via interpolation can be used also for the moredimensional case.
Example 2.3. For every differentiable radial function F on]R.N (Le., satisfying
F(x) = F(lxl)) such that F(O) = 0, the following inequality holds:
(2.9)
[
JRN
Ixl-ÄNp!F(x)IP dx ~ C [
[
JIRN JIRN
IF(x) - F(y)IP dxdy
Ix - ylN+Äp
GENERALISATIONS AND REFINEMENTS OF HARDY INEQUALITY
277
provided 1 < P < 00 and 0 < A< l/p.
Remark 2.3. (i) Notice that for N = 1 we obtain the Jakovlev-Grisvard inequality
(2.1).
(ii) Inequality (2.9) is a special case of a more general inequality which can be derived
analogously as in the one-dimensional case. We only replace the Hardy inequality (1.7)
by its N-dimensional analogue
(~N Wo(x)IF(xW dX) l/q ~ C(~N W(x)IV F(x)I PdX) l/p
(2.10)
which can be derived from the N-dimensional Hardy inequality (1.4) (see [4]). Inequality
(2.10) plays now the role of the first inequality in (2.2) and, combined with a trivial
imbedding, it leads to the corresponding N-dimensionalfractional order Hardy inequality
via interpolation.
Inequality (2.9) is its special case for p = q, Wo(x) = lxi-Np and W(x) == 1.
3. Hardy Inequalities of Fractional Order Via Direct
Methods
In this section we are looking mainly for an inequality of the form
1o
(3.1)
00
lu(x)IPwo(x) dx :::; CP
1 1 Ix - yl~(~)IP
00
0
00
lu(x) -
0
+P
W(x, y) dx dy
which is inequality (1.8) for q = r = p and (a,b) = (0,00). Partly guided by the
results of the previous section we will also consider the more general mixed norm
inequality of the form
(1
00
lu(xWwo(x)dx)l/ Q :::;
c(1 (1 '~~x~~,~i~~'P
OO
00
W(x,y)dXr/p dyf/q.
We will omit the proofs since most of the results can be found in [6].
(a) For the special case when the weight function W(x, y) in the right hand side
of (3.1) depends on Ix - yl, Burenkov and Evans [3] recently proved the following
interesting result:
Theorem 3.1. Let 0< p < 00, let w be a weight junction on (0,00) and define
v(x) :=
1
00
w(t) dt.
Suppose that there exists a constant c, 1 < c < 2, such that
v(t) :::; cv(2t)
Then for all u E LP(v)
1
00
lu(x)IPv(x) dx :::; CP
for all
11
00
00
t > O.
lu(x) - u(y)IPw(lx - yl) dxdy.
(b) For the case when the weight function W on the right hand side of (3.1) does
not depend on y, W(x, y) = W(x), the following result holds (slightly improving
a result from [8]):
H. HEINING, A. KUFNER AND L. E. PERSSON
278
Theorem 3.2. Let 1 < P < 00 and A ~ -1/p. Furlhermore, assume that the
junction u satisfies
l1
lim -
x-too X
x
0
u(t) dt = O.
Let Wo and Wl be weight junctions on (0,00) satisfying
B := ~~~(lX wo(t) dt)
Then, tor every ß ~ 0,
1
lu(x)IPwo(x) dx ~ CP
00
o
(1 w~-p'
00
1r
lu(x) - u(y)IP W(x) dy dx,
Ix - ylß
00
Jo
0
(t) dty-l < 00.
where W(x) = xß-1wo(x) + xß-1-PWl (x) and CP = 2P- 1 max(l, Cp) with Cp ~
BPP(p - l)l-p.
Example 3.1. (i) Applying Theorem 3.2 with wo(x) = x a - Ap and Wl(X) =
x a - AP+p , we obtain for ß ~ 0, A ~ -1/p and Q > AP - 1
(3.2)
1
00
o
lu(x)IPxa-Ap dx ~ CP
11
x
00
0
0
lu(x) - u(y)IP xß-I-Ap+a dy dx.
Ix - ylß
(ii) Applying Theorem 3.1 with w(t) = t a - Ap - 1 we find that for Q < AP we have
(3.3)
1
00
o
lu(x)IPxa-AP dx ~ CP
11
00
0
00
0
lu(x) - u(y)IP Ix _ yla dxdy.
Ix - yl1+ AP
Moreover, Theorem 3.1 cannot be usedfor any Q ~ AP (since then v(x) == 00). But
using part (i) we see that (3.3) holds also if AP ~ Q < AP + 1 : This fact follows
from (3.2) putting there ß = 0 and noting that x- 1- Ap + a ~ Ix - yl-l-AP+a for all
y,O < Y ~ x and -1 - AP + Q < O.
(iii) More generally, using Theorem 3.2 with ß = 0 and with W(x) strictly decreasing we obtain an inequality of the type
and this inequality cannot be obtained in general using Theorem 3.1, e.g. in the
case that the integral fxoo w(t) dt is divergent.
(c) Now we shall consider the mixed norm case with a special weight: W(x,y) =
w(x)vP/q(y) with w and v weight functions on (0,00). Define
V(y):=
Wp,q(Y) :=
(t l (~~j)
Y
The main result reads as follows:
l
Y
v(x) dx,
l/(p-l) dX) -q/p'
(t l
Y
v(x) dX) qv(y).
GENERALISATIONS AND REFINEMENTS OF HARDY INEQUALITY
279
Theorem 3.3. Let 1 < pS q < 00, A 2: -I/p, W>.(X) = Wp,q(X)X->.q and
(10 lu(xWw>.(x) dx f/q S 1 ~ K (10 (10 I~~x~ ~,~i~~P w(x) dx
00
00
00
r/
p
v(y) dy) I/q,
provided K = Cp,qq/(q - I)I/ql < 1.
Considering the case p = q in Theorem 3.3 we obtain
Corollary 3.1. 1/1< P < 00, A 2: -I/p, w>.(x) = wp,p(x)x->'P and
Cp := sup
r>O
(1
00
r
w>.(x)V-P(x)dx)
I/p
(
l
r
0
_
I
(w>.(x)v-p(X))1 p dx
)
I/pi
< 00,
then
(3.4) (IoOO,u(xWW>.(X)dxf/P S l~K(looo
10 '~~x~~,~i~~P w(x)v(y)dxdy)I/P
00
provided K = Cpp/(p - I)Ifp' < 1.
Consequently, (3.4) may be regarded as a fractional order Hardy inequality of the
type (1.8) with the weight W(x,y) = w(x)v(y) on the right hand side.
Remark 3.1. Applying Corollary 3.1 with w(x) == 1, v(y) == 1 we find that if 1 < P < 00,
,x > l/p, then
(3.5)
( {OO lu(x)IPx->'Pdx)I/P S ,xp+p-1 ({OO (CO lu(x) -u(y)I P dXdy)I/P,
Jo
Jo Jo
,xp - 1
Ix - yll+>'P
cf. inequality (2.1). Moreover, it is not difficult to see that (3.5) holds with 0 < ,x < 1
and with the constant ()../2)/(1 - ,x) for p = l.
(d) For the case of a general weight function W (x, y) we have finally the following
result (far p = q):
Theorem 3.4. Let W(x,y) be a non-negative measurable /unction on (0,00) x
(0,00), locally integrable in both variables separately. Let 1 < P < 00 and A 2:
-I/p. (i) Denote
1 f'"
) I-p
W(x) = (~Jo WI-p ' (x, t) dt
and wo(x) = W(x)x->.P. 1/
C p := SUp
r>O
(1
00
r
W(x) dx
Xp{>'+I)
)1/P(l WI-p (X)X>'P dx
r -
0
I
I
)
I/pi
< 00
H. HEINING, A. KUFNER AND L. E. PERSSON
280
and K = Cpp/(p - l)l/ pl < 1, then /or u E LP(wo)
(10 wo(x)lu(x)I Pdx)l/ P ::; l~K(fooo 10 1~~x~~I~i~;PW(X,Y)dXdyf/p·
00
(3.6)
00
(ii) Denote
W(y) =
(t
i Y W 1- p' (t,y) dtf-P
and wo(x) = W(x)x- AP . 1/
C p := ~~~(lOO y~A~~) dy) l/p (ir w 1 - P' (y)yApl dy) I/p' < 00
and K = Cpp/(p - l)l- pl < 1, then (3.6) holds with K replaced by K.
Remark 3.2. More results and extensions can be found in [6]. E.g. it is possible to
prove an Orlicz norm version and a multidimensional fractional order Hardy inequality
(see Section 2, Example 2.3).
(e) Up to now, we dealt in this section with inequalities ofthe type (1.8). Therefore,
let us finally consider an inequality of the type (1.9).
Theorem 3.5. Let 1 < r,p < 00. Let w(x) and W(x,y) be weight /unctions on
(a, b) and (a, b) x (a, b), respectively. Denote
V(x) =
and suppose that
(3.7)
C:=
l
x
w 1 - P' (t) dt
- V(y)lr/ p'
) l/r
(Jarb JarblV(x)Ix_yj1+Ar
W(x,y)dxdy
<00.
Then
b
b
(l l
I~~x~ ~,~i~~,r W(x,y) dxdy
f/r ~
b
C(l lu'(x)IPw(x) dx
f/
p
•
Prao/. By using Hölder's inequality we find that
h,r(u, W) =
::;
(l l Ix b
b
(l l Ix - yl- a
b
a
b
1
A
YI- 1- Ar W(X,y)lfv x u'(t)w1/P(t)w-1/P(t)dtlr dxdy f/r
rW(x, y) li
ly
X
lu' (t)IPw(t) dt
Ir/PI lyi w I(t) dt Ir/pI dx dy )l/r
X
1
-p
::; c(l lu'(t)IPw(t)dtf/ p·
b
Remark 3.3. Condition (3.7) is sufficient for the validity of the fractional order Hardy
inequality (1.9). If inequalities (1.8) and (1.9) hold simultaneously, we have that
IIUllq,wo ::; C1Jr,A(U, W) ::; C2I1u'lIp,w,
which may be regarded as a certain "refinement" of the usual Hardy inequality (1.7).
Open quest ion. Most of the conditions mentioned in this section have been only suffident. Find necessary and sufficient conditions of the validity of the inequalities considered.
GENERALISATIONS AND REFINEMENTS OF HARDY INEQUALITY
281
4. An Extension of a Result of Muckenhoupt and Sawyer
Let 1 < p ~ q < 00 and let H 2 be defined by (1.11). In [12] Sawyer proved that
the inequality
(4.1)
(1 1
00
00
W o(x,y)[(H2 f)(x,yW dydx f,q
~ c(l OO
1
00
W(X,Y)fP(x,Y)dYdxf/ P
holds for all nonnegative functions f if and only if the following three conditions
are satisfied:
(4.2)
Co :=
sup
x>O,y>o
(
1
00 100
x
y
Wo(s,t)dtds)
X
(4.3)
(4.4)
11(1 1
x
Y
1 1 (1
00
t
8
00
I/q
x
(l l w
0
x
Y
WI-pl(17,T)dTdorWo(s,t)dtds
~ C8(l
00
l - p ' (s,t)dtds ) I/p' < 00,
0
X
1
Y
WI-P'(s,t)dtdsf/P,
lOOWo(17,T)dTdaY'wl-P'(S,t)dtds
~ cö'
(1 1
00
00
Wo(s, t) dt ds
y' /q'.
Muckenhoupt observed already in [10] that (4.2), which is an analogue of the
corresponding condition for the one-dimensional case, is necessary for (4.1) to
hold but not sufficient.
Before formulating our results let us introduce some notations:
As already mentioned BM(X) denotes a ball in]RM with center at origin and radius
lxi. We also consider its complement BZ.(x) = ]RM\BM(X). Moreover, we will
use polar coordinates and write x E ]RM as
x = xox'
with
Xo = lxi E (0,00),
where EM-I denotes the unit sphere in ]RM.
Moreover, for f = f(~, "'), ~ E ]RM, '" E ]RN, we define
(4.5)
F(~o,"'o) =
x' E ~M-I,
h h f(~oe,"'o",')~f;1-I",:-1
d",' de
E M - 1 EN-l
and note that the Hardy operator H 2 from (1.13) can be written as
(4.6)
l lYO F(~o,"'o)d"'od~o
(Hd)(x,y) = 0
xo
0
with x = xox', Y = Yoy', Xo = lxi and Yo = lyl· Notice that the function
(H2 f)(x, y) is radial with respect to both variables, Le., it depends only on Xo = lxi
and Yo = lyl·
Now, our extension of the Muckenhoupt-Sawyer result reads:
282
H. HEINING, A. KUFNER AND L. E. PERSSON
Theorem 4.1. Let 1 < P :::; q < 00. Let Wo = WO(X,y) and W = W(x,y) be
weight functions on jRM x JRN and let W = W(x, y) be radial with respect to both
variables. Then the Hardy-type inequality
(4.7)
(LM LN Wo(x, y)[(H f) (x, yW dy dx flq
: :; c(LM LN W(x,y)fP(x,y) dydx f/P
2
holds for all measurab1e nonnegative functions f = fex, y) if and on1y if the fo1lowing three conditions are satisfied
(4.8)
A:=
sup ({
{
a>O,ß>O Axl?a J1yl?ß
Wo(x,y) dy dxflq x
X
([ 1
Ixl:Sa lyl:Sß
(4.10)
{
{
({
J B<it(x) J Bf,(y)
(
J B<it({) J Bf,('fJ)
I
wl-p (x, y) dy dx
)I/pl
< 00,
Wo(a,T)dTdar'WI-pl(~'1J)d1Jd~
:::; c l ( {
(
JB<it(x) JBf,(y)
WO(~'1J)d1Jd~r'fq'
for every x E jRM and y E JRN .
Proof. Since W(x, y) is radial we can write W(x, y) = W(xox', YOy') = W I (xo, yo).
We denote
WI(xo,Yo) =
Wo(xox',yoy')dy'dx',
l l
EM - 1 E N - 1
and rewrite the conditions (4.8)-(4.10) in the following way:
(4.8')
sup
({oo /.00 WI (~o, 1JO)~tt"-I1Jb"-1 d1Jo d~o) Ilq x
XO,Yo>O Jxo
Yo
(l 01
X
X
0
0
YO
I-pi
WI
M-I
(~o, 1Jo)~o
N-I
)I/pl
1Jo
d1Jo d~o
< 00,
GENERALISATIONS AND REFINEMENTS OF HARDY INEQUALITY
283
(4.10')
(i) Sufficiency. By rewriting the left hand side of (4.7) in polar coordinates and
using the above mentioned Muckenhoupt-Sawyer result together with Hölder's
inequality we obtain
kM kN
Wo(x, y) ((Hd)(x, y)
=
(*)
:::;
dy dx
1 1 J J x~-lyö"-lWO(Xox',yOY')X
(l l F(~o, d~o)
1 X~-lYö"-lWl(XO'YO)(Jor F(~O,1]o)d1]od~o)
c (1 1 x~M
ya
00
00
0
q
00
{
o
x
=
r
{
0
EM _1 EN _1
XO
YO
{oo
Jo
00
00
1]0) d1]o
q dy' dx' dyo dxo
-
o (YO
Jo
-l)(l-p)
q
(1 1
00
o
{
X ( J.."
00
0
cq
q/P
Xo(M-l)(l-p) Yo(N-l)(l-p)W1 (xo, Yo ) x
(
J.."
E M - 1 EN-l
=
dYodxo
N -l)(l-p) x
X W 1 (xo, yo)FP(xo, Yo) dyo dxo )
-- c
q
,
,
,
,)P
)q/P
M 1 N 1
Xo - Yo - f(xox, YoY ) dy dx
dyo dxo
(l 1 x~-lyö"-lWl(XO'YO)X
OO
00
x (~
~
f(xox',yoY') dy' dx'Y dYOdxof/P
EM - 1 EN - 1
(**)
:::; c q
x ~
(l 101? x~-lyö"-lWl(XO'YO)X
OO
~
EM - 1 EN - 1
fP(xox',yoy')dy'dx'I~M-lIP/pll~N-lIP/pl dyodxof/ P
= Cql~M-llq/pll~N-llq/pl
(kM kN W(x, y)fP(x, y) dydx) q/p,
and (4.7) holds. Note that the conditions (4.8)' - (4.10)' may be regarded as
the conditions (4.2)-(4.4) for a special choice of weight functions. Therefore the
inequality (*) follows from (4.1) applied for the function F from (4.5) and the
Hardy operator from (4.6) with this special choice of weight functions. In (**) we
have simply applied the Hölder inequality for the "inner" integral J~
J~
.
L..M-l
L..N-l
284
H. HEINING, A. KUFNER AND L. E. PERSSON
(ii) Necessity. We suppose that (4.7) holds and choose
f(x, y) = W l - p ' (x, y)X(o,a) (lxI)X(o,ß) (Iyl)
with 0: > 0, ß > 0 and obtain that
Therefore, by dividing with
(4.8) follows since C ~ A.
In order to derive (4.9) and (4.10) we replace f by fW l - P' in (4.7) and rewrite
this inequality in the form
(4.11)
(kM LN Wo(x,y)[H (JW P')(x,yW dYdX)
: ; C(lM LN w (X,Y)fP(x,Y)dYdX) l/P,
2
l-
l/q
l - p'
which by duality can equivalently be written as
for all measurable functions 9 ~ O. By now applying (4.11) with the function f
GENERALISATIONS AND REFINEMENTS OF HARDY INEQUALITY
285
chosen as f(x, y) = X(O,a) (lxI)X(o,ß) (Iyl) we obtain that
{
{
Jlzl~a JIYI~ß
W o(x,y)[H2 (fW 1-P')(x,yW dydx
={
(
Jlzl~a JIYI~ß
:::; {
{
Wo(x, y) ( {
{
JI(I~lzl JI'1I~IYI
r
W 1- p' (~, 77) d77 d~
dx dy
W o(x,y)[H2 (fW 1- P')(x,y)]q dydx
JRM JRN
:::; c q(kM kN w1-p' (x, y)fP(x, y) dydx) q/p
=c q( {
(
Jlzl~a JIYI~ß
w1-p' dydx) q/p
and (4.9) holds. In a similar way (4.10) follows by applying (4.12) with g(x, y) =
X(a,oo) (lxI)X(ß,oo) (Iyl)· The proof is complete.
Remark 4.1. (i) Notice that in the proof of the necessity of the conditions (4.6)-(4.8)
we did not use the assumption that the weight W was radial.
(ii) Ohviously, the conditions (4.8)-(4.10) are necessary not only for the case p :5 q hut
also for the case q < p.
(iii) By analysing our proof of Theorem 4.1 we see that similar results can he derived
also for the adjoint Hardy type operator
(H2f)(x, y) = {
(
JBf.t(z) JBf.(y)
f(~, TI) d~ dTl,
as weH as for the "mixed" operators
or
5. On the Mixed Norm Case
We will here derive an inequality of the type (1.14) for the special case that the
weight functions Wo and W have "separated" variables, Le., when we can write
(5.1)
Then we can rewrite the inequality (1.14) as
(5.2)
(kM W 1(x) (LN W (y)[(Hd)(x, yW2 dy) ql/q2 dX)
: :; c(LM Ul(X) (LN U (y)fP2(y) dy Yl/P2 dxf/ P1 .
l/ql
2
2
286
H. HEINING, A. KUFNER AND L. E. PERSSON
Now we can proceed similarly as in [1]: Let us consider the inequalities
(5.3)
(kM W I (x) (Hg(X))ql dxflql ~ Cl (kM UI(X)gPl(X)dxf/Pl
for g = g(x) 2:: 0, xE ]RM, and
for h = h(y) 2:: 0, Y E ]RN, where H is the Hardy operator defined in (1.3) (of
course on ]RM instead of]RN in the case of inequality (5.3)).
First we can derive almost immediately a necessary condition:
Theorem 5.1. A necessary condition /or the validity 0/ the mixed norm inequality
(5.2) with "separated" weights W I , W 2, UI , U2 according to (5.1) is the validity 0/
at least one 0/ the inequalities (5.3) and (5.4).
Proof. By choosing in (5.2) /(s, t) = g(s)h(t) we get (H2I)(x, y) = (Hg) (x)
(Hh)(y). Moreover, the left hand side in (5.2) is then a product of the left hand
sides in (5.3) and (5.4), and, similarly, the right hand side in (5.2) is a product of
the right hand sides in (5.3) and (5.4). Thus, if both inequalities (5.3) and (5.4)
are not valid, then the inequality (5.2) cannot be valid either.
Note that in the foregoing assertion no explicit restrictions concerning the admissible values of the parameters PI, ql, P2 and q2 were made. However, for deriving
sufficient conditions for (5.2), some restrictions on these parameters are needed.
Theorem 5.2. Let 1 < Pi < 00,
that either
°< qi <
00, i
= 1,2, and assume in addition
(5.5)
or
(5.6)
Then a sufficient condition /or the validity 0/ the mixed norm inequality (5.2) is
that both 0/ the inequalities (5.3) and (5.4) hold.
Proof. (i) Suppose first that (5.5) holds and denote
Then the Hardy inequality (5.4) yields
GENERALISATIONS AND REFINEMENTS OF HARDY INEQUALITY
287
for every xE ]RM where C 2 > 0 is independent of h (and hence independent of x).
Since
(Hd) (x, y) =
r
h(x,''1) d'fJ
JBN(Y)
we can use (5.7) and estimate the left hand side of (5.2) by
Moreover, since condition (5.5) implies that QI/P2 ~ 1 we can use the Minkowski
integral inequality and obtain the following upper estimate of the last integral:
We also note that the Hardy inequality (5.3) yields
for every y E ]RN, where C2 is independent of y. By using this estimate in (5.8)
and applying again the Minkowski integral inequality, this time for P2/Pl ~ 1 (due
to (5.5)) we obtain that the left hand side of (5.2) is estimated from above by
which is the right hand side of (5.2) with C = C1 C2 •
(ii) Assume now that (5.6) holds. The proof only consists of modifications of the
proof presented in (i), Le., in this case we use the Minkowski integral inequality
at the beginning instead of at the end and apply the Hardy inequality (5.3) to the
function
288
H. HEINING, A. KUFNER AND L. E. PERSSON
Remark 5.1. (i) In particular, Theorem 5.1 states some sufficient conditions on Pi and
qi under which the operator H2 defined by (1.14) maps the mixed norm space
X = LPI (lRM , Ul; LP2 (lRN , U2»
continuously into the corresponding mixed-norm space
If we "interchange the order of the spaces" and consider, e.g., the operator H2 as an
operator from X into
Yl = LQ2(lRN , W2;LQl(lRN , Wl»
(which means that we have to consider the expression
(kN W2(Y) (kM Wl(x)((H2f)(X,y»Ql dxr
2
/
Q1
dy
f/ Q2
at the left hand side of (5.2» then we can weaken the assumptions on Pi and qi since we
can now avoid the application of one of the Minkowski integral inequalities in the proof
(e.g. in the case just mentioned we can replace the condition (5.6) in Theorem 5.2 by
the weaker condition "PI ~ q2").
(ii) Similar results can again be derived also for the associated Hardy operators
and H2 defined in Remark 4.1 (iii).
H:;, H2
Acknowledgement. The research of the second author was partially supported by the
Grant Agency of Czech Republic, grant No. 201/94/1066, and by the Grant Agency of
the Czech Academy of Science, grant No. 1019506, which is gratefully acknowledged.
References
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mixed norms, Analysis 15 (1995), 91-98.
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3. V. Burenkov and W. D. Evans, Weighted Hardy's inequality for differences and the extension
problem lor spaces with genemlized smoothness (to appear).
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Inequalities 7 (Oberwolfach, 1995) (C. Bandie, W. N. Everitt, L. Losonczi, W. Walter, eds.),
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Math. 35 (1) (1979), 69-83.
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DISCRETE INEQUALITIES OF WIRTINGER'S TYPE
GRADIMIR V. MILOVANOVIC and IGOR Z. MILOVANOVIC
Faculty 01 Electronic Engineering, Department 01 Mathematics, P.Q. Box 73,
18000 Nis, Yugoslavia
Abstract. Various discrete versions of Wirtinger's type inequalities are considered. A
short account on the first results in this field given by Fan, Taussky and Todd [10] as weH
as some generalisations of these discrete inequalities are done. Also, a general method
for finding the best possible constants An and B n in inequalities of the form
n
n
n
An :~=>kxi :5 L rk(xk - XkH)2 :5 Bn LPkxi,
k=O
k=l
k=l
where p = (Pk) and r = (rk) are given weight sequences and :z: = (Xk) is an arbitrary
sequence of the real numbers, is presented. Two types of problems are investigated and
several coroHaries of the basic results are obtained. Further generalisations of discrete
inequalities of Wirtinger's type for higher differences are also treated.
1. Introduction and Preliminaries
In the well-known monograph written by Hardy, Littlewood and P6lya [13, pp.
184-187] the following result was mentioned as the Wirtinger's inequality:
Theorem 1.1. Let f be a periodic function with period (211") and such that l' E
L 2(0,211"). 11 1r f(x) dx = 0 then
f:
J
21r
(1.1)
f(x)2 dx ~
o
J
21r
j'(x)2 dx,
0
with equality in (1.1) il and only il 1(x) = A cos x + B sin x, where A and B are
constants.
Also, this inequality ean be found in the monograph of Beekenbaeh and Bellman
[4, pp. 177-180] and, especially, in one written by Mitrinovic in eooperation with
Vasic [25, pp. 141-154], including many other inequalities of the same type. The
proof of W. Wirtinger was first published in 1916 in the book [5] by Blasehke.
However, inequality (1.1) was known before this, though with other eonditions on
1991 Mathematics Subject Classification. Primary 26D15j Secondary 41A44, 33C45.
Key woms and phrases. Discrete inequalitiesj Differencej Eigenvalues and eigenvectorsj Best
constantsj Orthogonal polynomials.
This work was supported in part by the Serbian Scientific Foundation, grant number 04M03.
289
G. V. Milovanovic (ed.), Recent Progress in lnequalities, 289-308.
© 1998 Kluwer Academic Publishers.
G. V. MILOVANOVIC AND I. Z. MILOVANOVIC
290
the function f. The French and Italian mathematicalliterature do not mention the
name of Wirtinger in connection with this inequality. A historical review on the
priority in this subject was given by Mitrinovic and Vasic [24] (see also [25-26]).
They have mentioned various generalisations and variations of inequality (1.1), as
weIl as possibility of applications of such kind of inequalities in many branches in
mathematics as Calculus of Variations, Differential and Integral Equations, Spectral Operator Theory, Numerical Analysis, Approximation Theory, Mathematical
Physics, etc. Under some condition of f, there are also many generalisations of
(1.1) which give certain estimates of quotients of the form
b
J w(X)f(X)2 dx
a
JJ w(x, y)f(x, y)2 dxdy
D
b
J f'(X)2 dx
a
where w is a weight function (in one or two variables) and D is a simply connected
plane domain.
There are various discrete versions of Wirtinger type inequalities. In this survey
we will deal only with such kind of inequalities.
The paper is organised as follows. In Section 2 we give a summary on the first
results in this field given by Fan, Taussky and Todd [10] as weIl as some generalisations of these discrete inequalities. In Section 3 we present a general method for
finding the best possible constants An and B n in inequalities of the form
n
n
n
An LPkX~ ~ L rk(xk - Xk+1)2 ~ B n LPkX~,
k=l
k=O
k=l
where p = (Pk) and r = (rk) are given weight sequences and x = (Xk) is an
arbitrary sequence of the real numbers. This method was introduced by authors
[19] and later used by other mathematicians (see e.g., [1] and [36]). In the same
section we give several corollaries of the basic results. Finally, generalisations
of discrete inequalities of Wirtinger's type for higher differences are treated in
Section 4.
2. Discrete Fan-Taussky-Todd Inequalities and Some
Generalisations
The basic discrete analogues of inequalities of Wirtinger were given by Fan, Taussky and Todd [10]. Their paper has been inspiration for many investigations in
this subject. We will mention now three basic results from [10]:
Theorem 2.1. If Xl, X2, . .. ,Xn are n real numbers and Xl = 0, then
n-l
(2.1)
""(
~ Xk - Xk+l
k=l
n
)2' 2
2
~ 4sm 2(2n -1) ~ Xk'
k=2
7r
""
DISCRETE INEQUALITIES OF WIRTINGER'S TYPE
291
with equality in (2.1) if and only if
. (k - l)'Ir
xk=Asm 2n-1'
k = 1,2, ... ,n,
where A is an arbitrary constant.
Theorem 2.2. 1fxo(= 0), X},X2, ... ,Xn , XnH(= 0) are given real numbers,
then
(2.2)
n
~(
L..J Xk - XkH
)2. 2
'Ir
n
~
2
~ 4sm 2(n + 1) L..JXk,
k=O
k=l
with equality in (2.2) if and only ifxk = Asin~, k = 1,2, ... ,n, where Ais
n+1
an arbitrary constant.
Theorem 2.3. 1f Xl, X2, ... , Xn , XnH are given real numbers such that Xl = xnH
and
(2.3)
then
(2.4)
The equality in (2.4) is attained if and only if
2k'lr
. 2k7r
Xk = Acos - - + Bsm-,
n
n
k = 1,2, ... ,n,
where A and B are arbitrary constants.
Let A be a real symmetrie matrix of the order n, and R be a diagonal matrix of
the order n with positive diagonal elements. For the generalised matrix eigenvalue
problem
(2.5)
A:z: = >'R:z:,
:z: = [Xl
the following results are weIl known (cf. Agarwal [1, Ch. 11]):
10 There exist exactly n real eigenvalues >. = >'v, v = 1, ... , n, which need not be
distinct.
20 Corresponding to each eigenvalue >'v there exists an eigenvector :z:V which can
be so chosen that n vectors :z:l, ... ,:z:n are mutually orthogonal with respect to
the matrix R = diag (ru, ... , T nn ), i.e.,
(:z:i)T R:z:i =
n
L rkkx~x{ = 0
k=l
(i i- j),
G. V. MILOVANOVIC AND 1. Z. MILOVANOVIC
292
In partieular, these vectors are linearly independent.
3° If A is a tridiagonal real symmetrie matrix of the form
b2
b2
a3
Hn(a,b) =
(2.6)
0
a1 b1
b1 a2
0
bn- 1
bn- 1
an
where a = (al, ... ,an), b = (b 1, ... ,bn- 1) and b~ > 0 for k = 1, ... ,n -1, then
the eigenvalues Av of the matrix A are real and distinct.
4° If R = I and the eigenvalues Av of Aare arranged in an increasing order, Le.,
Al ~ ... ~ An, then for any vector X E !Rn, we have that
(2.7)
n
where (x, y) = E XkYk is the scalar product of the vectors
k=l
Ynf·
In the case Al < A2 the equality Al (X, X) = (Ax, X) holds if and only if X is a
scalar multiple of Xl. Similarly, if An > An-1 the equality (Ax,x) = An(X,X)
holds if and only if X is a scalar multiple of x n .
Further, for any vector X orthogonal to Xl ((X, Xl) = 0), we have
(2.8)
If Al < A2 = A3 < A4, then a vector X orthogonal to Xl satisfies the equality
A2 (x, x) = (Ax, x) if and only if X is a linear combination of x 2 and x 3 •
5° If the real symmetrie matrix A is positive definite, Le., for every nonzero
X E !Rn, (Ax,x) > 0, then the eigenvalues Av (v = 1, ... ,n) are positive. In
a partieular case when R = I and A = Hn(a, b) is positive definite, then the
eigenvalues Av (v = 1, ... ,n) can be arranged in a strictly increasing order, 0 <
Al< ... < An.
Note that inequalities (2.1), (2.2) and (2.4) are based on the left inequality in (2.7)
(Le., (2.8)). The right inequality in (2.7) has not been used, so that in [10] we
cannot find some opposite inequalities of (2.1), (2.2) and (2.4). As special cases of
certain general inequalities, the opposite inequalities of (2.1), (2.2) and (2.4) were
first proved in [19] (see also [2]).
Using a method similar to one from [10], Block [6] obtained several inequalities
related to (2.1), (2.2) and (2.4), as weH as some generalisations of such inequalities.
For example, Block has proved the foHowing result:
DISCRETE INEQUALITIES OF WIRTINGER'S TYPE
293
Theorem 2.4. For real numbers Xl, X2, ... ,xn (= 0), xnH = Xl, the inequality
n
n
k=l
k=l
~)Xk - Xk+I)2 ~ 4sin ;n LX~
(2.9)
holds, with equality in (2.9) if and only if Xk = A sin(k1l" In), k = 1,2, . .. ,n, where
A is an arbitrary constant.
A number of generalisations of (2.1), (2.2) and (2.4) were given by Novotna ([27]
and [29]). We mention here three of them.
Theorem 2.5. For real numbers Xl, X2, ... ,Xn satisfying (2.3), the inequality
(2.10)
holds, with equality in (2.10) if and only if Xk = Asin((2k - 1)1I"/(2n)), k =
1,2, ... , n, where A is an arbitrary constant.
Theorem 2.6. Let n = 2m and let Xl, X2, ... , Xn , XnH = Xl be real numbers
such that (2.3) holds. Then
n
n
~(
)2 ~ 4·
11" ~ 2
. 11" (.
211"
. 11") (Xm +X2m )2 ,
~Xk-Xk+l
sm2 -~Xk+nsmsm--smn
k=l
k=l
n
n
n
with equality if and only if
Xk = Acos(2k1l"In) + Bsin(2k1l"ln),
k = 1,2, ... ,n,
where A and Bare arbitrary constants.
Theorem 2.1. For real numbers Xl, X2, ... ,Xn satisfying (2.3), the inequality
n-l
~(
~ Xk-Xk+l
k=l
n
)2 ~ 4·
11" ~ 2 2nsm
• 11" (.
11"
• 11") (
)2
sm2 2n~Xk+
2n sm;;:-sm 2n XI+X n
k=l
holds, with equality if and only if Xk = Asin((2k -1)1I"/(2n)), k = 1,2, ... ,n,
where A is an arbitrary constant.
Using some appropriate changes, Novotna [27] showed that inequalities (2.1), (2.2)
and (2.10) can be obtained from (2.4). She proved the basic Theorem 2.3 using the
real trigonometrie polynomials. Namely, she used the fact that for every number Xi
there exist the Fourier coefficients C k and C; (k = 0,1, ... , m; j = 1, ... , m - 1)
such that
1 ~ i ~ n.
G. V. MILOVANOVIC AND I. Z. MILOVANOVIC
294
For details on this method see for example [1].
New proofs of inequalities (2.1), (2.2) and (2.4) were given by Cheng [8]. His
method is based on a connection with discrete boundary problems of the SturmLiouville type
(2.11)
~(p(k -1)~u(k -1)) + q(k)u(k) + Ar(k)u(k)
u(O) = Au(I), u(n + 1) = ßu(n).
= 0,
k = 1, ... ,n,
For some details ofthis method see Agarwal [1, Ch. 11]. Another method of
proving these inequalities was based on geometrie facts in Euclidean space (cf.
Shisha [32]).
3. A Spectral Method and Using Orthogonal Polynomials
In this section we consider our method (see [19]) for determining the best constants
An and B n in the inequalities
n
n
n
(3.1)
An LPkX~ ~ L rk(xk - xk+d 2 ~ B n LPkX~,
k=l
k=O
k=l
under some conditions for a sequence of real numbers a: = (Xk), where p = (Pk)
and r = (rk) are given weight sequences. The method is based on the minimal
and maximal zeros of certain dass of orthogonal polynomials, which satisfy a
three-term recurrence relation.
For two N -dimensional real vectors
and
W= [Wl
N
we define the usual inner product by (z,w) = L ZkWk and consider the sums
k=l
n
and
F = L rk(xk - Xk+1)2
k=O
If we put VPk Xk = Yk (k = 1, ... ,n), then F and G can be transformed in the
form
n
F = L ~(v'Pk+1 Yk - v'jJkYk+t} 2 = (HN(a,b)y,y)
k=O PkPk+l
and
n
G = LY~ = (y,y),
k=l
where y E ]RN and HN(a, b) is a three-diagonal matrix like (2.6), with N = n or
N = n -1, depending on the conditions for the sequence a: = (Xk). Especially, we
will consider the following two cases:
1° Xo = Xn+l = 0 and Xl, ... ,Xn are arbitrary real numbers (N = n)j
2° Xl = 0 and X2, ••• ,X n are arbitrary real numbers (N = n - 1).
For such three-diagonal matrices we can prove the following auxiliary result ([19]):
295
DISCRETE INEQUALITIES OF WIRTINGER'S TYPE
Lemma 3.1. Let p = (Pk) and r = (Tk) be positive sequences and the matrix
Hn(a, b) be given by (2.6).
10 1/ the sequences a = (al, ... , an) and b = (bI, ... , bn - l ) are defined by
_ (TO + Tl , ... , Tn-l + Tn ) ,
aPI
b= (_~
(3.3)
';PIP2 , ... ,
Pn
_
Tn-l
)
.,fPn-IPn '
then the matrix Hn(a, b) is positive definite.
20 1/ the sequences a = (al, ... , an-d and b = (bI, ... , bn - 2 ) are defined by
a
(3.2)
__ (Tl + T2 , ... , Tn -2 + Tn-l "Tn-l)
b=
(
-
P2
Pn-l
Pn
T2
Tn-l
)
';P2P3 , ... , - ';Pn-IPn '
then the matrix H n - l (a, b) is positive definite.
We will formulate our results in terms of the monie orthogonal polynomials (7Tk)
instead of orthonormal polynomials as we made in [19]. Such an approach gives a
simpler and nicer formulation than the previous one.
The monie polynomials orthogonal on the realline with respect to the inner product (J, g) =
/(t)g(t)dJ.L(t) (with a given measure dJ.L(t) on IR) satisfy a fundamental three-term recurrence relation of the form
IR.
(3.5)
with 7To(t) = 1 and 7T_I(t) = 0 (by definition). The coefficients ßk are positive.
The coefficient ßo, whieh multiplies 7T-I(t) = 0 in three-term recurrence relation
may be arbitrary. Sometimes, it is convenient to define it by ßo = dJ.L(t). Then
the norm of 7Tk can be express in the form
IR.
(3.6)
An interesting and very important property of polynomials 7Tk(t), k ~ 1, is the
distribution of zeros. Namely, all zeros of 7Tn (t) are real and distinct and are
located in the interior of the interval of orthogonality. Let r~n), 11 = 1, ... , n,
denote the zeros of 7Tn (t) in an increasing order
(3.7)
Tin) < rJn) < ... < rAn).
It is easy to prove that the zeros r~n) of 7Tn (t) are the same as the eigenvalues of
the following tridiagonal matrix
o
o
296
G. V. MILOVANOVIC AND I. Z. MILOVANOVIC
which is known as the Jacobi matrix. Also, the monie polynomial 7rn (t) can be
expressed in the following determinant form
7rn (t) = det(tIn - J n ),
where In is the identity matrix of the order n. For some details on orthogonal
polynomials see [17] and [23].
Regarding to the conditions on the sequence x = (Xk), we consider now two
important cases:
CASE 10 (xo = x n +1 = 0). If we take ak-l = -ak and ..(iJk = -bk (i.e., ßk = bi >
0), k 2: 1, then we can consider the matrix H n ( -a, -b) = -Hn(a, b), defined by
(2.6), as a Jacobi matrix for certain dass of orthogonal polynomials (7rk)' Thus,
for every y E IRn we have
(Hn(a,b)y,y) = (-Hn(-a,-b)y,y) = (-Jny,y)
and
_T~n)(y,y) ~ (-Jny,y) ~ -Tin)(y,y),
where the zeros TS n ), lJ = 1, ... ,n, of 7rn (t) are given in an increasing order (3.7).
On the other hand, putting
7r~_l(t)f
7r*(t) = [ 7ro(t) 7ri(t)
and
en=[O
0
1f,
where 7r k(t) = 7rk(t)/II7rkll, we have (cf. Milovanovic [18, tl. 178])
t7r*(t) = J n7r*(t) + $n 7r~(t)en.
This means that for the eigenvalue t = TS n ) of J n , the corresponding eigenvector
is given by 7r*(TS n )). Notiee also that the same eigenvector corresponds to the
eigenvalue -TS n ) of the matrix -Jn . Therefore, the following theorem holds.
Theorem 3.2. Let P = (PkhENo and r = (rk)kENo be two positive sequences,
ri
rk-l + rk
ß _
k - -(k 2: 1),
Pk
PkPk+l
and let (7rk) be a sequence of polynomials satisfying (3.5). Then for any sequence
of real numbers Xo (= 0), Xl, ... , Xn , XnH (= 0), inequalities
n
n
n
2
(3.8)
An LPkXi ~ L rk(xk - xk+d ~ B n LPkxt
ak-l = -
k=l
k=O
k=l
. h A n -- -Tn(n) an d B n -- -Tl(n) ,wh ere Tv(n) , V -- 1 , ... , n, are zeros of
h old ,Wtt
7rn (t) in an increasing order (3.7).
Equality in the left (right) inequality (3.8) holds if and only if
C 7rk-l(t)
..jfik l1 7r k-lll'
Xk = -- .
k = 1, ... ,n,
where t = T$.n) (t = Tin)), l17rkll is given by (3.6) and C is an arbitrary constant.
Some corollaries of this theorem are the following results:
DISCRETE INEQUALITIES OF WIRTINGER'S TYPE
297
Corollary 3.3. For each sequence of the real numbers Xo (= 0), Xl, ... , Xn , Xn +1
(= 0), the following inequalities hold:
(
)
3.9
n
n
n
k=l
k=O
k=l
. 2
1f
~ 2
~(
)2
2
1f
~ 2
4sm 2(n + 1) ~xk :::; ~ Xk - Xk+1 :::; 4cos 2(n + 1) L..Jxk·
Equality in the left inequality (3.9) holds if and only if
.
k1f
n+
k = 1, ... ,n,
Xk = C s l n - 1,
where C is an arbitrary constant.
Equality in the right inequality (3.9) holds if and only if
xk=C(-I)k sin k1f 1 ,
n+
k = 1, ...
,n,
where C is an arbitrary constant.
Proof. For Pk = rk = 1 we obtain ak = -2 and fA = 1 for each k. Consequently,
the recurrence relation (3.5) becomes
Putting t + 2 = 2x and 1fk(t) = Sk(X), this relation reduces to the three-term
recurrence for Chebyshev polynomials of the second kind
Thus, we have (cf. Milovanovic [17, pp. 143-144])
(3.10)
_ S ( ) _ sin(k + 1)/J
k X . /J
'
sm
t+2
cos/J=x= - - ,
1fk (t ) -
2
and therefore the zeros of 1fn (t) are (in an increasing order)
(3.11)
r(n)
v
= -4sin2 /J v
2 '
/J _ (n + 1 - v)1I'
vn+l
'
v= 1, ... ,no
Thus, the best constants in (3.9) are
A
n
= _r(n) = 4sin2
n
11'
2(n+l)
and
_
(n) _
B n - - T1
• 2
n1l'
_
2
1f
-4sm 2(n+l)-4cos 2(n+l)'
298
G. V. MILOVANOVIC AND I. Z. MILOVANOVIC
Since IISkll = ../'Ir/2 for each k, using (3.10) and (3.11) we find the extremal
sequences for the left and the right inequality in (3.9). For example, for the right
inequality we have
from which follows
(k = 1, ... ,n),
where C is an arbitrary constant.
0
Remark 3.1. Theorem 2.2 is contained in Corollary 3.3.
In a more general case we can take
Pk = (a + bk)2
and
rk = (a + bk)(a + b(k + 1»,
with a, b ~ O. When b = 0 we obtain Corollary 3.3. However, if b i- 0, because of
homogeneity in (3.8), it is enough to put b = 1. In that case, we obtain the same
polynomials as in Corollary 3.3.
Corollary 3.4. For each sequence of the real numbers Xo (= 0), Xl, ... , X n , X n +1
(= 0), the following inequalities
(3.12)
n
n
k=l
k=O
~)k + a)2x~ :::; ~)k + a)(k + a + 1)(xk - xk+d 2
4sin 2 2(n: 1)
n
:::; 4cos 2 2( 'Ir 1) "'(
L.... k + a )2 Xk2
n+
k=l
hold, where a ~ o.
Equality in the left inequality (3.12) holds if and only if
Xk
C
. k'lr
= -ksm--1 ,
+a
n+
k = 1, ... ,n,
where C is an arbitrary constant.
Equality in the right inequality (3.12) holds if and only if
Xk
C(-I)k.
k7r
= k +a sm-,
n+ 1
k = 1, ... ,n,
where C is an arbitrary constant.
Remark 3.2. The corresponding inequalities for a = 0 were considered in [19).
DISCRETE INEQUALITIES OF WIRTINGER'S TYPE
299
Corollary 3.5. For eaeh sequenee 0/ the real numbers Xo (= 0), Xl, ... , Xn , Xn+1
(= 0), we have
n
n
n
An LX% ~ Lk(Xk - Xk+1)2 ~ B n LX%,
(3.12)
k=l
k=l
k=O
where An and B n are minimal and maximal zeros 0/ the monie Laguerre polynomial Ln(x), respeetively.
Equality in the left (right) inequality (3.12) holds i/ and only i/
Xk = CLk-l(X)j(k - I)!
(k = 1, ... ,n),
where X = An (x = B n ) and C is an arbitrary eonstant.
In this case we have (lk = -(2k + 1) and ßk = k 2, so that the relation (3.5)
becomes
7rk+l (t) = (t + 2k + l)7rk(t) - k27rk_l (t).
Putting t = -x and 7rk( -x) = (-l)k Lk(X), this relation reduces to one, which corresponds to the monie Laguerre polynomials orthogonal on (0, +00) with respect
to the measure dJ.t(x) = e- X dx. The norm of Lk(X) is given by IILkll = k!.
In a more general case we can take
(3.13)
rO = 0,
1
rk = B(s + 1, k)'
1
Pk = (k + s)B(s + 1, k)
(k ~ 1),
where s > -1 and B(p, q) is the beta function (B(P, q) = r(p)r(q)jr(p + q), r
is the gamma function). Then we have (lk = -(2k + s + 1) and ßk = k(k + s),
and the corresponding recurrence relation, after changing variable t = -x and
7rk(-X) = (-l)kL k(x), becomes
(3.14)
L k+1 (x) = (x - (2k + s + l))Lk(x) - k(k + s)L k_l (x),
where Lk(x), k = 0,1, ... , are the generalised monie Laguerre polynomials orthogonal on (0, +00) with respect to the measure dJ.t(x) = xBe- X dx. Thus, we have
the following result:
Corollary 3.6. Let s > -1 and let r = (rk)kENo and p = (pkhEN be given by
(3.13). For eaeh sequenee 0/ real numbers Xo (= 0), Xl, ... , x n , Xn+1 (= 0), we
have
(3.15)
n
n
n
An LPk X% ~ L rk(xk - Xk+1)2 ~ B n LPk X%,
k=l
k=O
k=l
where An and B n are minimal and maximal zeros 0/ the monie generalised Laguerre polynomial L~ (x), respeetively.
G. V. MILOVANOVIC AND 1. Z. MILOVANOVIC
300
Equality in the left (right) inequality (3.15) holds if and only if
CL~_I(X)
Xk = -..;77.(k;=-~1)=;:§!r~(k;=+===:=s)
(k = 1, ... ,n),
where x = An (x = B n ) and C is an arbitrary constant.
CASE 2° (Xl
= 0). Here, in fact, we consider the inequalities
n-l
n
(3.16)
An LPkX~ ~ L
k=l
k=l
n
Tk(Xk - Xk+1)2 ~ B n LPkX~,
k=l
for any sequence of the real numbers Xl (= 0), X2, ... , Xn .
Using Lemma 3.1 (Part 2°) we put N = n - 1,
(3.17)
(k ~ 1),
and also Uk-l = -ak, V1fk = -bk (k ~ 1). Taking
7I"*(t)= [7ro(t) 7ri(t)
...
7r~_2(t)f
and
en-I=[O
0 ...
1f,
where 7rk(t) = 7rk(t)/II7rk11, we have, as in the previous case,
t7l"*(t) = Jn - 1 7l"*(t) + Vßn-1 7r~_I(t)en-l,
but now
Hn-l(a,b) = -Hn-l(-a,-b) = -Jn - l -
where D n - l = diag (0, ... ,0,1). So, we obtain that
T n D n- l ,
Pn
from which we conclude that the eigenvalues of Hn-l(a,b), in notation Av = - Tv ,
V = 1, ... ,n - 1, are the zeros of the polynomial
(3.18)
The corresponding eigenvectors are 71"* (Tv).
Since l17rn-11l = l17rn-21IJßn-l, the polynomial (3.18) can be reduced to one represented in terms of the monie polynomials,
(3.19)
DISCRETE INEQUALITIES OF WIRTINGER'S TYPE
301
TheoreIll 3.7. Let P = (PkhEN and r = (rkhEN be two positive sequences, Uk-l
and ßk (k 2: 1) be given by (3.17), and let (1I"k) be a sequence of polynomials
satisfying (3.5). Then for any sequence of real numbers Xl (= 0), X2, ... , Xn ,
inequalities (3.16) hold, with An = min{-Ty } B n = max{-Ty}, where Ty, 1/ =
y
y
1, . .. , n - 1, are zeros of the polynomial R n - l (t) given by (3.19).
Equality in the left (right) inequality (3.16) holds if and only if
k = 2, ... ,n,
Xl = 0,
where t = -An (t = -Bn ), l11l"kll is given by (3.6) and C is an arbitrary constant.
Some corollaries of this theorem are the following results:
Corollary 3.8. For each sequence of real numbers Xl (= 0), X2, ... , Xn , the
following inequalities hold:
(3.20)
11"
• 2
n
~
n-l
2
~
2
2
11"
n
~
2
4 sm 2(2n _ 1) L..J xk :::; L..J(Xk - XkH) :::; 4cos 2n _ 1 L..J xk·
k=2
k=2
k=l
Equality in the left inequality (3.20) holds if and only if
. (k - 1)11"
xk=Csm 2n-1 '
k = 1, ... ,n,
where C is an arbitrary constant.
Equality in the right inequality (3.20) holds if and only if
_ C(-l)k . 2(k - 1)11"
Xk sm 2n -1 '
k = 1, ... ,n,
where C is an arbitrary constant.
Here we have (as in Corollary 3.3) that
1l"k(t) = Sdx) = sin(~ + 1)0 ,
smO
and
Rn-l(t) = Sn-l (x) - Sn-2(X) =
and therefore
Ty
• 2
1/11"
= -4sm 2n -1 '
t + 2 = 2x,
cos«2n - 1)0/2)
cos(0/2)
,
v=l, ... ,n-l.
G. V. MILOVANOVIC AND I. Z. MILOVANOVIC
302
Corollary 3.9. Let 8> -1 and let r = (rkhEN and p = (Pk)kEN be given by
(3.21)
1
1
rl = 0, rk+l = B(8 + 1, k)' Pk+l = (k + 8)B(8 + 1, k)
(k ~ 1).
For each 8equence 01 real number8 Xl (= 0), X2, ... , Xn , we have
n-l
n
L rk(Xk - Xk+l)2 ~ B n LPk X%,
k=l
k=2
(3.22)
where B n i8 a maximal zero 01 the monic generalised Laguerre polynomial L~:='~(x).
Equality in (3.22) hold8 il and only il
(3.23)
Xl = 0,
k L'k_2(Bn )
Xk = C( -1) r(k + 8 _ 1) ,
k = 2, ... ,n,
where C is an arbitrary con8tant.
Proof. Taking 7I'k( -x) = (_l)k Lk(x) , with (3.21) we obtain the reeurrenee relation
(3.14), so that the polynomial (3.19) beeomes
Rn-l(t) = 7I'n-l(t) - (n + 8 - 1)7I'n-2(t)
= (_l)n-1 (L~_l (-t) + (n + 8 - 1)L~_2( -t))
= (-l)ntL~:='~(-t).
Thus, B n is a maximal zero of the monie generalised Laguerre polynomial L~:='~ (x).
Evidently, An = O.
Since
1
7I'k-2( -Bn )
..jPk
1171'k-211
(k + 8 - 1)r(8 + l)(k - 2)!
r(k + 8)
(_1)k-2 L'k_2(Bn )
J(k - 2)!f(k + 8 - 1)
= (_l)k Jr(8 + 1) U
(B)
r(k + 8 - 1) k-2 n,
we obtain the extremal sequence (3.23) for which the equality is attained in
(3.22). D
Remark 3.3. A few members of the monic generalised Laguerre polynomials L~+l(x)
are
Lg+l(x) = 1,
L~+l(x) = x - (8 + 2),
L;+l(x) = x 2 - 2(8 + 3)x + (8 + 2)(8 + 3),
L;+l (x) = x 3 - 3(8 + 4)x 2 - (8 + 3)(8 + 12)x - (8 + 2)(8 + 3)(8 + 4).
It is not difficult to show that B3 = 8 + 2, B4 = 8 + 3 + y'S"'+3.
DISCRETE INEQUALITIES OF WIRTINGER'S TYPE
303
Remark 3.4. For s = 0 the inequality (3.22) reduees to (see [19])
n-l
L
n
(k - 1)(Xk - Xk+1)2 ~ B n
E xi,
k=2
k=l
where B n is a maximal zero of the monie generalised Laguerre polynomial L~_2(X).
Remark 3.5. If for every k we take Xk = (-1)k ak the inequalities (3.1) become
n
n
An LPklakl2 ~ L
k=l
n
Tklak + aHl1 2 ~ Bn LPkl a kl 2 .
k=O
k=l
Moreover, these inequalities are valid for eomplex numbers too.
At the end of this section we mention some results of Losonczi [15]. He eonsidered
inequalities of the form
ar L I kl ~ L'
n .
(3.24)
X
2
k=O
IXk ± xHml 2 ~
ßr L I kI
n
X
2,
k=O
where Xo, Xl, ... , X n are real or complex numbers, 1 ~ m ~ n, summation symbols
defined by:
n-m
I
L=L,
k=O
with Xn+l = ... = X n + m = 0,
",3 __
L..J
~
L..J
with X-m = ... = X-I = 0,
k=-m
",4 __
L..J
ar, ßt (i =
~
L..J
with X-m = ... = X-I = 0 = Xn+l = ... = Xn + m '
k=-m
1, 2, 3, 4) are constants and either the + or the - sign is taken. It is
easy to see that the cases i = 2 and i = 3 are the same apart from the notation
of the variables Xk. Hence there are 6 different cases in (3.24) corresponding to
i = 1,2 or i = 3,4 and the + or - sign. Losonczi found the best constants
and
in all cases and it was based on the determination of eigenvalues of some
suitable Hermitian matrices.
ar
ßt
Theorem 3.10. Let n and m be fixed natural numbers (1 ~ m ~ n) and r =
[n/m]. The inequalities (3.24) hold for every real or complex numbers Xo, Xl, ... ,
X n , with the best constants:
at = a; = at = a3" = 4sin2 2( 2r'lr+ 3) ,
ß+
- ß- ß+
- ß- 4 cos2 _
'lr_ .
2 2 3 3 2r + 3 '
+ ___
.2
'Ir
a 4 -a4 -4sm 2(r+2)'
+_ __
2
'Ir
ß4 -ß4 -4cos 2(r+2)'
304
G. V. MILOVANOVIC AND I. Z. MILOVANOVIC
Remark 3.6. In connection with extrem al properties of nonnegative trigonometrie polynomials Szegö [33] and Egervary and Szasz [9] proved that for every complex numbers
Xo, Xl, ... , Xn the inequalities
(3.25)
n
n-m
k=O
k=O
n
-r L Ixd ::; L (XkXk+m + XkXk+m) ::; rL IXkl 2
k=O
holds, with the best constant r = 2cos(7r/(r + 2)), where r = [n/m]. The case m = 1
was previously proved by Fejer [11]. It is clear that the inequalities (3.25) are related to
(3.24).
4. Inequalities for Higher Differences
In this section we give a short account on generalisations of Wirtinger's type
inequalities to higher difIerences. The first results for the second difIerence were
proved by Fan, Taussky and Todd [10]:
Theorem 4.1. 1f Xo (= 0), XI,X2, ... ,X n , xn+d= 0) are given real numbers,
then
(4.1)
n-l
n
k=O
k=l
,,(
)2' 4
7r
"2
L...J Xk - 2XkH + Xk+2 ~ 16sm 2(n + 1) L...Jxk,
with equality in (4.1) if and only if Xk = Asin~, k = 1,2, ... , n, where A is
n+I
an arbitrary constant.
Theorem 4.2. 1f Xo, Xl, ... , Xn , Xn+l are given real numbers such that Xo = Xl,
Xn+l = Xn and (2.3) holds, then
(4.2)
The equality in (4.2) is attained if and only if
Xk =
A
cos
(2k - I)7r
2n
'
k = 1,2, ... ,n,
where A is an arbitrary constant.
A converse inequality of (4.1) was proved by Lunter [16], Yin [36] and Chen [7]
(see also Agarwal [1]).
Theorem 4.3. 1fxo(= 0), XI,X2, ... ,Xn , xn+d= 0) are given real numbers,
then
n-l
(4.3)
n
E(Xk - 2XkH + Xk+2)2 :::; 16cos4 2(n: 1) {; xk,
with equality in (4.3) if and only if Xk = A( -l)k sin~, k = 1,2, ... , n, where
n+I
A is an arbitrary constant.
Chen [7] also proved the following result:
DISCRETE INEQUALITIES OF WIRTINGER'S TYPE
305
Theorem 4.4. I/ XO,Xl, ... ,xn ,xn +1 are given real numbers such that Xo = Xl
and Xn+l = X n , then
with equality holding i/ and only i/
Xk
. (2k - 1)1f
= A( - l) k Sln
,
n
k = 1,2, ... ,n,
where A is an arbitrary constant.
Proof. In this case, the n x n symmetrie matrix corresponding to the quadratie
form
n-l
F 2 = Z)Xk -
2Xk+1
+ Xk+2)2 = (Hn,2X, x)
k=O
is
2
-3
1
-3
6
-4
1
-4
6
1
-4
1
H n ,2 =
1
-4
1
6 -4 1
-4
6-3
1 -3 2
This matrix is the square of the n x n matrix
(4.4)
Hn
1
-1
-1
2
-1
= Hn,l =
-1
2-1
-1
2-1
-1
1
The eigenvalues of H n are
\ _ \ (H ) - 4
/\" -
/\"
n
-
cos
2 (n -
v + 1)1f
2n
'
v = 1, ... ,n,
and therefore, the largest eigenvalue of H n is
The corresponding eigenvector is x n = [Xl n
_ ( 1)". (2v - 1)1f
x"n -
-
Sln
2n
X2n
'
X nn ]T, where
v=1,2, ... ,n.
G. V. MILOVANOVI<:: AND I. Z. MILOVANOVIC
306
Thus, the largest eigenvalue of H n ,2 is
and the associated eigenvector is x n .
0
Remark 4.1. Notice that the minimal eigenvalue of the matrix H n (and also H n ,2) is
..\1 = O. Therefore, the condition (2.3) must be inc1uded in Theorem 4.2 and the best
constant is the square of the relevant eigenvalue
..\2 = 4 cos
2 (n -1)11"
2n
. 2 11"
= 4 sm 2n·
For any n-dimensional vector X = [Xl X2 ... xnf, Pfeffer [30] introduced a
periodically extended n-vector by setting xHrn = Xi for i = 1,2, ... ,n and rEN,
and used the mth difference of x given by x(m) = [~mXl ~mX2 ... ~mxn]T,
where
1 ~ i ~ n,
in order to prove the following result:
Theorem 4.5. If x is a periodically extended n-vector and (2.3) holds, then
with equality case if and only if x is the periodic extension of a vector of the form
Cl U + C2 v, where
and
have the following components
Uk
2k1r
= cos--,
n
Vk
. 2k1r
= Sin
--,
n
k = 1, ... ,n,
and Cl and C2 are arbitrary real constants.
Recently we have studied inequalities of the form (see [21])
n
(4.5)
An,m LX~ ~ L
k=l
n
Um
(~mXk)2 ~ Bn,m LX~'
k=lm
where lm = 1 - [m/2], Um = n - [m/2] and
k=l
DISCRETE INEQUALITIES OF WIRTINGER'S TYPE
307
2
2: (LlmXk) for m = 1 reduces to
u~
The quadratie form Fm =
k=l~
F l = x~ +
n-l
n-l
k=2
k=l
L 2x~ + x~ - 2 L XkXkH,
with corresponding tridiagonal symmetrie matrix H n = Hn,l given by (4.4).
Under conditions
X s = Xl- s ,
Xn+l- s = x n + s
we proved that the corresponding matrix of the quadratic form Fm is exactly
the mth power of the matrix H n = Hn,l so that the best constant in the right
inequality (4.5) is given by
7f
B n,m =4mcos 2m _2n
Evidently, An,m = O. However, by restrietion (2.3), the best constant in the left
inequality (4.5) is given by
. 2m 7f
A nm= 4m Sln
-2.
,
n
For other generalisations of discrete Wirtinger's inequalities for higher differences
see [6), [16), [31] and [34]. There are also generalisations for multidimensional
sequences and partial differences (see [6] and [28]). Finally, we mention that there
exist some types of non-quadratie Wirtinger's inequalities (cf. [6), [10] and [12])
as weH as discrete inequalities of Opial's type (cf. [3], [14], [20], [22], [35]).
References
1. R. P. Agarwal, Difference Equations and Inequalities - Theory, Methods, and Applications,
Marcel Dekker, New York - Basel - Hong Kong, 1992.
2. H. Alzer, Converses 01 two inequalities by Ky Fan, O. Taussky, and J. Todd, J. Math. Anal.
Appl. 161 (1991), 142-147.
3. _ _ , Note on a discrete Opial-type inequality, Arch. Math. 65 (1995), 267-270.
4. E. F. Beckenbach and R. Beliman, Inequalities, Springer Verlag, Berlin - Heidelberg - New
York, 1971.
5. W. Blaschke, Kreis und Kugel, Veit u. Co., Leipzig, 1916.
6. H. D. Block, Discrete analogues 01 certain integral inequalities Prac. Amer. Math. Soc. 8
(1957), 852-859.
7. W. Chen, On a question 01 H. Alzer, Arch. Math. 62 (1994), 315-320.
8. S.-S. Cheng, Discrete quadratic Wirtinger's inequalities, Linear Algebra Appl. 85 (1987),
57-73.
9. E. Egervary and O. Szasz, Einige Extremalprableme im Bereiche der trigonometrischen
Polynome, Math. Z. 21 (1928), 641-692.
10. K. Fan, O. Taussky and J. Todd, Discrete analogs 01 inequalities 01 Wirtinger, Monatsh.
Math. 59 (1955), 73-90.
11. 1. Fejer, Über trigonometrische Polynome, J. Reine Angew. Math. 146 (1915), 53-82.
308
G. V. MILOVANOVIC AND I. Z. MILOVANOVIC
12. A. M. Fink, Discrete inequalities 0/ generalized Wirtinger type, Aequationes Math. 11 (1974),
31-39.
13. G. H. Hardy, J. E. Littlewood and G. P6lya, Inequalities, 2nd Edition, Univ. Press, Cambridge, 1952.
14. C.-M. Lee, On a discrete analogue 0/ inequalities 0/ Opial and Yang, Canad. Math. BuH. 11
(1968), 73-77.
15. L. Losonczi, On some discrete quadratic inequalities, General Inequalities 5 (Oberwolfach,
1986) (W. Walter, ed.), ISNM Vol. 80, Birkhäuser Verlag, Basel, 1987, pp. 73-85.
16. G. Lunter, New proo/s and a generalisation 0/ inequalities 0/ Fan, Taussky, and Todd, J.
Math. Anal. Appl. 185 (1994), 464-476.
17. G. V. Milovanovic, Numerical Analysis, Part I, 3rd Edition, Naucna Knjiga, Belgrade, 1991.
(Serbian)
18. ___ , Numerical Analysis, Part II, 3rd Edition, Naucna Knjiga, Belgrade, 1981. (Serbian)
19. G. V. Milovanovic and I. Z. Milovanovic, On discrete inequalities 0/ Wirtinger's type, J.
Math. Anal. Appl. 88 (1982), 378-387.
20. ___ , Some discrete inequalities 0/ Opial's type, Acta Sei. Math. (Szeged) 47 (1984),
413-417.
21. ___ , Discrete inequalities 0/ Wirtinger's type tor higher differences, J. Ineq. Appl. 1
(1997) (to appear).
22. G. V. Milovanovic, I. Z. Milovanovic and L. Z. Marinkovic, Extremal problems tor polynomials and their coejJicients, Topics in Polynomials of One and Several Variables and Their
Applications (Th. M. Rassias, H. M. Srivastava and A. Yanushauskas, eds.), World Seientific,
Singapore - New Jersey - London - Hong Kong, 1993, pp. 435-455.
23. G. V. Milovanovic, D. S. Mitrinovic and Th. M. Rassias, Topics in Polynomials: Extremal
Problems, Inequalities, Zeros, World Seientific, Singapore - New Jersey - London - Hong
Kong, 1994.
24. D. S. Mitrinovic and P. M. Vasic, An inequality ascribed to Wirtinger and its variations and
generalization, Univ. Beograd. Publ. Elektrotehn. Fak. Sero Mat. Fiz. No 247 - No 273
(1969), 157-170.
25. D. S. Mitrinovic (with P. M. Vasic), Analytic Inequalities, Springer Verlag, Berlin - Heidelberg - New York, 1970.
26. D. S. Mitrinovic, J. E. Pecaric and A. M. Fink, Inequalities Involving Functions and Their
Integrals and Derivatives, Kluwer, Dordrecht - Boston - London, 1991.
27. J. Novotna, Variations 0/ discrete analogues 0/ Wirtinger's inequality, Casopis Pest. Mat.
105 (1980), 278-285.
28. ___ , Discrete analogues 0/ Wirtinger's inequality tor a two-dimensional array, Casopis
Pest. Mat. 105 (1980), 354-362.
29. ___ , A sharpening 0/ discrete analogues 0/ Wirtinger's inequality, Casopis Pest. Mat.
108 (1983), 70-77.
30. A. M. Pfeffer, On certain discrete inequalities and their continuous analogs, J. Res. Nat.
Bur. Standards Sect. B 70B (1966), 221-231.
31. I. J. Schoenberg, The finite Fourier series and elementary geometry, Amer. Math. Monthly
57 (1950), 390-404.
32. O. Shisha, On the discrete version 0/ Wirtinger's inequality, Amer. Math. Monthly 80
(1973), 755-760.
33. G. Szegö, KoejJizientenabschätzungen bei ebenen und räumlichen harmonischen Entwicklungen, Math. Ann. 96 (1926/27), 601-632.
34. J. S. W. Wong, A discrete analogue 0/ Opial's inequality, Canad. Math. BuH. 10 (1967),
115-118.
35. G.-S. Yang and C.-D. You, A note on discrete Opial's inequality, Tamking J. Math. 23
(1992), 67-78.
36. X.-R. Yin, A converse inequality 0/ Fan, Taussky, and Todd, J. Math. Anal. Appl. 182
(1994), 654-657.
CONVEXITY PROPERTIES OF SPECIAL
FUNCTIONS AND THEIR ZEROS
MARTIN E. MULDOON
Department 0/ Mathematics & Statistics, York University, North York,
Ontario M3J lP3, Canada
Abstract. Convexity properties are often useful in characterising and finding bounds
for special function and their zeros, as weIl as in questions concerning the existence and
uniqueness of zeros in certain intervals. In this survey paper, we describe sorne work
related to the gamma function, the q-gamrna function, Bessel and cylinder functions and
the Herrnite function.
1. Introduction
Many inequalities for special functions are statements about the positivity or
monotonicity of certain quantities. Some deeper results refer to higher monotonicity or even complete monotonicity, Le., the derivatives of successive derivatives
or difIerences alternate in sign. An intermediate kind of result refers to convexity. Convexity is often used to "characterise" certain special functions such as the
gamma function. Convexity properties are often useful in obtaining bounds for
zeros. In other cases, such properties can be used to prove existence or uniqueness
of zeros in certain intervals. In this expository paper, we give examples of these
ideas.
2. Gamma and Related Functions
2.1. THE GAMMA FUNCTION
The gamma function is usually defined, for Re z > 0 by Euler's integral
r(z) =
(2.1)
1
00
e-tt z - 1 dt,
or for z # 0, -1, -2, ... by the infinite product
(2.2)
-
1
r(z)
= ze"lZ II [1 + -] e- z / n .
00
z
n=l
n
1991 Mathematics Subject Classification. Primary 33-02j Secondary 33B15, 33ClO, 33C15.
Key woms and phrases. ConvexitYj Gamma functionj Bessel functionsj Cylinder functions; Zeros; Inequalities.
Research supported by grants from the Natural Sciences and Engineering Research Council
(Canada)
309
G. V. Milovanovic (ed.), Recent Progress in lnequalities, 309-323.
© 1998 Kluwer Academic Publishers.
M. E. MULDOON
310
The formula (2.2) is sometimes more useful than the integral formula (2.1). For
example, by taking logarithms in (2.2) and differentiating twice, we are led to
cF
-d2 log jr(z) I =
(2.3)
Z
1
L ( + n )2'
00
n=O
Z
whieh shows that
(2.4)
cF
dx 2 log Ir(x)1 2': 0,
x # 0, -1, -2, ...
and hence that log Ir! is convex on those intervals, including (0,00), where it is
defined.
It is well known that r(x) is far from being the unique solution of the difference
equation
(2.5)
f(x + 1) = xf(x) ,
x > 0,
whieh satisfies
(2.6)
f(l) = 1
so it is of interest is to "characterise" r(x), e.g., to give a {;ondition, additional
to (2.5) and (2.6) whieh will determine it uniquely. One of the simplest such
conditions is ([5], [3])
(2.7)
logf(x) is convex.
The characterisation based on (2.5), (2.6) and (2.7) is usually referred to as the
Bohr-Mollerup characterisation. Other such characterisations depend on the limiting behaviour of f or on the monotonicity of the product of r(x) and a suitable
function. For example, r(x) is the unique function f(x) which satisfies (2.5) and
(2.6) and for which (e/x)"' f(x) is ultimately monotonie [52].
Recall that a real-valued function is said to be convex on an interval I if the "slope"
[f(xd - f(X2)]/(XI - X2) is a nondecreasing function of X2 on I for each fixed Xl
in I. A function f for whieh f" exists is convex on I if and only if f"(x) 2': 0 on
I. The Bohr-Mollerup characterisation of the gamma function may be though of
as a consequence of the following results ([28], [33]) on monotonie functions.
Lemma 2.1. Let cp(x) -t 0 as x -t 00. Then the difference equation
(2.8)
f(x + 1) - f(x) = cp(x) ,
xo:::; x < 00
has at most one nondecreasing solution satisfying
(2.9)
f(xo) = Yo·
311
CONVEXITY PROPERTIES OF SPECIAL FUNCTIONS
In other words, a nondecreasing solution is determined by its value at Xo, or any
two nondecreasing solutions must diJJer by a constant.
Proof. Suppose that there were two nondecreasing solutions h(x) and h(x). Let
g(x) = h(x) - h(x). Then g(xo) = 0 and 9 is periodic with period 1. Let
Xo < x < Xo + 1. Then, for every positive integer N, g(x) = g(x + N) = h(x +
N) - h (x + N) ~ h (xo + N) - h (xo + N + 1) = h (xo + N) - h (xo + N) cp(xo + N) = -cp(xo + N). Also g(x) = g(x + N) = h(x + N) - h(x + N) ~
h(xo +N + 1) - h(xo + N) = h(xo +N) +cp(xo +N) - h(xo +N) = cp(xo +N).
In other words we have Ig(x)1 ~ cp(xo + N) for every N. Letting N -+ 00, we get
g(x) = o. 0
Theorem 2.2. Let f be a junction from (0,00) to (0,00) satisfying f(x + 1) =
xf(x) and f(l) = 1 and with the property that logf is convex. Then f(x) = r(x),
0< x < 00.
Proof. First of all, the logarithm of the gamma function satisfies
(2.10)
f(x + 1) - f(x) = log x
and
(2.11)
f(l)=O.
If there were another convex function h satisfying these equations, then ßr and
ßh would be nondecreasing solutions of
(2.12)
f(x + 1) - f(x) = log (1 + ~)
and by Lemma 2.1, they would differ by a constant. So logr and logh would differ
by a linear function and since r and h coincide at the integers, this linear function
would be identically zero. 0
2.2. q-GAMMA FUNCTIONS
Lemma 2.1 may be applied also to the q-gamma function q-gamma function defined
by [20]
00 1- qn+1
r q(x) := (1 - q)1-x
1
+' 0 < q < 1,
n=O - qn x
II
and
r q (x) := (q - 1)1-Xq2 x(x-1) II 1 - q-(n+x)
q
,
1
00
1_
-(n+1)
n=O
We have
rq(x) -+ r(x)
as q -+ 1,
q>1
.
312
M. E. MULDOON
and r q (x) satisfies the functional equation
qX -1
rq(x + 1) = - - 1 rq(x).
q-
(2.13)
We have
(2.14)
~
d X 2 logrq(x) =
{
00
qx+n
(logq)2 n~o (1- qx+n)2 '
00
+
qX n
(logq)2 n~o (1- qx+n)2 + logq,
0< q < 1,
q> 1.
Askey [4] has shown that, for 0 < q < 1, r q is the unique logarithmically convex
solution of the functional equation (2.13) which satisfies 1(1) = 1. This follows
flOm Lemma 2.1 since, in this case,
~<p(X) = log(l - (x + l)q) -log(l - x q) --+ 0,
x --+ 00
on using the mean-value theorem. However this does not work for q > 1. As Moak
[50] has shown, in this case, we must assume astronger condition. Moak shows
that one can get a characterisation in this case based on the property that
d3
dx 3 logrq(x) < 0
(2.15)
or on the assumption that
(2.16)
~
dx 2 logrq(x) > logq,
or that qx 2 /2r q (x) is logarithmically convex. In this form the characterization can
be established by using Lemma 2.1.
3. Bessel and Related Functions
The Bessel function of the first kind is defined by
(_1)n(zj2)2n+v
Jv(z) = ~ n!r(v + n + 1)
00
It satisfies the differential equation
(3.1)
Z2 y" + zy' + (Z2 - v 2) Y = 0,
and [60, p. 482] has all its z-zeros real when v ~ -1. A second solution of equation
(3.1) is given by
Yv(z) = Jv(z) cos.V7r - Lv(z)
smV7r
where an appropriate limit is taken when v is an integer, and the general solution
is given by
(3.2)
(3.3)
The functions Cv(z) are referred to as cylinder junctions.
In sequel, we will use the notation ivk and Cvk for the respective k-th positive zeros
of Jv(x) and Cv(x).
313
CONVEXITY PROPERTIES OF SPECIAL FUNCTIONS
3.1. STURM'S RESULTS AND EXTENSIONS
It was shown by Sturm [57] as long ago as 1836 that the positive z-zeros of the
function Cv(z) form a concave or convex sequence according as lvi is greater than
or less than 1/2. If we use the notation ß for forward differences (Le., ßJ.Lk
J.Lk+l - J.Lk), then the above results can be stated as follows
1
lvi> 2
(3.4)
and
1
lvi< 2·
(3.5)
This led Lee Lorch and Peter Szego [42] in 1963 to investigate the higher differences
of these zeros. They showed ([42, Theorem 2.1] and [43]) that
(3.6)
(-1)nßn+1 Cvk > 0,
n = 0,1, ... , k = 1,2, ... ,
1
lvi> 2
and conjectured that
(3.7)
would hold for lvi< 1/2. This was proved for 1/3 ~ lvi< 1/2 by Muldoon [51,
Corollary 4.2], but the conjecture is still unsettled even for lvi = O. Some results
of Dosl<i [7] indicate that the differences of any fixed order have the expected sign
for sufficiently large zeros.
The results of Lorch and Szego depend heavily on Nicholson's integral representation [60, p. 444],
(3.8)
81
J;(x) + Y;(x) = 2"
1f
0
00
K o(2xsinhT) cosh2vTdT.
Later [43] they were extended, using [23], to a dass of differential equations.
The Sturm comparison Theorem may be used to show that if the coefficient function cp(t) in the differential equation
(3.9)
y" +cp(t)y = 0
is decreasing (increasing) on an interval, then the sequence of zeros of a solution
of (3.9) on that interval is convex (concave). A particularly simple treatment of
this is given in a 1952 paper by Makai [48]. Since the functions y(t, v) = tl/ 2 Cv (t)
satisfy
(3.10)
y" + [ 1 +
1/4
t~
v2 ]
Y = 0,
314
M. E. MULDOON
we see that this provides a simple proof of the result of Sturm expressed by (3.4)
and (3.5).
Strangely, the positive zeros of the derivative of Cv which are greater than lvi
form a concave sequence regardless of the value of lvi. This is induded in work of
Vosmansky [59]. There have been several extensions of the Lorch-Szego results.
See [51] for references.
A further result [38] of this kind is that for fixed a: > 0 the sequence of positive
zeros of the derivative y' of a solution y of the generalised Airy equation
(3.11)
is concave. This is done by using the Sturm comparison theorem. In fact, much
more is true; see [51, Theorem 6.1].
Makai's formulation [48] of the Sturm comparison is also the principal tool in the
discussion [54] ofthe convexity of the sequence of positive zeros of f..LJv(t) + tJ~(t).
In [54], there is delineated a fairly extensive region of the (f..L, v) plane in which
this sequence of zeros is concave.
3.2. VARIATION WITH RESPECT TO ORDER
On can also ask about monotonicity, convexity, etc., of zeros of Bessel functions
with respect to order v. Lorch and Szego [44] have some results on the monotonicity with respect to v of quantities which indude the spacings between the
zeros. These results, like many of these considered in this section, depend on the
"Nicholson-type" formula due to Watson [60, p. 508],
(3.12)
:~ = 2c
1
00
K o (2csinht)e- 2vt dt,
which is valid for all zeros C = Cvk of cylinder functions throughout the interval in
which they are continuous functions of v. Because of the simple nature (positive,
decreasing, etc.) of Ko(t), the formula (3.12) has been used to remarkable effect
in several discussions of monotonicity, convexity, etc., of the zeros; see [9], [12-15]
and references. This formula was used by Elbert [9] to show that the real zeros of
the Bessel function Jv(z) are concave functions in their interval of definition, as
might be conjectured from an examination of the diagram in [60, p. 510]. Earlier,
some partial results ab out squares of zeros were found [39]. Actually, a method
due to Lorch, Muldoon and Szego [45, §3] shows that log Cvk is a concave function
of v on the interval where it is defined. This is weaker than the result of Elbert
on the concavity of Cvk itself.
In [53], it was shown how to derive (3.12) by a differential equations method. It
would be of interest to be able to do this for a dass of differential equations.
Elbert's use of (3.12) to prove the concavity of ivk proceeds as follows. Differentiating (3.12) and using integration by parts, we get
(3.13)
CONVEXITY PROPERTIES OF SPECIAL FUNCTIONS
315
where
(3.14)
He then uses
(3.15)
1 djvk
2
f(v, t) = 2t - - . -d [2vtanht + tanh t].
Jvl
V
.
[v+ '1]2 djvk
dv <
Jvk,
proved by Sturmian methods for -k < v ::; 0, and by using (3.12) for v ~ 0, to
show the f(v, t) > 0, so the result holds. Later A. Laforgia and the author [37]
used a modification of this method to extend the result to real zeros of cylinder
functions.
3.3. ZEROS AS CONTINUOUS FUNCTIONS OF RANK
One can discuss the variation of the positive zeros of Cv(x, a) with respect to any
of the three variables v, a or k (the rank of a zero). However, a and k are not really
independent; they may, in fact, be subsumed in a single variable", = k - a/rr. To
see this, we consider that for v ~ 0, the zeros of Cv(x, a), < a < rr are the roots
of the equation
Yv(x)/Jv(x) = cota.
°
The graph of the left-hand side of (2.1) consists of branches which increase from
-00 to +00 in the intervals (O,jvl) and (jvk , jv,k+1), k = 1,2, ... , between the
positive zeros of Jv(x). This is most easily seen by using the relation
d Yv(x) = Jv(x)Y~ (x) - Yv(x)J~(x) _
2
dx Jv(x)
J:(x)
- rrxJ:(x) ,
where the last equation follows from the Wronskian relation [60, p. 76]. As a
decreases from rr to 0, cot a increases from -00 to 00. Thus each zero of Cv(x, a)
increases from one positive zero jvk of Jv(x) to the next larger one jv,k+1' At the
same time a new first positive zero appears and increases from to jvl' Thus it
makes sense to define jVI< for any real ", ~ 0, by jvo = and jVI< = cVk(a) where k
is the largest integer less than ", + 1 and a = rr(k - ",). Thus jVI< is a continuous
increasing function of ", on [0,00). The positive zeros of Jv(x) correspond to
positive integral values of K, and jv,k-l/2 = Yvk, k = 1,2, ... where Yvk is the kth
positive zero of Yv(x). In [12] it was shown that jVI< is the unique solution of the
differential equation
°
:~ = 2j
1
00
°
Ko(2jsinht)e- 2vt dt,
which satisfies j(v) ~ 0 as v ~ -K,+. This is motivated by the formula [60, p. 408]
for the derivative of Cvk with respect to v, and the fact that if, for v > 0, Cvk is
the kth positive zero of Cv(x), then Cvk may be extended in a continuous way to
M. E. MULDOON
316
v < 0, and Cvk -* 0, as V -* -(k - a/1f). The equation (3.12) may be used to
show that jVI< is an infinitely differentiable function of K,.
This suggests the consideration of successive derivatives of zeros with respect to
1'\" much as Lorch and Szego [42] (and others) have considered differences with
respect to the rank k. This idea is pursued in [22]. It is shown [22, Corollary 3.3]
that for v > 1/2,
(3.16)
°
so that, in particular, jVI< is a concave function of K"
< I'\, < 00, for v > 1/2.
However, it is known [14, p. 1485] that jVI< is a convex function of 1'\"
< K, < 00,
for Sv< 1/2. By considering the chord joining two points on the graph of jVI<
as a function of 1'\" we thus obtain:
°
(3.17)
jVI< > ; -=- ~ (ivK - jvk) + jvk ,
k < K, < K,
v > 1/2,
jVI< < ; -=- ~ (ivK - jvk) + jvk ,
k < I'\, < K,
°Sv< 1/2.
and
(3.18)
°
These become equalities when v = 1/2.
If we put k = n, K = n + 1, we get
(3.19)
jVI< > (I'\, - n)(iv,n+l - jvn) + jvn ,
(3.20)
jVI< < (I'\, - n)(jv,n+l - jvn) + jvn ,0 Sv< 1/2.
V
> 1/2,
In the case where I'\, = n + 1/2, we get
(3.21)
Yv,n+l > (iv,n+1 + jvn)/2,
v> 1/2,
°
Yv,n+1 < (iv,n+l + jvn)/2,
Sv< 1/2.
Some numerical examples given in [22] show the sharpness of these inequalities.
(3.22)
3.4. TURAN-TYPE INEQUALITIES
Tunin-type inequalities are named from a convexity result of P. Turan for Legendre
polynomials:
Pn+l (x)Pn- 1(x) - P~(x) ;::: 0.
A corresponding result for Bessel functions was proved by O. Sza,sz:
Jv(X)JV+2(X) - J';+l (x) < 0,
v> -1, -00 < x < 00.
Lorch [40] has examined the question of similar relations for the zeros. Using
(3.12), he has shown that
(3.23)
ICv+f,k+r
Cvk
Cv+c5,k+h I < °
CV+c5+f,k+h+r
'
°
where v, E, 6, h ;::: 0, k, r = 0,1, ... ,E + r > and h + 6 > 0. This recovers the
result mentioned earlier that log Cvk is a concave function of v.
Similar results were found for zeros of derivatives of cylinder functions by Laforgia
[35].
CONVEXITY PROPERTIES OF SPECIAL FUNCTIONS
317
3.5. MISCELLANEOUS PROPERTIES OF ZEROS
The (generally complex) zeros ofthe even entire function z-V Jv(z), for unrestricted
real v, are located symmetrically with respect to both the real and imaginary axes
in the z-plane. Following [60, p. 497], we denote the zeros of this function by
±jv1, ±jv2, ±jv3,"" where Re(jvn) > 0 and I Re(jvdl :::; I Re(jv2) I :::; IRe(jv3)I :::;
.... If Re(±jvn) = 0, for any value of n, we choose jlm to have its imaginary part
positive.
It is instructive to consider the evolution of the zeros as v decreases. For v ~ -1,
they are all real. As v decreases through -1 the numbers ±jv1 approach the origin,
collide, and move off along the imaginary axis. As v is further decreased, to -2,
these zeros return to the origin [31], [55]. All of this suggests that to deal with
purely imaginary zeros, it is advantageous to consider the squares of the zeros.
We find then that jZ1 can be continued analytically from the interval (-1, (0) to
the interval (- 2, 00 ). In fact [32], jZ1 decreases to a minimum and then increases
again (to 0) as v increases from - 2 to -l.
We conjecture that jZ1 is convex for -2 < v < 00; cf. [12] where the convexity is
proved for 0 < v < 00 and conjectured for -1 < v < 00.
Here we show, as in [55], that -j;;12 is a convexfunction ofv on (-2,-1). To see
this we write the well-known formula [60, p. 502]
L j;;; = [4(v + I)r
00
1
n=1
in the form
L j;;; - [4(1 + v)t 1 .
00
(3.24)
- j;;12 =
n=2
The last term here is obviously convex on (-00, -1) having the positive second
derivative -(v + 1)-3/2 there. On the other hand,
is positive, and decreasing
on (-2, (0). Its second derivative there is given by
j;;;
2j"
6j/2
----:a
+ -'4-'
J
J
where j = jvn, and primes denote derivatives with respect to v. But, from [9],
j" < 0, for v > -no Hence j;;;, is a convex function of v on (-2, (0) for v > -2.
Thus all terms on the right-hand side of (3.24) are convex functions of von (-2, -1)
and this completes the proof.
Since -j;;12 is a positive convex function on (-2,-1) which approaches +00 as
v --t -2+ and v --t -1-, we get an alternative proof of the result [32, Theorem
3.1] that -jZ1 is unimodal on (-2, -1).
In [10], defining jV,K for continuous K, in the usual way, it is shown that it is concave
for v ~ Vo if jVQ,k > Vo + 1/2 with Vo ~ -1/2. An interesting graph is shown.
M. E. MULDOON
318
3.6. SOME CONVEXITY PROPERTIES OF BESSEL FUNCTION VALUES
So far we have been mostly concerned with convexity properties of the zeros of
Bessel and related functions. But there are also such results for functional values.
It is shown in [25, Lemma 2.3] that for each fixed ß (0< ß ~ 1), and each x > 0
(x =f: j.,k, k = 1,2, ... ), the function J.,+ß(x)/ J.,(x) decreases as v increases,
-(ß + 1)/2 ~ v < 00, v > -1. This shows that, for fixed x (> 0), the function
log J.,(x) is a concave function of v on its interval of definition, so long as v > -1.
Similarly, we find, using [25, Lemma 2.2], that for fixed x (> 0), the function
10gK.,(x) is a convexfunction ofv on (-00,00).
The Bessel function J.,(x) has infinitely many zeros on the positive real axis.
However the modified Bessel function [60, p. 77]
(z/2)2n+.,
00
I.,(z) = ~ n! r(v + n + 1)
does not vanish there. In [56] there is demonstrated the complete monotonicity
(hence the convexity of x-"e-XI.,(x).
Mahajan [47] generalised a result of Mitrinovic [49, pp. 240-241] by showing that
(x + l)a+lJa (x: 1) - x a+1Ja (;) >
(ir r(a1+ 1) .
In the special case a = -1/2, this becomes
7r
7r
(x + 1) cos --1 - x cos - > 1
x+
x
Mitrinovic had established the latter inequality for x 2: v'3 while Mahajan improved this range of validity to x > 1.407 .... In [41], it is shown that the largest
interval of validity of this inequality is (1,00). The method depends heavily on
properties of concave functions.
3.7. USES OF CONVEXITY IN DETERMINING NU MB ER OF ZEROS
It is shown in [46] that J.,(x) has two inflection points before its first positive
zero when A < v < 0 and none in 0 < x < j.,l for -1 < v ~ A where A =
-0.1993707809 ....
The proof is broken into three parts, establishing respectively the uniqueness, the
existence and the evaluation of A. Only the first two parts will be discussed here.
(i) Uniqueness of A. Using the differential equation (3.1) and a recurrence relation
[60, p. 45, (4)]
(3.27)
we see that the positive zeros of
(3.28)
J: (x) occur where
J"+1(x)
J.,(x)
x 2 - v2 + V
x
----'--:-'-:-'-=----
CONVEXITY PROPERTIES OF SPECIAL FUNCTIONS
319
In view of the Mittag-LeIDer partial fractions expansion [60, p. 498, (1)]
J V +1(x) = ~
2x
L...J·2
J v ()
X
k=l
Jvk - X 2 '
(3.29)
the positive roots of J~ (x) are the same as those of the equation
(3.30)
Gv(x) := 2
1
v
L
+
k=lJvk-x
(Xl
·2
2
2 - v
--2- =
1.
X
It is elear that
lim Gv(x) =
x-+O+
-1< v < O.
lim Gv(x) = +00,
X-+jvl-
Consequently the graph of y = Gv(x) is convex, with a unique minimum, on
0< x < jvl, -1 < v < 0, since G~(x) > 0, 0 < x < jvl, -1 < v < O. To verify
this we write
·2
3 2
6( 2
)
G"(x) = 4 ~ Jvk + X + V - V
' x i:- 0, jvk .
v
L...J (.2 _ 2)3
x4
k=l Jvk x
(Xl
Also, for each fixed x in the interval 0< x < jvl, Gv(x) is a decreasing function of
v, -1 < v S o. More precisely, for -1 < v < v+€ S 0, we have GV+f(x) < Gv(x),
o < x < jvl, since [60, p. 508] each zero jvk is an increasing function of v.
The zeros of J~ (x) occur where the convex graph of y = Gv(x) crosses the horizontalline y = 1. Now it is elear from the consequence
IL~ > 2 (v 2 -
v)
of Lemma 2 that there are no zeros for v elose to -1, there are no crossings for
these values of v. However as v increases, the convex curve referred to above
becomes lower and if it meets the line y = 1 will do so for a unique value A.
(Recall the uniqueness of the minimum of the U-shaped graph of y = Gv(x).)
(ii) Existence 01 A. The existence of such a A is established as follows. Suppose
that no such A exists. In that case we would have for all v satisfying -1 < v < 0,
Gv(x) > 1,
0< x < jVl .
Taking the limit as v --+ 0-, we would get
GO(x) 2:1,
O<X<jOl.
Now
Go(x) = 2
1
L
n=l JOn - x
(Xl
·2
2
is continuous on [0, jvl) so we would get Go(O) 2: 1. But [60, p. 502]
1
1
L -:z = 2·
n=l JO n
(Xl
Go(O) = 2
Hence we have a contradiction and so A exists as asserted.
320
M. E. MULDOON
4. Hermite Functions
The Hermite function H>..(t) can be defined (see, e.g., [24]) by
(4.1)
H>..(t) = _ sin1l"Ar(1 + A)
211"
f
n=O
r{(n - A)/2} (-2t)n
r(n + 1)
or, in terms of the confluent hypergeometric functions, by ([8])
(4.2)
2>" [
A1I" (A 1)
(A 1
H>.(t) = .fi
cos"2 r 2 + 2 1 F1 -2' 2; t
2)
+ 2tsin A; r(~ + 1) 1 F 1 (-~ + ~,~;
e)].
Formula (4.1) is to be understood in a limiting sense when A is an integer and
the constant multiplying the sum is chosen so that H>..(t) reduces to the Hermite
polynomials (with the notation of, e.g., [58]) in case A is a nonnegative integer.
Thus Ho(t) = 1, Hl (t) = 2t, H2 (t) = 4t 2 - 2, H 3 (t) = 8t 3 - 12t, etc. In the
polynomial case, the zeros of H>..(t) are real and located symmetrically with respect
to the origin. In [17] we study the real zeros of H>..(t) in the case where A is a
positive real number. The largest real zero of H>..(t) is of importance in the study
of subharmonic functions [24].
It turns out that, when n < A ~ n + 1, with n a nonnegative integer, H>..(t) has
n + 1 real zeros which increase with A. As A passes through each nonnegative
integer n a new leftmost zero appears at -00 while the right-most zero passes
through the largest zero of H n ( t) .
For each fixed A, H>..(t) is that solution of the Hermite equation
y" - 2ty' + 2AY = 0
(4.3)
which grows relatively slowly as t -t +00.
We consider also a solution of (4.3) which is linearly independent of H>..(z):
(4.4)
. A1I" (A
1
G>..(t) = -2>' [-sm-r
- + -1) IF1 (-A
_·t
.fi
2
2 2
2 '2'
2)
+ 2t COS A211" r (~ + 1) 1 Fl ( - - : + ~, ~; e) ].
The functions e- t2 / 2 H>..(t) and e- t2 / 2 G>..(t), which have the same zeros as H>..(t)
and G>..(t) are linearly independent solutions of the modified Hermite equation .
[:2 +2A+l-t2]y(t)=O.
(4.5)
It was shown by Durand [8] that
(4.6 )
2->".fi
t 2 [ 2(
2( ]
r{A+l)e- H>..t)+G>..t)
_2.1
.fi
-
0
00
e -(2)''+I)r+t tanh r
2
dT
Vsinh T cosh T
.
321
CONVEXITY PROPERTIES OF SPECIAL FUNCTIONS
The main result of [17] is a formula analogous to this for the derivative of a zero
of a solution of (4.5) with respect to A.
We consider the zeros of H>.,(t), A > 0, or, more generally, the zeros of linear
combinations of H>., and G>., or zeros of the equation
cosaH>.,(t) - sinaG>.,(t) = 0.
(4.7)
In [17], we show
(4.8)
dc = ..fo
dA
2
where
1
co
e-(2).+1)T cp(c"'tanh 7')
0
d7'
'"sinh 7' cosh 7'
,
cp(x) = eX erfc (x) .
2
Here erfc is the complementary error function:
2 (CO _t 2
erfc (x) = ..fo 1x e
dt.
Now, from [1, 7.4.2]
<jJ(x) = ~ {CO e-t2-2xt dt
..(i 10
so <jJ is completely monotonie on (-00,00), i.e., (_I)n<jJ(n)(x) 2: 0, n = o,1, ....
This observation enables us to establish that C'(A) is completely monotonie on
its interval of definition. In partieular, C(A) is concave function of lambda on its
interval of definition.
References
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Series 55, Washington, 1964.
2. S. Ahmed, A. Laforgia and M. E. Muldoon, On the spacing of the zeros of some classical
orthogonal polynomials, J. London Math. Soe. 25 (1982), 246-252.
3. E. Artin, The Gamma Function, Holt, Rinehart and Winston, New York, 1964 [German
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7. Z. Dosl<i, Higher monotonicity properties of special junctions: application on Bessel case
lvi < 1/2, CMUC 31 (1990), 233-241.
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Theory and Applieations of Special Ftmetions (R. Askey, ed.), Aeademic Press, New York
and London, 1975, pp. 353-374.
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81-88.
10. A. Elbert, L. Gattesehi and A. Laforgia, On the concavity of zeros of Bessel junctions, Appl.
Anal. 16 (1983), 261-278.
322
M.E.MULDOON
11. A. Elbert and A. Laforgia, On the zeros 0/ derivatives 0/ Bessel functions, Z. Angew. Math.
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38.
INEQUALITIES IN CIRCULAR ARITHMETIC:
A SURVEY
LJILJANA D. PETKOVIC
Faculty 01 Mechanical Engineering, Beogradska 14, 18000 Nis, Yugoslavia
MIODRAG S. PETKOVIC
Faculty 01 Electronic Engineering, P.O. Box 73, 18000 Nis, Yugoslavia
Abstract. Many relations and properties in circular complex arithmetic involving estimates of various kind and inclusions reduce to analytical inequalities. In this paper we
give a collection of inequalities in the complex realm which are connected to the priority
of circular arithmetic operations, diametrical outer approximations by disks and circular
complex functions.
1. Introduction
With practical computational problems, a standard question should be "what is
the error in the results?" As already pointed out by Wilkinson [24], a considerable amount of the applied procedure is to improve the approximate results and
also to give error bounds for the improved approximations. The demands of the
computer age with its arithmetic of finite precision have dictated the need for a
structure which is referred to as interval arithmetic (Moore [9]). In particular,
error bound procedures for solving certain problems in the complex realm require
complex interval arithmetic (see Alefeld and Herzberger [1] for details). This type
of arithmetic was introduced by Gargantini and Henrici in 1973 in connection
with the simultaneous inclusion of complex zeros of polynomials [7]. Later, circular complex arithmetic was applied as an efficient tool for computer methods
for the range of functions (see the book [23] by Ratschek and Rokne and references cited there), for the construction of iterative methods for the inclusion of
complex zeros of polynomials (e.g. [5], [7], [20-21], [28]), analysis of numerical stability of iterative processes (e.g. [6], [13], [22])), inclusive calculus of residues [17],
and also for solving some problems in technical disciplines where circular complex
arithmetic appears as a natural extension of complex arithmetic in the presence
of uncertain quantities and providing the construction of convenient self-verifying
methods.
Except a few new operations, complex circular arithmetic does not require a special
technique; it uses elementary properties of complex analysis and some methods for
1991 Mathematics Subject Classification. Primary 30A10, 65G10j Secondary 30ElO.
Key words and phrases. Circular arithmeticj Inequalities in the complex domainj Inclusive
approximations.
This work was supported in part by the Serbian Scientific Foundation, grant number 04M03
325
G. V. Milovanovic (ed.), Recent Progress in Inequalities, 325-340.
© 1998 Kluwer Academic Publishers.
326
LJ. D. PETKOVrC AND M. S. PETKOVrC
obtaining outer approximations. The basis of this type of arithmetic are, actually,
estimates and inequalities in the complex realm. But, many examples in various
applications show a strong connection in the opposite direction, tao. For this
reason, it seems quite reasonable to cite R. D. Richtmeyer (Math. Comp. 22
(1968), p. 21) who said that "... internal analysis is in a sensejust a new language
for inequalities". Examples presented in this paper could be regarded only as a
short collection of selected problems which demonstrate a tight connection of the
two mentioned subjects, without the ambition to cover most different subjects in
interval mathematics.
2. Circular Complex Interval Arithmetic
In this section we give some basic properties of circular complex arithmetic and
circular functions. Let C E C be a complex number and r a nonnegative real
number. The set Z = {z : Iz - cl :::; r} = {c; r} is called a circular internal or
disk with the center c and the radius r. The set of circular intervals is denoted
by K(C). The notations mid (Z), rad (Z) and diam(Z) will be sometimes used to
denote center, radius and diameter of a disk Z. Disk Z is called a zero-internal
if 0 E Z. Two circular intervals Zl = {Cl; rl} and Z2 = {C2; r2} are equal if and
only if Cl = C2 and rl = r2. For two disks Zl = {cI;rd and Z2 = {c2;r2} the
following is valid:
ICI - c21 :::; rl - r2
{:}
ICI - c21 ~ rl + r2
{:}
Zl ~ Z2,
Zl n Z2 = 0.
Let Z = {c; r} and w E C. The following operations define circular complex arithmetic:
w ± Z = {w ± z : z E Z} = {w ± c; r},
w·Z={w·z: zEZ}={w·c;lwlr},
Zl ±Z2 = {Cl ±c2;rl +r2},
Z -l - {w - 1 . z E Z} - { 2 c 2 . 2 r 2 } 0 d Z - {Co r}
- ~ .
- Icl - r ' Icl - r ' y::. ,
•
The above operations are exact. Multiplication of disks was introduced by Gargantini and Henrici [7] in the following manner:
Zl . Z2 := {CIC2; ICllr2 + IC21rl + rlr2}.
According to this and the inversion Z-l division of disks is given by
Zl ._ Z . Z-l _ {
CiC2
. ICllr2 + hlrl + r l r 2 } 0 d Z
' y::. 2·
Z2·- 1 2 IC212 -r22'
IC212 -r2 2
Let GI and G2 be some subsets of C and f : GI 1-+ G 2 a mapping. By T(G I ) we
denote a family of circular subsets of GI and by T(G 2 ) a family of closed subsets
of G 2 • A mapping
T(Gt} 1-+ T(G 2 ), defined by
r :
r(Z) = {f(z) : z E Z} =
U{f(z)} for every Z E T(G
zEZ
I ),
INEQUALITIES IN CIRCULAR ARITHMETIC: A SURVEY
327
is called a closed uni ted extension of f. 1* possesses a subset property:
A ~ B
(A, BE T(G 1 )) :::} j*(A) ~ j*(B).
A mapping F : H f-t E (H, E ~ K(C)) is called a circular complex interval
junction. An interval function F : H f-t E is inclusive isotone if
Z ~ W :::} F(Z) ~ F(W)
(Z,W EH).
Let 0 be a closed region in the complex plane. A disk W such that n ~ W is
called an inclusive disk for O. Covering n by W is called a circular inclusive
approximation or shorter I-approximation. If W 1 and W 2 are two circular 1approximations of 0, we will say that W1 is a better approximation than W 2 if
rad (W1 ) ::; rad (W2 ).
I-approximations ofregion 1* (Z), obtained by mapping a given disk Z by a closed
complex function f, are of special importance. For an arbitrary complex number
(, a closed complex function f, and a disk Z, we define function
R(() = max If(z) - (I·
zEZ
Since If(z) - (I ::; R(() (z E Z) it follows that 1*(Z) = {f(z) : z E Z} is
contained in the disk V = {(; R( ()). Disk V is an inclusive disk for 1* (Z).
Let us denote A(() = area{V} = 7rR(()2 and let inf{ A(() = A((s). Then Vs =
{(s; R( (s)) is the smallest disk which contains 1* (Z). In other words, Vs is the
best I -approximation of 1* (Z). We denote the best I-approximation of 1* (Z) by
Is(f*(Z)). Then If(z) - (si::; R((s) holds (because of the inclusion 1*(Z) ~ Vs ).
This defines the so-called enclosing condition (see [19] for more details).
The diameter 2R( (s) of the smallest disk Vs can be at best equal to the diameter
(1)
of 1*(Z). Since 1*(Z) is closed, it suffices to take points on the contour r of Z for
finding the diameter, that is,
Disk Vd = {(s; D /2} is called the diametrical inclusive disk of the closed region
1*(Z) or D-form if the enclosing condition, given by the inequality
(2)
D
If(z) - (si::; "2
(z E Z),
holds. For this disk we will also use the notation Id(f*(Z)). D-form for some
elementary functions will be considered in Section 4.
The circular complex functions in D-form can be of certain importance as illustrated in [10], [14], [17-18] and [27]. For example, we emphasise an important
LJ. D. PETKOVrC AND M. S. PETKOVrC
328
problem in the optimisation theory which consists in determination maxzEZ If(z)l,
Z E K(C), for which it is obviously valid
If(z)lzEz::; maZx If(z)l::; ImidI(f*(Z)) I +radI(f*(Z)).
(3)
zE
Let Z be a given disk and F(Z) an inclusive disk for the closed region f*(Z) with
the center fee) and a radius R. For F(Z) = {fee); R}, whose center is the image of
the center of Z, we say that possesses the centered form or C-form. This form has
greater practical importance than D-form which requires two difficult problems to
be solved: (i) determination of the diameter of f*(Z) by (1) and (ii) checking of
the enclosing condition (2).
3. Particular Inequalities
In this section we give and prove some inequalities important for the determination
of the priority of operations in the circular arithmetic (see [11]).
Lemma 3.1. Let PI, ... ,Pn (n 2: 2) be real numbers such that Pk E (0,1), k =
0,1, ... ,n. Then
n
n
k=1
k=1
II (1 - Pk) + II (1 + Pk) > 2.
(4)
Proof. The inequality (4) can be easily proved by induction, but we give an interesting proof based on the idea of Trajkovic [26]. Let GI and G 2 be polynomials of
the n-th order with the zeros -PI, ... ,-Pn and PI, ... ,Pn, respectively, that is
GI(x) = (x + pt} ... (x + Pn) = x n + glXn-1 + ... + gn,
G 2 (x) = (x - pt} ... (x - Pn).
Since numbers PI, ... ,Pn are positive, we have gl,g2, ... ,gn > 0. It is easy to
show that
G2 ()
X = Xn - glX n-I + g2X n-2 - g3X n-3 + ... + ( - l)n gn·
Now we have
n
n
k=1
k=1
II (1 - Pk) + II (1 + Pk) = G2(1) + GI (1) = 2(1 + g2 + g4 + ... ) > 2. 0
Lemma 3.2. Let xE [0,1]. Then the inequality
holds for a E (0,1). The opposite inequality holds for a > 1. The equality appears
for a = and a = 1.
°
INEQUALITIES IN CIRCULAR ARITHMETIC: A SURVEY
329
The proof of this elementary inequality which has often appeared in literat ure is
simple and will be omitted.
The preceding lemmas are necessary for the proofs of the theorems in the sequel.
For a disk Zk = {ckjrd which do not contain 0 (shorter non-zero disk) we will
use the notation Zk = cd1jPk}, Pk = rkJlckl, 0< Pk < 1.
Theorem 3.1. Let Z1,'" ,Zn (n ~ 2) be non-zero disks such that the inequality
n
TI (1 + Pk) < 2 holds. Then we have
k=1
Proof. Directly applying the inversion and the product of disks (see Section 2) we
find
The inequality (4) can be written as
n
n
k=1
k=1
II (1 - Pk) > 2 - II (1 + Pk) or
n
n
n
n
k=1
k=1
k=1
k=1
2
II (1 - p~) > II (1 + Pk) [2 - II (1 + Pk)] = 1- [II (1 + Pk) - 1] ,
from which directly follows the assertion of the theorem.
0
From Theorem 3.1 we see that the product of the inversion of disks gives a better
result (a smaller disk) than the inversion of the product of disks.
Let Z = {cjr} = {peilljr}, rJp = P < 1 be a non-zero disk and let k E N. The
region {Z1/k: Z E Z} consists ofthe k disjoint regions 8 0 ,81 , •.. ,8k- 1 which are
not disks and which are of the same form. The k-th root of a disk in the centered
form, denoted by Z~/k, is defined in the paper [16] as
k-1
(5)
Z~/k:= U wJm) ,
m=O
wJm) = p1/k{ exp (/' + :m1T)j 1- (1 _ p)1/k}.
LJ. D. PETKOVIC AND M. S. PETKOVIC
330
Starting from the equality I/z 1 / k = (I/Z)l/k, we can introduce the following
two interval extensions: I/Z 1 / k and (I/Z)l/k. For the centered form we will use
notations
k-l
I/Z~/k =
U G(m)
k-l
U H(m).
(I/Z)~/k =
and
m=O
m=O
Disks Go, GI, ... ,Gk - l are identical and have the same radius. The same is valid
for the disks Ho, H 1 , ... , Hk-I. Let ra = rad (Gm) and rH = rad (Hm ).
Theorem 3.2. Let Z = {c;r} = {peill;r}, r/p =p < 1 be a non-zero disko Then
ra < rHo
Proof. First we have
pl/k (p r)l/k
ra =
- p2/;
- (pl/k - (p - r)l/k)2,
From Lemma 3.2 for a = I/k and x = r/p, we obtain
( pr)l/k - (r)2/k
r)l/k]2 ,
I- p
=1- [
1-( I- p
I1- (pr)211/k <2I-
that is
(p2 _ r2)I/k < p2/k _ (pl/k _ (p _ r)l/k)2,
wherefrom directly follows ra < rHo 0
As a consequence of Theorem 3.2 it follows that it is better to extract the root and
then make inversion, than contrary ([23]).
4. Diametrical Inclusive Approximations
In this section some inequalities concerning diametrical inclusive approximations
of zl/m = {z : zm E Z, 0 i Z, mE N}, and log Z (0 i Z) will be regarded. Let
Z = {(; r} and let us assurne that the disk Z does not contain the origin, that is,
p := r /1(1 < 1. As it was shown in [16), the construction of the diametrical disk
for the range zl/m reduces to the following problem:
Let D = {uo; d/2} be the disk with the center
uo :=
and the radius
d
rO
(1 + p)l/m + (1 _ p)l/m
:="2 =
2
(1 + p)l/m - (1 _ p)l/m
2
'
and let G be the image (one of m, see (5)) ofthe disk {I;p}, pE (0,1), under w =
zl/m with argz l / m E [-m- l arcsinp, m- l arcsinp). The question arises whether
G is completely contained inside disk D. Before exposing the main theorem we
give the following inequality:
Lemma 4.1. Let m ~ 2 and pE [0,1). Then
(6)
INEQUALITIES IN CIRCULAR ARITHMETIC: A SURVEY
Proof. By using binomial formula, we find
(l+p)l/m =
f C~m)pk,
(l_p)l/m =
k=O
331
f C~m)(-I)kpk.
k=O
Hence it follows
2
(1 + p)l/m - (1 - p)l/m = - p + 2
m
(7)
L a>.pA,
00
>.=2
where for k = 1,2, ...
1
k
a2k = 0, a2k+1 = (2k + 1)!m2k+ l }1[(2S -I)m - l][2sm - 1].
(8)
The function f(p) = sin(m- l arcsinp) will be developed into the power series.
Since
1
(arCSin p ) ,
f '(p ) -~cos
mV 1 - p2
m
p
p
p
f "(P) =- 2 1 .
Sln (arCSin ) +
cos (arCSin ) ,
m (1 - p2)
m
m(l- p2)3/2
m
we can form a differential equation
(1 - p2)f"(P) - pf'(p) + ~ f(p) = 0
m
with the initial conditions f(O) = 0, 1'(0) = l/m, 1"(0) = O.
(9)
00
Putting f(P) = 2: b>.p>', we solve the equation (9) by the weH known method of
>.=0
series. In this manner, taking into account the initial conditions, we get
k
II[(2s - I)2 m 2 - 1]
1
b2k =0, b1 =-, b2k +1 = 8=1(2k + I)!m 2k +1
m
(10)
(k = 1,2, ... ).
Hence, we have
. (arCSin p )
1
~
>.
f(p) = sm
m
= m p + L..- b>.p ,
(11)
>.=2
where the coefficients b>. are given by (10).
By using the developments (7) and (11) we obtain
(1
+
•
p) I/rn -
(1 -
p
· (arcsm
p) I / r- n
2 sm
m )
00
"
= 2 'L..-(a2k+l
- b2k+1 )p2k+1 .
k=1
To prove the relation (6) it suffices to show that a2k+1 - b2k+l 2: 0 for every m 2: 2
and k = 1,2, .... This is obvious since from (8) and (10) it follows
k
TI {[(2s - l)m - I][2sm - 1J - [(2s - 1)2 m 2 - In
a2k+l - b2k+l = 8=1
(2k + 1)!m 2k +1
~:'-_-------;-,-----:-:--,:-;--:-:-------
LJ. D. PETKOVIC AND M. S. PETKOVIC
332
Le.,
a2k+1 -
1
b2k+1 = (2k + I)!m 2 k+ 1
!!
k
(m - 2)[(2s - I)m - 1) ~ O.
0
Now we give our main result:
Theorem 4.1. If the inequality
sinmcp
UoP
->sincp - ro
(12)
holda, then the disk D = {uo;d/2} completely contains the range G = {I;pp/m
and presents the diametrical disk for this range.
Proof. Let us define the function h( cp) = sin mcp I sin cp. First we find h' (cp) =
x(cp)/sin 2 cp, where x(cp) = mcosmcpsincp - coscpsinmcp. Since x(O) = 0 and
x'(cp) = (1 - m 2 ) sin cp sin mcp ~ 0 (because sin mcp E [0, sin(mm- 1 arcsinp)] =
[O,p)), it follows that x(cp) < O. Hence, h'(cp) < O. Besides, h(O) = m and
we conclude that the function h(cp) is monotonically decreasing on the interval
(0,m- 1 arcsinp). For this reason the inequality (12) will be proved if we show
that
sin (m . arcsin p) > UoP
sin
arcsinp) - ro '
(.k
.k
which reduces to
(13)
(1 + p)l/m - (1 - p)l/m ~ [(1 + p)l/m + (1 _ p)l/m) sincrc~np).
Hence, by Lemma 3.2 (for a = 11m, m ~ 2 and p E [0,1)), the inequality
(6) proved in Lemma 4.1 directly folIows. Therefore, the assertion of Theorem 4.1 is proved and disk D = {uo; ro} is the diametrical disk for the region
G = {I;pp/m. 0
The presented problem was firstly considered in [16), and then posed as an open
problem by Lj. Petkovic [12). McCoy and Kuijlaars [8) first solved this problem
using an another approach although the inequality (13) appears in their paper too.
The mentioned problem was also considered in [18).
In the sequel we will consider a construction of the diametrical disk for the region
of the function f(z) = log z over a non-zero disk Z. Since log z presents a many
valued function, we will consider in the sequel only the principal value of log z
assuming that log z = log Iz I + i arg z, arg z E [0, 211'). Therefore, speaking about
the diametrical disk for log Z, we will regard only one set log Z = {log Izl +i arg z :
z E Z, 0 ~ arg z < 211'} which is called the region of the principal value or, shorter,
p. v. region.
The diametrical disk Id(log Z) was determined by Börsken in [4). Here we obtain
the same result using partly the result of Börsken but with a new simple estimation
approach which is based on the application of circular arithmetic. First, by using
(1), we find the diameter of the region 10gZ, 0 i Z = {z;p}. We again use the
notation p = pllzl « 1).
INEQUALITIES IN CIRCULAR ARITHMETIC: A SURVEY
333
Theorem 4.2. Let Z = {Zj p} and p = pilzi < 1, that is, 0 ft Z. Then
l+p
IlogZl -log z2I zt.z2Ez ~ log -1-'
-p
Proof. We have
Ilogzl -logz2Izt.z2Ez = Ilog(z + peio:) -log(z + peiß )10:,ßE[0,21r)
= Ilog(l + peio:') - log(l + peiß') 10:' ,ß'E[O,21r)
l° +
I
- l° +
eio:'
p
=
<
1
p
1
, , dt -
teto:
l
P
eiß'
° 1 + tet'ß' dt
eio:' , -eiß'
,
teto:'
1 + te tß '
I dt.
Using the operations of the interval arithmetic and (3), we find
I1 +
eio,'
eiß'
teio:' - 1 + teiß'
1
I
I= t1 I1 + 1
1
teiß' - 1 + teio:'
1
I
I 1 I{lj t}
1
{lj t}
~ t 1 + t{Oj I} - 1 + t{Oj I} = t 1- t 2 - 1- t 2
_ 11{0.
- t
2t
' 1 - t2
}I- 1 -
2
-
_
1
I
1
t 2 - 1 + t + 1 - t'
According to this we estimate
rI
10 +
1
eiß'
eio:'
teio:'
1 + teiß'
dt
r
~ 10 (-11
+ -11 ) dt = log 11 + P,
+t
- t
- P
which proves the theorem. 0
Therefore, the diameter of the region log Z is given by
d
= diam(log Z) = zt.z2EZ
max l10g Zl - log z21 = log 11 + P = log Ilzll + p.
- p
Z - P
From the geometrical construction and some facts given in [10] it is simple to show
that the center of the diametrical disk for log Z is the point
A = log vlzI2 - p2 + i argz.
Obviously, the disk {Aj d/2} will be the diametrical disk for the p.v. region oflog Z
if the disk {uo j ro} is the diametrical disk for the p. v. region {log IZ I+ i arg Z : Z E
{ljp}}, where
(14)
1
uo = 2 1og (1 - p2),
d
1
1 +p
ro = - = -log--.
2
2
1- p
Similarly as in the case of the region zl/m, it suffices to consider the mapping of
the disk {I j p}, 0 ~ p < 1 under the transformation z t-+ log z = u + i v in the
aim to obtain diametrical disks for the region log Z. Before that, we expose the
following Blaschke's result [3]:
334
LJ. D. PETKOVrC AND M. S. PETKOVrC
Theorem A. 11 the curvature 01 the simple closed smooth boundary w(O) 01 a
region G is strictly positive and has exactly 2'\ extreme points, then the contour
w(O) has at most 2'\ interseetions with any circle. Tangential interseetions are
counted as double intersections.
Sometimes, Theorem A enables us to check (2) in an elegant and simple way
proving that the curvature of the curve w(O) is greater than the curvature of a
possible inclusion disko Such an approach has been demonstrated in [8] for the
range zl/m.
Now we can formulate the following assertion:
Theorem 4.3. The disk D = {uo;ro}, where Uo and ro are given by (14), completely contains the p.v. region Go := {log Izl + i argz : z E {1;p}} il and only il
the inequality
d
1
l+p
Ilog( - AkEZ ::; - = -log-2
2
I-p
(p = pilz!)
holds.
Proof. Let G be the image of the disk Z = {1;p} under the transformation w =
logz. Obviously, the boundary rG of Gis given by
w(O) = 10g(1 + peiO) ,
0 E [0,211").
Let D denote the disk Iw - cl ::; R with
The mapping w = log z sends the points z = 1 ± p to the points w = log (1 ± p) =
c ± R, so that G can not have diameter less than 2R. We shall prove in the same
manner as in [3], that D is the diametrical disk for log Z.
r G is tangential to the circle D in the points c ± R. To prove that r G lies inside
D we compute its curvature.
The curvature K, of the curve w(O) in the complex plane is given by
Im (iiJw)
K,
= Iwl 3
'
where dots denote differentiation with respect to O. For w(O) = log z(O) with
z(O) = 1 + peiO, we compute
.
ipeiO
w(O) = z(O)
Hence
K,
and
..
peiO
w(O) = - z(O)2.
(0) = 1 + pcosO
plz(O)I'
INEQUALITIES IN CIRCULAR ARITHMETIC: A SURVEY
335
wherefrom we see that the curvature is strictly positivej therefore the domain G
is strictly convex. Further , we compute
. (0) = -p sin O(p + cos 0)
K
Iz(O)1 3
.
We see that k(O) has precisely four simple zeros in [0,271"), at 9 = 0,71" and
± arccos (-p). Since circle D is tangential to w(9) in the two points there are
no more points of intersection. Hence w(9) lies either completely inside or completely outside D.
It remains to show that in the point c + R the curvature of w(9) is greater than
the curvature 1/ R of D. Thus we want to show that k(O) > 1/ R, or
(15)
1+p
log-1- > 2p,
-p
for 0 < P < 1. Let h(P) = log((1 + p)(1 - p)) - 2p. Since h(O) = 0 and h'(P) =
2p2/(1- p2) > 0 for 0< p < 1, we conclude that the inequality (15) holds and the
proof of Theorem 4.3 is completed. D
Regarding the domain Z = {(j r} with p = r /1(1, it is easy to construct the
diametrical disk for the range log {(j r}.
5. Circular Complex Functions
In this section we will give some inequalities which salve some important problems
related to the centered forms of interval polynomials and analytical functions in
circular complex arithmetic. These inequalities have already been considered in
[2], [24-25] and [29]. We emphasise that, during the last two decades, J. Rokne
and H. Ratschek have achieved a great contribution to this subject (see the book
[23] and references cited there).
In this paper we give somewhat different proofs of the mentioned inequalities
connected to the centered forms of circular complex functions. Some of them are
proved using TrajkoviC's ideas given in his diploma's work [26] directed by the
authors of this paper. First we will regard the polynomial centered forms which
have a great practical importance (see [2], [23-25]).
Let
(akEC, k=O,1, ... ,n)
p(z) = anz n + ... + alZ + ao
be a complex polynomial and Z = {Cj r} a disko Denote by p* the closed united
extension of the polynomial p, that is,
p*(Z) = {p(z) : z E Z}
(Z E K(C)).
The region p* (Z) is not a circular interval in general. In this section we will prove
inequalities which are important for finding I-approximations for p* (Z) and for
the proof of inclusive isotonicity of the centered form of analytical functions.
LJ. D. PETKOVIC AND M. S. PETKOVIC
336
The power, Taylor's and Homer's C-form, denoted by
Ps(Z) = {p(c)jRs}, PT(Z) = {p(c)jRr}, PH(Z) = {p(C)jRH},
have the centers in p(c) and the radii given by
n
(16)
Rs = rad (anZ n + ... + alZ + a o) = L
(17)
R
k=l
= ~ Ip(k)(c)lrk
T
L..J
k=l
k!
lakl((lcl + r)k -Iclk),
'
RH = rad (( ... ((anZ + an-dZ + a n -2)Z + ... + al)Z + a o)
(18)
n
n
k=l
j=k
= r L [(Ici + rl-ll L ajd-kll·
(See [23-24]).
Let us compare the introduced inclusive approximations for the region p*(Z). For
this purpose we give the following simple lemma.
Lemma 5.1. Let p(x) = ao + alX + ... + anx n be a polynomial with the real
coefficients and x E~. If
(i = 0,1, ... ,n),
(19)
then p(x) ~ 0 for every x ~ O.
Indeed, since ai = p(i) (O)/i!, according to (19) it follows ai ~ 0 (i = 0,1, ... ,n)
so that obviously p(x) ~ 0 for every x> o.
Theorem 5.1 ([29]). Let be given a complex polynomial p(z) = ao + alZ + ... +
anz n . Then, for the I-approximations ZH = {cHjRH}, obtained by applying
Homer's scheme, and ZT = {cTjRT}, obtained by applying Taylor's C-form of
the region p*(Z), the inclusion ZT ~ ZH holds.
Proof. Since CT = CH = p(c), it is sufficient to show that RT ~ RH, where the
radii RT and RH are given by (17) and (18). Let q(r) = RH - RT, that is,
n
n
n
(k) ( )
q(r) = L {[(Ici + rl-lcl(lcl + r)k-IJl L ajd-kl} - LIP k! C Ir k .
k=l
j=k
k=l
Hence
q(i)(r) = ~ [ k!. (ici + r)k-i _ (k -:- I)! Icl(lcl + r)k-i-l] I~ ajd- k
L..J.
(k - z)!
(k - z - I)!
~
k=.
J=k
I
_ ~IP(k+i)(c) k
L..J
k!
r,
k=O
I
INEQUALITIES IN CIRCULAR ARITHMETIC: A SURVEY
337
for i = 1,2, ... ,n - 1. Since q(n) (r) = 0 it follows that q(r) is a polynomial of
degree n - 1. Besides, we find
q
(i)
o_kl_
_ ~(
k!
_ (k - I)! ) k-il~
(0) - ~ (k _ i)!
(k _ i-I)! Ici
~ aJd
e=:)
e=:)
It, t G:: D1-
0
k=,
IClk-ilt.ajd-kl-IP(i)(C)1
~ i! I~
t.ajd-il-IP(i)(C)1
= i!
ajd-'
J='
(j
(c)1
(pU) ('li
It (D I-It ~
ajd- i
(i)
J=k
= i! ~
= i!
Ip
i)!ajd-il = O.
J='
Since q(O) = 0 and q(i) (0) ~ 0, on the basis of Lemma 5.1 it follows that q(r) ~ 0
for every r > o. 0
Theorem 5.2 ([29]). For a disk Z = {Cj r} the following chain of inclusions
P*(Z) ~ PT(Z) ~ PH(Z) ~ Ps(Z).
holds. The equality PT(Z) = PH(Z) = Ps(Z) appears for C = O.
The inclusion P*(Z) ~ PT(Z) is obvious from the definition of Taylor's form. The
inclusion PT(Z) ~ PH(Z) is proved in Theorem 5.1. According to the subdistributivity
Zl (Z2 + Z2) ~ Z l Z2 + Z2Z3,
which holds in interval arithmetic (see, e.g. Alefeld and Herzberger [1]), and
Horner's scheme we have
PH(Z) = (... (anZ + an-dZ + ... + adZ + ao
~ anZ n + an-l Zn-l + ... + a1Z + ao = Ps(Z).
Now we will consider inequalities which are related to the inclusive isotonicity of
the centered form of analytical functions which includes the polynomial centered
form as a special case.
Let f be an analytic function in a given disk Z = {Cj r} and let f* (Z) = {! (z) :
Z E Z} be the exact complex-valued set in the complex plane. Taylor's circular
centered form of f is defined by
(20)
The following theorem, proved in [2], asserts that this form is inclusive isotone.
338
LJ. D. PETKOVIC AND M. S. PETKOVIC
Theorem 5.3. If Zl = {Wl; rt} and Z2 = {W2; r2} are disks in the complex plane
such that Zl ~ Z2, then
We will prove this theorem in a somewhat different way which needs the following
two lemmas.
Lemma 5.2. Let x f-t g(x) be areal junction having all derivatives over the
interval (0, R). If g(i)(O) ~ 0 for all i = 0,1,2, ... , then g(x) ~ 0 for every x ~ O.
The proof of this lemma comes from the fact that all terms g(k) (O)xk /k! of the
Maclaurin series of the function gare nonnegative.
Lemma 5.3. Let W f-t f(w) be an analytic junction in the disk Iw - w21 :::; r2 and
let IWl - w21 :::; r2 - rl. Then
(21)
and
(22)
~ If(k)(W2)llw - W Ik- i
If (i)(W 1 )1 -< L...J.
(k _ i)!
2
1
(i = 1,2, ... ).
k=,
Proof. Using the Taylor expansion of f(w) at the point W = W2 we get
(23)
Putting W = Wl, we obtain
which proves the inequality (21).
Furthermore, starting from (23), we find
so that
Hence, applying absolute values, we get the inequality (22).
0
INEQUALITIES IN CIRCULAR ARITHMETIC: A SURVEY
339
Proof of Theorem 5.3. By virtue of (1) and (20) it suffices to prove the implication
(24)
IWI - w21 ::; r2 -
Tl
=> If(wd - f(W2)1 ::;
f: If(k)k~W2)IT~ - f: If(k~!WI)ITlk.
k=l
k=l
Let us introduce
(25)
where >. = IWI - w21 ::; r2 - Tl, that is T2 ~ rl + >.. Obviously, if U(TI) ~ 0 for
every Tl > 0 then the inequality (24) will hold too as a consequence.
According to (25) (setting Tl = 0) and the inequality (21), we get
U(O) =
f: If(k~!W2)1 >.k -lf(WI) - f(W2)1 ~
O.
k=l
For i = 1,2, ... , from (25) and (22), we find
Therefore, u(i)(O) ~ 0 for each i = 0,1, .... Besides, the real nmction u can be
developed into Taylor's series on the interval (0, T2 - >') and whence, in regard
to Lemma 5.2, we obtain u(rd ~ 0 for 0 ::; rl ::; T2 - >., and Theorem 5.3 is
proved. D
Acknowledgement. This paper was initiated by Prof. D. S. Mitrinovic five years ago.
Although Prof. Mitrinovic did not work in the field of interval mathematics he feIt that
inequalities and estimations in a general sense He in the essence of this topic. He has
permanently encouraged the authors in their work on interval mathematics and, for these
reasons, the authors are very grateful to him.
References
1. G. Alefeld and J. Herzberger, Introduction to Interval Computation, Academic Press, New
York,1983.
2. P. G. Bao and J. Rokne, Inclusion isotonity of circular complex centered forms, BIT 27
(1987), 502-509.
3. W. Blaschke, Kreis und Kugel, 2. Auflage, De Gruyter, Berlin, 1956.
4. N. C. Börsken, Komplexe Kreis-Standardfunktionen, Freiburger Intervall-Berichte 2 (1978),
1-102.
5. I. Gargantini, Parallel Laguerre iterations: Complex case, Numer. Math. 26 (1976), 317323.
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LJ. D. PETKOVIC AND M. S. PETKOVIC
6. ___ , The numencal stability 0/ simultaneous iteration via square-rooting, Comput. Math.
Appl. 5 (1979), 25-31.
7. I. Gargantini and P. Henrici, Cireular anthmetic and the determination 0/ polynomial zeros,
Numer. Math. 18 (1972), 305-320.
8. T. L. McCoy and A. B. J. Kuijlaars, Answer to a query conceming the mapping w = zl/m,
Indag. Math. 4, 479-481.
9. R. E. Moore, Interval Analysis, Prentice-Hall, New Jersey, 1966.
10. Lj. D. Petkovic, On some approximations by disks, Doctoral dissertation, University of
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11. ___ , A note on the evaluation in cireular arithmetic, Z. Angew. Math. Mech. 66 (1986),
371-373.
12. _ _ , Query 359, Notices AMS 33 (1986), p. 629.
13. ___ , The analysis 0/ the numerical stability 0/ iterative methods using internal arithmetic,
Computer Arithmetic and Enclosure Methods (L. Atanassova, J. Herzberger, eds.), North
Holland, 1992, pp. 309-317.
14. ___ , On optimal including cireular approximation tor range 0/ a complex exponential
junction, Z. Angew. Math. Mech. 72 (1993), 109-116.
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Taylor senes, Z. Angew. Math. Mech. 61 (1981),661-662.
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Interval Computation 1 (1993), 34-50.
19. ___ , Verijication methods /or inclusion disks, Reliable Computing 1 (1995), 403-410.
20. M. S. Petkovic, On the Halley-like al90rithms /or the simultaneous approximation 0/ polynomial complex zeros, SIAM J. Numer. Anal. 26 (1989), 740-763.
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Berlin, 1989.
22. M. S. Petkovic and L. V. Stefanovic, The numerical stability 0/ the generalized root iterations
/or polynomial zeros, Comput. Appl. Math. 10 (1984), 97-106.
23. H. Ratschek and J. Rokne, Computer Methods /or the Range 0/ Functions, Ellis Horwood,
Chichester, 1984.
24. J. Rokne and T. Wu, The cireular complex centered form, Computing 28 (1982), 17-30.
25. ___ , The cireular complex centered form, Computing 30 (1983), 201-211.
26. M. Trajkovic, Inequalities in internal analysis, Diploma Work, Faculty of Electronic Engineering, Nis, 1993. (Serbian)
27. M. Trajkovic, Lj. D. Petkovic and M. S. Petkovic, Diametrical inclusion disk /or zn, Z.
Angew. Math. Mech. 75 (1995), 775-782.
28. X. Wang and S. Zheng, A /amily 0/ parallel and interval iterations /or jinding all roots 0/ a
polynomial simultaneously with rapid convergence (I), J. Comput. Math. 1 (1984), 70-76.
29. T. Wu, The cireular complex centered form, Master thesis, Department of Computer Science,
Calgary, 1981.
PROPERTIES OF ISOMETRIES AND APPROXIMATE
ISOMETRIES
THEMISTOCLES M. RASSIAS
National Technical University 01 Athens, Department 01 Mathematics
Zagralou Campus, 15780 Athens, Greece
Abstract. In the present paper an analysis of quasi-isometrie mappings and almost
isometries of function algebras is provided. In addition the A. D. Aleksandrov problem
of eonservative distanees is studied and new open problems are diseussed.
1. Quasi-isometrie Mappings
In his analysis of rotation and strain, F. John [16] eonsiders mappings x = I(x)
of dass Cl defined on an open subset Gof IRn in whieh the Jacobian matrix f' is
nonsingular. The element of length ds for the vector dx is given by
(1.1)
where
(1.2)
denotes the metric tensor of the mapping I, and the superscript T indieates the
transpose. The Euclidean norm of a vector ~ will be denoted by I~I. For the norm
lai of matrix a, we use
(1.3)
lai -_ sup {la~l.
1IT' ~ E IRn} .
Clearly the matrix 9 is positive and symmetrie. Thus, there exists a unique positive
matrix 1 + e such that (1 + e)2 = g. It follows that the matrix
(1.4)
c = f' . (1 + e)-l
is orthogonal, that is c- 1
cT . By (1.4), we see that the Jacobian matrix is
uniquely expressed in the form
(1.5)
I' = c(l + e),
1991 Mathematics Subject Classijication. Primary 51K05.
Key words and phrases. Isometries; Approximate isometries; Quasi-isometries; Strain; Function algebras; Commutative Banach algebras; Gelfand formula; Gelfand transform; Aleksandrov
problem.
341
G.v. Milovanovic (ed.), Recent Progress in lnequalities, 341-379.
© 1998 Kluwer Academic Publishers.
342
TH. M. RASSIAS
where c is orthogonal and 1 + e is positive and symmetric.
Following a suggestion of K. O. Friedrichs, John defines e in (1.4) as the strain
matrix and c as the rotation matrix at the point x. It should be noted that this
definition of the strain matrix differs from the standard one found in textbooks
on elasticity, which would be e(1 + e/2) instead of e. The notion of strain plays a
central role in the nonlinear theory of perfectly elastic solids, where it is assumed
that the stresses caused by adeformation depend only on the strains. A scalar
measure for the strain at a point x is the norm le(x)1 of the matrix e(x).
John deals with mappings such that
(1.6)
le(x)1 < c
for x in G
with a fixed c, with 0 < c < 1. A basic tool is the fact that for the mappings
satisfying (1.6), we will have
1 - c < II (y) - 1(x) I < 1 + c
Iy -xl
'
whenever the ellipsoid of revolution with foci y, x and eccentricity (1- c)/(1 + c)
is contained in the open set G. Such mappings might be called "loeally quasiisometrie".
F. JOHN'S CLASS 1e,G OF MAPPINGS
Let G c ~n be an open set and c a fixed number with 0 < c < 1. We say that
1 : G -t ~n belongs to the dass 1e,G if, for each c' with c < c' < 1 and each x in
G, there exists a positive number 8 = 8(c,x) such that
(1.7)
(1 - c') Iz - xl ~ I/(z) - l(x)1 ~ (1 + c') Iz - xl
when Iz - xl ~ 8.
From the right hand inequality of (1.7), we see that the 1 belonging to 1e,G satisfy
a Lipschitz condition in the neighbourhood of each point of G. This implies not
only that they are continuous but also, according to a theorem of Rademacher [25],
that they are differentiable almost everywhere. The left-hand inequality implies
that each 1 in 1e,G is one-to-one. By compactness, we have:
Lemma 1.1. With 1 in 1e,G, let 1 > c' > c and let S be a eompaet subset 01 G.
Then there exists a positive number ~ = ~ (c' ,S) sueh that (1. 7) holds lor x and
z in S when Iz - xl ~ ~.
Lemma 1.2. Given 1 in 1e,G, let the closed straight segment ('Tl, () with end points
'Tl and ( belong to G. Then
(1.8)
I/(() - 1('Tl) I < (1 + c) I( - 'Tl1·
Proof. With c' > c, denote the segment ('Tl, () by S. From Lemma 1.1 we can find
a number ~ such that (1.7) holds for Iz - xl ~ ~. Now divide ('Tl, () into m equal
segments by points xO = 'Tl, Xl, ... , x m = ( at a distance
lxi _ xi-li = I( - 'Tl I < ~
m
PROPERTIES OF ISOMETRIES AND APPROXIMATE ISOMETRIES
343
appart (1 ~ j ~ m). Then
m
11(() - 1(17)1 ~
m
L 11 (xi) - 1 (xi - I < (1 + c') L lxi - x
1)
i=l
j - 1 1,
i=l
so that I/(() - 1(17)1 ~ (1 + c') I( - 171. Letting c' -+ c, we obtain the inequality
(1.8). 0
Lemma 1.3. Given 1 in fe,G, let 17 and ( be two points ofG such that the ellipsoid
of revolution
(1.9)
with foci 17, ( and eccentricity (1 - c)j(1 + c), belongs to G. Then
(1.10)
Proof. The right-hand inequality in (1.10) follows from Lemma 1.2, since the segment (17,() is within the ellipsoid. Since G is open we can find an c', c < c' < 1,
such that the ellipsoid S given by
(1.9 /)
is also contained in G. Now let ~ = ~(c/, S) be chosen in accordance with
Lemma 1.1 and let m be a fixed integer with
l+c'
m > 2117 - (I (1 _ c') ~ .
A set A of m + 1 points xo, ... , x m will be called admissible if
(A) xo = 17, x m = (, x j ES, j = 0,1, ... ,m,
(B) Ixj - xj - 1 1~ ~ for j = 1, ... , m,
(C)
II (x
j ) -
1 (xi - 1 ) I ~ ~(1 ;- c') .
Admissible sets A exist, for, let AO denote the set obtained by dividing the segment
(17, () into m equal parts. Then
lxi _ xi-li = 117 - (I < (1 - c') ~ < ~
m
by Lemma 1.1
2 (1 +e /) -
,
344
TH. M. RASSIAS
so AO is admissible. Also, the set of admissible A is closed and bounded in
xO , ... , xm space. Define cp by
m
cp(xo, ... ,xm) = :E II (xi) -/(xi-l)l·
i=l
We find in particular that
m
cp (xo, ... ,xm) <:E (1 + c') lxi - xi-li = (1 + c') 177 - (I,
i=l
e
and since cp is continuous there exists an admissible set (xO, ... , xm ) = (eO, ... , m )
at which cp (xO , ... , xm) assurnes its smallest value. By the above estimate of cp
we have cp (eo, ... ,em) < (1 + c') 177 - (I. For any k = 0,1, ... , m, it follows from
(1.7) that
W-771 + le - (I ~ :E lei - ei-li ~ (1- c/)-l:E II (ei) - 1 (ei-I) I
i
i
= (1 - c') -1 cp (eO, ... , e < (1 + c /)(1 - c') -1 177 - (I.
m)
e
Hence, all the k are interior points of the ellipsoid S.
i < r will lie in S. If r < 6./2, we see
For sufficiently small r the ball
by (1.7) that the map x = I(x) is one-to-one and continuous in that ball. It
follows by Brouwer's theorem that 1 (ei) is an interior point of the image set of
1 (ei) < ß there is a unique x with
the ball. Thus, there is a ß such that for
eil< rand I(x) = x. If ß < (1- c') 6./2, it follows from (C) and (1.7) that
Ix -
and similarly that
Ix - e I
Ix -
I
Ix - ei+ll < 6.. If, in addition,
(eo, ... ,ei - l ,x,ei+1, ... ,em) will be admissible. Since the set
(eo, ... , ei , ... , em ) minimises the nmction cp, we will have
then the set
o ~ cp (eO, . .. , x, . .. , em) - cp (eO , . . . , ei , . .. , em)
= Ix-I (ei-I) I + Ix-/(ei+l)I-I/(ei)-/(ei-l)I-I/(ei)-/(ei+l)1
PROPERTIES OF ISOMETRIES AND APPROXIMATE ISOMETRIES
Ix -
for each X in the ball
1 (ei)
iH
1 (ei) ::/: 1 (e ), the balls
I < ß which satisfies (D). Therefore, if 1 W-
345
i)
-:f.
Ix - 1 (ei-i) I < II W) - 1 (ei-i) I ~ (1 - c') ~ ,
Ix - 1 (eHi ) I < II (e) - 1 WH) I ~ (1 - c') ~
and
Ix - 1 (ei) I < ß
can have no point in common. This implies that also 1 (ei-i) -:f. 1 WH) or
that 1 (ei) lies on the open segment with end points 1 (ei-i) and 1 WH). If
on the other hand, 1 (ei-i) = 1 (ei) -:f. 1 (eHi), we can replace ei by x with
i <
I(x) = x, where x lies on the open segment (J (ei-i) ,I WH)).
The resulting set (eo, ... ,x, .. . m ) will again be admissible with the same value
for cp as the set (eo, ... ,ei, ... ,em ) and hence also minimal. In this way, we
can obtain a minimising admissible set of points (xo, ... ,xm ) in which all the
1 (xi) are distinct and each 1 (xi) lies on the open segment joining 1 (Xi-i) and
1 (x iH ). For such a set, we see that
Ix - e I r,
,e
m
m
i=O
i=o
I( -1]1 ~ L IxiH - xii< (1- c/)-i L If (xHi ) - f (xi) I
= (1- c/)-ill(() - 1(1])1.
When c' ~ c, we obtain the required inequality, the left side of (1.10). 0
The ellipsoid given by (1.9) has the semi-minor axis a = k(c) 11] - (I, where
.,fi
k(c) = -1- .
-c
(1.11)
This ellipsoid is contained in a a-neighbourhood of the line segment (1], (). Thus,
(1.10) holds for any pair 1], ( such that the a-neighbourhood of the segment lies
in G. From this remark, the next theorem immediately follows.
Theorem 1.4. Given 1 in Ie.a, let S be a convex subset of G of diameter D. If
the k(c)D-neighbourhood of S belongs to G, then (1.10) holds for each pair 1], (
in S.
In particular, when G
an open ball, say G = {x E IRn :
XO < r}, then
inequality (1.10) will hold for any pair 1], ( in the concentric ball:
is
Ix - I
(1.12)
Since 1 E Ie.a, a boundary point of the image of the set where
be the image of a point with
= ß, so that
Ix - xOI
Ix - xOI ~ ß must
TH. M. RASSIAS
346
Thus, for each y in Rn satisfying
c)r
Iy - f ( 0) I (1 - c)ß = 1(1+ -2k(c)
,
(1.13)
X
:::;
there is a unique X with Ix - xOI :::; ß such that f(x) = y.
Note. For a quantity A and some positive quantity B, the notation A = O(B) will mean
that there is a universal constant M (Le., depending only on the dimension n) such that
IAI:::;MB.
Lemma 1.5. Put B (xo,ß) = {x E Rn: Ix-xol:::;ß}, where ß = r/(1 + 2k(c)).
Let 1 : B (xO, ß) -t Rn be a mapping such that (1.10) is true for all 'fJ and , in
B (xO, ß). Then there exists an orthogonal n x n matrix 'Y such that
(1.14)
I(x) = 1 (XO) + 'Y (x - xO) + O(cß)
lor all x E B (xO, ß) .
Proof. For x E B (XO, ß), we have by (1.10) that
II(x) -1(xO)I:::; (1+c) Ix-xol:::; (1+c- l )cß,
so that (1.14) holds with 'Y = I when c is bounded away from zero, Le., for
= 0(1). It is therefore sufficient to prove (1.14) when 0 < c < some universal
constant.
Choose Xl, ... , x n in Rn, so that the dot products indicated satisfy
Cl
(1.15)
Let X denote the matrix whose columns are the vectors xi - xO and let Y be the
matrix whose columns are 1 (xi) - 1 (xO), j = 1, ... ,n, k = 1, ... ,no By (1.15)
we have
(1.16)
Using (1.10) and (1.15), we find that
11 (xi) - 1 (x k) 12 = lxi - x k 12 (1 + O(c))
= lxi - x k 12 + 0 (cß2) ,
j, k
= 0, 1, ... , n.
Next we make use of the vector identity:
2A. B = A 2 + B 2 - (A - B)2
to obtain
2 (J (x j ) - 1 (XO)) . (J (x k ) - 1 (xO))
= 11 (x j ) - 1 (xO) 12 + 11 (x k) - 1 (xO) 12 - 11 (xi) - 1 (x k ) 2
= Ixj - xOl2 + Ix k - xOl2 -Ixj - xkl2 + 0 (cß2)
1
= 2 (x j - XO) . (x k - xO) + 0 (cß2)
= 2ß2 8jk + 0 (cß2) .
PROPERTIES OF ISOMETRIES AND APPROXIMATE ISOMETRIES
347
Hence
(1.17)
so that Y is non-singular for c less than a suitable universal constant. Thus, the
matrix Y X-I is also non-singular. We put
YX- 1 = "'(E,
where "'( is orthogonal and E is positive and symmetrie. By (1.16) and (1.17),
E 2 = ETE = (E T"'(T) bE) = ("'(Ef("'(E) = (YX- 1
Hence
that is
E = 1+ O(c),
f (YX- = 1+ O(c).
1)
Y = "'(EX = "'(X + o (cß) ,
f (xi) - f (XO) = "'( (xi - XO) + O(cß).
Now let x be any point in B (xO, ß) and put
Then, since "'( is an orthogonal matrix, we have
so that, again by the above vector identity,
2<p(x) . "'( (xi -xO) = 2 (J(x) - f (XO)) . "'( (xi - xO) - 2 (x - xO) . (xi - xO)
= 2 (J(x) - f (XO)) . (J (xi) - 1 (xO))
- 2 (x - xO) . (xi - xO) + 0 (cß2)
= If(x) - f (xO) 12 + 1I (xi) - f (XO) 12 -If(x) - f (xi) 12
-Ix - xOl2 -lxi - xOl2 + Ix - xil2 + 0 (cß2)
= 0 (cß2) .
By (1.15), {ß- 1 (xi - XO)}, 1 ~ j ~ n, is an orthogonal basis for ~ and since,
"'( is an orthogonal matrix, it follows that {ß-l"'( (xi - xO)} is also an orthogonal
basis. Thus we can represent <p by
n
<p(x) = L Ciß- 1"'( (xi - xO) ,
i=1
where ci = <p(x) . ß-l"'( (xi - XO), so that
n
n
1<p(xW = LC~ = ß- 2L (<p(x)· (xi - XO))2 = 0 (c2ß2).
i=1
i=1
348
TH. M. RASSIAS
Hence (1.14) holds when x E B (xo,ß).
0
We can apply Lemma 1.5 to mappings of the class IE,G in the open set G. For any
x in G, let r(x) denote the distance from x to the boundary of G. If
(1.18)
r(x)
ß ~ 1 + 2k(c) ,
then inequality (1.10) will apply to any pair 'f}, ( in a ball of radius ß about x.
Hence there will exist a linear mapping of the form
(1.19)
Lx,ß(Y) = f(x) + rx,ß(Y - x),
where rx,ß is orthogonal, such that
(1.20)
If(Y) - Lx,ß(y)1 < Ancß
for
Iy - xl < ß·
Note that, by Lemma 1.5, An is a universal constant.
Lemma 1.6. Given an nxn matrix /L and a vector z in ~n, suppose that I/Ly+zl <
M for some M > 0 and all Y in a closed ball of radius ß in ~n. Then the norm
I/LI < M/ß·
Proof. Without loss of generality, we mayassurne that the ball is centered at
the origin and that ß = 1. Let Yo be a unit vector with I/LYol = I/LI, so that
1/L(-Yo)1 = I/LI· By hypothesis, we have
If (/LYo) . z ~ 0, then I/LI = I/LYol < M. If (/LYo) . z < 0, replace Yo by -Yo with the
same result.
Small variations in x and ß with Iy - xl < ß will cause a change in Lx,ß(Y) by
at most 2A n ßc, and hence by Lemma 1.6, in a change of rx,ß by at most 2A nc.
Note that two orthogonal matrices have the same determinant (i.e. the same sign)
if the norm of their difference is less than 2. Thus, if we assurne that
(1.21 )
then the determinant of rx,ß, which can have the values 1 or -1 only, does not
change under small changes in x or in ß. If the open set G is connected, it follows
that the determinant of rx,ß is independent of x and ß for x E G. In the special
case when f is difIerentiable at xE G, we have Irx,ß - f'(x)1 < Anc + 0(1) < 1 for
sufficiently small ß. Hence, the determinant of f'(x) will have the same sign as
that of rx,ß' Thus, when G is connected and c is less than the universal constant
I/An, the determinants of all the rx,ß have the same sign as the determinant of f'
in case f is differentiable. We shall assurne from now on that this sign is positive.
PROPERTIES OF ISOMETRIES AND APPROXIMATE ISOMETRIES
349
Definition. For x in G, let r(x) be the minimum distance from x to the boundary
of G. We say that G has d as an inner radius and D as an outer radius if there is
a point xo in G (called a center point) such that each z in G can be joined in G
to
by a curve of dass Cl and of length 8 ~ D along which
xo
(1.22)
r(x)
~ (~)
1
z
ds.
Example. Let G be a convex open set of Rn containing a ball of radius d with
center Xo and contained in a ball of radius D with the same center xo. Then d
and D are the inner and outer radii, respectively, of G.
Indeed, any point z in G can be joined to Xo by the straight segment (XO, z) which
~ D. Let x = txO + (1 - t)z, with 0 ~ t ~ 1, be a
will have length 8 =
point of this segment. Then, since G is convex, the function r(x) is concave, so
that
r(x) ~ tr (XO) + (1 - t)r(z) ~ tr (xO) ~ td.
Ixo - zl
That is
dlx - zl
r(x) ~ Ixo_ zl =
(d)
S 1ds.
z
z
Theorem 1.7. 8uppose that I belongs to the class Ie,G, where the open set G has
an inner radius d and an outer radius D with center point xo. Then there exists
an orthogonal matrix 'Y such that
(1.23)
lor all z in G, where B n is a universal constant.
Prool. By definition, z can be joined to xO by a Cl-curve of length 8 ~ D along
which (1.22) holds. Using arc length s along the curve as the parameter with
x(O) = xO, x(8) = z, we have
r(x(s)) ~ (~) (8 - s)
(1.24)
for 0 ~ s ~ 8.
Let q be any number with 0 < q < 1. For m = 0,1,2, ... put Sm = (1 - qm) 8,
x m = x(sm), r m = qmd, ßm = qmßo, where ßo = d/(1 + 2k(c:)). By (1.24), we
have
(1.25)
r (x m )
~ (~) (8 - sm) = dqm = rm,
lxi - xml ~ ISi - sml = Iqi - qml 8.
By Lemma 1.5, there exists for each integer m an orthogonal matrix
(1.26)
,.,r such that
TH. M. RASSIAS
350
,r
where Lm(y) = 1 (x m ) + . (y - x m).
Now choose q so that sm - Sm-1 = ßm and q = 8/(8 + ßo). The two closed balls
with centers x m- 1, x m and radii ßm-1, ßm, respectively, will have the ball
in common. For all y in this smaller ball, we have by (1.26) that
(1.27)
From (1.27), we will obtain an upper bound for I,m - ,m- 1 1in the following way.
When
xm +xm - 1 1 ßm
I y2
< 2'
we have
Anc(ßm + ßm-1) ~ ILm(y) - Lm- 1(y)1
= I/(x m ) + ,m(y _ xm) _ l(xm-1) _ ,m-1(y _ xm-1)1
= I(,m - ')'m-1)(y) + I(x m ) _ l(x m- 1) + ')'m-1(x m- 1) _ ,m(xm)l.
Now we use Lemma 1.6 to find that
(1.28)
I rn - rn-li< A
,
')'
-
nC
ßm ßm/
+ ßm-1 = 2An C (1 + q-1) ,
2
since q = ßm/ßm-1.
Inequality (1.27) is valid for y = x m , and since ßm = ßoqm we get
(1.29)
IL m (x rn ) - L m- 1 (xm)1 $ A nc{ßm-1 + ßm) = A nc(1 + q)qm-1 ßo.
Let j ~ m be an integer. By (1.29), (1.28) and (1.25) we have
ILm(xi)-L m- 1 (xi)
I = ILm{xm)-Lm_!Cxm) + (rm_')'m-l) (xi-x m) I
$ A nc(1 + q)qm-l ßo + 2A nc (1 + q-l) (qm - qi) 8
$ A nc(1 + q)qm-l (ßo + 28) .
Summing this inequality from m = 1 to m = j and using the fact that q =
8/(8 + ßo), we have
i
II (xi) -Lo (xi) 1= ILi (xi) -Lo (xi) 1$ (:E qm-l) (A nc{l+q)(ßo+28))
m=l
$ Anc(ßo + 28) (~ ~ :) = Anc (ßo
D2
D2
;028)2
$ 9Anc ßo = 9Anc (1 + 2k(c)) d·
PROPERTIES OF ISOMETRIES AND APPROXIMATE ISOMETRIES
351
The inequality (1.25) implies that ,lim xi = z. Sinee 1 and L o are eontinuous
3-t00
funetions,
D2
If(z) - Lo(z)1 ~ 9A n c (1 + 2k(c)) d'
(1.30)
Now z is an arbitrary point of G and
,0
,0.
where
is orthogonal and independent of z. When c is bounded away from 1,
the function k(c) = .fij(1 - c) is bounded and (1.23) holds with , =
For c
dose to 1 it holds with
I. 0
In view of the example given above, we have the
,=
Corollary 1.8. Let GeRn be an open convex set contained in a ball of radius
D with center xo in G and let G contain a concentric ball of radius d. Assume
that 1 belongs to the dass Ie,G. Then the inequality (1.23) is satisfied for all z in
G, where B n is a universal constant.
We eite the following additional theorem of John [16].
Theorem 1.9. Let GeRn be an open convex set containing a ball of radius
d with center xo and contained in a concentric ball of radius D. Assume that 1
belongs to Ie,G. Then, for'f/ and ( in G, we have
(1.31)
(1 - Cn:D) 111- (I ~ 1/(11) - f(()1 ~ (1 + c) 111- (I,
where Cn is a universal constant. In particular, the mapping x = I(x) is a homeomorphism when c < dj(cnD).
John [17] extended some of these ideas and results to mappings between Banaeh
spaees. Given Banaeh spaces X and Y, let 1 : G -+ Y be a mapping from the
open set G C X into Y. Given x E G, he defines
D+ 1 = lim sup If(z) - f(x)1
'"
and
z-t'" zi.z
Iz - xl
D; 1 = lim inf If(z) - l(x)1 ,
Iz - xl
ealled the upper and lower scalar derivatives of 1 at the point x. In addition,
z-tz zi.'"
the mapping is required to be regular, i.e. loeally homeomorphic. The quantity
11 = max {I log D~ fI, I log D; II} is ealled the strain at the point x.
A regular mapping 1 : G -+ Y is ealled quasi-isometrie if it satisfies
m
= inf{D; I: x E G} > 0
TH. M. RASSIAS
352
and
M = sup{D:-/: x E G} < 00.
It turns out that if m = M = 1 and G is connected as well as open, then 1 is an
isometry.
For any two distinct points x and z of G, we may form the scalar difference quotient
ß 1 = I/(z) - l(x)1 .
Iz - xl
x,z
By use of a lemma of F. Nevanlinna, John demonstrated the following theorems
resembling the mean value theorem of calculus.
Theorem 1.10. 11 1 is a homeomorphism 01 a eonvex open set G c X onto a
convex set in Y, then
inf { D; 1 : x E G} ~ ß 1 ~ sup { D:- 1 : x E G} .
x,z
Theorem 1.11. 11 I: B ~ Y is a quasi-isometrie mapping, where B is an open
ball in X 01 radius ß, then the inequalities
m~ß/~M
x,z
hold lor all x, z in the eoneentrie ball 01 radius (mi M)ß.
Theorem 1.11 may be looked upon as a generalisation of Theorem 1.9. However,
it appears that no analog on the stability Theorem 1. 7 or of the Corollary 1.8 has
been proved for general Banach spaces.
2. Almost Isometries of Function Algebras
The Banach-Stone theorem relates an isometry T between the complex function
spaces C(8d and C(82 ) to a homeomorphism h: 8 1 ~ 8 2 between the Hausdorff
spaces 8 1 ,82 , where 9 = TU) is given by g(t) = F (h- 1 (t)). When we consider the
Banach algebras C(8d and C(82 ) produced by pointwise multiplication, we see
the relationship between isometry and algebraic isomorphism for these algebras.
The term function algebra is used to denote a subalgebra of such an algebra. More
precisely, following the book by Stout [36, p. 36], we cite the
Definition. Let 8 be a locally compact Hausdorff space and denote by Co(8)
the Banach algebra of continuous complex-valued functions on 8 which vanish at
infinity. A junction algebra A is a sub algebra of Co(8) which strongly separates
the points of 8, Le., if 0 =F s =F t then there exists an 1 in A such that 1(0) =F
I(s) =F I(t).
Nagasawa [24] generalised the Banach-Stone theorem in a significant way by
demonstrating:
PROPERTIES OF ISOMETRIES AND APPROXIMATE ISOMETRIES
353
Nagasawa's Theorem. Two function algebras A and B are algebraieally isomorphie if and only if they are isometrie.
This theorem has been generalised in various ways by a number of authors over
the years since 1959. One of these generalisations has to do with perturbations
(also called deformations) of Banach algebras, a comparatively new and growing
field. A good introduction to this subject is the book by Jarosz [15]. See also the
review of this book by R. Rochberg, MR 86k:46074.
Given two Banach algebras A and B, we might say that B is a small (metric)
perturbation of A if there is a linear map T : A --t B such that for some small
c > 0 and all 1 in A we have
(2.1)
(1- c) 11/11:::; IITfll :::; (1 + c) 11/11,
so that the norm is almost preserved. Another kind of (algebraic) perturbation is
to require that T be a bounded surjective linear map of A onto B such that
(2.2)
IIT(fg) - T(f)T(g) 11 < c'lI/lIlIgll
for some c' and all I and 9 in A.
Relations between these definitions are studied in perturbation theory (also called
deformation theory). If these inequalities look familiar, it is because the first
resembles F. John's definition of quasi-isometry for the case of a linear T, while
the second being B. E. Johnson's definition of approximately multiplicative linear
mappings.
Later in this section, we will consider the relation between the two definitions in
the case of function algebras.
When dealing with mappings T from a Banach algebra A to another, B, which
are linear, bounded and surjective, there are equivalent ways of expressing the
inequalities (2.1) for small c. For example Jarosz, among others, uses the inequality
(2.1')
in place of (2.1). Clearly this inequality is invariant under multiplication of T by
a positive real number. If we put 'T/ = c/(2 + c), we mayassume without loss of
generality that IITII = 1 + 'T/, and it follows easily that for all 1 in A,
(2.1")
(1 - 'T/) 11/11 :::; IITIII :::; (1 + 'T/) 11111·
Also (2.1') follows readily from (2.1"). Thus, we have the following:
Remark. For a bounded linear surjective mapping T : A -+ B between the Banach
algebras A and B, the inequalities (2.1') and (2.1") are equivalent, where 0 < TJ < 1,
1J = e/(2 + e) and 1 + e = (1 + 1J)/(1 -TJ).
TH. M. RASSIAS
354
COMMUTATIVE BANACH ALGEBRAS
We will need to cite some standard material concerning Banach algebras (see e.g.
Gamelin [10] or Goodearl [11]). Unless otherwise stated, we shall be dealing with
complex commutative Banach algebras with a unit element, denoted by 1 or e.
Definitions. The spectrum sp(x) of an element x of a Banach algebra Ais the
set of complex numbers A such that Al - x has no inverse in A. If A does not have
a unit, then sp (x) is defined as the spectrum of x in the algebra Al obtained from
A by adjoining a unit element in the standard way. The spectral radius r(x) of an
element x in A is the number
r(x) = sup{IJLI : JL E Sp (x)} .
Gelfand's formula for the spectral radius of each x in A is
r(x) = lim IIxn ll l / n .
n--+oo
The "characters" of a Banach algebra are the homomorphisms of A onto the
complex numbers, or otherwise stated, the set of all nonzero multiplicative linear
functionals on the Banach space A. This set will be denoted by X(A). For each t/J
in X(A) it is known that t/J(1) = 1 = 1It/J1I. Thus, we may think of X(A) as being
a subset of the unit sphere of the dual Banach space A"', and we will topologise
X(A) by using the weak*-topology of A"'. Thus, a net {t/Jß} in X(A) will converge
to t/J in X(A) when lim t/Jß(a) = t/J(a) for all a in A.
ß
Theorem 2.1. For t/J in X(A), let ker (t/J) = {x E A : t/J(x) = O}. Then the
mapping
t/J -+ ker (t/J )
defines a one-to-one mapping of X(A) onto the set M(A) of all the maximal ideals
of A. Moreover,
Sp (x) = {t/J(x) : t/J E X(A)}
and
r(X) = sup{It/J(x)1 : t/J E X(A)} .
In view of Theorem 2.1, we shall follow the custom of identifying each maximal
ideal M in M(A) with the character t/J that it determines. By definition, the
maximal ideal space is M(A) with the topology defined above for X(A).
Theorem 2.2. The maximal ideal space M(A) for the Banach algebra A is a
compact HausdorJJ space.
Definition. The Gelfand transform of x in A is the complex-valued function x
on M(A) defined by x(t/J) = t/J(x), for all t/J in M(A).
We note that x is continuous according to the topology of M(A) defined above.
Thus, the assignment r(x) = x gives a mapping r : A -+ C(M(A)), called the
Gelfand transform of A.
The notation A is also used for the Gelfand transform of A.
Citing Goodearl [11, p. 28] and Gamelin [10, p. 11], we state:
PROPERTIES OF ISOMETRIES AND APPROXIMATE ISOMETRIES
355
Theorem 2.3. Let A be a complex commutative Banach algebra with a unit. Then
the Gelland translorm r : A -+ C(M(A)) has the lollowing properties:
(i) r is a continuous complex algebra map and IIrll = 1.
(ii) IIr(x)1I = r(x) lor each x in A.
(iii) r(A) separates the points 01 M(A).
(iv) r : A -+ C(M(A)) is an isometry il and only il IIx211 = IIxl1 2 lor all x
in A.
Definition. A unilorm algebra on a compact Hausdorff space S is a uniformly
closed (i.e., closed under the topology given by the sup norm) subalgebra of C(S)
which contains the constants and separates the points of S. With the norm 11/11 =
sup{l/(s)1 : s in S}, the uniform algebra becomes a Banach algebra.
Definition. If A is a function algebra on the locally compact space S, a subset
E of S is called a boundary for A if for each I in A there is a point So in E such
that I/(so)1 = sup{l/(s)1 : SES}.
Theorem 2.4 (Shilov). 11 Ais a function algebra on the locally compact HausdorJJ
space S, then there exists a minimal closed boundary lor A, consisting 01 the
interseetion 01 all the closed boundaries lor A.
For a proof of Shilov's theorem see Stout [36, pp. 37-39].
Definition. This minimal closed boundary is called the Shilov boundary for A
and will be denoted by BA.
Definition. A point So E S is called a strong boundary point for the function
algebra (A, S) if, for every neighbourhood V of So, there is an I in A such that
11/11 = I(so) = 1 and I/(s)1 < 1 when s ~ V. The point So is called a peak point if
there is an I in A such that I(so) = 1 and I/(s)1 < 1 for s So.
t
Definition. The collection of the strong boundary points is called the Choquet
boundary for A. It will be denoted by Ch (A).
A prime example of the Shilov boundary occurs for the disc algebra, where S is
the closed unit disc {z : Izl :::; I} in the complex z-plane and A is the subalgebra
of C(S) consisting of all functions in C(S) which are analytic in the interior of
the disco Using the maximum modulus theorem, we see that the boundary of the
disc is the Shilov boundary for the disc algebra. Also in this example the Choquet
boundary coincides with the Shilov boundary. However, in general, the Choquet
boundary is a proper subset of the Shilov boundary. This is illustrated by an
example given by Stout [36, p. 40, Example 7.8], as follows. Let
S = {(z, t) : z E C, tE IR, Izl:::; 1, Itl :::; I},
and let A = {f E C(S) : l(z,O) is analytic in z for Izl < I}. The point (z,t) is a
peak point when t
0. Also (z,O) is a peak point when Izl = 1, for we can take
I()(z,t) = ze- i() which peaks at z = ei(). However, the points (z,O), Izl < 1, are
not peak points, by the maximum modulus theorem. In this example the Shilov
boundary is all of S, while the Choquet boundary appears to be the set S \ D,
where D is the disc D = {(z, t) : Izl < 1, t = O}. This illustrates the fact that
Choquet boundary is a dense subset of the Shilov boundary for uniform algebras.
t
356
TH. M. RASSIAS
METRIC AND ALGEBRAIC PERTURBATIONS OF FUNCTION
ALGEBRAS
Following Jarosz [14], we cite a standard definition of an algebraic c:-perturbation
of a Banach algebra (A, .). If x is another associative multiplication defined on
the Banach space A such that
(2.3)
111 x 9 - I· gll ~ c:llll1l1gll
for all 1,g in A,
then x is called an algebraic c:-perturbation of (A, .).
We note that if T : A -t B is a bounded linear surjective mapping between
the underlying Banach spaces of the algebras A and B then T defines another
multiplication on A by the formula
1 x 9 = T- 1 (T1Tg)
for all 1,g in A.
On the other hand, a bounded linear surjective mapping T : A -t B will be called
an c:-metric perturbation of A if IITIIIIT-111 ~ 1 + c:. As indicated by the above
Remark this is equivalent to the requirement that
(1 - 1]) 11111 ~ IIT111 ~ (1 + 1]) 11111
for all 1 in A,
where 1] = c:/(2 + c:).
The following result of Jarosz [14] concerning Banach algebras will be used later
in the proof of his theorem relating the two kinds of perturbations given below as
Theorem 2.8.
Theorem 2.5. Let A be a Eanach space and let· and x be two multiplications
on A with identity elements e and e, respectively. Assume that (A,·) is a Eanach
algebm. Suppose that lor each c: with 0 < c: < 1 we have
(2.4)
1111 x gll-111· glll ~ c: IIll1llgl1
10r all 1,g in A.
Then there exists a function c : lRt -t lRt with lim c( c:) = 0 such that
e-+O
(2.5)
lIe- 1 x 1 x 9 - I· gll ~ c(c:) IIll1llgl1
lor all 1,g in A,
where e- 1 is the inverse 01 e in the algebm (A, x).
Prool. In the inequality (2.4), replace 1 by exp(ßf) and 9 by exp( -ßf) . g, where
ß is a complex number and exp denotes the exponential function in the Banach
algebra (A, .). We get
IlIgll-lIexp(ßf) x [exp( -ßf) . g]1I1 ~ c: IIglili exp( -ßf)1I . 11 exp(ßf)11
< c: IIgll exp (2Ißllll1D .
PROPERTIES OF ISOMETRIES AND APPROXIMATE ISOMETRIES
357
Hence
Now let H be an arbitrary linear functional of norm one on the Banach space A.
Then from the last inequality we have
ß
ß2
1H(e x g) - (1!)H
(f x 9 - ex (f. g)) + (2f )H( ... ) -···1
::; IIgll [cexp(2IßIII/ID + 1].
Thus, the modulus of the entire function
ß
ß2
'I/J(ß) = H(e x g) - (1!)H (f x 9 - ex (f. g)) + (2! )H( ... ) _ ...
on the unit disc {ß E C : IßI < 1} does not exceed IIgll [cexp(211/ID + 1]. Hence
the first derivative 'I/J' of'I/J at ß = 0 also has modulus not greater than this constant:
I'I/J'(O) I = IH(f x 9 - ex (f. g))l ::; IIgll [cexp(211/ID + 1] .
Since H is an arbitrary linear functional of norm 1 it follows that
(2.6)
111 x 9 - ex (f. g)1I ::; IIgll [cexp(211/ID + 1].
Choose elements I,g in A with 11/11 = IIgll = 1. Let p = -(10gc)j2 > O. From
(2.6), we obtain
11I xg-e x (f·g)1I =11 (pI) x
(iD -ex ((pI). (!)) 1
::; p-l [cexp(2p) + 1] = - -41 .
ogc
Put Cl = cI(c) = -4/logc and notice that cI(c) ~ 0 as c ~ 0 and also that
(2.7)
II/xg-ex(f·g)ll::;cIII/lIlIgll forall/,ginA.
Let us estimate the norm of the identity element e of the algebra (A, x). From
the hypothesis (2.4) we have:
Illell - 11 eil I = Ille x eil - Ile . eil I ::; cllell·
Since (A,·) is a Banach algebra, lIell = 1, so that
_
1
lIell ::; 1 - c .
TH. M. RASSIAS
358
Put 1 = 9 = e in (2.7) to obtain
(2.8)
By (2.4) we have
(2.9)
111 x gll ~ (1 +c) 1I/1111gll·
We shall now assurne that
(2.10)
Then from (2.8) it follows that e x (e . e) is an invertible element of the algebra
(A, x). We take e- l = (e· e) x (e x (e· e))-l.
It follows from (2.10) that if u in A satisfies
lIe - ull ~ Cl lIell 2 <
I!
C'
then
00
n times
...
lIu- 1 1i ~ lIell+ LII'(e-u) x .. · x (e-u)'11
n=l
00
n=l
where we have used inequality (2.10). Thus we have
or
(2.11)
again by (2.10). Hence, writing e- l = (e· e) x (e x (e· e))-l and using (2.8), we
see that we may take u = e x (e· e) in (2.11). Thus, from (2.9), (2.11) and the
estimate lIeli < (1 - c)-l, we find that
(2.12)
Ile-111 = 11 (e· e) x (e x (e· e))-lll = II(e. e) x u-lll
~ (1 + c)(1 - c)-3 [1 - (1 + c)cI (1 - c)-2r l = k(c) ,
PROPERTIES OF ISOMETRIES AND APPROXIMATE ISOMETRIES
359
providing that Cl (1 + c) (1 - c) -2 < l.
Now we use (2.9), (2.7) and (2.12) to obtain
Ile- 1 x 1 x 9 - I· gll = lIe- 1 x 1 x 9 - e- x ex (J. g)1I
= lIe- x[J x9 - ex (J. g)]11
1
1
$(1+c)lIe- 1 1111I x g- ex (J·g)1I
$ (1 + c) k(c) c11111111gll.
Since Cl = -4/ logc and with k(c) defined by (2.12), it is apparent that we may
define c(c) = (1 + c) k(c) Cl to establish Lemma (2.5). 0
The following theorems of Jarosz [14] were first proved by Rochberg [35] for the
special case in which the Shilov and Choquet boundaries of A coincide and each
point of 8(A) is a G6 set.
Theorem 2.6. Let A be a function algebra and suppose that x is an algebraic
c-perturbation 01 A with 0 < c < 1. Then the multiplication x is commutative,
the Gelland translorm r 01 the algebra (A, x) is a continuous isomorphism 0/ A
onto a closed subalgebra B 0/ C(M(A x )) and we have
(2.13)
(1 - c)1I11I $ IIr(J)1I $ (1 + c)1I11I
/or all 1 in A.
Proof. We shall need the following:
Lemma 2.7. Let 0 < c < 1 and let x be an algebraic c-perturbation 0/ the function
algebra A. Then the spectral radius r(J) in (A, x) 0/ each 1 in A satisfies
(1 - c)1I11I $ r(J) $ (1 + c)lllll.
Proo/ 0/ Lemma. Since A is a function algebra, we have
A. By definition, we have from (2.3) that
IIPII = 11/11 2 for alliin
111 x 1 - 1 . 111 $ clll1l 2 ,
so that
(1 - c)1111I 2 $ 111 x 111 $ (1 + c)1111I 2 .
By induction, we find that the inequality
(1-"c)2 n- 111111 2n $111 x 1 x ... xiII $ (1 +c)2 n- 111111 2n
....
,..
2 n times
,
holds for 1 in A. By using the Gelfand formula for the spectral radius, we obtain
the inequality (2.14). 0
Pro%/ Theorem 2.6. From Lemma 2.7 together with a theorem of Hirschfeld and
Zelazko [13], it follows that the multiplication x is commutative since 0 < c < l.
By Theorem 2.3, we have r(x) = r(x) and Theorem 2.6 follows from Lemma
2.7. 0
The result just demonstrated showed that starting with an algebraic c-perturbation
we arrived at ametrie perturbation. Going the other way is considerably more
difficult. The next theorem of Jarosz shows that indeed a metric perturbation
gives rise to an algebraic perturbation for function algebras.
360
TH. M. RASSIAS
Theorem 2.8. Let A and B be function algebras with units eA and eB, respectively. Let T : A -+ B be a bounded linear surjective mapping which satisjies the
condition
(2.1')
Then there exists a bounded linear surjective mapping t : A -+ B such that if
fog = t-1(tftg), then
IIfog-f·gll ~c(c)lIflillgll
foralll,g inA,
where c(c) -+ 0 as c -+ O. Also ifT(eA) = eB, then t
= T.
The proof of Theorem 2.8 requires the
Lemma 2.9. Let T be a bounded linear surjective mapping of the function algebra
A onto the function algebra B where IITII = 1 and IIT- 1 1I ~ 1 + c < 3/2. Then
there exists a dense subset D of the Shilov boundary BA for A such that corresponding to each s in D there exists a u in the Shilov boundary BB for B such
that
(2.15)
IT(J)(u)1 ~ (11(s)l- 2cllilD (1 + c)-l
for all 1 in A.
Proof. A net (gß) of elements of Bis called a peaking net at a point u in ChB
when IIgßII = gß(u) = 1 for all ß, and (gß) tends uniformly to zero outside each
neighbourhood of u.
We denote by Du the subset of BA consisting of all points So admitting a net
(gß) C B peaking at u and a net (sß) C BA converging to So and such that
Now IITII = 1 and the set BA is compact, so that Du is not void.
For each 1 in A and for a suitable net (1]ß) of complex numbers of modulus one,
we have
so that
.
1-c+ll(so)1
hm
.
ß sup IITf + 1]ß9 ß 11 ->
1 +c
Hence, by the definition of the net (gß), we find that
IT(J)(u) I ~ (1- c + Il(so)l) (1 + c)-l - 1
or
IT(J)(u) I ~ (1I(so)l- 2c) (1 + c)-l
for alliin A with 11111 = 1.
It follows that (2.15) is true for all 1 in A, any u in ChB and any So in Du.
PROPERTIES OF ISOMETRIES AND APPROXIMATE ISOMETRIES
361
It remains to prove that the set
il = U{ilu : u E ChB}
is dense in 8A. Assurne the contrary. Then there exists an open subset V of 8A
such that V n il is empty. Choose a positive 6 < 1 - 2€ and an
in A such that
Illdl = 1 and Ift(s)1 < 6 for s in 8A \ V. Choose U1 in ChB such that
ft
and let (gß) eBbe a net peaking at U1' For some net (~ß) of complex numbers
of modulus one, we will have
Hence
lir sup 11ft + ~ßT-1gßII ~ 2 -
€,
and by the definition of ft, there exists a net (sß) C V such that
(2.16)
Because V is a compact subset of 8A we mayassume that the net (sß) converges
to some So in V. But then by (2.16) and the definition of ilUl C il, we find that
So E V n il. This contradicts the assumption that V n il is empty, and the lemma
is proved. 0
Proolol Theorem 2.8. Let us assurne, without loss of generality, that IITII = 1 and
IIT- 111 = 1 + €. By Lemma 2.9 and (2.15) we have, for any 1, 9 in A,
IIT1· Tgil = sup {I(T f)(u) . (Tg)(u)1 : u E 8B}
~ sup {(ll(s)l- 2€lIllD (lg(s)l- 2€llglD : sEil} (1 + €)-2
= (lIlgll - 4€ IIll1llgl1 + 4€211111I1gll) (1 + €)-2,
so that
(2.17)
IIT1· Tgll-lIlgll ~ -4€ IIllll1gl1 (1 + €)-2.
Now put Tl = T- 1/I1T- 111, ft = T 1 11 and gl = T 1- 1g. From Lemma 2.9, as in
the above argument, we get
IIlgll-IIT1- 11' T 1- 1gl1 = IIT1ftT1g111-lIft . gll1
> -4€(1 + €)-21Iftllllg111
~ -4€ 1111111gll.
TH. M. RASSIAS
362
Thus, we have
IIlgll-IITj· Tgll2: -4c 1I/IIIIgii + (1ITl- l I· T1lglI-IITI' TglI)
and
II/glI-IITI' Tgll2: -4c 1I/IIIIgil + 11 (2c + C2 ) TI· Tgll2: -4c 1I/1I1Igll·
(2.18)
The inequalities (2.17) and (2.18) imply that
IIITI· Tgll-lIlglil :::; 4c 11/1111glI·
By definition I x 9 = T-l(TI· Tg), so
11I x gll = IIT- l (T I· Tg)11 :::; IIT- l IlIITI· Tgil = (1 + c) IITI· Tgil ,
and
I1I x gll-lI/gll :::; (1 + c) IIT I· Tgll-II/gll
= IIT I· Tgil - IIlgll + c IIT I· Tgil
:::; 5c 1I/1I1Igll.
Similarly we find that II/gll -111 x gll 2: -5c 1I/IIIIgil and we have
1111 x gll-lI/glil :::; 5c 1I/1I1IglI·
(2.19)
Denote by e the unit of the algebra A with the new multiplication x. By (2.19)
and Theorem 2.5 we have for some c'(c) -+ 0 as c -+ 0 that
lIe Al x I x 9 - Igil :::; c'(c) 1I/1I1IglI,
(2.20)
where e A1 is the inverse of e A in the algebra (A, x). Define the operator S : A -+ A
by
Clearly,
(2.21)
IIISIII -11/111 = IlieAl x 111-11/111
= Ilie Al x f x eil - 11 fell I + I 11 fell - IIf x eil I·
By (2.20)
and by (2.19)
Illf . eil - Ilf x eil I :::; 5c IIfllliell,
PROPERTIES OF ISOMETRIES AND APPROXIMATE ISOMETRIES
363
so, from (2.21), we find that
(2.22)
IIIS/II-II/III ::; (c' + 5c) 11 eil 11/11·
From the definition of the "cross product" , we have e= e x e=T- 1 (T (e)· T (e)),
so that T (e) = T (e) . T (e). Therefore T (e) = eB. Since lIeBIl = 1, we have
lIell = IIT-I(eB)1I ::; IIT- 1 11 = 1 + c, and, from (2.22), we get
(2.23)
IIiS/II-II/III::; (c' + 5c) (1 + c) li/li.
Hence
(2.24)
Let T = TS. By hypothesis IITIIIIT- 1 11 = 1 + c, so by (2.24) we have
Also, T(eA) = TS(eA) = T (e A1 x eA) = T (e) = eB.
Thus, we may apply the same argument as above to T in place of T and find that
11/0 9 - /gll ::; c(c) 1I/lllIgll ,
for all /, gin A, where /0 9 = T- 1 (T / . Tu).
Finally, if T(eA) = eB then T = T. 0
Several other generalisations of the Banach-Stone theorem are cited in the survey
article by Fleming and Jamison [9]. For example, M. Cambern in the year 1967
proved that if SI and S2 are locally compact Hausdorff spaces and T : CO(SI) -+
CO(S2) is a linear homeomorphism with li/li < IIT/II < MII/II and with M < 2,
then SI and S2 are homeomorphic. He also demonstrated astability result, namely,
that if IITIIIIT- 1 11 = 1 + c, then there is an isometry U : Co(Sd -+ CO(S2) where
IIT - UII -+ 0 as c -+ O. The year 1973, B. Cengiz generalised Cambern's theorem
by dealing with subspaces. He showed that if A and B are extremely regular
subspaces of Co(Sd and CO (S2), respectively, and if T is a linear homeomorphism
satisfying IITIIIIT- 1 11 < 2, then SI and S2 are homeomorphic. Rochberg [34]
proved a similar theorem and he also showed that there is an isometry U : A -+ B
such that IIT/ - U/II < c(JI.) li/li, where JI. = IITIIIIT- 1 11 and c(JI.) -+ 0 as JI. -+ 1.
Finally, Lovblom [20] removed all assumptions of linearity for T and proved the
following stability result:
Theorem 2.10. Let SI and S2 be compact metric spaces and let Br(C(Sj)) denote
the closed ball in C(Sj) with center 0 and radius r (j = 1,2). Let
satisfy T(O) = 0 and the inequalities
(2.25)
(1 - c) 11/ - gll ::; 11 TU) - T(g) 11 ::; (1 + c) 11/ - gll
364
TH. M. RASSIAS
for all f, 9 in BI (C(81 )). Then there exists an isometry
such that IIT(f) - U(f)11 < c52 (c:), when f E B 1 - Ö1 (c) (C(8t), where c51 (c:) -t 0
and c52 (c:) -t 0 when c: -t O.
This result immediately suggests the possibiIity of generalisations.
Two unsolved problems:
(I) Generalise Lovblom's theorem to the case where 8 1 and 8 2 are locally compact
or else completely regular topological spaces.
(11) Prove such a theorem for a mapping T between unit balls of two more general
Banach spaces which satisfies (2.25).
3. Isometries and Conservative Distances
Given two metric spaces EI and E 2 and a mapping T : EI -t E 2 , what do we
really need to know about T in order to be sure that T is an isometry? Consider
the following situation. For some fixed number r > 0 suppose that T preserves
the distance r, Le., for all x, y in EI with d 1 (x,y) = r we have d 2 (Tx,Ty) = r,
where di (-,·) denotes the metric in the space Ei (j = 1,2). Then r is called
a conservative distance for the mapping T. The basic problem of conservative
distances is whether the existence of a single conservative distance for some T
implies that T is an isometry of EI into E 2 • It is also called the Aleksandrov
problem, since it was formulated for the case EI = E 2 by Aleksandrov [2]. When
EI and E 2 are normed vector spaces we mayassume without loss of generality
that the number r = l.
For the case when EI = E 2 = R.n with the usual Euclidean metric, Beckman and
Quarles [3] obtained the following result.
Theorem 3.1. 1f the number 1 is a conservative distance for the transformation
T : R.n -t R.n, where n ~ 2, then T is a surjective isometry.
They also pointed out that this theorem fails for n = 1 by means of the counterexample T : R. -t R. which moves each integral point one unit to the right and leaves
all other points fixed.
We shall first give their proof for the case n = 2, which we assume now.
Lemma 3.2. 1f v'3 - 1 < d(x, y) < v'3 + 1, then Tx "# Ty.
Proof. Clearly, the vertices of an equilateral triangle of unit side are mapped by T
to the vertices of a congruent triangle, and a unit circle (Le., having unit radius)
is mapped into a unit circle. Consider the unit circles Cl, C2 with centers at x
and y. Construct an equilateral triangle of unit side with two of its vertices on Cl
and the third on C2 • If Tx = Ty then the three vertices of an equilateral triangle
of unit side would He upon the unit circle having Tx = Ty as center, which is
impossible. D
PROPERTIES OF ISOMETRIES AND APPROXIMATE ISOMETRIES
365
Lemma 3.3. The distance J3 is preserved by T.
Proof. Consider a rhombus ABCD formed by two equilateral triangles ABD and
CBD of unit side. Then the diagonal points A and C must be mapped by T into
a single point or else onto two points at a distance v'3 apart. Lemma 3.2 shows
that the distance J3 is preserved. 0
Lemma 3.4. All integral distances are preserved by T.
Proof. Given a regular hexagon of unit side, its consecutive vertices must be
mapped by T onto the vertices of a congruent hexagon. We see this by observing the overlapping congruent rhombi. Hence, two diametrically opposed vertices,
which are a distance 2 appart, are mapped by T onto points also at a distance 2
apart. Thus the distance 2 is preserved by T. This figure may be extended by the
addition of equilateral triangles of unit side to form more overlapping rhombi in
order to show that T preserves all distances n E N. 0
Lemma 3.5. All distances of the form m/2 n , where m, n are positive integers,
are preserved by T.
Proof. Lemma 3.2 is easily extended to show that, if
m{v'3 -1) ~ d(x,y) ~ m{v'3+ 1),
then Tx i Ty. Consider the isosceles triangle with base of length m and the other
sides of length 2m. By Lemma 3.4 the vertices of this triangle are mapped onto
the vertices of a congruent triangle. Now a circle of radius m is mapped onto a
circle of the same radius, so the system of two intersecting circles, each of which
has one of the sides of length 2m of the triangle as diameter, is mapped into a
congruent system. The point x of intersection of the circles which coincides with
a vertex of the isosceles triangle is mapped into the corresponding point of the
congruent image system. The other point y of intersection of the circles, which is
the midpoint of the side of length m, is mapped into the corresponding point of
intersection of the congruent system. Since
In
mvT5.
In
m{v3
-1) < d(x,y) = 2 - < m{v3 + 1),
it follows that Tx i Ty. Thus the midpoint of the side of length m is mapped
onto the midpoint of the line segment joining the images of its end points. Hence,
the distance m/2 is preserved by T. By an extension of this argument, we find
that the distances m/2 n are preserved. 0
Proof of Theorem 9.1 for the plane. The set [m/2 n j m, nE N] is dense in lR.t.
Thus given two points x, y of the plane, there exists a sequence {Yk} of points of
the plane with each distance d(x, Yk) of the form m/2n , such that
lim d(x, Yk) = d(x, y).
k-too
TH. M. RASSIAS
366
Given n E N, there is a k E N so large that both Yk and y will lie on a circle of
radius 1/2n . It follows that TYk and Ty lie on a circle of that same radius. Hence
lim TYk = Ty. Since d(x, Yk) is preserved we also have
k-too
lim d(Tx, TYk) = lim d(X,Yk) = d(x,y).
k-too
k-too
Finally,
d(Tx, Ty)
= k-too
lim d(Tx, TYk) = d(x, y),
so that T is an isometry of the plane onto itself.
0
We now turn to the proof of Theorem 6.1 for all dimensions n > 2. Here, instead
of using the methods of Beckman and Quarles [3], we will give the proof due to
Bishop [7]. The next lemma, cited by Bishop, is due to P. Zvengrowski (Appendix
to Chapter 11 of the book by Modenov Parkhomenko [23]).
Lemma 3.6. 1fT: Rn ~ Rn preserves the distance rand m is an integer greater
than one, then d(Tx,Ty) ~ mr whenever d(x,y) ~ mr.
Proof. Since m > 1 we may join the point x to the point y by achain of points
x = XO,Xl, •.. ,X m = y, where d(Xj_l,Xj) = r, j = 1,2, ... ,mo The image by T
of this chain is a configuration of the same type since T preserves the distance r.
Using the triangle inequality we see that d(Tx,Ty) ~ mr. 0
Following Bishop [7], we demonstrate the following lemmas.
Lemma 3.7. Suppose that there are both arbitrarily large and also arbitrarily
small distances which are preserved by T : Rn ~ Rn. Then T is an isometry.
Proof. Given any two points x, y in Rn, let a = d(x, y). Let b > 0 be chosen so
that T preserves the distance a + b. Let z be the point at distance a + b from x
such that y is on the segment xz. Put u = d(Tx, Ty) and v = d(Ty, Tz). Also let
y' be the point on the segment TxTy with d (Tx, y') = a. Now suppose that r is
a distance preserved by T and let m, k be the integers such that
(m - l)r < a ~ mr
and
(k - l)r < b ~ kr.
By Lemma 3.6, we have u ~ mr and v ~ kr. Since -a < -(m - l)r, we get
u - a < mr - (m - l)r = rj similarly v - b < r. Now T preserves the distance
a + b, so that a + b = d(Tx, Tz) ~ d(Tx, Ty) + d(Ty, Tz) = u + v. Consequently,
a - u ~ v - b < T and b - v ~ u - a < r. Thus la - ul < r and Ib - vi < r. By
hypothesis, we can take r arbitrarily small. Therefore,
d(Tx,Ty) = u = a = d(x,y),
so T is an isometry.
0
PROPERTIES OF ISOMETRIES AND APPROXIMATE ISOMETRIES
367
Lemma 3.8. IIT:!Rn -t !Rn preserves distance r, then T also preserves distance
2r n -_ r [2(n + 1)] 1/2
n
Hence, there are arbitrary large distances which are preserved by T.
Prool. Consider the figure formed by two n-dimensional simplices having a common face and with each of their edges of length r. Let one vertex of the common
face be the origin 0 of the coordinates. The other vertices of this face will be
given by el, ... ,en-l, these vectors all having length r. The extreme points of the
figure will be represented by vectors 10 and 12, which are reflections of each other
in the hyperplane of 0, el, e2, ... ,en-l' Since the lengths of the vectors ei - ej
and Ik - ej are also equal to r for i '" j, we find that for i,j = 1, ... ,n -1, i '" j,
k = 0,2,
(3.1)
(ei, ei) = r 2 ,
r2
(ei, ej) = (Jk, ei) = 2
(i '" j) ,
where (.,.) denotes the inner product. Observe that (Jo + 12)/2 lies on the hyperplane of the common face and is symmetrically placed with respect to all the
vertices of this face, so it is their centroid:
10 + h el + ... + en-l
2
n
Thus, we have
.f
JO -
f -.f
f
2/ _ 2(el + ... + en-l - nh)
2 - JO + 2 2,
n
and it follows that
2 [
.f - f.f
f 2 ) -_ 4r
1
2 (n - 1)(n - 2) - n (n - 1)] -_ 2r 2 (n + 1) .
(JO
2')0 n - +n +
2
n
2
n
That is, the distance between the extreme points of the figure is
dUo,h) = rJ2(n: 1). 0
By Lemmas 3.7 and 3.8, it remains only to prove that T also preserves arbitrarily small distances in order to prove Theorem 3.1. We have found that the
above figure {O, el, ... ,en-l, 10, h}, consisting of the vertices of two simplices
with a common face and all their edges of length r, is transformed by T into
a congruent figure. Now we will generate more figures with this same property by chaining together equilateral simplices face to face. Put /I = en-l
and define Ik recursively in the following way: Ik+l is the reflection of Ik-l
in the hyperplane of {O, el, ... ,en -2, !k}. In this manner, we construct figures
{O,el,'" ,e n-2,/o,/I, ... ,lk}whicharemappedontocongruentfiguresbyT. By
construction, we have (Jk+1 - Ik-l,ej) = 0 for k = 1,2, ... and j = 1, ... ,n - 2.
By combining these equations we see that (Jk - Im, ej) = 0 for any two positive
integers k, m of the same parity. Thus, the projection into the two dimensional
plane through 0 perpendicular to {eI, ... ,en -2} carries Ik into gk, where the gk are
equally spaced on a circle centered at O. All of the distances dUo, Ik) = d(go, gk)
will be preserved by T because we can build such a figure around two points at
such a distance.
368
TH. M. RASSIAS
Lemma 3.9. The distance d(fo, 15) = rln 2 - 2n - 41/n 2 is preserved by T. When
n ::::: 3, this distance is less then r, so by iteration T preserves arbitrarily small
distances.
Proof. By (3.1) we have (ei,ej) = (fo,ej) = r 2/2 for i"l j while (ei,ei) = r 2. By
direct calculation it follows that
I
el + ... + e n -2
n- 1
JO -
is perpendicular to each ej, j = 1, ... ,n - 2. Thus
go = 10 -
el + ... + e n -2
1·
n-
Hence
n-2
n-2
(go, go) = (n-l)-2( (n - 1)10 - Lei, (n - 1)10 - L ej)
i=1
j=1
n-2
n-2
n-2
= (n-l)-2 [(n - 1)2r2 - 2(n - 1) L (fo, ei) + ( L ei, L ej)],
i=1
i=1
j=1
which by use of (3.1) reduces to
(3.2)
Thus, the gk all lie on a circle of radius rn-I with center 0. Let 2ß denote the
angle gOOgl. Since d(go, gd = d(fo,ld = rand d(O, gd = d(O,g2) = rn-I, the
triangle gOOgl is isosceles with base r. Hence
sinß= -r- = ~-1
-2r n -1
2n '
/n+l
cosß= V~·
Clearly, the base of the tri angle gOOgl may be written as r = 2r n -1 sinß. Considering the triangle gOOg5, the corresponding angle is 5ß and the base of this
tri angle is given by
(3.3)
To calculate this we use
sin 5ß = Im( exp( i5ß)) = Im( cos ß + i sin ß)5
= (sin ß) [5 cos 4 ß - 10 cos 2 ß sin 2 ß + sin 4 ß]
= (_r _) [5(n + 1)2 _ 10(n2 - 1) + (n - 1)2]
2r n -1
= (
2r
)
2n rn-I
4n 2
4n 2
(4 + 2n - n 2) .
4n 2
PROPERTIES OF ISOMETRIES AND APPROXIMATE ISOMETRIES
369
Hence by (3.3) we have
and Lemma 3.9 has been proved. Theorem 3.1 for n > 2 follows from Lemmas
3.7,3.8 and 3.9. 0
Beckman and Quarles [3] gave the following example to show that Theorem 3.1
above is not true for Hilbert space. Let H denote the real Hilbert space consisting
00
of all infinite sequences of real numbers x = (Xl, X2, ... ), where LX; converges.
j=l
There is in H a countable dense set of points which will be denoted by {yk}.
Consider a mapping 9 : H ~ {yk} such that d(x,g(x)) < 1/2. Define h: {yk} -+
H by h(yk) = a k , where a k = (a~, a~ , ... ) and aj = 8jk /../2, where 8jk is Kronecker
delta. Now let T : H ~ H be the mapping T = hg. Suppose that xl and x 2 are
two points with d (Xl, x 2 ) = 1. Then clearly 9 (Xl) # 9 (x 2) and hence Tx l # Tx 2.
Therefore d (Tx l , Tx 2 ) = 1. Thus, T preserves the distance one, but is not an
isometry, for if Xl and x 2 are any two points of H then d (Tx l , Tx 2 ) is either 0
or 1.
Of course the T of this counterexample was not continuous. What happens if
we impose conditions on T of continuity and/or surjectivity? One answer was
provided by Mielnik and Rassias [22], as follows
Theorem 3.10. With the real classical Hilben space denoted by H (H = !Rn,
1 :$ n :$ (0), let f be a homeomorphism 01 H onto H which preserves the distance
r > O. Then f is an isometry.
Proof. Clearly, f maps the sphere S(a,r) = {x EH: d(x,a) = r} into the sphere
S(f(a), r). By hypothesis, the mapping is continuous and injective. Also, the
image S· = f(S(a, r)) must be all of the sphere S(f(a), r). For, let Z be the complement of S(a, r) and Z· the complement of S·. Now Z is disconnected, being the
union of two open sets, the interior and exterior of S(a, r). The homeomorphism
f will map Z onto Z·. If S· C S(f(a), r) were a proper subset of S(f(a), r), its
complement Z· would be connected, which is impossible since Z· = f(Z). Thus
f(S(a, r)) = S(f(a), r). Under the assumption of the theorem, the restriction of f
to any sphere S(a, r) has a conservative angular distance of n/3, since the points
x, y of S(a, r) with d(x, y) = rand the center a of the sphere form an equilateral
triangle. Thus, according to Theorem I, page 336, of Mielnik [21], it follows that
f maps the sphere S(a,r) isometrically onto S(f(a),r). Since any two points x,y
in H with d(x, y) :$ 2r He on some sphere S(a, r), this proves that f preserves all
distances s :$ 2r. By iterating the argument, we find that all distances s :$ 4r are
conserved by f. By an induction argument, it follows that every distance s > 0 is
conserved by f. 0
Rassias and Semrl [26] studied the problem of conservative distances for mappings
between real normed vector spaces X and Y. They used the following definitions.
370
TH. M. RASSIAS
Definition. A mapping I : X ~ Y is said to have the distance one preserving property (DüPP) if for all x, Y in X with IIx - Yll = 1 it follows that
II/(x) - l(y)1I = 1.
Definition. A mapping I : X ~ Y is said to have the strong distance one
preserving property (SDOPP) ifit has (DüPP) and in addition, if 11/(x)- I(y) 11 = 1
implies Ilx - yll = 1 for all x, Y in X.
Using the latter definition they proved the following.
Theorem 3.11. Given real normed spaces X and Y, at least one 01 which has
dimension> 1, let I : X ~ Y be a surjective mapping having the SDüPP. Then
I is injective and satisfies the condition:
(3.4)
11I/(x) - l(y)lI- IIx - Ylil < 1
lor all x, Y in X.
Also, I preserves distance n in both directions lor n in N.
Proof. Observe that both X and Y have dimension> 1. For suppose dirn Y > 1.
Then there exist vectors x, y, z in Y such that IIx - Yll = IIx - zll = IIY - zll = 1.
The mapping is surjective and has (SDüPP). Hence there exist vectors Xl, Yl, Zl
in X with IIXI - Ydl = IIXI - zlll = IIYl - zlll = 1, so that dirn X > 1. The same
argument applies if we start with the assumption that dirn X > 1.
Next we show that I is injective. H not, then we could find X, Y in X with X '" Y
such that I(x) = I(y). Choose z in X so that IIx - zll = 1, lIy - zll '" 1. Then
we would have II/(y) - l(z)1I = II/(x) - l(z)1I = 1. But then lIy - zll = 1, a
contradiction, so I is bijective, and both I and 1-1 preserve unit distance.
In proving inequality (3.4) we will use the following notations. With X in X and
r > 0, K(x,r) = {z: Ilz-xll < r}, K(x,r) = {z: Ilz-xll ~ r}, C",(n,n+1] = {z:
n< IIz - xII ~ n + 1}. Given X in X and n in N with n > 1, let z be an element
of K(x,n). Since dirn X > 1 we can find a sequence X = XO,Xl, ... ,X n = Z such
that Ilxj - xj-lll = 1 for j = 1, ... ,n. Thus
II/(z) - l(x)11 ~
n
L II/(xj) - I(Xj-l)1I = n.
j=l
Hence
I (K(x, n)) C K(f(x), n) .
The same argument applies to 1-1 in place of I to obtain
1-1 (K(f(x),n)) c K(x,n).
Thus, for all X in X and n = 2, 3, ... , we have
I (K(x,n)) = K(f(x),n).
Now I is bijective, so for X in X and n in N with n > 1 we have
(3.5)
I (C",(n, n + 1]) = CJ(",)(n, n + 1].
PROPERTIES OF ISOMETRIES AND APPROXIMATE ISOMETRIES
371
In order to show that (3.5) also holds for n = 1, we fix x in X and choose any z
in Cx (l, 2]. Then I(z) E K(f(x), 2). Clearly u = z + (z - x)/liz - xii is contained
in C x (2,3]. By (3.5), we have I(u) E C/(x)(2,3], so that
11/(x) - l(u)11 > 2.
(3.6)
If II/(x) - l(z)11 < 1, then
II/(x) - l(u)11 :s 11/(x) - l(z)1I + 11/(z) - l(u)11 :s II/(x) - l(z)11 + 1 :s 2,
which contradicts (3.6). Thus we have
I (Cx(1,2]) C C/(x)(1,2].
Since the similar result will hold for 1-1, it follows that I (Cx (1,2]) = C/(x)(1,2],
and we conclude that (3.5) is true for all n in N. Finally we prove that
I (K(x, 1)) = K (f(x), 1).
As above, it is sufficient to prove the inclusion
I (K(x, 1») C K (f(x), 1) ,
since the similar result will then hold for 1-1. If this inclusion is not true then for
some integer n ~ 1 we would have I(z) E C/(x)(n,n + 1] for some z in K(x,l).
Since C/(x)(n,n+ 1] = I(C x (n,n+1]), then I(z) E I(C x (n,n+1]) and z E
Cx (n, n + 1] for some n ~ 1, which is a contradiction. Thus
I (K(x, 1)) c K (f(x), 1)
and we have
(3.7)
I (K(x, 1)) = K (f(x), 1) .
The fact that (3.5) holds for all positive integers n, together with (3.7) implies
that (3.4) is true. Indeed, given x and Y in X, let n + 1 be the integral part of
IIY - xII, so that Y E Cx(n,n + 1] if n ~ 1, while if n = 0 then either Y E K(x, 1)
or else lIy - xii = 1 and (3.4) becomes trivial. In the non-trivial cases we find by
n + 1 and -(n + 1) -lix - yll < -n,
(3.5) or (3.7) that n < II/(x) - l(y)11
from which (3.4) follows.
It remains to prove that land 1-1 both preserve the distance n for each n E N.
Make the induction assumption that I preserves the distance n. For n = 1 this is
true by hypothesis. Let x and z satisfy Ilz - xII = n + 1, so that z E Cx(n, n + 1].
Hence, I(z) E C/(x)(n, n + 1] so that 11/(z) - l(x)11 n + 1. Put
:s
:s
:s
I(z) - I(x)
u = 11/(z) _ l(x)1I + I(x),
TH. M. RASSIAS
372
Since lIu - f(x) 11 = 1 we have IIv - xII = 1. Now if lIu - f(z)1I < n we would have
IIv - zll < n, and since IIv - xII = 1 it would follow that IIz - xII < n + 1, which is
a contradiction. Hence, Ilu - f(z)11 ~ n, so that
n
~ lIu - f(z)1I = Ilf(X) - f(z) + II~~;~ =~~:~IIII
= Ilf(x) - f(z)1I (1-lIf(x) - f(z)II- 1 )
= IIf(x) - f(z)lI- 1.
Note that IIf(x) - f(z)1I > 1, for otherwise, since K (f(x) , 1) = f (K(x, 1)) we
would have IIz - xII ~ 1, a contradiction. Therefore IIf(x) - f(z) 11 ~ n + 1, and it
follows that IIf(z) - f(x)II-= n + 1. This completes the induction proof. The case
for f- 1 is proved similarly. 0
Theorem 3.12. Given real normed spaces X and Y, where one of them has
dimension greater than one, let f : X --+ Y satisfy the Lipschitz condition
(3.8)
Ilf(x) - f(y) 11 ~ IIx - ylI·
Assume also that f is a surjective mapping with the (SDOPP). Then f is an
isometry.
Proof. By Theorem 3.11, f preserves the distance n in both directions for all
n E N. Given two distinct x and Y in X, take m E N with IIx - yll < m. If
IIf(x) - f(y) 11 "lllx - yll, then byassumption (3.8) we would have
(3.9)
IIf(x) - f(y) 11 < IIx - ylI·
Put z = x + m(y - x)/lIy - xII, so that IIz - xII = m and IIz - ylI = m - IIY - xII.
Hence IIf(z) - f(x)1I = m and by (3.8) we have IIf(z) - f(y) 11 ~ m-IlY - xII. But
by (3.9),
Ilf(z) - f(x)1I ~ IIf(z) - f(y)1I + Ilf(Y) - f(x) 11 < m -IlY - xII + lIy - xII = m,
which is a contradiction. Thus, (3.9) is false and we have IIf(x) - f(y) 11 = IIx - yll
for all x, y in X. 0
Theorem 3.13. Let X and Y be real normed spaces, one of which has dimension
greater than 1 and one of which is strictly convex. If f : X --+ Y is a surjective
mapping having the (SDOPP), then f is an isometry of X onto Y.
Proof. By Theorem 3.11, fis bijective and both f and f- 1 preserve the distance n
for all positive integers n. We also know that both spaces have dimension greater
than one. We may assume, without loss of generality, that Y is strictly convex.
Our first objective is to show that f preserves the distance I/n for each n E N.
Take any x and y with IIx-YIi = I/n and choose z in X with IIx-zll = lIy-zll = 1.
Put u = z + n(y - z), v = z + n(x - z), s~ that IIv - xII = n -1 and IIv - zll = n.
PROPERTIES OF ISOMETRIES AND APPROXIMATE ISOMETRIES
373
Thus we have Ilf(x) - f(z)11 = 1, IIf(v) - f(x)11 = n - 1 and Ilf(v) - f(z)1I = n.
Since Y is strict1y convex,
1
n-l
f(x) = - f(v) + f(z) ,
n
n
1
n-l
f(y) = - f(u) + f(z).
n
n
Now Ilu - vii = 1, so that
1
Ilf(x) - f(y)11 = n-11If(v) - f(u)1I = -.
n
Thus, f preserves the distance l/n for all n E N.
For any two distinct points x, y in X, choose positive integers m and n such that
m/n < Ilx - ylI with m ;::: 2. As dimX > 1, we can find a sequence of vectors
x = Zo, Zl, ... ,Zm = Y such that IIzj - zj-lll = l/n (j = 1, ... ,m). Consequently,
m
Ilf(x) - f(y)11 ::;
L IIf(zj) - f(Zj-l)11 = m/n.
j=l
Hence, IIf(x) - f(y)11 ::; m/n < Ilx - yll for all x, y in X, so that f satisfies all the
hypotheses of Theorem 3.12 and f is an isometry of X onto Y. 0
Rassias and Semrl [26] also gave the following counterexamples.
(a) In the statement of Theorem 3.11, the property (SDOPP) cannot be replaced by (DOPP). Let 9 : [0,1) -+ [0,1) x IR be defined by g(O) = and
g(t) = (t, tan ('Ir (t - 1/2») for
< t < 1, and put f(t) = g(t - [t]) + ([t],O),
where [tl denotes the integral part of t. Then f is a surjective mapping from IR
onto IR2 which preserves the distance n for all n E N. However f does not satisfy
(3.4).
°
°
°< <
°<- <-
(b) The inequality (3.4) is sharp. Choose 6 with
function gE(t) : [0,1]-+ [0,1] by
cl
gE(t) =
{
1-6
when
1- 6 t
(
6
1
) +2- 6
6
t
1/2 and define the
1 - 6,
when 1 - 6 < t ::; 1 .
Let hE : IR -+ IR be given by hE(s) = [s] + gE(S - [s]). Thus hE is a monotonically
increasing function which for n E N satisfies:
(3.10)
Is - tl = n
Is - tl ::; n
if and only if IhE(s) - hE(t)1 = n } .
if and only if IhE(s) - hE(t)1 ::; n
Moreover IhE(s) - he(t)1 ::; c1ls - tl.
Let G[O, 1] be the space of real-valued continuous functions on the closed interval
[0,1] with the usual norm IIxll = max{lx(t)1 :
t ::; I}. Define the function
<PE : G[O, 1]-+ G[O, 1] by <PE (x)(t) = hE(x(t). Clearly<pE is bijective with
°: ;
<p;l(X)(t) = h;l(X(t)).
374
TH. M. RASSIAS
Notice that for any two elements x and y in C[O,l] we have IIx - yll = n if and
only if there is a to in [0,1] with Ix(to) - y(to)1 = n and Ix(t) - y(t)1 :::; n for
all t in [0,1]. By (3.10) this holds if and only if Ihe(x(to)) - he(y(to))1 = n and
Ihe(x(t)) - he(y(t)) I :::; n for t in [0,1], which is equivalent to IIcPe(x) - cPe(y)11 = n.
Therefore, cPe preserves distance n in both directions for each positive integer n.
If we take x(t) == 1 - e and y(t) == 1 as elements of C[O, 1], then Ilx - yll = e,
cPe(x)(t) = e, cPe(y)(t) = 1. Hence IlIcPe(x) - cPe(y)II-llx - ylll = 1- 2e. Since we
may choose e arbitrarily small, the inequality (3.4) of Theorem 3.11 is sharp.
COMMENTS
The example of Beckman and Quarles [3] quoted above to show that Theorem
3.1 fails for n = 1 used a discontinuous mapping. Rassias [29] showed that the
same result holds even for continuous mappings by the following example. Let
T : IR --+ IR be defined by T(x) = [x] + (x - [x]) 2 , with [x] =integral part of x.
Clearly T(x+ 1) -T(x) = 1 for all x in IR, and so T preserves the distance one. It is
also easy to check that T is continuous and that T(n) = n, so that T is surjective,
but obviously T is not an isometry.
Rassias raised the question (cf. [29], [31], [33]) as to whether Theorem 3.1 could
hold for mappings T from IRn into IRffi, where m > n. The answer seems to be
negative according to examples found so far. In particular, Rassias [29] constructed
examples of mappings f : IR2 --+ IR8 and 9 : ]R2 --+ IR6 which preserve the distance
one but are not isometries.
Rassias [29] also proved that for each integer n > 1 there exists an integer k(n) such
that m > k(n) implies that there exists a map T : IRn --+ IRffi which preserves the
distance one but is not an isometry. In these counterexamples T is not continuous.
It would be interesting to find a continuous counterexample which would settle the
quest ion for all pairs n, m with m > n (see also Rassias [28], [30], [32] and Rassias
and Sharma [27]).
What about non-Euclidean spaces of finite dimension? Let Ln denote a Lobacevskii space of n dimensions and let sn-l(x,a) be the (n - l)-sphere in Ln with
center x and radius a. Guc [12] proved that if for some fixed a > 0, a bijective
mapping f : Ln --+ Ln (n ~ 2) satisfies f(sn-l(x,a)) = sn-l(f(x),a) for all x
in Ln, then f is a motion. Kuz'minyh [18] proved that if there exist two positive
numbers a and b, such that if the mapping f : Ln --+ Ln has the property that
d(x,y) = a always implies that d(f(x),f(y)) = b, then b = a and f is an isometry.
What happens if we require, instead of one conservative distance for a mapping
between normed vector spaces, two conservative distances? An answer in a generalised form to this quest ion was given by Benz and Berens [6] who proved the
following:
Theorem 3.14. Let X and Y be real normed spaces such that dirn X > 1 and Y
is strictly convex. Given any fixed integer m > 1, suppose that f : X --+ Y is a
mapping such that for all x, y in X with IIx-YIi = 1 we have Ilf(x) - f(y)lI:::; 1
and for all x, y in X with IIx-yll = m we have IIf(x) - f(y)11 ~ m. Then f is an
isometry.
PROPERTIES OF ISOMETRIES AND APPROXIMATE ISOMETRIES
375
These authors also use the example cited earlier of r/> : IR -+ IR defined by r/>(x) =
[x] + (x - [X])2 to show that the condition dirn X > 1 is necessary, and they use the
example of X = Y = IR2 , endowed with the norm IIxll = max(lIxlll,lIx211), where
x = (Xl, X2), ! (X) = (4)( xt), 4>( X2)) to show that the condition of strict convexity
of the space Y is needed.
MORE COUNTEREXAMPLES
Ciesielski and Rassias [8] have examined cases of mappings of a finite dimensional
space into another using the often used metries:
n
dM(X,y) =max{lxj -Yjl : j = 1, ... ,n}
or
dE(X,y) = ~)Xj -Yjl,
j=l
as well as the usual Euclidean metric denoted by dE(X, y). These authors give
counterexamples for various mappings preserving the distance one, using these
metrics.
Example A. A continuous surjective mapping ! : (IRn , dM) -+ (IRn, dM) which
satisfies (DOPP) need not be an isometry. Let 4> : IR -+ IR be given, as above, by
4>(t) = [tl + (t - [t])2 and define
!(Xl, ... , Xn )
= (r/>(xt), r/>(X2), ... , 4>(xn )).
Then with the metric dM, ! is a continuous surjective mapping of (IRn , dM) onto
itself which satisfies (DOPP) but is not an isometry.
Example B. Let ! : (IR2 , dM ) -+ (1R2 , dM) be defined as in Example A with
n = 2. Let h : (IR2 , dM ) -+ (IR2, dE) be defined by means of the matrix
h_
1 [
1
- V2 -1
It is easily seen that this orthogonal transformation maps unit balls in metric dM
into balls of radius V2 in the metric d. Now let the mapping 9 : (1R2 , dE) -+
(IR2, dE) be the composite mapping
Then 9 is continuous and satisfies (DOPP), but is not an isometry.
Unfortunately this example has nodirect generalisation for mappings from (IRn , dE)
into itself when n > 2. This is because the balls in metries dM and dE are of the
same shape only for n = 1 and n = 2.
These authors also considered mappings ! from 1R2 onto 1R2 with different metries
in the domain and range of !, and demonstrated the following result among others.
TH. M. RASSIAS
376
Theorem 3.15. There is no mapping f : (IR2, dM) --+ (IR?, dE) satisfying the
condition (DOPP).
Proof. Let such an f satisfy (DOPP). Let Adenote the set
{(O,O), (0,1), (1,1), (1,0)}.
When x 'I Y for x and y in A, then dM(X, y) = 1. Then by (DOPP) we would have
dE (f(x), f(y)) = 1 for x 'I y with x and y in A. But this is a contradietion since
every subset of (IR2 , dE) having this property consists of at most three points. 0
Benz [5] has formulated a "fundamental principle" in geometry whieh might be
thought of as an "inverse program" to Klein's Erlanger program. The Erlanger
program begins with a group of transformations of a set of geometrie objects and
asks what are the invariants and invariant notions of the group. The program
of Benz on the other hand starts with an invariant (e.g., distance between two
points, angles between two intersecting lines, cross ratio of four points, etc.) or an
invariant notion (line, plane, circle, orthogonality, etc.) and looks for the functions
preserving that invariant or invariant notion, thus establishing functional equation
problems.
To see how Benz deals with the case of distance preservance, let M and W be
nonempty sets and let d: M x M --+ W be a mapping. The tripIe (M, W, d) will
be called a distance space and for x, y in M, d( x, y) the distance from x to y (in
that order). Given a distance space (M, W, d), let S be a fixed subset of Mx M.
The problem of distance preservance is to find all functions f : M --+ M such that
the functional equation d (f(x), f(y)) = d(x, y) holds for all (x, y) in S.
Example (I). M = IR2 , W = IR and, with x = (Xl,X2), y = (Yl,Y2), d(x,y) =
J(Xl - Yl)2 + (X2 - Y2)2, S = ((x,y) E M x M : d(x,y) = I}. If we put f(x) =
(4)(Xl,X2),1/!(Xl,X2)) then the above functional equation of distance preservance
becomes
[4>(Xl + COSX3,X2 + sinx3) - 4>(Xl,X2)]2
+ [1/!(Xl + COSX3,X2 + sinx3) -1/!(Xl,X2)]2 = 1
for all Xl, X2, X3 in llt
This functional equation has solutions
{
4>(Xl, X2) = Xl COS t - X2 sin t + a,
1/!(Xl,X2) = Xl sint + X2 cost + b
{
4>(Xl, X2) = Xl COS t + X2 sin t + a,
1/!(Xl, X2) = Xl sin t - X2 COS t + b,
and
where a, b and t are constants. This of course is an analytieal expression of the
theorem of Beckman and QuarIes, cited above as Theorem 3.1. See also Benz [4].
PROPERTIES OF ISOMETRIES AND APPROXIMATE ISOMETRIES
377
S = {(X,y) E M x M : d(x,y) = 1}.
The functional equation dU (x), f (y)) = d( x, y) for (x, y) in Snow becomes
Solutions cp, 'l/J are given by:
CP(Xl,X2)=axl+b,
'l/J(Xl,x2)=a- 1x 2+ c ,
CP(Xl,X2) = aX2 +b,
'l/J(Xl,X2) = a-1xl +c.
ExaIllple (111). Let M be the set of lines in ]R3, let W = ]R and let d(x, y) be
the usual distance ofthe lines x and y. Put S = ((x,y) E M x M : d(x,y) = 1}.
In this case, the functional equation of distance preservance has not been solved.
However, a partial result by Lester [19] is the following:
TheoreIll 3.16. Suppose that M is the set of lines of]R3 and that f : M --+ M is
bijective and preserves the distance one in both directions. Then f is a congruent
mapping of]R3 .
ExaIllple (IV). M = ]Rn, n ~ 3, W = ]R, and
while S = {(x, y) E M x M : d( x, y) = O}. The functional equation of distance
preservance has not been solved for this case either. However, Aleksandrov [1] has
proved an important result in this connection:
TheoreIll 3.17. Every bijection of M that preserves the distance 0 in both directions is a Lorenz transformation, up to a dilatation.
ExaIllple (V). Let k be a fixed real number, let M = ]Rn, n ~ 2, W = ]R, and
d(x,y) be defined as in Example (IV), but with
S={(x,Y)EMxM: d(x,y)=k#O}.
In this case all solutions of the functional equation of distance preservance are
Lorenz transformations. When k < 0 or when n = 2 this was proved by W. Benz
and when k > 0 by J. Lester. For the demonstrations of these results together
with other examples from various geometries, see the book by Benz [4].
Acknowledgement. I wish to express my thanks to Professor Gradimir V. Milovanovic
for the kind invitation to participate in the Dragoslav S. Mitrinovic Memorial Conference. In addition, I would like to thank Professor Milan D. Mihajlovic for preparing the
manuscript in '!EX form.
378
TH. M. RASSIAS
References
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2. ___ , Mappings of families of sets, Soviet Doklady 11 (1970), 116-120; 376-380.
3. F. S. Beckman and D. A. Quarles, On isometries of Euc/idean spaces, Proc. Amer. Math.
Soc. 4 (1953), 810-815.
4. W. Benz, Geometrische Transformationen unter Besonderer Berücksichtigung der Lorentztransformationen, BI-Wissenschafts-Verlag, Mannheim - Wien - Zurieh, 1992.
5. ___ , On a general principle in geometry that leads to junctional equations, Aequationes
Math. 46 (1993), 3-10.
6. W. Benz and H. Berens, A contribution to a theorem of Ulam and Mazur, Aequationes
Math. 34 (1987), 61-63.
7. R. Bishop, Characterizing motions by unit distance invariance, Math. Mag. 46 (1973),
148-151.
8. K. Ciesielski and Th. M. Rassias, On the isometry problem for Euc/idean and non-Euc/idean
spaces, Facta Univ. Sero Math. Inform. 7 (1992), 107-115.
9. R. J. Fleming and J. E. Jaminson, Isometries in Banach spaces: A survey, Analysis, Geometry and Groups (H. M. Srivastava and Th. M. Rassias, eds.), Hadronic Press Inc., Palm
Harbor, Florida, 1993, pp. 52-123.
10. T. W. Garnelin, Uniform Algebras, Prentice-Hall, Ine., Englewood Cliffs, New Jersey, 1969.
11. K. R. Goodearl, Notes on Real and Complex C-Algebras, Shiva Publishing Ltd., Nantwich,
Cheshire, England, 1982.
12. A. Gue, On Mappings that Preserve a Family of Sets in Hilbert and Hyperbolic Spaces,
Candidate's Dessertation, Novosibirsk, 1973.
13. R. A. Hirsehfeld and W. Zelazko, On spectral norm Banach algebras, Bull. Aead. Polon.
Sei. 16 (1968), 195-199.
14. K. Jarosz, Metric and algebraic perturbations of junction algebras, Proe. Edinburgh Math.
Soe. 26 (1983), 383-391.
15. ___ , Perturbations of Banach Algebras, Leet. Notes Math. 1120, Springer Verlag, Berlin
- New York, 1985.
16. F. John, Rotation and strain, Commun. Pure Appl. Math. 14 (1961), 391-413.
17. ___ , On quasi-isometrie mappings I, Commun. Pure Appl. Math. 21 (1968), 77-110.
18. A. V. Kuz'minyh, On a characteristic property of isometrie mappings, Soviet Math. Dokl.
17 (1976), 43-45.
19. J. Lester, On distance preserving transformations of lines in Euc/idean three space, Aequationes Math. 28 (1985), 69-72.
20. G. Lovblom, Isometries and almost isometries between spaces of continuous junctions, Israel
J. Math. 56 (1986), 143-159.
21. B. Mielnik, Phenomenon of mobility in non-linear theories, Commun. Math. Phys. 101
(1985), 323-339.
22. B. Mielnik and Th. M. Rassias, On the Aleksandrov problem of conservative distances,
Proe. Amer. Math. Soe. 116 (1992), 1115-1118.
23. P. S. Modenov and A. S. Parkhomenko, Geometrie Transformations Vol. 1, Aeademic Press,
New York - London, 1965.
24. M. Nagasawa, Isomorphisms between commutative Banach algebras with an application to
rings of analytic junctions, Kodai Math. Sem. Rep. 11 (1959), 182-188.
25. H. Rademacher, Über partielle und total DiJJerenzbarkeit von Funktionen mehrer Variabeln
und über die Transformation von Doppelintegralen, Math. Ann. 79 (1919), 340-359.
26. Th. M. Rassias and P. Semrl, On the Mazur-U/am theorem and the Aleksandrov problem
for unit distance preserving mappings, Proe. Amer. Math. Soe. 118 (1993), 919-925.
27. Th. M. Rassias and C. S. Sharma, Properties of isometries, J. Natural Geometry 3 (1993),
1-38.
28. Th. M. Rassias, Is a distance one preserving mapping between metric spaces always an
isometry?, Amer. Math. Monthly 90 (1983), 200.
PROPERTIES OF ISOMETRIES AND APPROXIMATE ISOMETRIES
379
29. _ _ _ , Some remarks on isometrie mappings, Facta Univ. Sero Math. Inform. 2 (1987),
49-52.
30. _ _ _ , On the stability of mappings, Rend. dei Sem. Mat. Fis. Milano 58 (1988), 91-99.
31. _ _ _ , The stability of linear mappings and some problems on isometries, Proc. Intern.
Conf. Math. Analysis and Its Applications, 1985, Pergamon Press, Oxford, 1988, pp. 175184.
32. _ _ _ , Mappings that preserve unit distanee, Indian J. Math. 32 (1990), 275-278.
33. _ _ _ , Problems and Remarks, Aequationes Math. 39 (1990), 304 and 320-321.
34. R. Rochberg, Almost isometries of Banaeh spaees and moduli of planar domains, Pacific J.
Math. 49 (1973), 455-466.
35. _ _ _ , Deformations of uniform algebras, Proc. London Math. Soc. 39 (1979), 93-118.
36. E. L. Stout, The Theory of Uniform Algebras, Bogden and Quigley, Inc., Publishers, Tarrytown-on-Hudson, New York and Belmont, California, 1971.
INEQUALITIES FOR THE ZEROS OF AN
ORTHOGONAL EXPANSION OF A POLYNOMIAL
G. SCHMEISSER
Mathematisches Institut der Universität Erlangen-Nümberg, 91054 Erlangen,
Germany
Abstract. Turan pointed out the importanee of studying the loeation of the zeros of a
polynomial in terms of the eoefficients of an orthogonal expansion. He himself obtained
numerous results for the Hermite expansion. Later Specht showed in aseries of papers
that analogous theorems hold for any expansion with respeet to a system of polynomials
orthogonal on the realline. His work stimulated various further studies. We give a survey
on this topie with special emphasis on some results from an unpublished manuseript of
Specht and new eontributions by the author.
1. Introduction
The late Professor D. S. Mitrinovic extended his eager interest in all kind of inequalities also to estimates for the zeros of a polynomial. His book [13] includes
various elegant inequalities in case the polynomial is given as
(1)
j(z) = ao + alZ + ... + an zn .
His more recent book [12] with Milovanovic and Rassias contains a section entitled
"Zeros in a Strip" which deals with orthogonal expansions. The present article can
be considered as a supplement to that contribution. Let us first give a motivation
why orthogonal expansions gained especial interest.
In a keynote speech at the congress of Hungarian mathematicians in 1950, P. Thran
[24] discussed the location of the zeros of a polynomial in terms of its coefficients.
This is an old subject which was formerly part of Algebra but has moved to
Analysis in our century. Fundamental contributions are due to Descartes, Newton,
Fourier, Sturm, Cauchy, Hermite, Laguerre and various other mathematicians. In
their studies, a polynomial is usually assumed to be given in the form (1), which
Thran called the Vieta expansion of j.
It turns out that the coefficients ao, al, ... ,an are a fairly good information for
estimating the distances of the zeros of j from the origin. However, they are not
so suited for estimating distances from a line - a problem as it arises in connection
with one of the greatest challenges in mathematics: the Riemann hypothesis. For
1991 Mathematics Subject Classification. Primary 30C15j Secondary 30AlO, 42ClO.
Key words and phrases. Inequalitiesj Zerosj Bound of zerOSj Orthogonal Expansionj Algebraic
polynomialsj Orthogonal polynomialsj Norm estimates.
381
G.v. Milovanovic (ed.), Recent Progress in lnequalities, 381-396.
© 1998 Kluwer Academic Publishers.
G. SCHMEISSER
382
an explanation, Thnin pointed out that the fundamental polynomials Zll (v =
0,1, ... ) of the Vieta expansion have concentric circles about the origin as level
curves. In order to get good estimates for the distances of the zeros from a line C
one should rather consider an expansion
n
j(z) =
I: blll/JlI(z)
11=0
with fundamental polynomials l/JII(Z) (v = 0,1, ... ) whose level curves have a
tendency to be approximately parallel to C. In the case that C is the real line,
Thran proposed to use the expansion
n
j(z) =
I: bIlHII(z) ,
11=0
where
(v = 0,1, ... )
are the Hermite polynomials. He announced a variety of results which showed
that there is a striking correspondence between estimates for the moduli of the
zeros in terms of the coefficients ao, ... ,an and estimates for the imaginary parts
of the zeros in terms of the coefficients bo, ... ,bn . Towards the end of his lecture,
Thnin asked if this analogy is an isolated phenomenon for the Hermite expansion
or holds for other orthogonal expansions as weIl.
Proofs of the statements in [24] and offurther results were given in [25-26]. They
make decisive use of the particular properties of the Hermite polynomials.
Later Specht [18-21] showed that most of Tunin's results for the Hermite expansion hold analogously for any expansion with respect to a system of polynomials
orthogonal on the real line. Nevertheless, the Hermite expansion plays a distinguished role since it can be generated by a composition of the Vieta expansion with
a fixed special polynomial. This allows a simple systematic approach to some of
Tunin's results and to several new ones [20]. Specht's research stimulated various
further studies [27], [7-9]. Here we want to give a survey on this topic with special
emphasis on some results from an unpublished manuscript of Specht [22], which
existed already in 1964, and refinements by the author.
2. Notations and Agreements
Let J.L be the distribution function of a positive Borel measure on the real line.
Let the support of the measure be an infinite set and let all the moments J.Ln :=
J~oo x n dJ.L(x) (n = 0,1, ... ) exist and be finite. It is weIl known [4] that in this
situation there exists an infinite sequence of polynomials l/Jo(z), l/J1 (z), ... with
each l/Jn(z) being of exact degree n such that
(2)
I:
l/Jm (x)l/Jn (x) dJ.L(x) =
°
if and only if
m-:f. n.
INEQUALITIES FOR THE ZEROS OF AN ORTHOGONAL EXPANSION
383
To make the various results easily comparable, we shall always assume that the
polynomials f/Jn(z) (n = 0,1, ... ) are monie. Then they are uniquely determined
by f..L. Furthermore, there exist real numbers a1, a2, . .. and positive numbers
'Yo = 1, 'Y1 , 'Y2, . .. such that
(3)
{
f/Jo(z) == 1,
f/J1 (z) = z - a1 ,
'Y
f/Jn+1 (z) = (z - a n+1)f/Jn(z) - _n_ f/Jn-1 (z)
(n = 1,2, ... ) .
'Yn-1
Conversely, by a result of Favard [6], there exists for any system of polynomials
defined by (3) with real numbers an and positive numbers 'Yn bo = 1, n =
1,2, ... ) a positive Borel measure with a distribution function f. L such that (2)
holds. Moreover, if f..L is normalized by J~oo df..L(x) = 1, then
i:
f/J~(x) df..L(x) = 'Yn·
Hence f/J~(z) := 'Y;;1/2 f/Jn(z) (n = 0,1, ... ) is a system of orlhonormal polynomials.
It is known that the zeros of f/Jn(z) (n = 1,2, ... ) are real and those of two
consecutive polynomials f/Jn(z), f/Jn+1(z) separate each other. Denoting by Jn
the smallest compact interval that contains the zeros of f/Jn(z), we introduce the
distance function
(4)
for
z E C.
Obviously,
(5)
Hence any upper bound for dn(z) is also an upper bound for 11m zl.
In what follows, we shall discuss the location of the zeros of a polynomial
(6)
where f/Jo(z), f/J1 (z), . .. is a system of orthogonal polynomials given by (3). If not
specified otherwise, the coefficients ao, . .. ,an are assumed to be complex numbers.
3. An Analogue of the Cauchy Bound
n
Let f(z) = L: a"z" be a polynomial of degree n. It was observed by Cauchy in
,,=0
1829 that the associated polynomial
n-1
(7)
la,,1 z" -Ianl zn
L
,,=0
has exactly one positive zero p[f], whieh is abound for the moduli of the zeros of
f. Obviously, it is the best possible bound that depends only on the moduli of the
coefficients of f since it is attained for the polynomial (7). As usual, we call p[f]
the Cauchy bound of f.
The following theorem was obtained by Specht [18] with dn (() replaced by 11m (I·
As stated, it was established in [22] and independently in [8].
G. SCHMEISSER
384
Theorem 1. Let f be a polynomial given in the form (6). Denote by p the Cauehy
n
bound of the assoeiated polynomial E avz v . Then eaeh zero ( of f satisfies the
inequality dn (() ~ p.
1'=0
Theorem 1 provides an inclusion of the zeros of f by a region which resembles a
racecourse (see Fig. 1).
FIG. 1: The racecourse region
A proof of Theorem 1 is not difficult. As an auxiliary result we use [23, Theorem 3.3.5] in the following supplemented form.
Lemma 1. Let 4Jn-1 and 4Jn be two eonseeutive monie orthogonal polynomials.
Denote by 6, . .. , en the zeros of 4Jn. Then
for
where
for
n
and E AI'
1'=1
v = 1, ... ,n
= 1.
Proof of Theorem 1. The lemma readily implies that
for
v = 1, ... ,n.
Hence for v < n, we find, using (5), that
(8)
j
4JV(Z) 1= l4Jv(z) I·· ·14Jn-d ) 1
l 4Jn(z)
4JvH (z)
4Jn(Z)
z
1
1 (1
)n-v
dn(z)
~ dvH(z) ... dn(z) ~
Now let ( be an arbitrary zero of f which is not contained in J n . Then
-an4Jn(() =
n-1
L av4Jv(() .
1'=0
INEQUALITIES FOR THE ZEROS OF AN ORTHOGONAL EXPANSION
385
Using (8), we get
or equivalently,
n-1
L la,,1 dn (()" .
lanl dn(()n :5
,,=0
From this we conclude that d n (() does not exceed the positive zero of lanl x n -
n-1
L la,,1 x". Therefore d n (() :5 p.
,,=0
0
By the above mentioned property of the Cauchy bound, any upper bound for the
n
moduli of the zeros of L avz" that depends only on
,,=0
a::
1: : 1 ' 1: : 1 ' ... , 1
1
1
is an upper bound for p. Therefore Theorem 1 in conjunction with some of the
standard estimates for the zeros of a Vieta expansion (see [11, §27], [12, Sec. 3.3.1],
or [15]) implies estimates for the zeros of an orthogonal expansion.
Corollary 1. Let ( be any zero 0/ the polynomial (6). Then
d n (() :5 max { 1,
d n (():5
(
dn (() :5 2
?;
~ 1:: I} ,
2) 1/2
n
1: : 1
'
1/(n-,,)
max
0~,,~n-1
1/n
1a" 1
1
11 /(n-1)
d n (():5 1:: 1
+ ::
d n (():5
(n 1a" I)
max
0~,,~n-1
,
an
+ ... +
a::
1
1
1
,
1/(n-,,)
an
It may be surprising that in the estimates by Theorem 1 and Corollary 1 the
constants a" and "(,, (v = 1,2, ... ), which determine the system of orthogonal
G. SCHMEISSER
386
polynomials, do not appear explicitly. The reason is that Theorem 1 holds for a
much wider dass of expansions. In fact, Lemma 1 has an analogue for any two
monic polynomials ifJn-1 and ifJn which have weakly interlacing zeros (see [15] for
details). For example, Theorem 1 holds for Newton expansions, where
ifJn(z) =
n
II (z - ~v)
(n = 1,2, ... )
v=l
and 6, ~2, . •. is a sequence of real numbers. Obviously, the Vieta expansion
n
~ avz v is a special case of a Newton expansion, and in this situation the extended
v=O
form of Theorem 1 reduces to the dassical result of Cauchy.
4. A Norm Estimate
Another result of Specht [18] whose proof makes decisive use of the properties of
orthogonal polynomials states that each zero ( of (6) satisfies the inequality
(9)
IIm(1 ~
L
n-1 IV 1a v 12
v=o In-1 an
Here the constants 10, 11, . .. do appear explicitly. This might suggest that (9)
is more appropriate for orthogonal expansions. However, (9) and the estimates
flowing from Theorem 1 cannot be compared for all possible orthogonal systems.
By choosing 11, ... "n-2 as very small positive numbers and In-1 as a very large
one, we can construct an orthogonal system (3) for which (9) is superior to Theorem 1. On the other hand, if 11, ... "n-2 are large and In-1 is a very small
positive number, then Theorem 1 will yield better bounds.
An interesting property of (9) is that its right hand side may be expressed in
terms of a norm. In fact, if J.t is the distribution function of a Borel measure
associated with
ifJ1, ... as described in Section 2 and J~oo dJ.t(x) = 1, then, for
a polynomial of the form (6),
ifJo,
00
111111' := ([00 11(x)1 2 dJ.t(X))
1/2
=
(n~ IV la l )
v
2
1/2
defines a norm. It allows us to rewrite (9) as
(10)
11m (I ~ _ 1
Vln- 1
IIL -ifJnll
an
I'
This may be interpreted as a perturbation theorem. We know that ifJn has all its
zeros on the realline. Now (10) teIls us that, apart from a constant, the deviation
of flan from ifJn, measured by the norm, is an upper bound for the distances of
the zeros of 1 from the realline. Most of the results in the following sections will
be refinements of (9) or its equivalent form (10).
INEQUALITIES FOR THE ZEROS OF AN ORTHOGONAL EXPANSION
387
5. Refinements by Matrix Methods
Defining
'Yvßv.·--
( 1/ = 1, ... , n )
'Yv-l
an d
bv ..-- - a v
an
J
'Yv
'Yn-l
(1/ = 0, ... , n - 1) ,
we introduee the n X n matrices
Sn:=
[ a,
v7J;
v7J;
a2
-/ßn-l
~l'
an
0
0
Bn'~ [r ... ~ 1
0
b:~'
(11)
and denote by In the identity matrix of order n. It was observed by Specht [20],
[21] that the zeros of the polynomial (6) are the eigenvalues of the matrix F n .
More precisely,
(12)
(see also [1]). This relation may be looked upon as follows. If we let ao, al, ... ,
an-l tend to zero, then j(z) reduees to anc/>n(z), which is a polynomial with real
zeros. Simultaneously, the matrix Fn reduees to Sn, which is a real symmetrie
matrix. We may therefore eoneeive F n as aperturbation of the matrix Sn by B n .
This raises the question as to how the speetrum of a real symmetrie matrix may
ehange under a non-symmetrie perturbation. An answer ean be obtained from
Matrix Analysis. For instanee, there is a result of Bauer and Fike [2] whieh may
be stated and supplemented as follows [15].
Lemma 2. Let N be a normal matrix and A an arbitrary matrix, both 01 order n.
Denote by Al, ... , An the eigenvalues 01 N and by 11 . IIs the spectral norm. Then
A has all its eigenvalues in the union U 01 the disks
{z E C: IZ-Avl ~ IIA-NlIs}
(1/= 1, ... ,n).
Moreover, in each connected component olU, the matrices A and N have the same
number 01 eigenvalues (counted with multiplicities).
This leads us to the following refinement of (9).
Theorem 2. Denote by 6, ... , en the zeros 01 c/>n. Then every polynomial I 01
the lorm (6) has all its zeros in the union U 01 the disks
Vv:={ZEC: Iz-evl~r}
(1/= 1, ... ,n),
388
G. SCHMEISSER
where
(13)
r ..-
?; '::1 ::
n-l
1
12
Moreover, il k 01 these disks constitute a connected component 01 U, then their
union contains exactly k zeros 01 I.
Proof. In view of Lemma 2 it is enough to show that r is the spectral norm of
the matrix B n . Equivalently, we have to verify that r 2 is the largest eigenvalue
of the matrix B~Bn, where B~ denotes the conjugate transpose of B n . An easy
calculation shows that the entries of B~Bn are all zero except for the last element
in the diagonal, which turns out to be r 2 • This completes the proof. 0
As an easy consequence, we get the following improvement of inequality (9).
Corollary 2. Let ( be a zero 01 the polynomial (6). Then
(14)
L IV 1a 1
n-l
v=o In-l
v
2
an
Corollary 2 has been directly proved in [22], but it seems that Theorem 2 has not
been stated anywhere. Specht [21] used the representation (12) to prove the following theorem, which improves upon Theorem 2 provided that we are only interested
in estimates for the imaginary parts of the zeros. This result was independently
obtained in [8].
Theorem 3. Let ( be a zero 01 the polynomial (6). Then
6. Refinements Involving All the Zeros
As a refinement of (14), Specht [22] obtained the following result. A proof will
appear in [15].
Theorem 4. Let ZI, ... ,Zn be the zeros 01 the polynomial (6) in an arbitrary
order. Then
(15)
In Theorem 4, we mayorder the zeros as
(16)
INEQUALITIES FOR THE ZEROS OF AN ORTHOGONAL EXPANSION
389
The left hand side of (15) is a sum of non-negative terms and 'Yn-ldn(Zn)2 is one
of them. Hence dividing both sides by 'Yn-l, we see that (15) improves upon (14)
and is a refinement of (9).
Furthermore, if (16) holds, then we may estimate the left hand side of (15) from
below by
n
n
lI=k
lI=k
L 'Y1I-1dn(ZIY ... dn (zn)2 ~ L 'Y1I_ 1dn (Zk)2(n-II+l) ~ 'Yk_l dn(Zk)2(n-k+l),
where 1 ~ k ~ n. This allows us to establish the following individual bounds for
the zeros of f.
Corollary 3. Let Zl, ... , Zn be the zeros of the polynomial (6) ordered as in (16).
Then
(k = 1,2, ... ,n).
Giroux [7], who knew about Theorem 4, discovered an alternative inequality which
is also an improvement of (14) and involves all the zeros of f.
Theorem 5. Let Zl, ... , Zn be the zeros of the polynomial (6). Then
(17)
Equality is attained if and only if f(z) is of the form
with
or can be deduced /rom such a polynomial by replacing some of the zeros by their
conjugates.
As in the case of Theorem 4, we can again establish an individual bound for each
zero.
Corollary 4. Let Zl, ... , Zn be the zeros of the polynomial (6) ordered as in (16).
Then
(k = 1,2, ... , n) .
Yet another inequality comparable with (15) and (17) was established by Lajos
Lasz16 [9] who seems to have been unaware of Theorems 4 and 5. Using matrix
methods and employing an inequality of Schur, he obtained a result which may be
stated as folIows.
390
G. SCHMEISSER
Theorem 6. Let Zl, ... ,Zn be the zeros 01 the polynomial (6). Then
(18)
Theorem 6 improves upon (9) but it does not imply Theorem 3. As a consequence,
we get the following individual bounds.
Corollary 5. Let Zl, ... ,Zn be the zeros 01 the polynomial (6) ordered as
Then
_ 1 ((Im a n _
n - k +1
an
1)2 + _ 1 ~'Yv av 2)
2'Yn-l
an
l
l
1'=0
lor k = 1,2, ... ,n.
The estimate for Zn is not as good as that of Theorem 3. It should be mentioned
that in Theorems 3 and 6 and in Corollary 5 it is not possible to replace lImO I
by dnO on the left hand side. The reason is that abound which does not involve
the real part of an-dan cannot restrict the real parts of all of the zeros.
7. Refinements for Real Polynomials
As we have mentioned in the introduction, there is a elose correspondence between
estimating the zeros relative to the origin in terms of the coefficients of the Vieta
expansion and estimating them realtive to the real line in terms of the coefficients
of an orthogonal expansion. In the important case of a polynomial with real
coefficients, an orthogonal expansion has an additional property which the Vieta
expansion does not share. If in (6) the coefficients ao, ... ,an-l are zero, then 1
has n distinct real zeros. By a continuity argument, we easily conelude that the
distances of the zeros from the real line remain zero if these coefficients are real
and of sufficiently small modulus. This phenomenon does not show in our previous
estimates, except for a slight indication in Theorem 3.
Besides, the non-real zeros of areal polynomial appear in pairs of conjugates.
Therefore, the bounds of Corollaries 3-5 for k = n - 1 are upper bounds for the
distances of the zeros of 1 from the real line. In particular, that of Corollary 5
yields that (9) can be replaced by
1
IIm(1 ~ 2
~~
laan
'Yn-l
1'=0
V
2
1
This inequality also foHows from Theorem 3.
We now aim at a refinement which takes into account the phenomenon described
in the previous paragraph and the possible appearance of pairs of conjugate zeros
as weH. For this we need the following lemma whose proof can be found in [15].
INEQUALITIES FOR THE ZEROS OF AN ORTHOGONAL EXPANSION
391
Lemma 3. Let f/Jo, f/Jl,'" be a system 0/ monie orthogonal polynomials satisfying
(3). Denote by ~m,l, ... , ~m,m the zeros 0/ f/Jm and define
a,.,v(z) := f/J,.(z)f/Jv(z) - f/J,.(z)cPv(z),
f/Jm-l(~m,,.)
f/J~(~m,,.)
.
Then öm > 0 and
tor
Theorem 7. Let the polynomial (6) have real eoeffieients. Then, in the notation
01 Lemma 3, eaeh zero ( 0/ 1 satisfies the inequality
(19)
IIm(1 ~
(
t 2~ I2) 1/2 -
n
la v
v=o 'Yn-2 an
'Yn-l Ön-l
'Yn-2
provided that the radieand is non-negative, else 1 has n distinct real zeros whieh
separate those 0/ f/Jn-l.
Proof. If ( is a non-real zero of I, then f/Jn-l (()/(() - f/Jn-l (()/(() = O. In the
notation of Lemma 3, we have equivalently
-ana n,n-l(() =
n-2
L a a v,n-l(()
V
v=o
and so, as an obvious consequence,
lanl ~ ~ J
v=o
'Yv lavl' J'Yn- 2 1a v,n-l(() I.
'Yn-2
'Yv a n,n-l (()
Now applying the Cauchy-Schwarz inequality on the right hand side and using
Lemma 3 with m = n - 1 thereafter, we get that
From this we conclude that a non-real zero ( can exist only if the radicand in (19)
is non-negative and then inequality (19) holds.
It remains to prove the interlacing property. This is done by considering the
polynomials f/Jn-l(Z) and
n-l
I(t, z) := t
L avf/Jv(z) + anf/Jn(z) ,
v=o
tE [0,1],
392
G. SCHMEISSER
and using a eontinuity argument. Details are given in [15].
0
Theorem 7 has two interesting properties. It provides a sufficient eriterion for a
polynomial to have real distinct zeros and it yields bounds for the imaginary parts
of possible non-real zeros. For the seeond purpose, we may even replaee in (19)
the number 8n - 1 by zero.
In general, the numbers 8n are not easily available. However, for applieations of
Theorem 7, it is enough to know a non-triviallower bound for 8n . For this, we
ean proeeed as follows. It ean be seen with the help of the Gaussian quadrat ure
formula that
(j = 1, ... ,m).
If we know a finite interval that eontains the zeros of <Pm, then it is not diffieult to
establish a erude upper bound Km for the right hand side holding for j = 1, ... , m.
Then K;;.1/2 is a lower bound for 15m .
The classieal orthogonal polynomials satisfy a differentiation formula
with eonstants Am, B m , Gm, (m = 1,2, ... ) and a polynomial q(z) whose monie
form is 1, z, and Z2 - 1 for the Hermite, Laguerre, and Jaeobi polynomials, respeetively (see [4, p. 149], [23, § 4.5, § 5.1, § 5.5]). Henee the eonstant 15m mayaiso
be expressed as
In partieular, for the Hermite polynomials
n!
'Yn
(n = 1,2, ... ).
= 2n '
Noting that 2-" H,,(z) is monie, we easily deduee from Theorem 7 the following
result due to '!Unin [26].
n
Corollary 6. Let j(z) = L: b"H,,(z) be a Hermite expansion with real coefficients
satisfying
,,=0
n-2
L 2" v! b~ < 2 (n - I)! b!.
n
,,=0
Then j has n distinct real zeros.
INEQUALITIES FOR THE ZEROS OF AN ORTHOGONAL EXPANSION
393
8. Estimates in Terms of Some of the Coefficients
n
Let k, m, and n be integers satisfying O<k:5m:5n and let f(z} = E a"z" be a
,,=0
polynomial of degree n with a m # O. A fascinating result whieh originates from
work of van Vleck, Montel and Ballieu (see [11], [15]) states that there exists a
bound for k zeros of f whieh depends only on
Moreover, the best possible bound is given by the positive root of the equation
n-f.L
m-l-f.L
k-1 (
lam I zm - ~
m _ f.L) ( k _ 1 _ f.L ) la,.. I z'" = 0 .
One may ask if there is something analogous for orthogonal expansions. In the
case of polynomials with real coefficients, we are able to present two results in this
direction. Proofs are given in [15].
Theorem 8. Let f be a polynomial of the form (6) with real coefficients. 1f k is
an integer so that 1 :5 k :5 n and n - k is even, then f has at least k zeros in the
strip
k-2
{
zEC:IImzl:5 ( L~la"l
,,=0 'Yk-2
2)
1/(2n-2k+4) }
an
.
Theorem 9. For a given system 4>0,4>1, ... of orthogonal polynomials (3) and
integers 0 :5 k :5 n there exists a constant rkn with the following property. Every
polynomial (6) with real coefficients satisfying
'Yka~ > r kn
k-1
L 'Y"a~
,,=0
has at least k real zeros of odd multiplicity lying in the smallest interval Jm that
contains the zeros of 4>m, where m = [(n + k + 2}/2].
The constant r kn can be expressed as
r kn = (-.!:..
max IIQII!) - 1,
'Yk
Q
where the maximum extends over all monie divisors Q of degree k of 4>m.
Both theorems imply the known result that a polynomial
with real coefficients has always at least k real zeros and at least k + 1 real zeros
if n - k is odd.
G. SCHMEISSER
394
Note that the case k = n of Theorem 8 is covered by Theorem 7. The case k = n-l
of Theorem 9 yields a sufficient criterion for a polynomial to have all its zeros on
the realline, which should be compared with that flowing from Theorem 7. We
also find that r n-l,n = 8;;2 - 1. It would be desirable to establish a theorem for
orthogonal expansions with real coefficients, which embraces Theorems 7-9.
Finally, we mention that a Hermite expansion
f(z) = aoHo(z) + a1H1(z) + ... + akHk(z) + cHn(z)
(1 :::; k < n, ak "I- 0)
with complex coefficients has at least k zeros in a strip {z E C : IImzl :::; A} whose
width 2A depends only on ao, ... ,ak but not on n and c. For the trinomial
1 +H1(z) +cHn(z) the best possible bound A is known. For these results, we refer
to [10] and [16-17].
9. An Analogue of Descartes' Rule
If the coefficients of the polynomial (6) are real and ao, al, ... ,an-l are of very
small modulus, then f will have n distinct real zeros which are elose to those of
4Jn, no matter how the signs of the coefficients may be. Hence when we aim at an
upper estimate for the number of zeros in a semi-infinite interval I by an analogue
of Descartes' rule, we must require that I is devoid of zeros of 4Jn. Therefore the
following result of Obreschkoff [14] is a proper analogue of Descartes' rule with a
hypothesis that cannot be relaxed.
Theorem 10. Let f be a polynomial of the form (6) with real coefficients and let
a be areal number which exceeds the largest zero of 4Jn. Then the number of zeros
of f in the interval [a, 00) is not greater than the number of variations of sign in
the sequence ao, al, ... ,an, where vanishing coefficients are ignored.
10. Random Sums of Orthogonal Polynomials
There is an extensive literature on random polynomials [3]. Let
n
f(z) =
L a,,(w)z"
be a polynomial whose coefficients a,,(w) are independent real-valued standard
normal random variables. In various applications it is of interest to know the
expected number of real zeros E n . In 1943, Kac produced an integral formula for
E n and derived the asymptotic representation
(20)
2
E n = - logn + 0(1)
as
7r
n -t 00.
For refined considerations, we denote by En(a, b) the expected number of zeros f
in the interval [a, b] and by Pn(x) the density of the expected number of real zeros
at x E R, Le.,
En(a, b) =
l
b
Pn(x) dx.
There are also results for random sums of orthogonal polynomials. They have
been stated in terms of the normalized polynomials 4J~ := "{;;1/2 4Jn (n = 0,1, ... )
generated by (3). The following theorem is contained in [5, Sec. 3.1.4].
INEQUALITIES FOR THE ZEROS OF AN ORTHOGONAL EXPANSION
Theorem 11. Let
395
n
I(z) =
L av(w) q;~(Z)
v=O
be a random sum 01 normalized orthogonal polynomials, where av(w) are independent real-valued standard normal random variables. Then
v'3 V2G'(x) - G2(X) ,
Pn(x) = 61T
where
._ d
d (q;~+1(X))
G(x) .- dx log dx
q;~(x)
.
By asymptotic estimates, Das and Bhatt [3, p. 111] found for the Jacobi polyn0mials that
n
as
n-+oo.
E n ( -1,1) '" v'3
Comparison with (20) suggests again that orthogonal expansions are much more
adequate for questions of reality of zeros.
References
S. Barnett, A companion matrix analogue /or orthogonal polynomials, Linear Algebra App!.
12 (1975), 197-208.
[2] F. L. Bauer and C. T. Fike, Norms and exclusion theorems, Numer. Math. 2 (1960), 13714l.
[3] A. T. Bharucha-Reid and M. Sambandharn, Random Polynomials, Academic Press, Orlando, 1986.
[4] T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon & Breach, New York,
1978.
[5] A. Edelman and E. Kostlan, How many zeros 0/ a random polynomial are real?, BuH. Amer.
Math. Soc. 32 (1995), 1-37.
[6] J. Favard, Bur les polynomes de TchebycheJJ, C.R. Acad. Sei. Paris 200 (1935), 2052-2053.
[7] A. Giroux, Estimates 0/ the imaginary parts 0/ the zeros 0/ a polynomial, Proc. Amer. Math.
Soc. 44 (1974), 61-67.
[8] E. M. Gol'berg and V. N. Malozemov, Estimates /or the zeros 0/ certain polynomials, Vestnik
Leningrad Univ. Math. 6 (1979), 127-135 [Trans!. from Vestnik Leningrad Mat. Mekh.
Astronom. (1973), No. 7, 18-24].
[9] Lajos Laszl6, Imaginary part bounds on polynomial zeros, Linear Algebra App!. 44 (1982),
173-180.
[10] E. Makai and P. Turan, Hermite expansion and distribution 0/ zeros 0/ polynomials, Pub!.
Math. Inst. Hung. Acad. Sei. Sero A 8 (1963), 157-163.
[11] M. Marden, Geometry 0/ Polynomials, Math. Surveys 3, Amer. Math. Soc., Providence,
R.I., 1966.
[12] G. V. Milovanovic, D. S. Mitrinovic, and Th. M. Rassias, Topics in Polynomials: Extremal
Problems, Inequalities, Zeros, World Seientific Pub!., Singapore - New Jersey - London Hong Kong, 1994.
[13] D. S. Mitrinovic, Analytic Inequalities, Springer Verlag, Berlin - Heidelberg - New York,
1970.
[14] N. Obreschkoff, Über die Wurzeln von algebraischen Gleichungen, Jahresber. Deutsch.
Math.-Verein. 33 (1924), 52-64.
[15] Q. I. Rahman and G. Schmeisser, forthcoming book on polynomials, to be edited by Oxford
University Press.
[1]
396
G. SCHMEISSER
[16] G. Schmeisser, Optimale Schranken zu einem Satz über Nullste/len Hermitescher 1hnome,
J. Reine Angew. Math. 246 (1971), 147-160.
[17] ___ , Nullstelleneinschliepungen und Landau-Fejer-Montel Problem, Studia Sci. Math.
Hung. 7 (1972), 459-472.
[18] W. Specht, Die Lage der Nullste/len eines Polynoms, Math. Nachr. 15 (1956), 353-374.
[19] ___ , Die Lage der Nullstellen eines Polynoms, 11, Math. Nachr. 16 (1957), 257-263.
[20] ___ , Die Lage der Nullste/len eines Polynoms, 111, Math. Nachr. 16 (1957), 369-389.
[21] ___ , Die Lage der Nullste/len eines Polynoms, IV, Math. Nachr. 21 (1960), 201-222.
[22] ___ , Zur Analysis der Polynome, unpublished typed manuscript written not later than
1964.
[23] G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ. 23, 4th edn., Providence,
R.I., 1975.
[24] P. Turan, Sur I'algebre lonctionelle, Comptes Rendus du Premier Congr. Math. Hongr.
1950, Akad. Kiad6, Budapest, 1952, pp. 279-290.
[25] ___ , Hermite-expansion and strips lor zeros 01 polynomials, Arch. Math. 5 (1954), 148152.
[26] ___ , To the analytic theory 01 algebraic equations, Izvestija Mat. Inst. Bulg. Akad. Nauk.
3 (1959), 123-137.
[27] R. Vermes, On the zeros 01 alinear combination 01 polynomials, Pacific J. Math. 19 (1966),
553-559.
ERROR INEQUALITIES FOR DISCRETE HERMITE
AND SPLINE INTERPOLATION
PATRICIA J. Y. WONG
Division 01 Mathematics, Nanyang Technological University, Singapore
RAVI P. AGARWAL
Department 01 Mathematics, National University 01 Singapore, Singapore
Abstract. In this paper we shall develop a dass of discrete Hermite and spline interpolates in one and two independent variables. Further , we shall offer explicit error bounds
in loo norm for both cubic and bicubic discrete Hermite and spline interpolates.
1. Introduction
In 1971 Mangasarian and Schumaker 17] investigated some constrained minimisation problems in areal Euclidean space which were discrete analogs of minimisation problems in a Banach space. The solutions of these discrete problems
exhibited a spline-like structure, and were hence introduced as 'discrete splines'.
These discrete splines were further found [8] to playafundamental role in certain
best summation formulae for a finite sequence of real numbers. In the field of
approximation theory, these discrete splines have been characterised in the work
of Schumaker [9], Astor and Duris [5], and Lyche [6]. In contrast to continuous
splines where derivatives are involved, discrete splines only involve differences, and
hence have a wider range of applications. Motivated by this attractive aspect of
discrete splines, in this paper we shall develop a cubic discrete spline which is
different from those considered in [5-9]. Our work naturally complements several
known results for the continuous case [3-4], [11-15].
Let a, b (b > a) be integers. We shall denote the discrete interval N[a, b] =
{a, a + 1, ... , b}. Let a, b, c, dEN = {O, 1, 2, ... }. For the intervals N[a, b] and
N[c, dj, we let
= k1 < k2 < ... < km = b,
ki E N, 1 $ i $ m (~3)
p' : c = lt < l2 < ... < ln = d,
li E N, 1 $ i $ n (~3)
p:a
and
be uniform partitions of N[a, b] and N[c, dj with stepsizes
h = ki+l - ki (~3), 1 $ i $ m - 1,
and
h' = li+l -li (~3), 1 $ i $ n - 1,
1991 Mathematics Subject Classijication. Primary 41A15j Secondary 41A05.
Key words and phrases. Discrete Hermite interpolationj Discrete spline interpolationj Error estimates.
397
G. V. Milovanovic (ed.). Recent Progress in Inequalities. 397-422.
© 1998 Kluwer Academic Publishers.
P. J. Y. WONG AND R. P. AGARWAL
398
respectively. Further, we let T = pxp' be a reet angular partition of N[a, b] xN[c, dj.
The standard symbol d is used to denote the forward differenee operator with
stepsize 1.
For a given nmetion f defined on N[a, b + 1], we define the usualloo norm, Le.,
IIfll = max If(t)l· In the two-dimensional ease the norm 11 • 11 is defined
tEN[a,b+1j
analogously.
Our main eontribution in this paper is the derivation of explicit error estimates in
the norm 11 • 11 between
(i) funetion f(t) defined on N[a, b + 1] and its eubic diserete Hermite interpolate Hpf(t);
(ii) f(t) and its eubic diserete spline interpolate Spf(t);
(iii) funetion f(t, u) defined on N[a, b + 1] x N[c, d + 1] and its bicubic diserete
Hermite interpolate Hd(t, u); and
(iv) f(t,u) and its bicubie diserete spline interpolate STf(t,u).
The plan of this paper is as foHows. In Seetion 2, we define the Hermite spaee
H(p) whose elements are eubic polynomials in eaeh subinterval N[ki , ki +1] , 1 ~
i ~ m-2, and N[k m- l ,b+l], and express Hpf(t) in terms ofthe basic elements of
H(p). Next, we define the spline space S(p) C H(p) whose elements are also eubic
polynomials in eaeh subinterval N[ki , k i +1] , 1 ~ i ~ m - 2, and N[k m - l , b + 1].
For a given function g(t) E H(p), we provide neeessary and suflicient eonditions so
that g(t) E S(p). This leads to reeurrenee relations involving first order differenees
of g(t). Two representations of Spf(t) are also given, one in terms of the basic
elements of H(p), another in terms ofthe eardinal splines. The minimum eurvature
property of Spf(t) is investigated in Seetion 3. In Seetion 4, we use diserete Peano's
kernel theorem to determine explicit error estimates for IIf - Hpfll in terms of
maxtEN[a,b+1-ijldi f(t)l, 1 ~ j :$ 4. Not only do these results supplement, they
also improve the work of Agarwal and Lalli [2]. Seetion 5 contains the derivation of
explicit error bounds for IIf-Spfll in terms ofmaxtEN[a,b+1-ijldi f(t)l, 1 ~ j ~ 4.
Finally, the two-dimensional diserete Hermite and spline interpolation as weH as
their error analysis are respeetively diseussed in Seetions 6 and 7.
2. Discrete Hermite and Spline Interpolation
Definition 2.1. For a fixed p and JEN « h), let Ii(t) be defined on N[ki , ki +1 +
j], 1 ~ i ~ m - 2, and fm-l (t) be defined on N[km- lo b + 1]. H
(2.1)
2 ~ i ~ m - 1, 0 ~ I ~ j,
then we say that f(t) == Ul<i<m-lli(t) E D(j)[a,b]. The set D(j,l)([a,b] x [c,dj) is
analogously defined.
Relation (2.1) is also equivalent to
(2.2)
Ii(ki + I) = Ii-l (k i + I),
2 ~ i ~ m - 1, 0 ~ I ~ j.
Henee, it is noted that f(t) = Ul<i<m-di(t) is weH defined on N[a, b + 1].
ERROR INEQUALITIES FOR DISCRETE INTERPOLATION
399
Definition 2.2. For a fixed p, we define the set H(p) as
H(p) = {g(t) E D(l)[a, b] : g(t) is a cubic polynomial in each subinterval
N[ki , ki +1], 1 ~ i ~ m - 2, and N[k m -1' b + In.
Clearly, H(p) is of dimension 4(m - 1) - 2(m - 2) = 2m.
Definition 2.3. For a given function f(t) defined on N[a, b + 1], we say Hpf(t)
is the H(p)-interpolate of f(t), also known as the discrete Hermite interpolate of
f(t), if Hpf(t) E H(p) with Hpf(ki) = f(ki), AHpf(ki ) = Af(ki ), 1 ~ i ~ m.
It is clear that Hpf(t) can be explicitly expressed as
Hpf(t) =
(2.3)
m
L [f(ki)hi(t) + Af(ki)hi(t)] ,
i=l
where hi(t), hi(t), 1 ~ i ~ m, are the basic elements of H(p) satisfying
I~i,j~m,
and are given as follows:
h ·(t) -
•
3
- h(h + 1)
(t _ k· 1)(2) _
2
(t - k· 1)(3)
.h(h - I)(h + 1)
.-
(tEN[k i- 1 ,ki ], 2~i~m-I; tEN[km _ 1 ,b+I], i=m),
(2.4)
3
(2)
2
= h(h - 1) (t - ki+1) + h(h _ I)(h + 1) (t - kHd
(t E N[ki, ki+1],
(3)
1 ~ i ~ m -1),
= 0 (otherwise) ;
- . _ -(h - 2) _ .
(2)
1
_.
(3)
h.(t) - h(h + 1) (t k.- 1 ) + h(h + 1) (t k.- 1 )
(tEN[k i- 1 ,ki ], 2~i~m-I; tEN[km _ 1 ,b+I], i=m),
(2.5)
h +2
(2)
1
(3)
= h(h _ 1) (t - ki+1) + h(h -1) (t - kH1 )
(tEN[ki,ki+1]' I~i~m-I),
= 0 (otherwise) .
400
P. J. Y. WONG AND R. P. AGARWAL
Lemma 2.1. The following equality holds for 2:::; i :::; m - 2:
(2.6)
max
tEN[k, ,k'+l)
[lhi(t)1 + Ihi+l(t)1] = max{l, B([T*]), B([T* + I])} == M(h),
where
B( )
u
(2.7)
(2.8)
T* =
= u(h - u)(h 2 + h + 2 - 2u)
h(h+l)(h-l)'
~ (h 2 + 3h + 2- Vh 4 + 7h 2 + 4)
and [.] denotes the usual greatest integer function.
Proof. For 2 :::; i :::; m - 2 and t E N[k i , ki+l], from (2.5) we have
1- .
-I h(hh +_ 21) (t -k.+d
'
'
1
+ h(h 1_ 1) (t -k.+d
(2)
h.(t)1 -
(3)
= 1 h + 2 (T _ h)(2) +
1
(T _ h)(3) 1
h(h-l)
h(h-l)
(2.9)
1
= h(h _ 1) T(h - T)(h + 1 - T),
where T = t - k i E N[O, h], and also
-.
-I-(h +-1)
Ih.+ 1 (t)1 -
2)
h(h
(2.10)
(t
_
. (2)
' (3) 1
k.)
+ h(h 1+ 1) (t -k.)
1
= h(h + 1) TIT - II(h - T).
For T E N[O, 1], an addition of (2.9) and (2.10) gives
-
-
Ihi(t)1 + Ihi +1 (t)1 =
T(h - T)(h - 2T + 3)
(h + 1)(h _ 1)
from which it is obvious that the maximum occurs at T = 1. Thus,
(2.11)
max
TEN[O,l)
[lhi(t)1 + Ihi+1(t)1] = 1.
Next, for T E N[I, h], we sum (2.9) and (2.10), to get
(2.12)
-
-
Ihi(t)1 + Ihi+l(t)1 =
T(h-T)(h 2 +h+2-2T)
h(h + l)(h -1)
= B(T).
Theating T as a continuous variable in the interval [1, h], we differentiate B(T) with
respect to T and set dB/dT = 0, to obtain T = T* (see (2.8)). It can be verified
that T* E (1, h) and B(T) attains its maximum at T = T*. Hence, it follows from
(2.12) that
(2.13)
max
TEN[l,h)
[lhi(t)1 + Ihi+l(t)1] = max{B([T*]), B([T* + I])}.
Combining (2.11) and (2.13), we immediately obtain
max
TEN[O,h)
[lhi(t)1 + Ihi+l(t)I] = M(h).
ERROR INEQUALITIES FOR DISCRETE INTERPOLATION
401
Definition 2.4. For a fixed p, we define the set S(p) as
S(p) = {g(t) E n(2)[a,b] : g(t) is a cubic polynomial in each subinterval
N[ki , kiH ], 1 ~ i ~ m - 2, and N[km-l, b +
In.
Clearly, S(p) is of dimension 4(m - 1) - 3(m - 2) = m + 2.
Definition 2.5. For a given function j(t) defined on N[a, b + 1], we say Spj(t) is
the S(p)-interpolate 0/ j(t), also known as the discrete spline interpolate 0/ /(t), if
Spj(t) E S(p) with Spj(ki) = j(ki), 1 ~ i ~ m and 6,Spj(ki ) = 6,/(ki ), i = 1, m.
Lemma 2.2. Let 2 ~ i ~ m - 1 but fixed, and let p(t), q(t) be two cubic polyno-
mials defined on N[k i- l , ki + 2] and N[k i , ki+l + 1], respectively. Suppose that
and
(2.14)
(h - 1)(h - 2)6,q(ki+l) + 4(h - l)(h + 1)6,Yi + (h + l)(h + 2)6,p(ki- l )
=3(h - l)q(ki+d + 6Yi - 3(h + l)p(ki- l ).
Proo/. Let
3
Gl(t) = Laj(t-ki)(j)
3
and
j=O
G 2(t) = L bj(t - kd j )
j=O
be two cubic polynomials defined on N[k i- l , ki+2] and N[k i , kiH +1], respectively.
To have Gl(t) == p(t), we set
and
to get an algebraic system of four equations that determines the unknowns aj, 0 ~
j ~ 3 in terms of p(ki- l ), 6,p(ki-l), Yi and 6,Yi. SimiIarly, the unknowns bj , 0 ~
j ~ 3 are computed by requiring
and
such that G 2(t) == q(t). Now, 6,2p(k i ) = 6,2 q(k i ), if and only if, a2 = b2, which is
the same as (2.14).
P. J. Y. WONG AND R. P. AGARWAL
402
Lemma 2.3. For a given g(t) E H(p), we define Ci = g(ki ), ßCi = ßg(ki ), 1 ~
i ~ m. Then, g(t) E S(p), iJ and only iJ,
(2.15)
(h - 1)(h - 2)ßCi+1 + 4(h - l)(h + I)ßci + (h + 1)(h + 2)ßCi-1
3(h - l)ci+1 + 6Ci - 3(h + l)ci-l,
2 ~ i ~ m - 1.
Moreover, from (2.15) the unknowns ßCi, 2 ~ i ~ m -1 can be obtained uniquely
in terms of Ci, 1 ~ i ~ m, ßCI and ßcm •
Proof. From Lemma 2.2, the 'continuity' of ß2 g(t) is the same as (2.14) which is
equivalent to (2.15). The system (2.15) in matrix form can be written as
B(ßc) = w,
(2.16)
4(h - 1)(h + 1) (h - 1)(h - 2)
(h + 1)(h + 2) 4(h - 1)(h + 1)
(2.17)
(h - 1)(h - 2)
B=
(h + 1)(h + 2)
4(h - 1)(h + 1) (h - l)(h - 2)
(h + 1)(h + 2) 4(h - 1)(h + 1)
and w = [Wi],
(2.18)
{
WI = 3(h - l)c3 + 6C2 - 3(h + I)CI - (h + 1)(h + 2)ßCI,
Wi = 3(h - l)ci+1 + 6Ci - 3(h + I)Ci-1
(2 ~ i ~ m - 3),
Wm-2 = 3(h - l)cm + 6Cm -1 - 3(h + l)cm-2 - (h - 1)(h - 2)ßcm .
Since h ~ 3, it can easily be checked that the matrix B is strictly diagonally
dominant. Hence, the system (2.16) has a unique solution.
Lemma 2.4. For a given junction j(t) defined on N[a, b + 1], Spj(t) exists and
is unique.
Proof. For any given function g(t) defined on N[a, b + 1], Hpg(t) exists and is
unique. Further, by Lemma 2.3 for the given set of numbers Ci = j(ki ), 1 ~ i ~
m, ßCi = ßj(ki ), i = 1, m, there exist unique ßCi, 2 ~ i ~ m - 1 satisfying
(2.15). Now, let g(t) be such that g(ki ) = Ci, ßg(ki) = ßCi, 1 ~ i ~ m. Then,
again by Lemma 2.3, Hpg(t) E S(p). However, from Definition 2.5 this Hpg(t) is
actually the same as Spj(t).
Remark 2.1. From the proof of Lemma 2.4 and (2.3), it is clear that Spf(t) can be
expressed as
(2.19)
Spf(t) =
m
m-I
i=1
i=2
E f(ki)hi(t) + t:..f(kl)hl (t) + t:..f(km)hm(t) + E t:..Ci hi(t),
where t:..Ci, 2 ~ i :5 m - 1 satisfy (2.15).
ERROR INEQUALITIES FOR DISCRETE INTERPOLATION
403
Remark 2.2. It is possible to describe a basis for S(p), namely the 'cardinal splines',
{Si(t)}~t2, which are defined by the following interpolating conditions:
si(kj) = Dij, ßSi(a) = ßSi(b) = 0,
1:5 i,j:5 m,
Sm+1(ki) = 0, ßSm+l(a) = 1, ßSm+1(b) = 0,
1:5 i:5 m,
sm+2(ki) = 0, ßSm+2(a) = 0, ßS m+2(b) = 1,
1 :5 i :5 m.
Obviously, Spf(t) can be explicitly expressed as
m
(2.20)
Spf(t) =
L f(ki)Si(t) + ßf(kl)Sm+1(t) + ßf(km )sm+2(t).
i=l
Lemma 2.5 ([10]). Let A = [aij] be an n x n matrix such that
Then, (I ± A) is nonsingular and
11(/ ± A)-lll $ (1 -IIAII)-l,
where I is the identity matrix.
Theorem 2.1 (Discrete Peano's Kernel Theorem). Let E be a linear junctional
and E(P(t)) = 0 /or all polynomials p(t) 0/ degree (n - 1). Then, /or any /(t)
defined on N,
(2.21)
where
(
(n-l)
)
t-s-l+
={(t0, - s - 1)(n-l),
t ~ s + 1,
t< s + 1,
and E t (-) means the linear junctional E applied to the expression (.) considered as
a junction 0/ t.
Proof. Following as in [1], the discrete Taylor's formula with exact remainder can
be written as
(2.22)
f(t) = ~ (t -.a)(i) flif(a) + ~ (t - s _1)~-1) flnf(s).
~~!
L.J
(n - I)!
,=0
s=a
By using the linearity of E, and the fact that E annihilates all polynomials of
degree (n - 1), (2.21) immediately follows by applying E to both sides of (2.22).
404
P. J. Y. WONG AND R. P. AGARWAL
3. Minimum Curvature Property of Discrete Spline
In this section, we shall show that the discrete spline Spj(t) defined earlier has
the minimum 'least square variation', which is analogous to the continuous case
[3]. For this, for any two functions j(t), g(t) defined on N[a, b - 1], we introduce
the inner product
b-l
(f, g) = ~ j(t)g(t),
(3.1)
t=a
and denote
(3.2)
IIjll~ = (f, f).
Theorem 3.1. Let p, j(k i ), 1::; i ::; m and 6,j(k i ), i = 1, m be given, and let
V == {w(t) E N[a,b+ 1]: w(ki ) = j(k i ), 1::; i::; m; 6,w(k i ) = 6,j(k i ), i = 1,m}.
The variational problem 0/ finding the /unction p(t) E V which minimises 116,2wll~
over all w(t) E V has the unique solution Spj(t).
Proof. First we shall show that p(t) E V is a solution of the variational problem
if and only if the inner product
(6,2 p, 6, 2 8) = 0
(3.3)
for all functions 6(t) E Vo == {w(t) E N[a, b + 1] : w(ki ) = 0, 1 ::; i ::; m;
6,w(k i ) = 0, i = 1, m}, i.e., p(t) is a solution of the generalised Euler's equation
(3.3).
To prove the necessity part, we note that if p(t) E V, then p(t) + 0: 8(t) E V
for all real numbers 0: and 8(t) E Vo. Suppose that p(t) E V is a solution of the
variation al problem. Then, the function F(o:) = 116,2(p + 0:8)11~ should attain its
minimum at 0: = O. This simply means that
dFI
(3.4)
do: ",=0
=0.
Since
F(o:) = 116,2(p + 0:8)11~ = (6,2(p + 0:8), 6,2(p + 0:8))
= (6,2 p,6,2p) + 20:(6,2p, 6, 28) + 0:2(6, 28, 6, 28),
it follows that relation (3.4) is equivalent to (3.3).
Next, to prove the sufficiency part, we let p(t) E V be a solution of (3.3) and w(t)
be any function in V. Then, w(t) - p(t) E Vo and therefore (6,2 p, 6,2(W - p)) = O.
Using this fact, we find that
116,2wll~ = (ß 2w,6,2W)
= (ß 2(w _ p) + 6,2p, 6,2(W _ p) + 6,2p)
= (ß 2(w - p), 6,2(w - p)) + 2 (6,2(w - p), 6,2p) + (6,2 p,6,2p)
(3.5)
= IIß 2(w - p)lI~ + 116,2pll~
(3.6)
2: IIß2pll~
ERROR INEQUALITIES FOR DISCRETE INTERPOLATION
405
for any w(t) E V, i.e., pet) is a solution of the variational problem. Moreover, we
note that equality holds in (3.6) if and only if
On summing, the above relation gives
~w(t)
= ~p(t) + a,
tE N[a,b],
where a is a constant. Since ~(w -p)(b) = 0 (w(t) -pet) E Vo), we see that a = O.
Now, another summation yields
w(t) = pet) + ß,
tE N[a, b + 1],
where ß is a constant. Using (w - p)(b) = 0 (w(t) - pet) E Vo), we immediately
obtain ß = O. Hence, w(t) = pet), tE N[a, b+ 1] and so pet) is the unique solution
of the variational problem.
To complete the proof, we shall show that Spf(t) is a solution of the generalised Euler's equation (3.3).
Using summation by parts (2::=a v(t)~u(t) =
u(t)v(t)I~+1- 2::=a u(t+ 1)~v(t)), boundary conditions, and the fact that Spf(t)
is a cubic polynomial in each subinterval N[ki , ki +1] , 1 ~ i ~ m - 1, we find for
any 6(t) E Vo,
(~2Spf, ~26) =
b-i
rn-i ki+1- i
t=a
i=i
L ~2Spf(t) . ~26(t) = L L ~2Spf(t)· ~2d(t)
t=k.
~ ~ [ a' 8,/{') . M{') I::+> - '~' a'{t+ 1) . a' 8,f(t)]
= ~2Spf(t) . ~6(t)l~ - ~ [ ~3Spf(t) . 6(t + 1)1::+
-'J;;' .(.
1
+ 2) . a' 8,/{.)]
= - ~3Spf(t). 6(t + 1)1~
= _~3 Spf(b) . 6(b + 1) + ~3Spf(a) . 6(a + 1)
= -~3Spf(b) . 6(b) + ~3Spf(a). 6(a)
=0.
This completes the proof of the theorem.
As a particular case of (3.5), we obtain the following discrete analog of the 'first
integral relation' [3].
P. J. Y. WONG AND R. P. AGARWAL
406
Corollary 3.1. Let f(t) be defined on N[a, b + 1]. Then
= 1I~2 fll~.
Theorem 3.2. Let g(t) be defined on N[a, b + 3] with g(ki) = f(ki), 1 ~ i ~ m,
and ~g(ki) = ~f(ki), i = 1, m. Then,
1I~2Spfll~ + 1I~2Spf - ~2 fll~
(3.7)
b-l
1I~2(g - Spf)lI~
(3.8)
= ~)g - Spf)(t + 2) . ~4g(t).
t=a
ProoJ. By using the same type of summation by parts formula as in the proof of
Theorem 3.1, we obtain
b-l
1I~2(g _ Spf)lI~ = ~)g - Spf)(t + 2) . ~\g - Spf)(t)
t=a
b-l
= ~)g - Spf)(t + 2) . ~4g(t).
t=a
The following corollary is the discrete analog of the 'second integral relation' [3].
Corollary 3.2. Let f(t) be defined on N[a, b + 3]. Then,
b-l
11~2(f - Spf)lI~ = L(f - Spf)(t + 2) . ~4 f(t)
(3.9)
t=a
(3.10)
Proof. Equality (3.9) follows from (3.8) immediately. To prove (3.10), we apply
(E:=a
summation by parts
u(t + 1)~v(t) = u(t)v(t)I~+1 the boundary conditions in (3.9), to get
rn-I ki+l- 1
1I~2(f - Spf)lI~ = L
i=l
L
E:=a v(t)~u(t)), and
(f - Spf)(t + 2)· ~4f(t)
t=ki
rn-I
k
ki+l- 1
k.
t=ki
= L[(f-Spf)(t+1).~3f(t) 1 ~+1_ L ~3f(t).~(f-Spf)(t+1)]
i=l
b
rn-I
a
i=l
k
= (f - Spf)(t + 1)· ~3 f(t) 1 - L [~(f - Spf)(t)· ~2 f(t) Ik~+l
•
ki+l- 1
-
=(f - Spf)(t + 1) . ~3 f(t)
I: -~(f
L
~2 f(t) . ~2(f - Spf)(t)]
t=ki
- Spf)(t) . ~2 f(t)
I:
b-l
+L
t=a
~2 f(t) . ~2(f - Spf)(t),
ERROR INEQUALITIES FOR DISCRETE INTERPOLATION
407
Le.,
IIß2 (f - Spf)lI~ = (f - Spf)(b + 1) . ß 3/(b)
- (f - Spf)(a + 1) . ß 3/(a) + (ß2 I, ß2(f - Spf))
= (f - Spf)(b) . ß 3/(b) - (f - Spf)(a) . ß 3/(a) + (ß2 I, ß2(f - Spf))
= (ß2 I, ß2(f - Spf)).
4. Error Estimates for Discrete Hermite Interpolation
Theorem 4.1. Let I(t) be defined on N[a, b + 1]. Then,
111 - Hpfll ~ aj(h) tEN[a,b+1-j]
max
Ißj l(t)l,
(4.1)
1 ~ j ~ 4,
where the constants aj(h), 1 ~ j ~ 4, are given as lollows:
al
(h) = 2h
a2(h) = {
2+h - 7
h+1
'
h(h21_1) max{fh([tiJ), (h([ti + 1])},
1
h(h 2 _ 1) flt (2),
[ti] E N[2, h - 1],
otherwise,
. { 32
1 (2h + 1)2( h - 1) 2, (h - l)(h - 2) 2
~ h(h 1+ 1) mm
(2h -}
3) ,
where (h (u) = (h - l)u(u - l)(h - u)(2h + 1- 2u) and ti E (1, h - 1) is the root
01 the equation 8t3 - 3(4h + 3)t2 + (4h 2 + lOh + 2)t - 2h2 - h = 0;
[t; + 1] E N[2, h - 1],
otherwise,
<
1
min{116(h2+1)2, (h-2)2(h 2 -2h+3)},
- 2h(h+1)
where 02(U) = (h - l)u(u - l)(h - u)[h 2 + 1- u(h -1)] and t; E (2, h) is the root
01 the equation 4(h - 1)t3 - 6h 2t 2 + 2(h3 + 2h2 + l)t - h 3 - h = 0; and
a4(h) = {
2~ max{ 03([tj]) , 03([t; + 1])},
1
24 03 (2),
1
22
~ 384 (h - 1) (h + 1) ,
[t; + 1] E N[2,h -1],
otherwise,
P. J. Y. WONG AND R. P. AG ARWAL
408
where 03(U) = U(U - l)(h - u)(h + 1 - U) and tj = (h + 1)/2.
Proof. Without loss of generality, let p : 0 = a = k 1 < k 2 = b = h. Then, from
(2.3) we have
(4.2)
_
[3
HpJ(t) - J(O) h(h _ 1) (t
3
_h)
+ J(h) [ h(h + 1) t
(2)
(2)
+ h(h22_ 1) (t _h)(3)]
2
(3)]
- h(h2 _ 1) t
(2)
1
(3)]
h+2
+ ~J(O) [h(h _ 1) (t - h) + h(h _ 1) (t - h)
[-eh -
2) (2)
1
(3)]
+ ~J(h) h(h + 1) t + h(h + 1) t
.
Hence, on using Theorem 2.1 we obtain
(4.3)
1
h+l
J(t) - HpJ(t) = Ci _ I)!
Gj(t, s)~j J(s),
L
1 ~ j ~ 3,
8=0
where
Gj(t, s) = (t - s - 1)~-1)
(4.4)
) (j-l) [
- (h - s - 1 +
3
h(h + 1) t
(2)
2
(3)]
- h(h2 _ 1) t
_ Ci - l)(h - s _ 1)(j-2) [_ (h - 2) t(2) +
+
h(h+1)
1
h(h+1)
t(3)].
Noting that J(t) - HpJ(t) = 0 for t = 0,1, h, h + 1, we have
IIJ - HpJl1 = tEN[O,h+1]
max IJ(t) - HpJ(t)1 =
max IJ(t) - HP/(t)l·
tEN[2,h-l]
Further, for tE N[2, h-1] it is obvious that Gj(t, h+ 1) = Gj(t, h) = 0, 1 ~ j ~ 3.
Coupling all these, it follows from (4.3) that
(4.5)
1
h-l
.
IIJ - HpJl1 ~ ('J - 1)'. tEN[2,h-l]
max L IGj(t, s)l'
max
I~J J(t)l,
8=0
tEN[O,h+l-j]
for 1 ~ j ~ 3.
CASE: j = 1. Here, (4.4) gives
(4.6)
I
h(h 2 _ 1) <PI (t),
G1 (t,s) = {
1
h(h2 _ 1) <P2(t),
t ~ s,
t ~ s + 1,
ERROR INEQUALITIES FOR DISCRETE INTERPOLATION
409
where (PI (t) = t(t - 1)(2t - 3h - 1) and 4J2(t) = h(h2 - 1) - 4Jl (t). It is clear that
4Jl (t) ~ 0 and 4J2(t) ~ 0 for t E N[2, h - 1].
Hence, we find for t E N[2, h - 1],
h-l
h(h2 -1) L
t-l
h-l
IG1(t,s)1 = L 14J2(t)1 + L l4Jl(t)1
8=0
s=t
8=0
t-l
h-l
= L4J2(t) + ~)-4Jl(t))
s=t
8=0
= t 4J2 (t) + (h - t)( -4Jl (t)).
It can be verified that the maximum of the above expression occurs at t = h - 1,
Le.,
(4.7)
h-l
2h2 + h-7
max "'IG1(t,s)l=
h 1
=al(h).
tEN[2,h-l] ~
+
On substituting (4.7) into (4.5), we get (4.1) immediately.
CASE: j = 2. From (4.4), we find that
(4.8)
G 2 (t, s) = {
h(h}-I) 4J3(t) - (h -1- S)4J4 (t),
1
h(h2 -1) 4J5(t) - s 4J6(t),
t ~ s,
t ~ s + 1,
where
4J3(t) = t(t -1)(h - t)(h -1),
4J4(t) = t(t - 1)(3h - 2t + 1),
4J5(t) = (t - 1)(h - t)(h + 1- t)(h -1),
4J6(t) = (h - t)(h + 1- t)(2t + h -1).
Clearly, 4J3(t), 4J4(t) ~ 0, tE N[O, h] and 4J5(t), 4J6(t) ~ 0, tE N[I, h]. Further, it
is noted that G2(t, s) changes sign for t ~ s as weIl as for t ~ s + 1.
We shall make use of the following inequality
la - bl ~ max{a,b},
(4.9)
a,b ~ O.
Then, it follows from (4.8) that
(4.10)
IG2(t,s)1 ~ {
h(h21_1) max{4J3(t), Ih -1- sl4J4(t)},
t ~ s,
1
h(h2 -1) max{4J5(t), S4J6(t)},
t~s+1.
P. J. Y. WONG AND R. P. AGARWAL
410
Subsequently,
h-l
IG2(t,s)1
h(h2-1) L
8=0
t-l
h-l
8=0
8=t
~ Lmax{<p5(t),S<P6(t)} + Lmax{<p3(t), (h -1- S)<p4(t)}
= max{ t<P5(t),
t(2)
t~) <P6(t)} + max{ (h - t)<P3(t), (h -2t )(2) <P4(t)}
_ { ""2 <P6(t) +
(h - t)(2)
2
<P4(t),
t <P5(t) + (h - t)<P3(t),
[h - 1 ]
tE [0,1] U -2-' h
[ h-1]
tEl, -2-
= (h - l)t(t - l)(h - t)(2h + 1 - 2t) = fh (t),
tE N[O, h].
Treating t as a continuous variable in the interval [2, h - 1], we differentiate Ih (t)
with respect to t, and set dfh/dt = 0, or
8t 3 - 3( 4h + 3)t2 + (4h 2 + lOh + 2)t - 2h 2 - h =
°
to obtain t = ti E (1, h - 1). It is dear that fh (t) attains its maximum at t = ti.
Hence, it follows that
h-l
(4.11)
1
max L IG 2(t, s)1 = h(h2 1)
max fh(t) = a2(h).
tEN[2,h-l] 8=0
tEN[2,h-l]
The inequality (4.1) is now immediate on using (4.11) in (4.5).
Further , it is obvious that
max lh(t) ~ (h -1) { max t(2h + 1- 2t)} { max (t -l)(h - t)}
tEN[2,h-l]
tE[O,h]
tE[O,h]
= 312 (2h + 1)2(h - 1)3
and also
max lh(t) ~ (h - 1) { max t(t - I)} { max (h - t)(2h + 1- 2t)}
tEN[2,h-l]
tE[2,h-l]
tE[2,h-l]
Thus, we have the upper estimate
h-l
(4.12)
1
max L IG 2(t,s)1 = h(h2 1)
max fh(t) = a2(h)
tEN[2,h-l] 8=0
tEN[2,h-l]
ERROR INEQUALITIES FOR DISCRETE INTERPOLATION
411
1
m' {(2h + 1)2(h _1)3 (h _ 1)2(h _ 2)2(2h _ 3)}
- h(h2 - 1) m
32
'
<
=
CASE: j
(4.13)
1
. {(2h + 1)2(h - 1)2 (h -l)(h _ 2)2(2h _
h(h + 1) mm
32'
3)} .
= 3. Relation (4.4) provides
G3 (t,s) =
h - s -1 (2)
h(h 2 _ 1) t [(h(t) - (h - s - 2)4>s(t)],
+1
h(~2 -1) (h + 1- t)(2)[s4>g(t) - 4>lO(t)],
{
t ~ s,
t ~ s + 1,
where 4>7(t) = 2(h - t)(h - 1), 4>s(t) = 3h - 2t + 1, 4>g(t) = h + 2t - 1, 4>lO(t) =
2(t - l)(h - 1). We note that 4>i(t) ~ 0, 7 ~ i ~ 10, tE N[l, h].
Applying inequality (4.9), it follows from (4.13) that
(4.14)
IG 3(t, s)1 ~
{
Ih - s - Ilt(2)
h(h2 _ 1) max{ 4>7(t), Ih - s - 214>s (t)},
t ~ s,
(s + l)(h + 1 _ t)(2)
h(h2 _ 1)
max{ s 4>g(t), 4>lO(t)},
t ~ s + 1.
Henee, we find
h-l
t-l
h(h2 - 1) L IG 3(t, s)1 ~ L(s + l)(h + 1 - t)(2) max{ s 4>g(t), 4>lO(t)}
8=0
8=0
h-l
+ L(h - s - 1)t(2) max{ 4>7(t), (h - s - 2)4>s(t)}
8=t
= (h - l)t(t - l)(h - t)[h 2 + 1 - t(h - 1)]
=02(t),
tE N[l, h - 1].
°
Onee again, we treat t as a eontinuous variable in the interval [2, h - 1]. By setting
d02/dt =
or 4(h - 1)t3 - 6h 2t 2 + 2(h 3 + 2h 2 + l)t - h3 - h = 0, we obtain
t = t; E (2, h) whieh maximises 02(t). Therefore,
(4.15)
h-l
1
L IG3(t, s)1 = h(h2 1)
max
tEN[2,h-l] 8=0
-
max 02(t) = a3(h).
tEN[2,h-l]
The inequality (4.1) immediate follows on using (4.15) in (4.5).
To obtain an upper estimate on a3(h), we note that
max
tEN[2,h-l]
02(t) ~ (h - 1){ max t[h 2 + 1 - t(h -1)]}{ max (t -l)(h - t)}
tE[O,h]
1
= 16 (h
2
2
+ 1) (h - 1),
tE[O,h]
412
P. J. Y. WONG AND R. P. AGARWAL
as weH as
max
tEN[2,h-I)
02(t) ~ (h -1){ max t(t - 1)}{ max (h - t)[h 2 + 1- t(h -1)]}
tE[2,h-I)
tE[2,h-I)
= (h - l)(h - 2)2(h2 - 2h + 3).
It foHows that
1
h-I
max
2)G3(t, 8)1 = h(h2 - 1) tEN[2,h-I)
max 02(t) = a3(h)
tEN[2,h-I) 8=0
<
1
- h(h2 -
(4.16)
CASE: j
(4.17)
min{ (h 2 + 1)2(h - 1) (h _ l)(h _ 2)2(h 2 _ 2h + 3)}
1)
16'
. {(h + 1)2
2 2
}
= h(h 1+ 1) mm
16
' (h - 2) (h - 2h + 3) .
2
= 4. From [2, Theorem 5.1] we have
1
IIf - Hpfll ~ 4'
°
max
. tEN[2,h-I)
03(t)·
16. 4 f(t)l·
max
tEN[0,h-3)
As before, we set d0 3 /dt = to get t = t; = (h+ 1)/2 E [2, h -1] which maximises
03(t). Hence, (4.17) leads to (4.1). Further, it is obvious that
1
(4.18)
1
a4(h) = 4'. tEN[2,h-I)
max 03(t) ~ 4' max 03(t)
. tE[2,h-I)
*
1
= 4!1 03(t3)
= 384
(h - 1) 2(h + 1)2 .
Remark 4.1. The case j = 4 is given in [2, Theorem 7.3] as follows
11/ - Hp/li :5 3814 h4 tEN[a,b-3)
max
1~4 /(t)l·
Since
1
2
a4(h) :5 384 (h - 1) (h + 1)
2
1
4
:5 384 h ,
our result is an improvement.
5. Error Estimates for Discrete Spline Interpolation
Let f(t) be an arbitrary function defined on N[a, b+ 1]. We begin with the equality
(5.1)
In (5.1) the term (Hpf - SpJ)(t) belongs to H(p), and (Hpf - SpJ)(ki ) = 0,
1 ~ i ~ m, and 6.(Hpf - SpJ)(ki) = 0, i = 1, m. Hence, it follows from (2.3) that
(Hpf - SpJ)(t) =
(5.2)
rn-I
L 6.(Hpf - SpJ)(ki ) . hi(t)
i=2
rn-I
rn-I
i=2
i=2
= L [6.f(ki ) - 6.Spf(ki)] hi(t) = L eihi(t),
ERROR INEQUALITIES FOR DISCRETE INTERPOLATION
413
where ei = Af(ki) - ASpf(ki ). Denoting e = [ei], the relation (5.2) leads to
I(Hpf - Spf)(t)1 ~ lIell
(5.3)
= lIell
rn-I
max
L Ihi(t)1
max
[lhi(t)1 + IhHI (t)1] ,
tEN[a,b+1] i=2
tEN[ki,ki+d
29$;rn-2
where we have used the fact that Ihi(t)1 is nonzero only in the interval N[k i- I ,
ki +1] , 2 ~ i ~ m - 1. Using (5.3), it now follows from (5.1) that
(5.4)
I(J - Spf)(t)1 ~ I(J - Hpf)(t)1 + lIell
max
tEN[ki,ki+l]
[lhi(t)1 + Ihi+1(t)1] .
2$;i$;rn-2
In the right side of (5.4) we can make use of Theorem 4.1 and Lemma 2.1, and
hence what remains is to compute an upper bound for lIell.
Lemma 5.1. Let f(t) be defined on N[a, b + I]. Then,
(5.5)
lIell ~ bj(h) tEN[a,b+1-j]
max
IAj f(t)l, 1 ~ j ~ 4
where the constants bj(h), 1 ~ j ~ 4 are given in Table 5.1.
TABLE 5.1
j
1
2
3
4
bj(h)
3h 2
h 2 -4
h(h - 1)(7h + 1)
4(h 2 - 4)
h(h - 1)(3h2 - h + 2)
8(h 2 - 4)
h(h 2 - 1)(h2 + 2)
24(h 2 - 4)
Proof. Let r = [ri(J)]~~;2I be an (m - 2) x 1 vector defined by
(5.6)
r=Be,
where the matrix B is given in (2.17). Then, it follows that
(5.7)
B(Af) = w+r,
where w is defined in (2.18) and Af = [Af(ki)]~2I is an (m - 2) x 1 vector. From
Lemma 2.3, we have
(5.8)
ri(J) = (h - l)(h - 2)Af(kHI) + 4(h -l)(h + l)Af(ki)
+ (h + l)(h + 2)Af(ki - l ) - 3(h - l)f(kH d
- 6f(ki ) + 3(h + l)f(ki -d, 2 ~ i ~ m - 1.
414
P. J. Y. WONG AND R. P. AGARWAL
For 2 ~ i ~ m - 1, since ri(p) = 0 for all polynomials p(t) of degree (j - 1),
1 ~ j ~ 4, it follows from Theorem 2.1 that
1
(5.9)
ki+l +1
:L (ri}t(t - s - 1)~-1) ~j I(s),
ri(f) = (j _ I)!
S=ki_l
where by (5.8) we have
(ri)t(t - s - 1)~-1) =(j -1) [(h - l)(h - 2)(ki+1 - S - 1)~-2)
+ 4(h - l)(h + l)(k i - S - 1)~-2)
+ (h + l)(h + 2)(ki- 1 - S - 1)~-2)]
(5.10)
- 3(h - 1)(ki+1 - S - 1)~-1)
- 6(k i - S - 1)~-1) + 3(h + 1)(ki- 1 - S - 1)~-1).
It is obvious from (5.10) that (riMt - s _1)~-1) = 0 if s = ki+1, kiH + 1. Thus,
(5.9) reduces to
(5.11)
ri(f) = (j ~ I)!
ki+l- 1
:L (ri}t(t - s _1)~-1) ~j I(s), 2 ~ i ~ m - 1.
S=ki_l
We shall continue the proof only for j = 2 as the proof for other cases is similar.
From (5.10), we find
(ri)t(t - s - 1)+ =(h -l)(h - 2)(ki+1 - S - 1)~)
+ 4(h - l)(h + l)(k i - S - 1)~)
+ (h + l)(h + 2)(ki - 1 - S - 1)~)
(5.12)
- 3(h - 1)(ki+1 - S - 1)+ - 6(ki - S - 1h
+ 3(h + 1)(ki - 1 - S - 1)+, 2 ~ i ~ m - 1.
Hence, for s E N[ki , kiH - 1], it follows from (5.12) that
(riMt-s-1)+ = (h-1)(h-2)-3(h-1)(ki+1-s-1) = (h-1)(h-2-3T) = 1/11 (T),
where T = k iH - s - 1 E N[O, h - 1]. Since 1/11 (T) changes sign in the interval
N[O, h - 1], we apply the inequality (4.9) to obtain
11/11(T)I~(h-1)
max{h-2, 3T}.
Thus,
ki+l -1
h-1
S=ki
T=O
:L I(ri)t(t - s - 1)+1 = :L 11/11(T)1
(5.13)
~ (h -1) max{~(h - 2), ~ 3T}
= ~ h(h - 1)2.
2
ERROR INEQUALITIES FOR DISCRETE INTERPOLATION
415
For s E N[ki-l, ki - 1], from (5.12) we get
(riMt - s - 1)+ = (h - l)(h - 2) + 4(h - 1)(h + 1)
- 3(h - l)(ki+l - S - 1) - 6(ki - S - 1)
=(h + 1)[2(h - 1) - 3T] = 'l/J2(T),
where T = ki - S - 1 E N[O, h - 1]. Onee again 'l/J2(T) changes sign in the interval
N[O, h - 1], and so by using (4.9) we find
ki-1
h-l
L I(ri)t(t - s - 1)+1 = L 1'l/J2(T)1
T=O
(5.14)
:::; (h + 1) max {
t;, =
h-l
t;,
h-l
2(h - 1),
}
3T
= 2h(h - l)(h + 1).
Coupling (5.13) and (5.14), it follows from (5.11) that
ki+l- 1
Iri(f)I:::;
L l(riMt - s -1)+1· tEN[a,b-l)
max Iß 2j(t)1
S=ki_l
(5.15)
:::; [~2 h(h - 1)2 + 2h(h - l)(h + 1)]
max
tEN[a,b-l)
= -21 h(h - 1)(7h + 1) tEN[a,b-l)
max Iß 2 j(t)l,
Iß 2j(t)1
2:::; i :::; m - 1.
Now, we multiply both sides of (5.6) by the diagonal matrix 0
dii = l/a, a E IR+ to obtain
Or = OBe.
= [dij ], where
This implies that
(5.16)
Writing OB = 1+ A where A is an (m - 2) x (m - 2) matrix with the property
that IIAII < 1, it follows from Lemma 2.5 and (5.16) that
(5.17)
To obtain the smallest bound in (5.17), we shall maximise (1-IIAIDa over a E JR+.
For this, from (2.17) we find
416
P. J. Y. WONG AND R. P. AGARWAL
IIAII = (h + l)(h + 2) + 4(h - 1)(h + 1) _ 11 + (h - 1)(h - 2)
1
(5.18)
=
{
a
a
6h2
--1
a ~ 4(h 2 - 1),
1
~ 2h
2 '- 8,
a
a ~ 4(h 2 - 1).
a
Further, the condition IIAII < 1 is equivalent to 6h 2 ja - 1 < 1 (since 1 - (2h 2 8)ja< 1 for a ~ 4(h 2 - 1)) which gives a > 3h2 • Hence, in view of (5.18) we have
max
(1 - IIAll)a = max (1 - IIAII)a
aER+, IIAII<l
a>3h 2
(5.19)
= 2h 2 - 8.
Using (5.15) and (5.19) in (5.17), we get
1
1
lIell ~ 2h2 8' -2 h(h - 1)(7h + 1)
-
max
tEN[a,b-l]
Iß2 f(t)1
= h(h - 1)(7h + 1)
max Iß 2 f(t)l.
a~t:5b-l
4(h 2 - 4)
This completes the proof of (5.5) for j = 2.
Theorem 5.1. Let f(t) be defined on N[a, b + 1]. Then
IIf - Spfll ~ dj(h)
(5.20)
max
Ißj f(t)l, 1 ~ j ~ 4
tEN[a,b+1-j]
.
where
dj(h) = aj(h) + bj(h)M(h),
and aj(h), bj(h) and M(h) are given in Theorem 4.1, Lemma 5.1 and (2.6),
respectively.
Proof. An application of Theorem 4.1, Lemmas 5.1 and 2.1 in (5.4) yields the
inequalities (5.20) immediately.
6. Two-variable Discrete Hermite Interpolation
For a given T, we define
H(T) = H(p) EB H(p') (the tensor product)
= Span {hi(t)hj(u), hi(t)hj(u), hi(t)hj(u), hi(t)hj(u)} /::1 ~=l
g(t,u) E D(1,l)([a,b] x [e,d]) : g(t,u) is a two-dimensional
polynomial of degree 3 in each variable and in each subrectangle
=
[ki,ki+l] x [lj,lj+1]' [km-1,b+ 1] x [lj,lj+1]' [ki,ki+1] x [ln-bd+ 1],
1 ~ i ~ m - 2, 1 ~ j ~ n - 2
and [km- 1,b+l] x [ln-l,d+l].
ERROR INEQUALITIES FOR DISCRETE INTERPOLATION
417
Since H(r) is the tensor product of H(p) and H(p') which are of dimensions 2m
and 2n respectively, H(r) is of dimension 4mn.
Definition 6.1. For a given function f(t, u) defined on N[a, b + 1] x N[c, d + 1],
we shall denote ft'/' = ~r ~~ f(ki,lj), /L, v = 0,1, 1 ~ i ~ m, 1 ~ j ~ n. We say
Hrf(t,u) is the H(r)-interpolate of f(t,u), also known as the discrete Hermite
interpolate of f(t,u), if Hrf(t,u) E H(r) with ~r~~ Hrf(ki,lj) = ft'/, /L,V =
0, 1, 1 ~ i ~ m, 1 ~ j ~ n.
Clearly, Hrf(t,u) can be explicitly expressed as
m
(6.1)
Hrf(t,u) = L
i=l
n
L[f~jOhi(t)hj(u) + f~ihi(t)hj(u)
j=l
The following result provides a characterisation of Hrf(t,u) in terms of onedimensional interpolation schemes.
Lemma 6.1. Let f(t, u) be defined on N[a, b + 1] x N[c, d + 1]. Then,
(6.2)
ProoJ. By definition
= Hrf(t,u).
The proof of the second equality in (6.2) is similar.
Now let f(t, u) be an arbitrary function defined on N[a, b + 1] x N[c, d + 1]. From
Lemma 6.1, we have
f - Hrf = (f - Hp!) + Hp(f - Hp'!)
(6.3)
(6.4)
= (f - Hp!) + [Hp(f - Hp'!) - (f - Hp'!)] + (f - Hp'!)
= (f - Hp!) + [Hp' (f - Hp!) - (f - Hp!)] + (f - Hp'!).
Using these relations we shall deduce error estimates for two-dimensional discrete
Hermite interpolation.
418
P. J. Y. WONG AND R. P. AGARWAL
Theorem 6.1. Let f(t, u) be defined on N[a, b + 1] x N[c, d + 1]. Then,
Ilf - Hrfll ~ a4(h)
(6.5)
max
tEN[a,b-3]
uEN[c,d+1]
Ißt/(t, u)1
+ al(h)a3(h')
max
tEN[a,b]
uEN[c,d-2]
Ißtß!f(t, u)1
+ a4(h')
IIf - Hrfll ~ a4(h)
(6.6)
max
tEN[a,b+1]
uEN[c,d-3]
Ißt/(t, u)1
max
tEN[a,b-3]
uEN[c,d+l]
+ a2(h)a2(h')
Iß; ß;'f(t, u)1
max
tEN[a,b-l]
uEN[c,d-l]
+ a4(h')
and
(6.7)
IIf - Hrfll ~ a4(h)
max
tEN[a,b-3]
UEN[c,d+l]
Ißtf(t,u)l,
max
tEN[a,b+1]
uEN[c,d-3]
Ißtf(t, u)1
Ißt/(t, u)1
+a3(h)al(h')
max
tEN[a,b-2]
uEN[c,d]
Iß~ßuf(t,u)1
+ a4(h')
max
tEN[a,b+1]
uEN[c,d-3]
Ißtf(t, u)l·
Proof. It follows from (6.3) that
ICf - Hrf)(t,u)1 ~ ICf - Hpf)(t,u)1
(6.8)
+ I [HpCf - Hpl f) - Cf - Hpl f)] (t, u)1 + ICf - Hpl f)(t, u)l·
Applying Theorem 4.1 in (6.8) gives
ICf - Hrf)(t,u)1 ~ a4(h)
(6.9)
max
tEN[a,b-3]
uEN[c,d+l]
+al(h)
Ißt/(t,u)1
max
tEN[a,b]
uEN[c,d+l]
IßtCf-Hplf)(t,u)1
+ a4(h')
max
tEN[a,b+l]
uEN[c,d-3]
Ißtf(t,u)l·
Since ßtHp1f = Hplßd, using Theorem 4.1 again we get
(6.10)
IßtCf - Hplf)(t,u)1 ~ a3(h')
max
Ißtß!f(t,u)1
tEN[a,b]
uEN[c,d-2]
which on substituting into (6.9) yields (6.5). The proof of (6.6) and (6.7) is similar.
ERROR INEQUALITIES FOR DISCRETE INTERPOLATION
419
7. Two-variable Discrete Spline Interpolation
For a given 7, we define
8(7) = 8(p) EB 8(p') (the tensor product)
= Span {si(t)sj(u)};:i 2j 2 (see Remark 2.2)
:i
g(t,u) E D(2,2)([a,b] x [c,d]): g(t,u) is a two-dimensional
polynomial of degree 3 in each variable and in each subrectangle
=
[ki ,ki+1] x [lj,lj+1]' [km-bb+ 1] x [lj,lj+1], [ki ,ki+1] x [ln-1,d+ 1],
1 ~ i ~ m - 2, 1 ~ j ~ n - 2,
and [k m- 1,b+ 1] x [ln-1,d+ 1].
Since 8(7) is the tensor product of 8(p) and 8(p') which are of dimensions (m+2)
and (n + 2) respectively, 8(7) is of dimension (m + 2)(n + 2).
Definition 7.1. For a given function f(t, u) defined on N[a, b+ 1] x N[c, d+ 1], We
say 8 r f(t,u) is the 8(7)-interpolate of f(t,u), also known as the discrete spline
interpolate of f(t,u), if 8 r f(t,u) E 8(7) with ßrß~ 8 r f(k i ,lj) = ff,t where
/-L, /J, i and j satisfy
(7.1)
(I)
{ (2)
(3)
(4)
if /-L = /J = 0, then 1 ~ i ~ m, 1 ~ j ~ nj
if /-L = 1, /J = 0, then i = 1, m, 1 ~ j ~ nj
if /-L = 0, /J = 1, then 1 ~ i ~ m, j = 1, nj and
if /-L = /J = 1, then (i, j) = (1,1), (1, n), (m, 1), (m, n).
Remark 7.1. Since S(r) C H(r), in view of (6.1) Sr!(t,U) can be explicitly expressed
as
(7.2)
Sr !(t, u) =
m
n
E E [Sr !(ki, lj)hi(t)hj(u) + ßuSr/(ki , lj)hi(t)iij(u)
i=l j=l
In (7.2), the values ßrD..~8rf(ki,lj) where /-L, /J, i and j do not fulfil (7.1) exist
uniquely. Indeed, this is an immediate consequence of Lemma 2.3 and is stated as
follows:
Lemma 7.1. For a given g(t, u) E H(7), we define cti = ßr ß~g(ki, lj), /-L, /J =
0,1, 1 ~ i ~ m, 1 ~ j ~ n. The function g(t, u) E 5(7) if and only if cti, where
/-L, /J, i and j are such that
(7.3)
(1)
{ (2)
(3)
0, then 2 ~ i ~ m - 1, 1 ~ j ~ nj
if /-L = 0, /J = 1,
then 1 ~ i ~ m, 2 ~ j ~ n - 1j and
if /-L = /J = 1, then 2 ~ i ~ m - 1, j = 1, n and
~f /-L = 1,
/J =
1 ~ i ~ m, 2 ~ j ~ n - 1,
P. J. Y. WONG AND R. P. AGARWAL
420
satisfy the following relations
(7.4)
(h - l)(h - 2)C;:1,j + 4(h - l)(h + l)c;,j + (h + l)(h + 2)C;~1,j
= 3(h - l)C?:l,j + 6c?,j - 3(h + l)C?~l,j'
where v, i and j in (7.4) are such that if v = 0, then 2 S i S m - 1, 1 S j sn,
and if v = 1, then 2 S i Sm - 1, j = 1, n; and
(7.5)
(h' - l)(h' - 2)CILt,J+1
,1 + 4(h' - l)(h' + l)cIL,~
+ (h' + l)(h' + 2)CIL,~_
t,J
t,J 1
IL '?_l'
IL ,? - 3(h' + l)c't,)
= 3(h' - 1)c!:'?+1
+ 6c1,,)
'l.,)
where j.L, i and j in (7.5) are such that j.L = 0,1, 1 Si S m, 2 S j Sn - 1.
<;
Moreover, from (7.4) and (7.5) the unknowns
where j.L, v, i and j satisfy (7.3)
can be obtained uniquely in terms of
where j.L, v, i and j fulfil (7.1).
<;
Lemma 7.2. For any function f(t, u) defined on N[a, b+ 1] x N[c, d+ 1], Srf(t, u)
exists and is unique.
Proof. The proof is similar to that of Lemma 2.4.
Remark 7.2. In view of Remark 2.2, Sr J(t, u) can be explicitly expressed in terms of
cardinal splines as
Sr J(t, u) =
(7.6)
m
n
m
L L J~:? Si(t)Sj(u) + L [J~ll Sn+1 (u) + J~~ Sn+2(U)] Si(t)
i=l j=l
i=l
n
L
[Jt:?Sm+l(t) + J~?iSm+2(t)] Si(U) + Jt:tSm+1 (t)sn+l (u)
i=l
+ Jt:~Sm+l (t)Sn+2( u) + J~\ Sm+2 (t)sn+l (u) + J;;~nSm+2(t)Sn+2( u).
+
As a direct consequence of Remarks 2.2 and 7.2, we have the following result which
provides an important characterisation of Srf(t, u) in terms of one-dimensional
interpolation schemes.
Lemma 7.3. Let f(t,u) be defined on N[a,b+ 1] x N[c,d+ 1]. Then,
(7.7)
Proof. The proof is similar to that of Lemma 6.1.
Now let f(t, u) be an arbitrary function defined on N[a, b + 1] x N[c, d + 1]. From
Lemma 7.3, we have
(7.8)
f - Srf = (f - Spl) + Sp(f - Spl I)
= (f - Spl) + [Sp(f - Spl I) - (f - Spll)] + (f - Spl I)
(7.9)
= (f - Spl) + [Spl (f - Spl) - (f - Spl)] + (f - Spll).
Using these relations and Theorem 5.1 we shall deduce error estimates for twodimensional discrete spline interpolation.
ERROR INEQUALITIES FOR DISCRETE INTERPOLATION
421
Theorem 7.1. Let I(t, u) be defined on N[a, b + 1] x N[c, d + 1]. Then,
111 - STIli ~ d4 (h)
(7.10)
max
tEN[a,b-3]
uEN[c,d+1]
I~U(t,u)1
max
+d1 (h)d 3 (h')
tEN[a,b]
uEN[c,d-2]
l~t~~/(t,u)1
+ d4 (h')
111 - STIli ~ d4 (h)
(7.11)
max
tEN[a,b+1]
uEN[c,d-3]
I~tl(t, u)l,
I~U(t,u)1
max
tEN[a,b-3]
uEN[c,d+1]
+ d2 (h)d2 (h')
1~~~~/(t,u)1
max
tEN[a,b-l]
uEN[c,d-l]
+ d4(h')
max
tEN[a,b+l]
uEN[c,d-3]
l~tl(t,u)1
and
111 - STIli ~ d4 (h)
(7.12)
max
tEN[a,b-3]
uEN[c,d+l]
I~U(t,u)1
+ d3 (h)d1 (h')
max
tEN[a,b-2]
uEN[c,dj
I~: ~u/(t, u)1
+ d4 (h')
max
tEN[a,bH]
uEN[c,d-3]
I~tl(t, u)l·
Proof. The proof is similar to that of Theorem 6.1.
References
1. R. P. Agarwal, Difference Equations and Inequalities - Theory, Methods, and Applications,
Marcel Dekker, New York - Basel- Hong Kong, 1992.
2. R. P. Agarwal and B. S. Lalli, Discrete polynomial interpolation, Green's junctions, maximum principles, error bounds and boundary value problems, Comput. Math. Appl. 25 (1993),
3-39.
3. R. P. Agarwal and P. J. Y. Wong, Error Inequalities in Polynomial Interpolation and Their
Applications, Kluwer Academic Publishers, Dordrecht, 1993.
4. R. P. Agarwal and P. J. Y. Wong, Explicit error bounds lor the derivatives 01 spline interpolation in L2 norm" Appl. Anal. 55 (1994), 189-205.
5. P. H. Astor and C. S. Duris, Discrete L-splines, Numer. Math. 22 (1974), 393-402.
6. T. Lyche, Discrete Polynomial Spline Approximation Methods, Lecture Notes Math. 501:
Spline Functions, Springer Verlag, Berlin - Heidelberg - New York, 1976.
7. O. L. Mangasarian and L. L. Schumaker, Discrete splines via mathematical programming,
SIAM J. Control Optim. 9 (1971), 174-183.
422
P. J. Y. WONG AND R. P. AGARWAL
8. O. L. Mangagarian and L. L. Schumaker, Best summation lormulae and discrete splines,
SIAM J. Numer. Anal. 10 (1973), 448-459.
9. L. L. Schumaker, Constructive aspects 01 discrete polynomial spline lunctions, Approximation Theory (G. G. Lorentz, ed.), Academic Press, New York, 1973, pp. 469-476.
10. R. A. Usmani, Applied Linear Algebra, Marcel Dekker, New York, 1987.
11. P. J. Y. Wong and R. P. Agarwal, Explicit eN'Qr estimates lor quintic and biquintic spline
interpolation, Comput. Math. Appl. 18 (1989), 701-722.
12. P. J. Y. Wong and R. P. Agarwal, Quintic spline solutions 01 Fredholm integral equations 01
the second kind, Intern. J. Computer Math. 33 (1990), 237-249.
13. P. J. Y. Wong and R. P. Agarwal, Explicit error estimates lor quintic and biquintic spline
interpolation II, Comput. Math. Appl. 28 (7) (1994), 51-69.
14. P. J. Y. Wong and R. P. Agarwal, Sharp error bounds lor the derivatives 01 Lidstone - spline
interpolation, Comput. Math. Appl. 28 (9) (1994), 23-53.
15. P. J. Y. Wong and R. P. Agarwal, Sharp error bounds lor the derivatives 01 Lidstone - spline
interpolation II, Comput. Math. Appl. 31 (3) (1996), 61-90 ..
Contributed Papers
AN INEQUALITY CONCERNING SYMMETRIC
FUNCTIONS AND SOME APPLICATIONS
DORIN ANDRICA and LIVIU MARE
"Babes-Bolyai" University, Faeulty of Mathematies, Str. Kogalnieeanu 1,
Ro-3J,OO, Cluj-Napoea, Romania
Abstract. An inequality for symmetrie continuous functions E : In --t lR is proved in
Theorem 1.1 and a variant for Cl-differentiable functions is given in Theorem 1.2. Some
applications concerning inequalities between means or convex functions are presented in
the second section.
1. The Main Results
Let I ~ lR be an interval and In = I X ... xI, In ~ lRn • Consider (Sn, 0) the permutations group ofthe set {1,2, ... ,n} acting on In by ax = (Xu(l)"" ,Xu(n)),
where X = (Xl, ... ,xn ). Recall that a real-valued function E: In --t lR is symmetrie or Sn-invariant if for every X E In the relation E(ax) = E(x) holds, i.e.,
Eis constant on the Sn-orbits.
The main purpose of this section consists in proving of two general results on
symmetrie functions whieh will be very useful in obtaining some important inequalities.
Theorem 1.1. Let I ~ lR be an interval and let E : In -+ lR be asymmetrie
eontinuous funetion satisfying for every a = (al, ... , an) E In with al ~ a2 ~
. . . ~ an the inequality
(1)
<
E(al, ... ,an)(~)E
(al + a2 al + a2
)
2'
2 ,a3,···,an
Then for every a = (al, . .. , an) E In the following inequality
(2)
<
E(al, ... ,an)(~)E
(al + ... + a
al + ... + a )
n
n, ... ,
n
n
holds.
Proof. We prove by induction on k, 2 ~ k ~ n, that for every a = (al, ... ,an) E
In with al ~ a2 ~ ... ~ an the following inequality is satisfied
(3)
<
E(al,'" ,an)(~)E
(al+ ... +ak
al+···+ak
)
k
, ... ,
k
,ak+l,··· ,an'
1991 Mathematics Subject Classification. Primary 26D20, 26A51.
Key woms and phrases. Symmetrie functionsj Arithmetie, geometrie and harmonie meanSj Jensen's inequality.
425
G. V. Milovanovic (ed.), Recent Progress in Inequalities, 425--431.
© 1998 Kluwer Academic Publishers.
D. ANDRICA AND L. MARE
426
Taking into account the hypothesis (1) one obtains that the assertion is true for
k=2.
Let us suppose that (3) is verified for a fixed number k ~ 2 and denote
al + ... + ak
a = ---::----
(4)
k
Because a :::; a :::; ... :::; a :::; ß :::; ak+1 :::; ... :::; an it follows
(5)
<
E(a, ... ,a.,ß,ak+I, ... ,an)(~)E(xp, ... ,xp,Yp,zp,ak+2, ... ,an),
where the sequences (xp), (Yp), (zp) are defined by
Y2p+1 = X2p+1 = X2p+2 =
Y2p = Z2p = Z2p+1 =
(k - 1) X2p + Z2p
k
' P ~ 1,
X2p-1 + Z2p-1
2
'
P ~ 1.
Put Up = Z2p = Z2p+1, P ~ 0 and Uo = Zo = ZI = ß. We also denote vp = X2p-1 =
X2p, P ~ 1. Then
{
(6)
Vp+1
=
up =
(k - 1)vp + u p
k
'
vp + Up-I
2
where P ~ 1, VI = a, Uo = ß. From the relations (6) one obtains
j~1.
By adding these equalities for j = 1,2, ... ,P, it follows k(v p +1 - VI) = Uo - u p ; so
k(vp+1 - a) = ß - up. Therefore, kVp+1 = ka + ß - u p and using the first relation
of (6) one obtains
k -1
ka + ß
vp +1 = ~ vp + 2k .
Because of 0 :::; k 2~ 1 < 1, it immediately follows that the sequence (vp ) is convergent and
.
ka+ß
p = k
hm
v
p-too
+ 1·
Moreover
ka + ß
ka + ß
ka + ß
u p = kV p+1 - (k - 1)vp -t k k + 1 - (k - 1) k + 1 = k + 1 .
Using the continuity of the function E and the inequality (5) it follows for p -t 00
AN INEQUALITY CONCERNING SYMMETRIC FUNCTIONS
427
Taking into account that (3) is satisfied for the fixed number k, one obtains
and the assertion is proved by mathematical induction principle.
0
Theorem 1.2. Let I ~ 1R. be an open interval and let E : In -+ 1R. be asymmetrie
Cl-differentiable function satisfying for every a = (al, ... , an) E In with al :::;
a2 :::; ... :::; an the inequality
(7)
< äE
äE
-ä (a) (-) -ä (a).
~
Xl
X2
Then for every a = (al, ... , an) E In the inequality (2) holds.
Proof. Applying the mean value theorem for the function E and the segment [a, b]
wherea=(al, ... ,an),b= (
e E (a, b) with the property
al + a2 al + a2
2'
2
,a3,···,an
)
.
.
oneobtamsapomt
that is, the condition (1) in Theorem 1.1 is satisfied and the desired conclusion is
obtained. 0
Remark 1.3. Suppose that the function E : In -+ 1R. is symmetrie and eontinuous. To
verify the eondition (1) in Theorem 1.1 for E and a = (al, ... , an) E r, al :::; a2 :::;
a2--2al- an d t he f
'
T1D'
... <
_ an, eonSl'der ß = al +
2 a2 ' , = unetlon
rp: [0]
,,-+ ll\i.
glven by
rp(t) = E(ß - t, ß + t, a3, .. . , an). If the funetion rp is deereasing (inereasing) on [0,,] it
follows
that (1) is satisfied and in this ease one obtains the inequality (2).
If the funetion E : In -+ 1R. satisfies the hypothesis of Theorem 1.2 then the derivative
of function rp is given by rp'(t) = - 88E (u(t)) + 88E (u(t)) (:::;) 0 on [0, ,], where u(t) =
Xl
X2
~
(ß-t,ß+t,a3, ... ,an), eonsequently rp is deereasing (inereasing) on [0,,], and (1) is
verified.
Other results involving weighted-symmetric functions are given in the forthcoming
author's paper [3].
428
D. ANDRICA AND L. MARE
2. Applications
In this seetion the following standard notations will be used (see [4], [5]). For
1= (0,00), a = (al, ... ,an) E In let us eonsider
(arithmetie mean),
(geometrie mean) ,
(harmonie mean) ,
(mean of order 0: (0: > 0)).
Application 2.1. Let
be the kth symmetrie sum of al, ... ,an. It is easy to verify that for al ~ a2 ~
... ~ an the eondition (7) in Theorem 1.2 is satisfied. Therefore, Sk (ab· .. ,an) ~
Sk(An(a), ... ,An(a)), whieh is equivalent with the well-known MeLaurin' inequality
(8)
If k = n, (8) beeomes the arithmetie-geometrie means inequality Gn(a) ~ An(a).
Application 2.2. Consider
satisfying for al ~ a2 ~ ... ~ an the condition (7) in Theorem 1.2.
E(al, ... ,an) ~ E(An(a), ... ,An(a)), that is
Then
which represents the first part of W. Sierpinski' inequalities ([6], [5, pp. 21-25]):
(9)
Taking into account the following relations
AN INEQUALITY CONCERNING SYMMETRIC FUNCTIONS
429
one obtains that the first inequality in (9) is equivalent with the second one.
The first inequality in (9) is the best in the following sense:
(10)
Using (9) one obtains that a 2: n is a sufficient condition for (10). To show that
a 2: n is also necessary for (10) let us consider a = (l-c, 1 +c, 1, ... ,1), cE [0,1)
and it follows
n
.
( \h-c 2 - n - 2 + 2/
(1 - c 2 )
)'"<
Put t = 1 - c 2 and one obtains the equivalent inequality
t",jn <
n
/'
- n-2+2 t
t E (0,1].
Therefore, (n - 2)t",jn + 2t",jn-l ~ n. If a < n, then for t '\t 0 a contradiction
follows and consequently a 2: n.
Application 2.3. We shall use Theorem 1.1 to prove the inequality
(11)
which is a refinement of arithmetic-geometric means inequality since
(see [4, pp. 76-77]).
Consider E(al, ... ,an) = (ala2·· ·an )2 (ai + .. . a;,r· Suppose al ~ a2 ~ ... ~
al + a2
a2 - al
..
.
an and put ß =
2
,'Y =
2 . Followmg the Idea presented m Remark
1.3 let us consider the function 'P : [0, 'Y] -+ R, 'P(t) = E(ß - t, ß + t, a3,· .. , an).
An elementary computation shows that
and
'P' (t) = (a3 ... a n )2 4t (ß2 - t 2) (2t 2 + 2ß2 + a~ + ... + a~) x
x (-(n + 2)t 2 + (n - 2)ß2 - a~ - ... - a~) .
Because of 0 ~ t ~ 'Y < ß ~ a2 ~ a3 ~ ... ~ an one obtains 'P'(t) ~ 0 on [O,'Y]'
Le., 'P is decreasing on [0, 'Y]. Applying Remark 1.3 and Theorem 1.1 it follows
(11).
The inequality (11) is strongest in the following sense:
(12)
D. ANDRICA AND L. MARE
430
The sufficiency of condition 0: ~ 1 was proved above. For the necessity consider
al = 1 + x, a2 = 1 - x, a3 = a4 = ... = an = 1, where x E [0,1). Then
r J +
(\11- x 2 s:
2x 2n n' thus (1 - x 2)"'/n (2x 2 + n)I/2
s: Vn·
Let 1 : [0,1] -+ IR be the function given by I(t) = (1 - t)Ot/n(2t + n)1/2. Remark
that for every t E [0,1), I(t)
1(0) = Vn, Le., t = 0 is a maximum point of f.
On the other hand the derivative of 1 is
s:
f'(t) = -(1 - t)Ot/n-1 (2t + n)-1/2 (
If 0: < 1, then 0 <
1(- ~ j
1 + 20: n
[ 1-0:)
(2: + 1) +
t
0: - 1) .
< 1 and one obtains that 1 is strictly increasing on
[ 1-0:)
the interval 0, 1 + (20:)jn . Therefore, Vn = 1(0) < I(t), t E 0, 1 + (20:)jn '
a contradiction.
Application 2.4. For a given function 9 : I -+ IR let us denote
where al, ... , an EI.
Definition. The function 1 : I -+ IR is m-g-convex if for all al, a2 E I the following
inequality is verified:
(13)
l(aI) + l(a2) _ 1 (al + a 2 ) > . D(2)(
)
2
2
- m
9
al, a2 .
The function 1 : I -+ IR is M -g-concave if for all al, a2 E I the following relation
(14)
holds.
Let 1 : I -+ IR be a m-g-convex and M-g-concave continuous function on I, where
9 : I -+ IR is continuous and convex, M > m. Consider
n
m
E1(al, ... ,an) = L
I(ai) - m Lg(ai),
i=l
i=l
n
n
E 2(al, ... , an) = M L
g(ai) - L
i=l
i=l
I(ai).
AN INEQUALITY CONCERNING SYMMETRIC FUNCTIONS
431
It is clear that the functions Ei, E 2 : I -t IR are symmetrie, continuous and taking
into account (13), (14) it follows that Ei, E 2 satisfy the condition (1) in Theorem
1.1 with "~". From (2) one obtains
whieh represent refinements of the well-known Jensen' inequality.
An interesting situation studied in [1], [2] (see also [5, pp. 564-566]) is given by
the convex function 9 : I -t IR, 9 (t) = t 2 • In this case
and if I = [a, ß], then every function f E C 2 [a, ß] is m-g-convex and M -g-concave
on I, where
m = ~ min{f"(t) : tE [a, ß]}
and
M = ~ max{f"(t)
tE [a, ß]}.
The inequalities (15) becomes
(16)
>
m ~(a' _ a.)2
- 2L...J'
3'
n
i<j
whieh have many interesting applications (see [1] for instance).
References
1. D. Andrica and I. R~a, The Jensen inequality: refinements and applications, Anal. Numer.
TMor. Approx. 14 (1985), 105-108.
2. D. Andrica and M. O. Drimbe, On some inequalities in'llol'lling isotonic functionals, Anal.
Numer. TMor. Approx. 17 (1988), 1-7.
3. D. Andrica and L. Mare, An inequality conceming weighted-symmetric functions and applications, in preparation.
4. D. S. Mitrinovic, Analytic Inequalities, Springer Verlag, Berlin - Heidelberg - New York,
1970.
5. D. S. Mitrinovic, J. E. Pecaric and A. M. Fink, Classical and New Inequalities in Analysis,
Kluwer Academic Publishers, Dodrecht - Boston - London, 1993.
6. W. Sierpinski, Sur un inegaliU pour la moyenne arithmetiqe, geometrique et harmonique,
Warsh. Sitzungsber. 2 (1909), 354-357.
A NOTE ON THE SECOND LARGEST EIGENVALUE
OF STAR-LIKE TREES
FRANCIS K. BELL
Department 0/ Mathematics & Statistics, University 0/ Stirling, Stirling,
FK94LA, Scotland, United Kingdom
SLOBODAN K. SIMIC
Department 0/ Mathematics, Faculty 0/ Electrical Engineering,
University 0/ Belgmde, P. O. Box 35-54, 11120 Belgrade, Yugoslavia
Abstract. Star-like trees are trees homeomorphic to stars. In this paper we identify
those star-like trees for which the second largest eigenvalue is extremal- either minimal
or maximal - when certain conditions are imposed. We also obtain partial results on
the way in which the second largest eigenvalue of a simple dass of star-like trees changes
under local modifications (graph perturbations). Analogous problems for the largest
eigenvalue (known as the index of the graph) have been widely studied in the literature.
1. Introduction
Let A be the adjacency matrix of an undirected graph G, without loops or multiple
edges, and let cI>(G, A) = det(AI -A) be the characteristie polynomial ofG. Since A
is real and symmetrie its zeros are real, and they are referred to as the eigenvalues
of G. (See [2], abasie reference on graph spectra; other terminology follows [9].)
Let
AdG) ~ A2(G) ~ ... ~ An(G)
be the eigenvalues of G. Note that if G is connected then Al (G) is a simple
eigenvalue (i.e., an eigenvalue of multiplicity one), and it is called the index of G.
The second largest eigenvalue of a graph is an important graph invariant whieh
captures much information on graph structure, and it also has various applications
(see the recent survey paper by Cvetkovic and Simic [8]).
The main object of this paper is to identify those star-like trees (within a dass
with prescribed parameters) for whieh the second largest eigenvalue is extremal
(either minimal or maximal). In deducing these results we have addressed some
perturbation problems concerning the second largest eigenvalue. Similar problems
for the largest eigenvalue of more complex graphs have already been treated in the
literature (see, for example, [17], or the nicely presented survey paper by Cvetkovic
and Rowlinson [6]). In partieular, the solution of the index problem for star-like
trees is contained, along with other results, in [19].
1991 Mathematics Subject Classijication. Primary 05C50j Secondary 05C99.
Key words and phrases. Graph eigenvaluesj Second largest eigenvaluej Star-like treesj Graph
modifications.
433
G. V. Milovanovic (ed.J, Recent Progress in Inequalities, 433-443.
© 1998 Kluwer Academic Publishers.
434
F. K. BELL AND S. K. SIMIC
2. Preliminaries
We first give some recurrence relations for computing the characteristic polynomials of graphs.
Given any graph G, and a sub set U of the vertex set of G, let G - U denote the
graph obtained from G by deleting all vertices belonging to U. For convenience,
G - {u} is denoted by G - u, and G - { u, v} by G - u - v. The following theorem
is due to Heilbronner (see [2, p. 59]):
Theorem 2.1. I/G is a graph with a pendant edge uv, where v has degree 1, then
(2.1)
<T>(G, A) = A<T>(G - v, A) - <T>(G - u - v, A)
Remark. Formula (2.1) follows easily from one of the well-known formulas of Schwenk
[16], and will be sufIicient for our purposes in this note.
Let G and H be graphs with disjoint vertex sets, and let u and v be distinguished
vertices of G and H, respectively. The coalescence (or dot product) of these graphs,
denoted by G· H, is the graph obtained from G and H by identifying the vertices
u and v. The following result can be proved (see, for example, [2, p. 159] with an
obvious misprint):
Theorem 2.2. The characteristic polynomial 0/ the coalescence G· H is given by
(2.2)
<T>(G· H, A) = <T>(G - u, A)<T>(H, A) + <T>(G, A)<T>(H - v, A)
- A<T>(G - u, A)<T>(H - v, A).
An important ingredient in many situations is the classical interlacing theorem
(see, for example, [2, p. 19]):
Theorem 2.3. Given a subset U 0/ the vertex set 0/ a graph G, with IUI = k, the
/ollowing inequalities hold:
Ai+k(G) ::; Ai(G - U) ::; Ai(G)
(i=l, ... ,n-k).
Note that for k = 1 we have in particular:
A2(G - u) ::; A2(G) ::; AI(G - u).
We mention finally a result due to Smith (see [20], or [2, p. 79]) which plays an
important role in explaining some of the phenomena observed in Section 3.
Theorem 2.4. Let G be a graph with index Al. Then Al ::; 2 (Al < 2) i/ and
only i/ each component 0/ G is a subgraph (resp. a proper subgraph ) 0/ one 0/ the
graphs depicted in Figure 1, all 0/ which have index equal to 2.
() r-H--:: .. 1.....
... l. ..
A
+
FIG. 1: The Smith's graphs
THE SECOND LARGEST EIGENVALUE OF STAR-LIKE TREES
435
The above graphs will be referred to subsequently as Smith 's graphs.
3. A First Example
In this seetion we foeus attention on a very simple example. Let Pn,r be the graph
shown in Figure 2.
n+l
•1 •2
r
-
n
FIG. 2: The graph Pn,r
r
We may assume that r ::; nj21, and it will be eonvenient to make the further
assumption that n ~ 9. With the help of the programming paekage GRAPH (see
[3]), we were led to make the following eonjeeture:
Conjecture 3.1. Write >'2(r) = >'2(Pn ,r)' Ifn ~ 25 then
In proving this eonjeeture it will be eonvenient to denote r + 1 by r' . We start by
making the following observations:
(i): By Theorem 2.1, the eharaeteristic polynomial of Pn,r is given by
(3.1)
Reeall also that
(3.2)
(
<1> Pn ,
>. ) = {
Sin(n+l)tjSint
if 1>'1::;2, >'=2eost,
sinh(n+l)tjsinht if 1>'1~2, >'=2eosht.
From (3.1) and (3.2) we easily get
Thus >'2(Pn ,1) < >'2(Pn ,2) for n ~ 7. Notice also that, by the interlacing theorem
(Theorem 2.3), >'2(Pn,r) < 2 for eaeh r.
(ii): From (3.1) and (3.2), if r* > r, we easily obtain
in particular, setting r* = r' , we obtain
(3.3)
6.<1>(>') = <1> (Pn,rl , >') - <1> (Pn,r , >') = -<p(Pn - 2r - b >.).
F. K. BELL AND S. K. SIMIC
436
(iii): From the interlacing theorem, if v is any vertex of Pn,Tl then
max{(A2(Pn,r - v)} ~ A2(Pn,r) ~ min{(Al(Pn,r - v)} < 2.
v
v
In particular, taking v = n + 1, we obtain:
2COS(n
2: 1) ~ A2(Pn,r) ~ 2COS(n: 1)'
(iv): If n is sufficiently large, and 1 ~ r ~ 3, then A2(r) < A2(r').
This assertion was proved in (i) above in the case r = 1. For 1 < r ~ 3, let
ar = A2(Pn- 2r -t} (= 2cos(271"/(n-2r))), br = Al (Pn-2r-t) (= 2cos(71"/(n-2r))),
as suggested by (3.3). Note that a3 < a2 < b3 < b2 for all n > 8. If n is sufficiently
large then
<I>(Pn,rl,a r ) =<I>(Pn,r,ar ) > 0
(since lim <I> (Pn,r , ar ) = _r 2 + 3r + 2),
<I>(Pn,r"br ) =<I>(pn,r,br ) < 0
(since lim <I>(Pn,r,br ) = r 2 - 3r - 2).
n--+oo
n--+oo
Thus both <I> (Pn,rl ,A) and <I>(Pn,r, A) must vanish in the interval (ar, br ). Also,
<I>(Pn,r"A) - <I>(pn,r,A) > 0 in (ar,br).
Assurne first that r = 2 and n 2:: 13. Then, since A2 (Pn,2) = 2 cos(371" /2n), we have
A2(Pn,2) E (a2, b2). It follows that Pn ,3 has an eigenvalue in (a2, b2) greater than
A2(Pn,2). Since the largest eigenvalue of Pn,3 is greater than 2 (as can be seen, for
example, by considering Smith's graphs), we obtain that A2(Pn,2) < A2(Pn,3).
Now assurne that r = 3 and n 2:: 13. Note first that since <I> (Pn ,3 , 2) < 0 whenever
n > 8, if Pn ,3 were to have any eigenvalues in (b 3 , 2), there would have to be at least
two of them (possibly coincident), and it is easily checked that this would contradict the interlacing theorem. For A E (b 3 , 2), we have <I> (Pn ,4 , A) - <I> (Pn ,3 , A) < 0,
and therefore <I> (Pn,4 , A) < O. It follows that A2(Pn ,4) E (a3, b3), and hence, by the
same argument as before, that A2(Pn,3) < A2(Pn ,4).
(v): If n 2:: 18, and 4 ~ r ~ L(n - 2)/4J, then A2(r') < A2(r).
Let Cr = Al (Pn- 2r -t) (= 2cos(71"/(n - 2r))). If we now prove that
4> = <I>(Pn,rl,cr) = <I>(Pn,r,cr ) > 0,
we are done (since ß<I>(A) < 0 if A E (c r ,2». By (3.1) and (3.2), we get
4> =
. 2(
sm
(. ( 271" ) . (n+1)71")
1
71" ) sm n _ 2r sm n - 2r
-n - 2r
_ sin(~) sin(n - r + 1)71")).
n - 2r
n - 2r
It then folIows, after some simple trigonometry, that the sign of 4> depends on the
sign of the following expression:
sin (_r71" ) sin er + 1)71") (
----,--n-_2r--':----'--:-n_2r---,-sin (n ~71"2r)
1
sin (n ~71"2r)
_ cot(_r71" ) _ cot(r + 1)71")
n - 2r
n - 2r
)
.
THE SECOND LARGEST EIGENVALUE OF STAR-LIKE TREES
437
It is therefore sufficient to show that the function
1
fr(O) = --:--20 - cotrO - cot(r + 1)0
sm
is positive on the interval (0, tr /2(r + 1». This may be verified directly when r = 4
or 5. When r ~ 6 we can argue as follows. Since Icosxl :::; 1, and
~x
tr <
- sinx <
- x
we have that
tr(l
(x E [0,tr/2]),
1 1)
fr(O) ~ 20 ;: - ;: - r + 1 > O.
(vi): If r(n - 1)/41 :::; r :::; Ln/2J then A2(r') < A2(r) unless n is even and r = n/2,
in which case A2(r') = A2(r).
=
If r < n/2, let er
2cos(tr/(n - 2r)), as in (v) above. By (3.3), ~~(A) < 0
if A > Cr. By (iii) above, A2(Pn ,r) ~ 2cos(2tr/(n + 1», and the result follows,
because Cr :::; 2cos(2tr/(n + 1» for r ~ r(n -1)/41Remark. If n is not sufficiently large, Conjecture 3.1 must be modified: the maximum
value of A2(r) is obtained not for r = 4, but for r < 4. We give below some sampie
results obtained by the system GRAPH.
r\n
1
2
3
4
5
6
7
8
9
10
11
12
11
1.7709
1.8193
1.8142
1.7881
1.7531
1.7321
12
1.8019
1.8478
1.8478
1.8292
1.8019
1.7757
13
1.8271
1.8700
1.8733
1.8601
1.8392
1.8152
1.8019
22
1.9319
1.9543
1.9627
1.9623
1.9596
1.9558
1.9510
1.9452
1.9387
1.9319
1.9267
23
1.9372
1.9587
1.9664
1.9662
1.9640
1.9608
1.9557
1.9519
1.9463
1.9402
1.9345
1.9319
24
1.9419
1.9616
1.9696
1.9696
1.9678
1.9650
1.9616
1.9575
1.9527
1.9474
1.9419
1.9378
Note in particular that when n = 23 the maximum value of A2(r) is attained when
r = 3, whereas for n = 24, it is attained when r = 3 or r = 4 (now we can show
that the corresponding values from the above table are identical since being equal
to the largest root of the equation A6 - 6A4 + 9A - 3 = 0). However, we will now
demonstrate that for all n ~ 25 that the maximum value of A2(r) is attained when
r = 4. If r ~ 3, then at least one of Smith's graphs (from Figure 1) appears in
Pn,r as an induced subgraph, and this suggests applying the interlacing theorem
to the graphs Pn,r - 8 and Pn,r - (8 + 1), where 8 is chosen as follows:
438
F. K. BELL AND S. K. SIMIC
8,
s = { 7,
r + 2,
We obtain
A2(Pn,r) E [2COS (n
if r = 3,
if r = 4,
if r ~ 5.
=J,
2cos (n _: + 1)]
In the deduction of the upper bounds here, we need to ass urne that n ~ 29 when
r = 3, that n ~ 24 when r = 4, and that n ~ 3(r + 1) when r ~ 5. It follows that
[2 cos (1l' / (n - 8)), 2 cos (1l' / (n - 7))],
r = 3, n ~ 29,
{
r = 4, n ~ 24,
A2(Pn,r) E [2cos(1l'/(n -7»,2cos(1l'/(n - 6»],
[2 cos (1l' / (n - r - 2»,2 cos (1l' / (n - r - 1»], 5::; r ::; Ln/3J - 1.
We can verify directly that A2(Pn,3) ::; A2(Pn,4) whenever 24 ::; n ::; 28. We
know also from (v) above that A2(Pn,5) ::; A2(Pn ,4) whenever n ~ 24. From
(vi) we know that A2(Pn,r+l) ::; A2(Pn,r) whenever Ln/3J ::; r ::; Ln/2J. Finally,
notice that A2(Pn,2) ::; A2(Pn,r) for r ~ 3, because A2(Pn,2) = 2cos(31l'/2n) ::;
2cos(1l'/(n - 8» whenever n ~ 24.
Remark. Similar conjectures can be made for the other eigenvalues of Pn,r, e.g. that
for i:::; Ln/2J, Ai(Pn,r) has i peaks where the maximum is achieved.
4. Main Results
As suggested in the previous section, it is probably very difficult to trace how
the second largest eigenvalue of an arbitrary star-like tree behaves under local
modifications. We shall therefore confine ourselves in this section to finding those
trees from a given class for which the second largest eigenvalue is extremal.
Given nl ~ ... ~ nk, we shall henceforth denote by S(nl, ... , nk) the tree obtained
from the star K1,k (with k legs) by subdividing its i-th leg with ni -1 vertices. Let
n = nl + ... + nk + 1 and denote by Sn,k the set of all such trees with n vertices.
(The graph Pn,r in the previous section is therefore S(n - r, r - 1,1) E Sn+l,3')
In what follows, let SE Sn,k (k ~ 3), and let us denote by r the unique vertex of
S of degree k. By considering S - rand invoking the interlacing theorem, we get
i.e., all graphs from Sn,k are reflexive (see [13]). If nl = n2 then
(4.1)
It follows that, in cases where nl = n2, A2(S) does not depend on n3,'" , nk.
With more careful analysis we can get:
THE SECOND LARGEST EIGENVALUE OF STAR-LIKE TREES
439
Lemma 4.1. Given S = S(nl,'" ,nk) with nl = ... = nh > nh+1 ~ ... ~ nk,
then A2(S) = 2cos(rr/(nl + 1)) is 0/ multiplicity h - 1 i/ h ~ 2, and A2(S) <
2 cos(rr /(nl + 1)) i/ h = 1.
Proof. Write S = p. S*, where P = Pn1+l, S* = S(n2,'" , nk). By (2.2) we
easily get
Let A = Al(PnJ. It is enough to show that, for all h ~ 1,
cI>(S, A) = (A - A)h-l/(A)
for some polynomial f such that f(A) < O. This is true for h = 1 since
cI>(Pn1 - 1 , A) > 0
and
cI>(S* - T, A) =
k
II cI>(Pnn A) > O.
i=2
For h ~ 2, assurne (for an inductive proof) that cI>(S*, A) = (A - A)h-2 f* (A) for
some polynomial f* with f* (A) < O. The truth of the assertion for h then follows,
using (4.2), and this establishes the result. 0
In what follows, we assurne that, as in the lemma, S = S (nI, ... , nk), where nl =
... = nh > nh+1 ~ ... ~ nk (h ~ 1). As it would be very complicated to determine
the dependence of A2 (S) on all its parameters (the leg lengths nl, ... , nk), we shall
consider the effect of reducing by 1 the length of a longest leg (say the h-th leg)
and increasing by 1 the length of a shorter leg (say the j-th leg). To be precise, let
S' (= S(n~, ... ,n~)) be the graph obtained from S(= S(nl, ... ,nk)) such that
n~
=
ni -1,
{ ni + 1,
ni,
i = h,
i = j(= min{s I n s :::; nh - 2}),
i E {I, ... ,k} \ {h,j}.
(If no such j exists then there is no need to define S', as it would be isomorphie
to S whenever h < k.)
CASE 1: k ~ 5
We first note that A2(S') = A2(S) for h ~ 3. For h = 2, we have A2(S') < A2(S),
because, by Lemma 4.1, A2(S') < 2 cos (rr/(nl + 1)) = A2(S), So assurne h = 1 (Le.
nl ~ n2 + 1). Then observe first that A2(S) E [2cos(1l'/nl)' 2cos(rr/(nl + 1)). The
lower bound here is obtained by deleting the vertex adjacent to T whieh belongs
to the longest leg of S, and making use of the interlacing theorem and the fact
that k ~ 5; for the corresponding upper bound, see also Lemma 4.1. We similarly
have A2(S') E [2cos(rr/(nl -1)),2cos(rr/nl)), and it follows that A2(S') < A2(S)
unless A2(S') = A2(S) = A* (= 2cos(rr/nl)). The latter is not possible as can be
seen by using Theorem 2.2: we have
cI>(S', A) - cI>(S, A) = cI> (Pm, A)(cI>(H, A) - AcI>(H - T, A)),
F. K. BELL AND S. K. SIMI(~
440
where m = nl -nj-2 and H is the subgraph of G obtained by deleting the first and
the j-th leg. For A = A * we have ~(Pm, A *) "# 0 and ~(H, A *) - A *~(H - r, A *) <
O. To see why this last inequality holds, note that by one of the formulas of A.J.
Schwenk mentioned in Section 2, we have
~(H, A) - A~(H - r, A) = - 11 ~(Pn., A) . L ~~~;-\~).
8#I,j
t#l,j
n"
We therefore arrive at the following result:
Theorem 4.2. 11 k ~ 5 and SE Sn,k, then
P
k-p
k-l
A2(S(q+1,.~. ,q+i,~)):::; A2(S):::; A2(S(n-k,~),
(4.3)
where q = l (n - 1) / k J, p = n - qk - l.
Remar k. The graph S for which the lower bound in (4.3) is attained need not be unique.
For example, if k - p > 2 and q ~ 2, then
P
k-p
p~1
k-p-2
~...........-...
r
,,-"--..
A2(S(q+l, ... ,q+l,q, ... ,q))=A2(S(q+l, ... ,q+l,q, ... ,q,q-l).
In contrast, the upper bound is attained for a unique S - as given in (4.3).
CASE 2: k = 4
Inequalities (4.3) hold also for k = 4, but a different argument is required. To this
end, let S belong to Sn,4, and consider the subgraph S* of S obtained by deleting
all the vertices of the longest leg of S. Suppose first that S* is a supergraph of
one of Smith's graphs (i.e., the index of S* is at least 2). We then find, as before,
that
A2(S) E [2COS
(:1)' 2cos (nI: 1)]'
In this case it is easily seen that A2 (S) is a maximum if nl = n -7, and a minimum
if nl = (n - 1) /41. The other possibility is that S* is a proper subgraph of one
of Smith's graphs (Le., the index of S* is less than 2). We then find that, if nl is
sufficiently large,
r
A2 (S) E [2 cos (nI
~ 1)' 2 cos (:1 )] .
This time, A2(S) takes its maximum value if nl = n - 4. In order to find the
minimum value, note first that, by considering the possible proper subgraphs of
a Smith's graph, we must have n4 = 1 and n3 :::; 2. Moreover, if n3 = 2, then
n2 :::; 4; while if n3 = 1 then nl + n2 = n - 3, and consequently
n-3
-2- <
- nl <
- n-4.
This reasoning shows that (4.3) holds for k = 4, as claimed.
THE SECOND LARGEST EIGENVALUE OF STAR-LIKE TREES
441
CASE 3: k = 3
We shall see now that the right-hand inequality of (4.3) does not hold in this case.
Assurne first that nl = n2: we then have
minA2(S) = 2cos
(n -1)/3,
n=O (mod 3),
n=1 (mod 3),
(n + 1)/3,
n=2 (mod 3),
2)/2,
(D: 1)' where D= { (n(n -- 3)/2,
n=O (mod 2),
n=1 (mod 2).
(d: 1)' where d=
{ n/3,
and
maxA2(S) = 2cos
In order to deal with cases in which nl > n2, we consider two subranges for nl.
1° n/3 ~ nl < (n + 2)/2. For this range we have:
as can be seen by deleting the vertex adjacent to r in the longest leg (recall also
Lemma 4.1 for the rightmost point of interval).
2° nl ~ (n+2)/2. By deleting the vertex adjacent to r in the longest leg we obtain
A2 (S) ~ 2 cos (7r / nl). In order to get a tight lower bound we need to assurne either
that n3 ~ 3 or that n3 = 2 and n2 ~ 5. Then one of the connected subgraphs
obtained by deleting the vertex of the longest leg at distance 2 from r will be a
supergraph of a Smith's graph, and it will follow that A2(S) ~ 2cos(7r/(nl -1)).
(For n - 3 = 2 and n2 ~ 4, we have A2(S) ~ 2cos (ll-j(nl - 2)); for n3 = 1, see
Section 3.)
It is now possible, by analysis which is straightforward if somewhat tedious, to
obtain the graphs from Sn,3 with extremal values of A2, at least when n sufficiently
large. It turns out that the minimum value of A2(S) is achieved when the leg
lengths nl, n2, n3 are as equal as possible (Le. nl - n3 ~ 1). In contrast to the
situation when k ~ 4, the graph with maximum value of A2(S) is S(n - 5,2,2).
Note that the only candidates, after considering the intervals where the second
largest eigenvalue is located, are the graphs S(n - 5,3,1) and S(n - 5,2,2) and
the first of these can be eIiminated by the technique used in proving Theorem 4.2.
Remark. It is weIl known that in the set of all trees T with a prescribed number (;::: 3)
of vertices, the minimum value for A2(T) is attained by the star Kl,n-li and we have
A2(K1 ,n-d = o. According to [11], A2(T) ;::: 1 holds for all other trees T from this set.
On the other hand (see [11], and also [12]), for all trees T in the set,
It is shown in [14] that this bound is best possible, at least asymptotically (for large n).
The lower and upper bounds for the i-th eigenvalue of trees with a prescribed number of
vertices were studied in [1] and [10].
442
F. K. BELL AND S. K. SIMIC
5. Additional Remarks
We can gain some insight into the phenomena of Section 3 by setting these results
in the framework of graph perturbations (see [15] for more details). We shall need
the following formula (see [6]):
(1)
~(Gi,A) = ~(G,A) (A - f A~j
j=l
.),
/.LJ
where Gi is the graph obtained from G by attaching a pendant edge at the i-th
vertex, /.LI, • .. ,/.Lm are the distinct eigenvalues of G, and aij = IIPjeill. (Here Pj
is the projection matrix corresponding to /.Lj in the spectral decomposition of the
adjacency matrix of G.)
In [4], the authors made use of this formula in obtaining the estimate
(5.1)
for A1(r) (= A1(Gr)), and went on to give a partial explanation of some perturbation phenomena for unicyclic graphs. According to [19], Al (Pn,r) is unimodal in r
for fixed n, and this follows also from the above formula. To see this, it is enough
to note from [2] that, for the path Pn :
. ( ijrr
x·(i) -_ ~
--sm
- - ) (i,j = 1, ... ,n),
J
n+1
n+1
where XCi) = (xii), ... ,x~)) is the normalised eigenvector corresponding to the
eigenval~e /.Li. The result follows, since aji is equal to xji) to within sign. By
contrast, a formula similar to (5.1) for A2(r) (= A2(Gr)), such as
could not explain the behaviour of the second largest eigenvalue of the graphs Pn,r
(for fixed n).
References
1. J. Chen, Sharp bound 0/ the kth eigenvalue 0/ trees, Discrete Math. 128 (1994), 61-72.
2. D. Cvetkovic, M. Doob, and H. Sachs, Spectra 0/ Graphs - Theory and Application, Second
edition, 1982; Third edition, Johann Ambrosius Barth Verlag, 1995, Deutscher Verlag der
Wissenschaften - Academic Press, Berlin - New York, 1980.
3. D. Cvetkovic, L. Kraus, and S. Simic, Discussing graph theory with a computer I, Implementation 0/ graph theoretic algorithms, Univ. Beograd. Publ. Elektrotehn. Fak. Sero Mat.
Fiz. No 716 - No 734 (1981), 49-52.
4. D. Cvetkovic and P. Rowlinson, Spectra 0/ unicyclic graphs, Graphs Combin. 3 (1987), 7-23.
5. _ _ , Further properties 0/ graph angles, Scientia (Valparaiso) 1 (1988), 41-51.
6. _ _ , The largest eigenvalue 0/ a graph - a survey, Linear and Multilinear Algebra 28
(1993), 45-66.
THE SECOND LARGEST EIGENVALUE OF STAR-LIKE TREES
443
7. D. Cvetkovie and S. Simie, Graph theoretical results obtained by the support 0/ the expert
system "GRAPH", BuH. Aead. Serbe Sei. Arts, Cl. Sei. Math. Natur., Sei. Math., No. 19
101 (1994), 19-41.
8. _ _ _ , The second largest eigenvalue 0/ a graph (A survey), FILOMAT (Formerly: Zb.
Rad.) 9 (1995), 449-472.
9. F. Harary, Graph Theory, Addison Wesley, Reading, MA, 1969.
10. Y. Hong, The kth largest eigenvalue 0/ a tree, Linear Algebra Appl. 13 (1986), 151-155.
11. ___ , Sharp lower bounds on the eigenvalues 0/ a trees, Linear Algebra Appl. 113 (1989),
101-105.
12. A. Neumaier, The second largest eigenvalue 0/ a tree, Linear Algebra Appl. 46 (1982), 9-25.
13. A. Neumaier and J. J. Seidel, Discrete hyperbolic geometry, Combinatorica 3 (2) (1983),
219-237.
14. L. D. Powers, Bounds on graph eigenvalues, Linear Algebra Appl. 111 (1989), 1-6.
15. P. Rowlinson, Graph perturbations, Surveys in Combinatories (A. D. KeedweH, ed.), Cambridge University Press, Cambridge, 1991, pp. 187-219.
16. A. J. Schwenk, Computing the characteristic polynomial 0/ a graph, Graphs and Combinatories (R. Bari and F. Harary, eds.), Springer Verlag, Berlin - Heidelberg - New York, 1974,
pp. 153-172.
17. S. K. Simie, Some results on the largest eigenvalue 0/ a graph, Ars Combin. 24A (1987),
211-219.
18. ___ , On the largest eigenvalue 0/ unicyclic graphs, Publ. Inst. Math. (Beograd) 42 (56)
(1988), 13-19.
19. S. Simie and V. Koeie, On the largest eigenvalue 0/ some homeomorphic graphs, PubJ. Inst.
Math. (Beograd) 40 (54) (1986), 3-9.
20. J. H. Smith, Some properties 0/ the spectrum 0/ a graph, Structures and Their Applications
(R. Guy, H. Hanany, N. Sauer, J. Schönheim, eds.), Gordon and Breach, Seience Publ., Inc.,
New York - London - Paris, 1970, pp. 403-406.
REFINEMENTS OF OSTROWSKI'S AND
FAN-TODD'S INEQUALITIES
MOMCILO BJELICA
University 0/ Novi Sad, Technical Faculty "M. Pup in", 23000 Zrenjanin,
Yugoslavia
Abstract. An inequality of A. M. Ostrowski and an inequality of K. Fan and J. Todd
are refined.
A. M. Ostrowski ([2]) in 1951, proved the next result: Let a = (al, ... ,an) and
b = (b 1 , .•. ,bn ) be two sequences 0/ non proportional real numbers. Let x =
(Xl, ... ,Xn ) be arbitrary sequence 0/ real numbers such that
(1)
Let
n
n
Laixi = 0,
i=l
LbiXi = 1.
i=l
n
A = L ar,
i=l
B =
n
n
i=l
i=l
L br, C = L aibi .
Then
A
n
L
x7 ~ -A-B---C-2 '
i=1
(2)
with equality i/ and only i/
(3)
Theorem 1. Let real numbers Xi, 1 :::; i :::; n, satisfy (1), and let
(4)
Abi - Cai
Yi = AB _ C2'
1:::; i :::; n .
Then the numbers aXi + (1 - a)Yi, 1 :::; i :::; n, satisfy (1) and
(5)
1991 Mathematics Subject Classijication. Primary 26D15.
Key words and phrases. Inequalities; Refinements of inequalities.
445
G.v. Milovanovic (ed.), Recent Progress in Inequalities, 445-448.
© 1998 Kluwer Academic Publishers.
M. BJELICA
446
n
n
= 1 or L: x~ = L: yr·
i=l
i=l
The second inequality in (5) becomes equality if and only if 0: = 0 or Xi = Yi,
The first inequality in (5) becomes equality if and only if 10:1
1 ~ i ~ n.
Proof. First note that
Therefore,
n
n
n
L [x~ - y;] = L [x~ - 2XiYi + y;] = L(Xi - Yi)2 ~ 0,
i=l
i=l
i=l
what proves (2).
The first inequality in (5)
n
n
LX~ ~ L
[0:2x~ + 20:(1 - O:)XiYi + (1- 0:)2 y;] ,
i=l
i=l
(1 - 0: 2 )
L X~ ~ (20: - 20: + 1 - 20: + 0: L yr,
Le.,
n
n
2
2)
i=l
i=l
is equivalent to (2).
The second inequality in (5) follows from
n
n
L [O:Xi + (1 - 0:)Yi]2 = L [0:2(Xi - Yi)2 + 20: (XiYi - yr) + Y;]
i=l
i=l
= 0:2 t(Xi - Yi)2 + AB ~ C 2
i=l
A
~AB-C2' 0
Let X = Rn,
n
~
fex) = L.J Xi
i=l
2
and
x+y
F(x) = -2- .
The monotonicity condition
fex) ~ f(F(x))
is equivalent to (2),
REFINEMENTS OF OSTROWSKI'S AND FAN-TODD'S INEQUALITIES
447
Le.,
n
3L
.
.=1
Ln Ab· - Ca· Ln (Ab' - Ca·)2
X.'
, +
•
•
• -.
'AB - C2
. (AB _ C2)2
x~ > 2
.=1
2
= AB _ C2
.=1
n
n
i=l
i=l
(L Abixi - L CaiXi)
- AB-C2'
Thus, we obtain sequence of successive approximations
x, F(x), ... ,Fn(x), ... ,
converging to y, which interpolate (2),
n
L
A
x~ = f(x) ~ f(F(x)) ~ ... ~ f(Fn(x)) ~ ... ~ f(y) = AB _ C2 .
i=l
The "opposite" sequence
x,
Xl
= X + (X - y), ... , X n = X n-1 + (X n-1 - y) , ...
can be used in proving (2). If we suppose that
(2)
n
A
LX~ ~ AB-C2'
i=l
then
f(x) ~ f(xd ~ ... ~ f(x n ) ~ ... ,
what is a contradiction to lim f(x n ) = +00.
n-too
K. Fan and J. Todd ([2]) in 1955, proved the next inequality: Let a = (ab' .. ,an)
and b = (bI, . .. ,bn ), n ~ 2, be two sequences 0/ real numbers such that aibj f:. ajbi
for all i f:. j. Then
M. BJELICA
448
Theorem 2. Let a = (al, ... , an) and b = (bi, ... , bn ), n ;::: 2, be two sequences
of real numbers such that aibj =f. ajbi for all i =f. j. If 101 ~ 1, then
A
-,---...,<
AB-C2
Proof. H we take
1~i~n,
the proof is similar to the proof of Theorem 1.
o
References
1. M. Bjelica, Fixed Point and Inequalities, Ph. D. Thesis, University of Belgrade, Belgrade,
1990.
2. D. S. Mitrinovic (in cooperation with P. M. Vasic), Analytic Inequalities, Springer Verlag,
Berlin - Heidelberg - New York, 1970.
3. Z. M. Mitrovic, On a generalization 01 Fan-Todd's inequality, Univ. Beograd Pub!. Elektrotehn. Fak. Sero Mat. Fiz. No. 412 - No. 460 (1973), 151-154.
ON THE STABILITY OF THE QUADRATIC
FUNCTIONAL EQUATION AND RELATED TOPICS
STEFAN CZERWIK
Institute of Mathematics, Silesian University of Technology, Gliwice, Poland
Abstract. In this paper we consider the problem of the stability of a quadratic equation
in some abstract space in the sense of Hyers-Ulam-Rassias. This is a generalisation of an
idea originally stated by S. M. Ulam for the linear (Cauchy) equation. Similar problems
for other equations are actually investigated by several mathematicians.
1. Introduction
The problem of the stability of functional equations has been posed by S. M. Ulam.
Answering that question, Hyers [1] has proved the following result:
Theorem 1. Given two Banach spaces (X, 11 . 11), (Y, 11 • 11) and a real positive
number c, assume that a function f : X ~ Y satisfies the inequality
IIf(x + y) - f(x) - f(y) 11 ~ c
for all x, y EX. Then there exists exactly one additive mapping A : X ~ Y such
that
IIf(x) - A(x) 11 ~ c
holds for all x EX.
From that time many other related questions have been studied. Professor Rassias
[9] gave a generalised solution to Ulam's problem for so called approximately linear
mappings and in [10] he generalised the Ulam-Hyers stability theorem considering
the most general Ulam-Hyers sequence (see also [11] and nice book devoted to that
subject [12]).
Also problems of stability of polynomial functions, homogeneous functions, convex
functions have been considered by many authors (see e.g. [1-3], [6]).
Let G B , s = 1,2 be groups. A function f : Gi ~ G2 fulfilling the following
equation
(1)
f(x + y) + f(x - y) = 2f(x) + 2f(y)
for all x, y E Gi is called the quadratic function and (1) the quadratic functional
equation.
The problem of the Hyers-Ulam stability of the quadratic equation (1) has been
studied in [2] and more general case, which I call Rassias type of stability, by the
author in [4-5].
1991 Mathematics Subject ClasBification. Primary 39C05.
Key words and phrases. Functional equationsj StabilitYj Hyers-Ulam-Rassias sequencesj Quadratic mappings.
449
G.v. Milovanovic (ed.), Recent Progress in lnequalities, 449-455.
© 1998 Kluwer Academic Publishers.
s. CZERWIK
450
2. Stability of Quadratic Functions
1. Let ll4 be the set of all nonnegative real numbers. Assurne that X is a
commutative semigroup with zero in which the following low of cancellation holds
(2)
a + b = b + c implies a = b for all a, b, c E X.
In X is defined a multiplication by nonnegative real scalars satisfying
(3)
(4)
(5)
(6)
a( a + b) = aa + ab,
aa+ ßa = (a + ß)a,
a(ßa) = (aß)a,
la = a,
for all a, bE X and a,ß E ll4.
Moreover, let (X, D) be a metric space such that
(7)
(8)
d(x + y, x + z) = d(y, z) for all x, y, z E X,
d(tx, ty) = d(x, y) for all x, y E X, tE ll4.
In X we define
(9)
IIxll := d(x, 0),
x E X.
A commutative semigroup with zero and metric d satisfying the conditions (2)-(8)
we call a quasi-normed space.
To construct an example of such space, let us consider a normed space Y. Given
sets A, BeY and a number t E IR (the set of all real numbers) we define
A + B := {x E Y : x = a + b, a E A, bEB}
and
tA := {x E Y : x = ta, a E A}.
Let CC(Y) denote the space of all non-empty compact convex subsets of Y. Put
d(A,B) :=inf{t>O: ACB+tK, BCA+tK},
where K is the closed unit ball in Y and A, B are non-empty closed bounded
subsets of Y. The function dis a metric called the Hausdorff metric induced by
the metric of the space Y.
The space CC(Y) with Hausdorff metric is a quasi-normed space. Moreover, if Y
is a Banach space, then CC(Y) is a complete metric space.
2. Let E be a group and let h : E x E -+ ll4 be a given function. We denote
H(x, y) := h(x, y) + h(x, 0) + h(y, 0) + h(O, 0),
K(x, y) := 2h(x, y) + h(x + y, 0) + h(x - y, 0)
for all x,y E E.
Now we shall prove the following
ON THE STABILITY OF THE QUADRATIC FUNCTIONAL EQUATION
451
Lemma 1. Let E be a group and let (E1 , d) be a quasi-normed space. If the
function F : E --t E 1 satisfies the inequality
(10)
d[F(x + y) + F(x - y), G(x) + G(y)] ~ hex, y) for alt x, y E E,
then we have
(11)
(12)
d[F(x + y) + F(x - y) + 2F(0), 2F(x) + 2F(y)] ~ H(x, y),
d[G(x + y) + G(x - y) + 2G(0), 2G(x) + 2G(y)] ~ K(x, y),
for alt x, y E E.
Proof. We get for x, y E E,
d[G(x + y) + G(x - y) + 2G(0), 2G(x) + 2G(y)]
+ y) + G(x - y) + G(O), 2F(x + y) + G(x - y)]
+ d[2F(x + y) + G(x - y) + G(O), 2F(x + y) + 2F(x - y)]
+ d[2F(x + y) + 2F(x - y), 2G(x) + 2G(y)]
~ hex + y,O) + hex - y,O) + 2h(x,y) = K(x,y),
~ d[G(x
Le., the inequality (12). The inequality (11) can be proved in the same way.
Let N denote the set of all natural numbers.
0
Lemma 2. Let E be a group and E 1 be a quasi-normed space. If G, F : E --t E 1
satisfy the inequality (10), then
k-l n-l
(13)
~ k 2 (n-l) L
L(k - m)H(mk 8 x, k 8 x)k- 28 ,
m=18=0
k-l n-l
(14)
~ k 2 (n-l) L
L(k - m)K(mk 8 x,k 8 x)k- 28 ,
m=18=0
for all xE E and n,k E N, where k 2: 2.
Proof. The proof follows by induction on n.
Now we can establish the following
0
Theorem 2. Let E be an Abelian group and let E 1 be a complete quasi-normed
space. Assume that inequality (10) is satisfied. Let for so me integer k 2: 2 and
m = 1, . .. , k - 1 the series
00
(15)
L
8=0
h(mk 8 x, k 8 x)k- 28 ,
S. CZERWIK
452
00
(16)
h(k 8x, 0)k- 28 ,
L
8=0
be convergent tor all X E E. 1/, moreover,
liminf h(knx, k n y)k- 2n = 0 tor all x, y E E,
(17)
n-4OO
then there exists exactly one quadratic /unction A : E -+ E l such that
k-l
(18)
d[A(x) + F(O), F(x)] :s k- 2 L
00
L(k - m)H(mk 8, k 8x)k- 28 ,
m=l8=0
k-l
(19)
d[2A(x) + G(O), G(x)] :s k- 2 L
00
L(k - m)K(mk 8, k 8x)k- 28 ,
m=l8=0
tor all xE E.
Proof. We define
(20)
We shall prove that {An(x)} is a Cauchy sequence for every X E E. In fact, by
Lemma 2 and (8) we have for n > rand X E E
d[An(x),Ar(x)] = k- 2n d[F(k n x), k 2(n-r) F(Fx)]
:s k- 2r IIF(0)11 + k- 2n d[F(k n x) + [k 2(n-r) - l]F(O), k 2(n-r) F(F x)]
k-l n-r-l
:s k- 2r IIF(0)11 + k- 2 L
L
m=l
8=0
k-l
00
:s k- 2r IIF(0)11 + k- 2 L
(k - m)H(mk 8+r , k 8+r x)k- 2(8+r)8
L(k - m)H(mk 8,k 8x)k- 28 .
m=l8=r
Hence we get the conclusion. Therefore, there exists the limit
(21)
A(x):= lim An(x)
n-4OO
for all xE E.
Now we shall check that ({k- 2n G(k n x)} is a Cauchy sequence)
(22)
2A(x) = lim k- 2n G(k n x), xE E.
n-4OO
Indeed, we have by (10) and (9)
d[2A(x), k- 2n C(k n x)] :s d[2A(x), 2· k- 2n F(knx)]
+ d[2. k- 2n F(knx), k- 2n G(k n x) + k- 2n C(0)]
+ d[k- 2n G(k n x) + k- 2n C(0), k- 2n C(k n x)]
:s 2d[A(x), An(x)] + k- 2n h(k n x, 0) + k- 2n IlG(0)1I·
ON THE STABILITY OF THE QUADRATIC FUNCTIONAL EQUATION
453
Hence applying (17) and (21) it follows that
lim d[2A(x), k- 2n C(k n x)]
for all xE E,
n--+oo
i.e., we obtain (22).
The function A is quadratic. In fact,
Letting n -t 00 in view of (16) and (22) we obtain the equality
A(x + y) + A(x - y) = 2A(x) + 2A(y)
for all x, y E E.
The estimations (18) and (19) one can establish directly from (13) and (14), respectively. Now we shall prove that the function A is unique. To this end let
assume that there exist two quadratic functions Ci : E -t E 1 , i = 1,2, such that
k-1
d[Ci(x) + F(O), F(x)] :s; k-2ai L
00
L(k - m)H(mk S , k 8x)k- 28 ,
m=18=0
for x E E and i = 1,2, where ai 2: 0, i = 1,2, are real constants.
It is a simple exercise to verify that for i = 1,2
xE E, nE N.
Now we get for x E E
k-1
:s; (al + a2)k- 2 L
00
L(k - m)H(mk s +r , k 8+n x)k- 2(8+n)
m=18=0
k-1
= (al + a2)k- 2 L
00
L(k - m)H(mk S, k 8x)k- 28 .
m=18=n
In view of the convergence of the series (15), the right hand side of the last
inequality can be made as small as we wish taking n sufficiently large. Hence
Cdx) = C 2(x) for all x E E and the proof of the theorem is completed. 0
Theorem 3. Let E be an Abelian group divisible by k E N, k 2: 2 and let E 1 be
a complete quasi-normed space. Let for m = 1, ... ,k - 1 the series
L h(mk-Sx, k- 8x)k 28 ,
00
(23)
8=1
454
S. CZERWIK
(24)
Lh(k- Sx,0)k 2S ,
s=1
00
be convergent for all xE E. Moreover, if
liminf h(k-nx, k- n y)k 2n = 0
n-+oo
for all x, y E E,
and F(O) = 0, then there exists exactly one quadratic function B : E --t EI such
that
k-l
00
L(k - m)H(mk-Sx, k- Sx)k 2s ,
m=1 s=1
d[B(x),F(x)] ~ k- 2 L
k-l
00
L(k - m)K(mk-Sx, k- Sx)k 2s
m=1 s=1
d[B(x),G(x)] ~ k- 2 L
for all xE E.
In the next part we present a result concerning the case F(O) #- O. By IR we denote
the set of the real numbers.
Theorem 4. Let E be an Abelian group divisible by k E N, k ~ 2 and let EI be
a Banach space. Let the functions F, G : E --t EI satisfy the inequality
IJF(x + y) + F(x - y) - G(x) - G(y)1I ~ h(x, y)
for all x, y E E.
Suppose that the senes (23) and (24) are convergent /or all x E E and the condition
(25) is satisfied. Then there exists exactly one quadratic function 9 : E --t EI such
that
k-l
(27)
IIg(x) + F(O) - F(x)1I ~ k- 2 L
00
L(k - m)H(mk-Sx, k- Sx)k 2s
m=1 s=1
and
k-l
(28)
112g(x) + G(O) - G(x)11 ~ k- 2 L
00
L(k - m)K(mk-Sx, k- Sx)k 28
m=18=1
for all xE E.
Moreover, if E is a linear topological space and F is measurable (i.e., F- 1(U) is
a Borel set in E for every open set U E Ed or the function R :3 t --t F(tx) is
continuous for every fixed x E E, then
g(tx) = t 2g(x),
(29)
x E E, tE R
Proof. Denote
/(x) := F(x) - /(0), q(x):= G(x) - G(O)
for x E E.
ON THE STABILITY OF THE QUADRATIC FUNCTIONAL EQUATION
455
Then by Lemma 1 and (26) we have
Ilf(x + y) + f(x - y) - 2f(x) - 2f(y)11 ~ H(x, y)
for all x, y E E. Hence (see Theorem 2)
g(x):= lim k 2n f(k- nx ),
n-+oo
x E E,
is correctly defined quadratic function satisfying the conditions (27) and (28). The
proof of the uniqueness follows the argument as for the proof of Theorem 2.
To end the proof let L be any continuous linear functional defined on the space
EI. Let cp : IR ~ IR be given by
cp(t) := L[g(tx)],
x E E, tE IR,
where x is fixed. Then cp is a quadratic function and, moreover, as the pointwise
limit of the sequence
CPn(t) = k 2n L[f(k- ntx)],
tE IR,
is also measurable and hence has the form cp(t) = t 2cp(1) for t E llt Therefore for
every t E IR and every x E E
L[g(tx)] = cp(t) = t 2cp(1) = L[t2g(x)],
which implies the condition (29). This completes the proof.
0
Remark. If h = const, we have Ulam-Hyers type of stability, whereas for h(x, y) =
Ilxll v + lIyllV, x, y E E (E-normed space) Rassias type of stability.
References
1. M. Albert and G. A. Baker, Ftmctions with bounded n-th diJJerence, Ann. Polon. Math.43
(1983), 93-103.
2. P. W. Cholewa, Remarks on the stability 0/ /unctional equations, Aequationes Math. 27
(1984), 76-86.
3. S. Czerwik, On the stability 0/ the homogeneous mapping, C. R. Math. Rep. Aead. Sei.
Canada 14 (1992), 268-272.
4. _ _ , On the stability 0/ the quadratic mapping in normed spaces, Abh. Math. Sem. Univ.
Hamburg 62 (1992), 59-64.
5. _ _ , The stability 0/ the quadratic junctional equation, Stability of Mappings of HyersUlam Type (Th. M. Rassias and J. Tabor, eds.), Hadronic Press Colleetion of Original
Articles, Hadronic Press, Ine., Palm Harbor, FL, 1994, pp. 81-91.
6. G. L. Forti, The stability 0/ homomorphisms and amenability with applications to /unctional
equations, Abh. Math. Sem. Univ. Hamburg 57 (1987), 215-226.
7. R. Ger, Almost approximate/y convex junctions, Math. Slovaca 38 (1988), 61-78.
8. D. H. Hyers, On the stability 0/ the linear junctional equation, Proe. Nat. Aead. Sei. U.S.A.
27 (1941), 222-224.
9. Th. M. Rassias, On the stability 0/ the linear mapping in Banach spaces, Proe. Amer. Math.
Soe. 72 (1978), 297-300.
10. _ _ , On a modified Hyers-Ulam sequence, J. Math. Anal. Appl. 158 (1991), 106-113.
11. Th. M. Rassias and P. Semrl, On the behaviour 0/ mappings which do not satisfy Hyers-Ulam
stability, Proe. Amer. Math. Soe. 114 (1992), 989-993.
12. Th. M. Rassias and J. Tabor (eds.), Stability 0/ Mappings 0/ Hyers-Ulam Type, Hadronic
Press Colleetion of Original Articles, Hadronic Press, Ine., Palm Harbor, FL, 1994.
13. J. Ratz, On approximate/y additive mappings, General Inequalities 2, ISNM 47, Birkhauser
Verlag, Basel, 1980, pp. 233-251.
A DIRICHLET-TYPE INTEGRAL INEQUALITY
W. N. EVERITT
School of Mathematics and Statistics, University of Birmingham, Edgbaston,
Birmingham BiS 2TT, England, UK
Abstract. This note concerns the derivation of an integral inequality associated with a
Sturm-Liouville differential expression. The inequality results from the Dirichlet formulae
for the differential exression, and the lower bound of the self-adjoint differential operator
determined by the Neuman boundary condition at the regular end-point.
1. Introduction
In this note we discuss some properties of the integral inequality
(f E D),
(1.1)
where the domain D ~ L 2 (0, 1) and the number 0: is non-negative.
The maximal domain for which the inequality can be considered is defined by
(1.2)
D := {J: (0,1]--+ lR I fE AC1oc(0, 1]; fand
xi' E L 2 (0, I)}.
However in applications this definition may be replaced by the smaller domain
(1.3)
We consider the inequality (1.1) on the maximal domain (1.2) and thereby give
some explanation for the introduction of the class of functions AC1oc(0, 1].
The non-negative number 0: in the inequality (1.1) is taken to be best possible,
i.e., the largest number for which the inequality, on the chosen domain, is valid.
Of course the inequality is always valid when 0: = and thus interest is in the
possibility when 0: > 0. With 0: so determined there remains the problem of
characterising all the cases of equality in (1.1), other than the null function, if any.
°
1991 Mathematics 8ubject Classification. Primary 26DI0, 34B24j Secondary 34L05.
Key W07US and phrases. Integral inequalitYj Sturm-Liouvillej Differential operator.
457
G.v. Milovanovic (ed.), Recent Progress in Inequalities, 457-463.
© 1998 Kluwer Academic Publishers.
w. N. EVERITT
458
2. Methods
The theory of differential operators allows us to prove the existence of inequalities of the form (1.1), Le., to decide if the number 0: is positive or zero, and to
characterise all the possible cases of equality.
The method is to consider differential operators generated in the Hilbert function
space L2(0, 1) by the Lagrange symmetrie differential expression M, where
(x E (0,1]),
(2.1)
with the domain of M defined by
(2.2)
D(M) := {J: (0,1]--+ C
I I, I' E AC1oc(0, In·
The spectral properties of these operators are associated with the solutions of the
linear differential equation
(2.3)
M[y] = >.y on (0,1],
where>. is the spectral parameter, Le., >. E C.
The general theory of these operators is developed in the classie text by Naimark
[5]; see in partieular Chapters V and VI.
3. A General Integral Inequality
The analysis of the inequality (1.1) depends upon a general result of Amos and
Everitt given in [1]; see in particular [1, Thm. 2]. We show below that all the
conditions of this theorem are satisfied by the differential expression M of (2.1)
as defined on the interval (0,1]; here a = and b = 1, with a as the singular
end-point and b as the regular end-point. This application requires the reversal of
the roles played by the end-points a and b in the results quoted from [1], but the
theorem is equally valid in this case.
With the inequality domain D defined as in (1.2), compare with [1, Eq. (2.5)], let
the domain D(T) of the self-adjoint operator T be defined by, compare with [1,
Eq. (2.6)],
°
(3.1)
D(T) = {I: (0,1]--+ C
I I, I' E ACloc(O, 1], 1'(1) = 0
and I, M[/] E L 2 (O, I)}
and let the operator T be defined by
(3.2)
TI:= M[/]
(J E D(T)).
Suppose now that the following conditions two are satisfied for the differential
expression M, as given by (2.1):
(i) the Diriehlet condition
(3.3)
D(T) cD
A DIRICHLET-TYPE INTEGRAL INEQUALITY
459
is satisfied, compare with [1, Eq. (2.9)]j we note that this condition implies that
the differential expression M is strong limit-point at the singular end-point 0, see
[1, p. 243], i.e.,
(3.4)
lim Ig' =
:1:-+0+
°
(J,g E D(T))j
this in turn implies that the differential operator T, as give in (3.2), is self adjoint
in the Hilbert function space L 2 (0, 1), see [5, Chap. V]j
(ii) the self-adjoint operator T, as defined by (3.2), is bounded below in the space
L 2 (0, 1), i.e., there exists areal number J.L such that (here ( . ,. ) denotes the inner
product for L 2 (0, 1))
(3.5)
(T I, f) ~ J.L(J, f)
(J E D(T))j
this in turn implies that the spectrum a(T) ~ IR of the operator T is also bounded
below, on the realline IR, by the same number J.Lj in (3.5) we suppose that J.L is the
best possible, i.e., the largest, number for which the inequality holdsj since a(T)
is a closed set of the real line this terminology gives J.L E a(T).
We show below that the domains D, D(T) and the operator T satisfy both the
conditions (i) and (ii).
These results imply that an application of [1, Thm. 2] may be made and so we
obtain
(3.6)
(J E D).
This yields the required inequality (1.1) with the number a = J.L.
The additional results of [1, Thm. 2] give information on the cases of equality
in (3.6)j if J.L is an eigenvalue of the operator T then all the non-trivial cases of
equality are prescribed by the eigenspace of J.Lj if J.L is not an eigenvalue but is in
the continuous spectrum of T then there are no non-trivial cases of equality, i.e.,
the only case is the null function on [0,1].
Thus the analysis of the original inequality depends upon obtaining information
about the operator T and its spectrum a(T).
4. Properties of the Operator T
Lemma 4.1. The differential expression M, 0/ (2.1), is strong limit-point at 0+,
see (3.4), and the operator T is Dirichlet, see (3.3), on (0,1].
Proof. It is sufficient to prove these results for the case when 1 is real-valued on
(0,1].
On integration by parts, with 1 E D(T),
(4.1)
11{t2J'(t)2+t2/(t)2}dt=t2/(t)J'(t)
I: -1
+1
= -x2/(x)J'(x)
1
{(t 2J'(t))'-ef(t)2}dt
1
M[/](x)/(x) dx.
w. N. EVERITT
460
If now T is not Dirichlet on (0,1] then there exists 1 E D(T) such that the lefthand side of (4.1) tends to +00 as x tends to 0+; in turn this result implies that
-x21(x)f'(x) also tends to +00. Thus for all x sufficiently near to 0+ there exists
a positive number K such that I(x)f'(x) < -Kx- 2; if we integrate this result
over (x, 1] then ~ l(x)2 - ~ 1(1)2 > Kx- 1 - K, again for all x near to 0+. This
last result is inconsistent with 1 E L 2 (0, 1), and this yields the Dirichlet property.
If now for real I,g E D(T), then a similar integration by parts gives the result,
compare with (4.1), gives
(4.2)
1
1
{t 2f'(t)g'(t) + t 2I(t)g(t) }dt = -x21(x)g'(x) +
1
1
M[g] (t)f(t) dt.
The left-hand side of (4.2) tends to a finite limit as x -+ 0+, from the Dirichlet
property, as does the integral on the right-hand side; thus for some real number
k we have limx 2 f(x)g'(x) = k as x tends to 0+. If k f:. 0 suppose that k > 0;
then for all x near to 0+ we have I(x)xg'(x) ~ ~ X-i, but this is inconsistent with
I, xg' E L 2 (0, 1); there is a similar argument if k < O. Thus k = 0 and the strong
limit-point result (3.4) is seen to hold. 0
From these results we can now establish the identity
(JED(T));
(4.3)
this shows that the operator T is bounded below in L 2 (0, 1), and that the exact
lower bound {l of (3.5) satisfies
(4.4)
{l ~ O.
5. Spectral Properties of the Operator T
To establish that the original inequality (1.1) holds on the domain D with a number
0: that is positive we have to establish that the lower bound {l of the spectrum
u(T) is positive. This result is achieved by applying the Liouville transformation
to the differential equation (2.3); details of this transformation may be found in
[3, Chap. X, Sect. 9] and in [4, Sect. 12].
In the case oft he differential equation (2.3), and in the notation of [4], we transform
the equation (2.3) by the introduction ofnew independent and dependent variables,
respectively, as follows
(5.1)
X(x) =
1- =
11
'" t
dt
-log(x)
(x E (0,1]) and Y(X) = X1/ 2y(X).
It may be verified that the original equation is transformed into
(5.2)
-Y"(X) + (~+exP(-2X))Y(X) = AY(X)
(x E [0,00));
A DIRICHLET-TYPE INTEGRAL INEQUALlTY
461
also, and this is of significance for the cases of equality in (3.6), the separated
boundary condition y'(l) = 0, invoked in the definition of D(T) in (3.1), i.e.,
/'(1) = 0, is transformed into
Y(O) + 2Y'(0) = O.
(5.3)
The significance of the boundary value problem represented by the equation (5.2)
and the boundary condition (5.3) is to be seen in the following results:
(i) the equation (5.2) is regular at the end-point 0, and is limit-point in
L 2 (0,00) at the end-point +00;
(ii) the operator S: D(S) C L2(0,00) -t L2(0,00) defined by
D(S) := {F: [0,00) -t elF, F' E ACloc[O,OO), F(O) + 2F'(0) = 0,
FE L 2 (0, 00), -F" + (1/4 + exp (-2X))F E L 2 (0,
oo)}
and
(5.4)
SF:= -F" + U +exp(-2X))F
(F E D(S))
is self-adjoint in L 2 (0, 00);
(iii) the operators T in L 2 (0, 1) and S in L 2 (0, 00) are unitarily equivalent;
(iv) a(T) = a(S);
(v) the spectrum of S is purely continuous on (1/4, 00) and is discrete below the
point 1/4, i.e., there are only eigenvalues below 1/4 and the only possible
limit point, if any, is at 1/4;
(vi) there is a smallest discrete eigenvalue, say >'0, and >'0 ~ O.
As references for these results:
(i) see ([5, Sect. 17.5] and [6, Sect. 2.20];
(ii) see [5, Sect. 24.2];
(iii) see [4];
(iv) see [4];
(v) see [5, Sect. 24.2] and [6, Sect. 5.7];
Additional analysis shows that:
(vi) the spectrum of S is purely continuous on [1/4,00);
(vii) there is only one eigenvalue of S in the interval [0,1/4], i.e., >'0, and this
eigenvalue is simple, i.e., the eigenspace is one-dimensional and generated
by the single, real-valued eigenfunction 'l/Jo, say, which is the unique solution
of the differential equation, see (2.3), M[y] = >'oy on (0,1] with y E D(T).
To establish these results it is necessary to study explicitly the form of the spectrum
a(S); this can be done along the lines of the example considered in [6, Sect. 4.14].
w. N. EVERITT
462
6. The eigenvalue .xo
We have
Lemma 6.1. The eigenvalue AO satisfies
0< AO :5 1/4.
(6.1)
Proof. From (4.3) we obtain, with 1 = t/Jo,
(6.2)
1{x2t/J~(X)2 +
1
x2t/JO(x)2} dx = (Tt/Jo, t/Jo) = AO
1
1
t/JO(X)2 dx.
Prom this result it is clear that if AO = 0 then t/Jo would be null on (0,1] and this
gives a contradiction.
The upper bound of 1/4 for AO follows from the results in the previous section. 0
7. The Inequality
We may now state
Theorem 7.1. For the inequality (1.1) let the domain D be defined by (1.2); then
the inequality is valid with the best-possible number a = AO > 0, i. e.
(J E D).
(7.1)
All cases 01 equality are determined by
(7.2)
I(x) = At/Jo(x)
(x E (0,1])
where A E IR.
Proof. The proof follows from the general equality given in [1, Thm. 2]. All the
conditions required to apply this theorem have now been established or referenced
in the preceding sections. 0
8. The Domain of the Inequality
The maximal domain D of the inequality, see (1.2), is determined essentially by
the coefficients ofthe differential expression M (see (2.1», which is singular at the
end-point 0+ and regular at the end-point 1. A singular end-point, in inequalities
of this form, often allows the introduction of elements into the domain of the
inequality that have singular behaviour themselves at such a point. We can see
the result in this particular examplej let 1 E D be defined by
I(x) := (x 1/ 2 Iog(x»-1
(x E (0,1/2])
and by construction let 1 E C(2)[1/2, 1]. It may be seen by calculation that 1 E D
and yet 1 has a singularity at the end-point 0+. The domain D o, for applications,
excludes such singular elements from the inequality.
A DIRICHLET-TYPE INTEGRAL INEQUALITY
463
9. N umerical Results
A numerical value for AO can be found from the computer program Sleign2; see [2];
an approximate value is AO ~ 0.243. The program also indicates that the extremal
nmction 1/;0 is likely to have no zeros in the interval (0,1].
Acknowledgement. The author is grateful to Professor Vivian Hutson, University of
Sheffield, England, UK for posing the problem considered in this paper. The integral
inequality arises from a problem in mathematical biology concerning the determination
of stability for a reaction-diffusion system with non-linear diffusion.
References
1. R. J. Amos and W. N. Everitt, On a quadratie integral inequality, Proe. Royal Soc. Edinburgh 78A (1978), 241-256.
2. P. B. Bailey, W. N. Everitt and A. Zettl, The Sleign2 Computer Program for the Automatie
Computation of EigenIJalues, [To be entered into the Public Domain in September 1995].
3. G. Birkhoff and G.-C. Rota, Ordinary Differential Equations, Ginn and Company, Boston,
1960.
4. W. N. Everitt, On the transformation theory of ordinary second-order linear symmetrie
differential equations, Czechoslovak Math. J. 32 (107) (1982), 275-306.
5. M. A. Naimark, Linear Differential Operators, 11, Ungar Publishing Company, New York,
1968.
6. E. C. Titchmarsh, Eigenfunction Expansions, I, Oxford University Press, 1962.
ON THE HYERS-ULAM-RASSIAS STABILITY
OF MAPPINGS
P. GA.VRUTA.
Department of Mathematics, Technical University, P- ta Hora tiu Nr. 1,
1900 - Timi soara, Romania
Abstract. We give an answer to a question of Hyers and Rassias [5] concerning the
stability of mappings.
For a survey about the stability of mappings see [5]. In this note we denote by
(G, +) an Abelian group and by (X, II'I!) a Banach space.
In [1] we obtain the following general theorem concerning the stability of mappings.
Theorem 1. Let be <p : G x G -+ [0, +(0) so that
(1)
1
L 2 + <p(2 x, 2 y) <
00
cll(x, y) :=
n
1
n
n
00
n=O
for all x, y E G and f : G -+ X a mapping so that
(2)
Ilf(x + y) - f(x) - f(y)11 ~ <p(x, y)
for all x, y E G.
Then there exists an unique additive mapping T : G --t X so that
(3)
Ilf(x) - T(x)11 ~ cll(x, x)
for all xE G.
Moreover,
(4)
T(x) = lim f(2 n x) ,
n-*oo
2n
x EG.
A generalisation of this theorem was given in [2] (see also [3]).
1991 Mathematics Subject Classijication. Primary 39B72, 41A35j Secondary 47H19.
Key words and phrases. Stability of mappingsj Additive mappingj Banach space.
465
G.v. Milovanovic (ed.), Recent Progress in Inequalities, 465-469.
© 1998 Kluwer Academic Publishers.
P.GAVRUTA
466
Theorem 2. Let be k ::::: 2 integer and <p : G x G -t [0,00) so that
(5)
~k(X,y):=
1
L k n+ <p(knx,kny) < 00
00
1
n=O
lor all x,y E G.
11 1 : G -t X is a mapping so that (2) holds, then there exist an unique additive
mapping Tk : G -t X so that
(6)
xE G,
where
k-l
(7)
'l1k(X,y):=
L ~dx,my),
x,y E G.
m=l
Moreover,
(8)
xE G.
Recently, Jung [6] obtain independently this result when <p is asymmetrie function.
In the following, we consider a particular case:
(i) G = E a normed space with the norm 11 . 111;
(ii) <p(x, y) = H(lIxlll, Ilylll) for x, y E E,
where H : [0,00) x [0,00) -t [0, 00) is a homogeneous function of degree P E [0, 1).
Then we have
hence
It follows
H(l,m) 11 x II P1
~k (x, mx ) = k _ kp
hence
(9)
x EE,
where
(10)
1
k-l
8k(H) = k _ kp
H(l,m).
L
m=l
ON THE HYERS-ULAM-RASSIAS STABILITY OF MAPPINGS
467
We consider now a mapping f : E -+ X so that
IIf(x + y) - fex) - f(y)1I ~ H(llxlh, Ilylll)
for all x, y E E. From Theorem 2 it follows that for every k ~ 2 integer there is
an unique additive mapping Tk : E -+ X so that
x EE.
We take here x t-+ 2n x and obtain
xEE.
For n -+ 00 it follows
xEE,
hence
xEE,
k ~ 3.
Thus we have the following result:
TheoreID 3. Let be p E [0, 1) and H : [0,00) x [0,00) -+ [0,00) a homogeneous
junction 01 degree p. 11 f : E -+ X is a mapping so that
(11)
IIf(x + y) - fex) - f(y) 11 ~ H(lIxlh, lIylll)'
X,y E E,
then there is a unique additive mapping T : E -+ X so that
Ilf(x) - T(x) 11 ~ o(H)lIxlli,
xE E,
where
(12)
For the particular case
8> 0;
we prove [2] that
ok(Ho) > o2(Ho),
s,t E [0,00),
for
k> 2,
and in [4] we prove that o2(Ho) is sharp.
The problem is if we can to have H so that
We take
Hds,t) = min(sP,tP),
° <
<p
1;
s,t E [0,00).
P. GAVRUTA
468
In this case
and we shall prove that
(13)
k ~ 2.
The inequality (13) it is equivalent with
1 + (k - l)(k + l)P < k· kP.
(14)
Since the function h : [0,00) -+ IR., h(x) = x P is strictly concave we have
h(AX + (1 - A)Y) > Ah(x) + (1 - A)h(y)
for A E (0,1) and x,y E (0,00), x =I y.
We take here
x = 1,
and obtain
k- 1
( -1 + --(k
+ 1)
k
k
y=k+1
)P > -1 + --(k
k - 1
+ 1)p
k
k
that is (14).
Thus (13) holds and hence the generalisation in Theorem 2 is not trivial.
In this case in Theorem 3 we have 8 = 1 and we prove that this 8 is sharp.
We take f : IR -+ IR
f(x) =
{
xP
_( -x)P
if
if
x ~ 0,
x< 0.
We prove that
If(x + y) - f(x) - f(y)1 ~ min(lxl P, lylP)
(15)
for all x, y E III
If x, y ~
°
it follows
°
If x ~ 0, y < we have two cases.
10 x + y ~ 0. Then
where u = -y, v = x + y. It follows
If(x + y) - f(x) - f(y)1 ~ u P = lylP
ON THE HYERS-ULAM-RASSIAS STABILITY OF MAPPINGS
and
469
If(x + y) - f(x) - f(y)1 $ v P $ Xp.
2° x + Y < 0. Then
where Q: = -x - y. Hence
If(x + y) - f(x) - f(y)1 $ x P
and
If x < 0, Y <
If(x + y) - f(x) - f(y)1 $ Q:P $ (-y)P = Iylp.
°the proof is dear.
In this case T(x) = 0, If(x)1 = IxIP, x E Ilt
Thus, using Theorem 3, we have the following result.
Theorem 4. Let p E [0, 1) and f : E -t X so that
IIf(x + y) - f(x) - f(y)II $ min(IIxllf, IIylln
tor all x, y E E.
Then there exists a unique additive mapping T : E -t X so that
IIf(x) - T(x) 11 $ IIxllf,
xEE,
and this inequality is optimal.
Remark. This result give an answer to the last question of Hyers and Rassias [5].
References
1. P. Gävrutä, A generalization 01 the Hyers-Ulam-Rassias stability 01 approximate/y additive
mappings, J. Math. Anal. Appl. 184 (1994), 431-436.
2. P. Gävrutä, M. Hossu, D. Popescu, C. Cäpräu, On the stability 01 mappings, Bull. Appl.
Math. Techn. Univ. Budapest 83 (1994), 169-176.
3. P. Gävrutä, M. Hossu, D. Popescu, C. Cäpräu, On the stability 01 mappings and an answer
to a problem 01 Th. M. Rassias, Annales Math. Blaise Pascal (1995), 55-60.
4. P. Gävrutä, On the approximately linear mapping, (submitted).
5. D. H. Hyers and Th. M. Rassias, Approximate homomorphism, Aequationes Math. 44
(1992), 125-153.
6. S. M. Jung, On the Hyers-Ulam-Rassias stability 01 approximately additive mappings, J.
Math. Anal. Appl. 204 (1996), 221-226.
FUNCTIONS WITH QUASICONVEX DERIVATIVES
VIDAN GOVEDARICA and MILAN JOVANOVIC
Faculty 0/ Electrical Engineering, Patre 5, 78000 Banjaluka, Bosnia and Hercegovina
Abstract. The necessary and suflicient conditions for quasiconvexity are given for the
derivative of real-valued function, defined and continuously differentiable on I = [a, b) C
llt Also, some inequalities are presented in this paper.
1. Introduction
It is well-known that functions with monotonie derivatives are convex or concave.
The continuously differentiable functions with convex derivatives have been studied
in [6]. We will consider a more general dass, Le., functions with quasieonvex
derivatives.
Recall that 9 : I --t IR is quasieonvex if and only if for all x, y E I and t E (0,1)
g(tx + (1 - t)y) ~ max{g(x) , g(y)}.
If gis quasieonvex, then -g is quasieoncave. A function 9 is said to be quasimonotonie if it is both quasieonvex and quasiconcave [3]. Clearly, if 9 is convex, it is
quasieonvex, but not conversely.
We use the following result:
Theorem 1 ([3], [5]). The continuous function 9 is quasiconvex on [a, b] if and
only if there exists a point c E [a, b] such that 9 is nonincreasing on [a, c] and
nondecreasing on [c, b].
2. Conditions for Quasiconvexity
Theorem 2. Let f : [a, b] --t IR be a continuously difJerentiable function. The
following three conditions are equivalent:
(a) f' is quasiconvex;
(b) There exists a point c E [a, b] such that f is concave on [a, cl, and convex
on [c, b];
(c) For all x, y E [a, b] the inequality
1991 Mathematics Subject Classification. Primary 26A51, 26D1Oj Secondary 90C26.
Key woms and phrases. Quasiconvex functionsj Inequalities.
471
G. V. Milovanovic (ed.), Recent Progress in Inequalities, 471-473.
© 1998 Kluwer Academic Publishers.
472
V. GOVEDARICA AND M. JOVANOVIC
(1)
f(x) - f(y) ~ max{!,(x),!,(y)}
x-y
holds.
Proof. From Theorem 1 we immediately obtain (a) <===}(b). Now let!, be quasiconvex. By the Mean-Value Theorem we have
f(x) - f(y) = !'(() ~ max{f'(x), !,(y)},
x-y
since () E (x, y). Thus (a) ~ (c). Conversely, suppose that the condition (c)
holds, but
f'(Zo) > max{!,(x),!,(y)} = m,
for some x, y E [a, b], x < y, and Zo E (x, y). The continuity of f' implies that
there are Zl,Z2 E [x,y] such that Zo E (Zl,Z2), !,(zt) = !'(Z2) = m, f'(z) > m for
all z E (Zl,Z2). On the other hand, for some () E (Zl,Z2) we have
f'(()
= f(Z2) - f(zt} ~ m,
Z2 - Zl
which is a contradiction. This proves (c) ~ (a).
0
Corollary 1. Let 9 be continuous on I. Then 9 is quasiconvex if and only if, for
every x, y E I,
(2)
-1-1
y- x
z
Y g(t) dt ~ max{g(x), g(y)}.
Remark 1. When 9 is a convex function, the inequality (2) is a direct consequence of
the famous Hadamard's inequality (see [4]).
Remark 2. If 9 is a continuous quasiconvex function, the inequality
gC;Y) +min{g(t)ltE [x,Y]}:5 y:xlY g(t)dt,
x<y,
holds, but it does not define the quasiconvexity (see [2]).
Remark 3. The corresponding characterisation of quasiconvex functions in several variables can be given by (2). This is based on the fact that a function is quasiconvex on
a convex set C ~ !Rn if and only if its restrietion to each line segment in the set C is
quasiconvex.
3. Some Inequalities
Example 1. Let f' be quasimonotone, then by (1)
min{f'(x), f'(y)} :5 f(x) - f(y) ~ max{!,(x), f'(y)},
x-y
so that, e.g. for f(x) = log x, x ~ 1, y = 1, we have
x-I
- - ~logx~x-l.
x
FUNCTIONS WITH QUASICONVEX DERIVATIVES
473
Example 2. f(x) = sinx has a quasiconvex derivative on [0,211"], hence
sinx - siny
----.:::... ~ max{cosx,cosy}.
x-y
Moreover, on [0,11"] and on [11",211"], l' is quasiconcave so that
.
sinx - siny
mm{cosx,cosy} ~
~ max{cosx,cosy}.
x-y
Example 3. Let f(x) = xne- x , n 2:: 2, x 2:: 0, and let 0< h ~ n, then f(n+h) >
f(n - h) [1]. On the other hand, by the quasiconvexity of l' on [n - n 1/ 2 ,+oo),
and of (log f)' on lR.t- we get
2h2 )
f(n + h) ~ ( 1 + n _ h f(n - h),
and
f(n + h) ~ e 2h2 /(n-h) f(n - h),
Vn< h < n.
We can also obtain, in a different way,
f(n + h) ~ e h2 /(n-h) f(n - h),
0 ~ h < n.
Example 4. In [4, pp. 362], we can find the inequality
f(x + 1) - 2f(x) + f(x -1) > !,,(x),
where x E (a + 1, b - 1), b - a > 2 and 1'" is increasing on [a, b]. Clearly, I" is
convex. If f" is quasiconvex continuous, we get
O· f"(x - 1) < 0 f"(x + 1) < 0
{ '
f(x + 1) - 2f(x) + f(x - 1) <
- ,
- ,
2max{!"(x-1),I"(x+1)}; otherwise.
References
1. G. Klambauer, On a property 01 xne- x , Amer. Math. Monthly 95 (1988), 551.
2. M. Longinetti, An inequality lor quasi-convex lunctions, Appl. Anal. 13 (1982), 93-96.
3. B. Martos, Nonlinear Programming, Theory and Methods, Akademiai Kiado, Budapest,
1976.
4. D. S. Mitrinovic (in cooperation with P. M. Vasic) , Analytic Inequalities, Springer Verlag,
Berlin - Heidelberg - New York, 1970.
5. V. A. Ubhaya, Quasi-convex optimization, J. Math. Anal. Appl. 116 (1986), 439-449.
6. A. 1. Vorob'eva and A. M. Rubinov, On functions with convex derivatives, Prim. Funkc.
Anal. Teor. Pribl. (1990), 33-38. (Russian)
ON THE LOCAL APPROXIMATION BY
QUASI-POLYNOMIALS
YU. KRYAKIN
Odessa State University, 2 Petra Velikogo, 270000 Odessa, Ukraine
Abstract. A new proof of the multidimensional analogue of Whitney theorem is given.
Some new estimates of Whitney constants are also obtained.
1. Introduction
Let f(x) be a measurable function on 1[0,1]. Define
wn(f) =
sup
x,x+nhEI
IßU(x)l.
In 1957 H. Whitney proved the following classical result in approximation theory.
Theorem A. For any continuous function f(x) on land for any integer n ~ 1,
there exist a polynomial P n - 1 (x) of degree at most n - 1 and a positive constant
W n, such that
sup If(x) - P n- 1 (x)1 :::; Wnwn(f).
xEI
In [20] much attention had been given to the estimations of the constants Wn .
Following Bl. Sendov we will call these constants as Whitney's constants. The
evaluation of Whitney's constants is an exceedingly difficult unsolved problem.
There is no conjecture about value W n . Therefore, finding estimations of W n for
some concrete methods of approximation is of great importance. In the proof of
his theorem, Whitney used an interpolation polynomial over the uniform mesh:
P(i/(n -1)) = f(i/(n - 1)), i = 0,1, ... ,n - 1. His proof was rather complicated
and did not allow one to control the growth of the constants as n tends to infinity.
However, he obtained some lower estimates and gave upper and lower bounds for
the interpolation constants W~ and for the constants of best approximation W n
for small values of n.
Whitney established the inequalities
1 :::; W~,
1/2:::; W n :::; W~ < 00,
and gave the following estimates:
1991 Mathematics Subject Classification. Primary 41AI0j Secondary 41A17, 41A44.
Key words and phrases. Interpolation polynomialsj Whitney's constantsj Quasi-polynomials.
475
G.v. Milovanovic (ed.), Recent Progress in lnequalities, 475-480.
© 1998 Kluwer Academic Publishers.
YU. KRYAKIN
476
n
1
2
W n 1/2 1/2
W~
1
1
4
3
8/15 7/10
16/15 14/9
1/2
1
5
3.25 1/2
3.25 1
10.4
10.4
A single value denotes a sharp constant, a pair - the upper and lower bounds.
Further progress in obtaining good estimations of Whitney's constants is connected to works of BI. Sendov. Sendov proposed a numerical method based on
linear programming for estimating Whitney's constants. Based on this work, he
conjectured that the constants are bounded: W n by one and W~ by two (see [13]).
At the time, Sendov's conjecture seemed rather bold - the existing estimates of
the constants for large n were very pessimistic (Wn < cn 2n , Brudnyi [2], 1964).
In 1985 the earlier known estimates were lowered substantially. In the span of
a single month there were published the papers of Ivanov and Takev [5] (Wn ::;
c· n In n), Binev [1] (Wn ::; C· n) and Sendov [14] (Wn ::; const). In 1986 Sendov
proved [15] that Wn ::; 6. In these works the interpolating polynomials over a
mesh: i / (n + 1), i = 1, ... , n, were used. The further development of this theme:
W n ::; 3, independently Sendov [16], Brudnyi [4], Kryakin [6], W n ::; 2, Kryakin
[9]. Sendov and Brudnyi did not publish their proofs. We used polynomials that
are "interpolating in the mean". They are defined by the conditions
tin (f(x) - Qn-l(X)) dx = 0, i = 1, ... ,n.
Jo
For n = 1,2 these polynomials were used by Storozenko [18] for the proof of
Whitney's theorem in LP (p ~ 1) classes. The use of these polynomials permits
a simple proof and allows to obtain some new results. New estimations of Whitney's constants in integral metric were given in [7], [11]. It was proved [12] that
interpolating constants W~ do not exceed 5. One of the peculiarities of the use of
polynomials interpolating in the mean is the relative simplicity of obtaining lower
bounds. In contrast to interpolation constants, for example, the constants in the
case of approximation in mean are bounded below by one and this estimate is
sharp. In addition the exact estimates of constants for small values of n have been
obtained [8], [10]:
Thus, the strongest result now is the following:
Theorem B. For any continuous function f(x) on land each integer n ~ 1, we
have
with
- {2',
Wn <
-
1
if n ~ 5,
if n ::; 4.
In this paper we give the multi dimensional generalisation of Theorem B. At first the
multidimensional generalisations of the theorem of H. Whitney were obtained by
LOCAL APPROXIMATION BY QUASI-POLYNOMIALS
477
Brudnyi [3] for classes LP, p::::: 1, and C. Sendov and Takev ([17], [19]) proved that
the constants for approximation by quasi-polynomials do not depend on degree
of quasi-polynomials. In this paper we improve the result of Sendov-Takev and
obtain the sharp theorem if the degree of the quasi-polynomial does not exceed 4.
In addition we prove a new estimate of Whitney's constants in integral metric.
2. Notations and Main Result
Let IRd denote the d-dimensional Euclidean space, and II d be the unit cube in IRd .
The elements of IRd will be denoted by x = (Xl, ... ,Xd), Y = (YI, ... ,Yd), ... and
we will write X ~ Y if the corresponding inequalities hold for Xi and Yi; Also, let
X· Y be the vector {Xi· Yi}.
Put
Pn(x) =
nl
nd
i=O
;=0
2: h,i(XI) X~ + ... + 2: fd,i(Xd) X~,
where fk,i(Xj) are continuous functions in the variables Xl, ... ,Xj-l, Xj+1, . .. ,Xd,
and
~hf(x) = ~~~el ••• ~~~eJ(x),
where ei is the i-th coordinate vector.
Denote by Qn-l quasi-polynomial which is determined by the conditions
(1)
l
(j+I)/n i
(J(x) - Qn-I(X)) dx; = 0,
j/ni
i = 1, ... ,d, j = 0,1, ... ,n; - 1.
A constructive definition of Qn-l was given in [19-20]. We will discuss it below.
Theorem 1. For any f E C (lId) the following inequality
sup lJ(x) - Qn-l(x)1 ~ Wn(d)
xEll d
sup
x,x+nhEll d
l~hf(x)1
holds, with
if n::::: 5,
if n ~ 4.
Theorem 1 improves the results of Brudnyi (Wn(d) ~ n 2nd ), Sendov and Takev
(Wn (2) ~ 49), Takev (Wn(d) ~ d! 6d). It is easy to show that the constants in
Theorem 1 are sharp for n ~ 4.
3. Constructions of the Polynomials
Let D i be an operator of differentiation in the direction of ei. Denote by Di l the
inverse operator of D i ,
YU. KRYAKIN
478
and by Li the operator of interpolation
n,
Li! = "Lf(Xi,jfni) lj(ni,xi),
j=O
where Ij (ni, x) is the Lagrangian algebraic polynomial of degree ni
lj(ni, kjni) = Ok,j,
k,j = 0,1, ... ,ni'
Put Ci = DiLiD;l and observe that
CiCj = CjCi.
It is easy to see, that for an integrable function g,
l
<i+1)/n,
(g - Cig)dxi = 0,
j/ni
j
= 0,1, ... ,ni-I.
Denote
C(nl) = Cl,
C(nl, n2) = Cl + C2 - Cl C2,
C(nl, n2, n3) = Cl + C2 + C3 - ClC2 - ClC3 - C2C3 + ClC2C3,
i = 1, ... ,d.
Put Qn-l = C(n)f and verify that conditions (1) are true
4. Proof of Theorem 1
We shall prove Theorem 1 by iterations of the one-dimensional result. Suppose
that the theorem is valid in jRd-l. Putting 9 = f - Qn-l and taking into account
that ~U(x) = ~~g(x) we have
s~p Ig(x)1 = 9 (x~, x~) ~ W nd I~~~ed 9 (x~, Xd) I·
Using our assumption we get
S?P I~~~edg (Xd, Xd) 1 = S?P IG (Xd, Xd) I
Xd
Xd
~ Wn1 ... Wnd_1 I~~rel
... ~~t: ed-l G (Xd' Xd)
=Wn1 '" Wnd_ll~~·f(x")I·
1
LOCAL APPROXIMATION BY QUASI-POLYNOMIALS
479
5. Estimates in L p
Theorem 2. For f E LP (lid), P 2': 1, the inequality
holds, where
d
pd=II~I,
i=1
n·t
IId(h, n) =
d
II (1 - nihi) I.
i=1
For d = 1, Theorem 2 was proved by Sendov [13-14]. The prooffor d 2': 2 is similar
to the proof in C (lid). Here, we consider the case d = 2. Put 9 = f - Qm-l,n-l
and use the one-dimensional result. Then, we obtain
11
o Ig(x,y)IPdx ~ 10m
11/m l
0
dh
0
1 - mh
16.h,og(x,y)IPdx
and
References
1. P. Binev, O(n) bounds of Whitney constants, C.R. Aead. Bulgare Sei. 38 (1985),1315-1317.
2. Ju. A. Brudnyi, On a theorem of local best approximations, Kazan. Gos. Univ. Ueen. Zap.
124 (1964), 43-49. (Russian)
3. _ _ _ , Approximation of junctions of n variables by quasi-polynomials, Izv. Akad. Nauk
SSSR Sero Mat. 34 (1970), 564-583. (Russian)
4. _ _ , The Whitney Constants, Baku, 1989, pp. 24. (Russian)
5. K. Ivanov and M. Takev, O(nln(n)) bounds of constants 0/ H. Whitney, C. R. Aead. Bulgare
Sei. 38 (1985), 1129-1131.
6. Yu. V. Kryakin, Whitney constants, Mat. Zametki 46 (1989), 155-157. (Russian)
7. _ _ , On the theorem of H. Whitney in spaces L p , 1 ~ p ~ 00, Math. Balkaniea (N.S.) 4
(1990), 258-271.
8. _ _ _ , Exact constants in the Whitney theorem, Mat. Zametki 54 (1993), 34-51. (Russian)
9. _ _ , On a theorem and constants of Whitney, Mat. Sb. (N.S.) 185 (1994), 24-40 (Russian) [English trans!. Russian Aead. Sei. Sb. Math. 81 (1995), 281-295].
10. _ _ , On the local approximation by polynomials and quasi-polynomials, Dok!. RAN (to
appear). (Russian)
11. Yu. V. Kryakin and L. G. Kovalenko, Whitney constants in the classes L p , 1 ~ p ~ 00,
Izv. Vyssh. Uehebn. Zaved. Mat. 1992,69-77 (Russian) [English trans!. in Soviet Math. (Iz.
VUZ) 36 (1992)].
12. Yu. V. Kryakin and M. D. Takev, Whitney interpolation constants, Ukrain. Mat. Zh. 47
(1995), 1038-1043. (Russian)
13. B!. Sendov, On the constants of H. Whitney, C. R. Aead. Bulgare Sei. 35 (1982), 431-434.
480
YU. KRYAKIN
14. ___ , The constants 01 H. Whitney are bounded, C. R. Aead. Bulgare Sei. 38 (1985),
1299-1303.
15. ___ , On the theorem and constants 01 H. Whitney, Constr. Approx. 3 (1987), 1-11.
16. BI. Sendov and V. Popov, The Average Moduli 01 Smoothness, John Wiley & Sons, Ine.,
New York, 1988.
17. BI. Sendov and M. D. Takev, A theorem 01 Whitney's type in ]R2, PLISKA Stud. Math.
Bulgar. 11 (1991), 78-85.
18. E. A. Storozenko, Approximation ollunctions by splines that are interpolation al in the mean,
Izv. Vyssh. Uehebn. Zaved. Mat. 1976 no. 12 (175), 82-95. (Russian)
19. M. D. Takev, A theorem 01 Whitney type in ]Rn, Construetive Theory of Funetions (Varna,
1987), Bulgar. Aead. Sei., Sofia, 1988, pp. 441-447.
20. H. Whitney, On lunction with bounded n-th differences, J. Math. Pures AppI. 36 (1957),
67-95.
LOGARITHMIC CONCAVITY OF DISTRIBUTION
FUNCTIONS
MILAN MERKLE
Faculty 01 Electrical Engineering, University 01 Belgrade, P. O. Box 95-54,
11120 Belgrade, Yugoslavia
Abstract. We give suflicient conditions for a probability distribution function to be logarithmically concave. The limiting behaviour of corresponding inequalities is discussed.
1. Introduction
The importance of the concept of logarithmic convexity or concavity is weH known
(see [5], [3] or [1]). The analogy between log-convexity and log-concavity is not
so simple as the one between plain convexity and concavity. For example, although the sum of log-convex functions is log-convex, a parallel statement for
log-concave functions does not hold. Further, log-convexity is astronger property
than convexity, whereas log-concavity is weaker than concavity. A typical continuous probability distribution function is neither convex nor concave on (-00, +00).
However, it turns out that many continuous distribution functions that are in
frequent use in Probability theory are log-concave. A result similar to ours is obtained in [3], but under more restrictive conditions and with an involved proof.
Using the relation between log-concavity and Schur-concavity [3], we can produce
some interesting inequalities. For example, if Xl, ... ,Xn are independent random
variables, identically distributed with a log-concave distribution function F, then
_
Xl
+ ... + X n
x= -=------=-:.
(1)
n
Some related results, regarding Schur-convexity with respect to parameters of
distributions are obtained in [4] and in references therein.
2. Main Results
J:
Theorem 1. Let F(x) =
f(t) dt, where f is an integrable, positive and twice
differentiable function on (a, b), lor b > a. Assume the lollowing:
(i) f is monotone on (a, b) or it has a unique maximum in (a, b).
(ii) 11 there is c E (a,b] such that f is increasing on (a,c), then fis log-concave
on (a,c).
1991 Mathematics Subject Classification. Primary 26A51, 60E15.
Key woms und phrases. ConvexitYi Schur-convexitYi Logarithmic convexity and concavitYi Distribution functions.
481
G.v. Milovanovic (ed.), Recent Progress in Inequalities, 481-484.
© 1998 Kluwer Academic Publishers.
M. MERKLE
482
Then F is log-concave on (a,b).
Proof. Let x E (a,b) and suppose that f'(x) :::; O. Then
F(x)F"(x) - F I2 (X) :::; F(x)F"(x) = F(x)f'(x) :::; O.
(2)
If f'(x) > 0 for some x, then by the assumptions, there exists an interval (a, c),
a < c :::; b such that f'(x) > 0 for all xE (a, c) and f is log-concave on (a, c). Now,
the inequality F F II - P 2 < 0 is equivalent to
P(x)
F(x) :::; jt(x) = R(x).
For x = a we have F(a) = 0 and lim R(x) ;::: O. Further, FI(x) = f(x) and for
",-+a+
a < x:::; c,
RI(x) = 2f(x)f'(x)2 - p(x)r(x) = f(x) + f(x) f'(x)2 - f(x)r(x) > f(x)
jt(x)2
jt(x)2
by log-concavity of f. From these facts it follows that F(x) :::; R(x) for x E (a, cl.
Therefore, we proved that (2) holds for every x E (a, b) and F is a log-concave
function. 0
Under conditions of Theorem 1, Jensen's inequality applied on [x, x + h) c (a, b)
yields
(3)
F(x + Ah) ;::: F1-A(x)FA(x + h) ,
0:::; A :::; 1.
The next theorem is related to the sharpness of (3).
Theorem 2. Let pt be continuous on (a, +00). If conditions of Theorem 1 are
satisfied for b = +00 then for every h > 0 and A E [0,1]
(4)
[im (F(x + Ah) - F1-A(x)FA(x + h)) = O.
"'-++00
Proof. Let us remark that if conditions of Theorem 1 hold on (a, +00) then
lim f(x) = o. If we denote u(x) = log F(x) then
"'-++00
o
II() _ F(x)F"(x) - p2(X)
p2(X)
;::: U X F2(X)
;::: F2(X)
P(x)
= F2(X) --+ 0 as x --+ +00.
Let x E (a, +00) and let h > 0 and A E [0,1) be fixed. Let Xo = x + Ah. From the
Taylor formula with the integral form of the remainder it follows that
u(x) = u(xo) + ul(xo)(x - xo) +
1'"
"'0
u(x + h) = u(xo) + UI(Xo)(x + h - xo) +
(x - t)u"(t) dt,
l
"'+h
"'0
(x + h - t)ull(t) dt.
LOGARITHMIC CONCAVITY OF DISTRIBUTION FUNCTIONS
483
From these equalities we get
(1 - 'x)u(x) + 'xu(x + h) - u(x + 'xh)
=(I-'x)
l
x + Ah
x
(t-x)u"(t)dt+,X
lx+h
x+Ah
(x+h-t)u"(t)dt.
By letting x ~ +00 and h, ,X being fixed, we conclude that
lim ((1 - ,X) u(x) + ,Xu(x + h) - u (x + 'xh)) = 0,
x--t+oo
or, equivalently,
lim
x--t+oo
F(x + 'xh) = 1.
FI-A(x)FA(x)
Now (4) follows easily by boundedness of F.
0
3. Schur-concavity
It is weIl known (see [3]) that the function
n
<P(XI,'" ,xn ) =
II F(Xi)
i=l
is Schur-concave if and only if F is log-concave. In the setup of distribution
functions, if Xl, ... , X n are independent random variables with the common distribution function F which is log-concave, then
whenever (Xl, ... , x n ) )- (YI, ... Yn). Inequality (1) is a particular case of this
result, since (Xl, ... , X n ) )- (x, ... , x).
4. Examples
From Theorem 1 it follows that if a density is either nonincreasing or unimodal
and log-concave, then the corresponding distribution function is log-concave. We
give some examples.
10 Anormal density
f(x) =
1
. /iC exp
av 27l"
(
-
(x - f-L)2)
2
'
a
f.L E lR, a > 0
is log-concave on lR and so is its distribution function.
20 A Gamma density
f(x) =
N" e- AX X",-l
r(a)
,
0:>0, ,X>O, x~O
M. MERKLE
484
is unimodal and log-concave for Cl > 1, and it is decreasing for 0 < Cl S 1.
Therefore, for every Cl > 0 the Gamma distribution function is log-concave. In
particular, exponential and chi squared distribution functions are log-concave.
3° Reciprocal Gamma density (see [2])
c
fex) = r(x) ,
x> 0,
is log-concave because the Gamma function is log-convex one. The corresponding
distribution function is log-concave.
4° Let
2xe x2
fex) = ~1'
e
-
(0 S x S b) .
The corresponding distribution function is
Then F"(x)F(x) - F'2(X) < 0 for 0 < x < Xo, where Xo ~ 1.256. So, for
b S xo, the function F is log-concave. The observed density is increasing and it
is log-concave for 0 < x < 1/v'2 ~ 0.707. This example shows that conditions of
Theorem 2 are not necessary for F to be log-concave.
References
1. E. Artin, The Gamma Function, Holt, Rinehart and Winston, New York, 1964 [Translation
from the German original from 1931].
2. A. Fransen and S. Wrigge, Calculation of the moments and the moment generating function
for the reciprocal gamma distribution, Math. Comp. 42 (1984), 601-616.
3. A. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, Academic Press, New York, 1979.
4. M. Merkle and Lj. Petrovic, On Schur-convexity of some distribution functions, Pub!. Inst.
Math. 56 (70) (1994), 111-118.
5. D. S. Mitrinovic, Analytic Inequalities, Springer Verlag, Berlin - Heidelberg - New York,
1970.
SHARPENING OF CAUCHY INEQUALITY
ZIVOJIN MIJALKOVIC
Higher Technical School, Beogradska 20, 18000 NiI, Yugoslavia
MILAN MIJALKOVIC
The Bank 0/ Nova Scotia, Systems Development, Toronto, Ganada
Abstract. A sharpening of the Cauchy A-G inequality is given. Also, some weighted
generalisations are considered.
It is known that for a finite series a = (al, a2, ... , an) of positive numbers from the
interval [m, n] the foHowing Cauchy A-G inequality (see [1-2])
holds, with equality if and only if al = a2 = ... = an. There are several proofs of
this inequality (see [1-2]). Twenty years ago, the first author of this paper found
a new proof of A-G inequality, as weH as one sharpening of this inequality using
convex functions, and sent it to Professor Mitrinovic. As a replay, that was very
prompt, Professor Mitrinovic accepted the submitted paper and also he expressed
an assumption and opinion that the number of inequalities which are sharper then
A-G is uncountable. This fact was an inspiration for the authors of this paper.
Here we present results that confirm the assumptions made by Prof. Dragoslav S.
Mitrinovic.
We consider a new function
(0: ~ 0)
and we intend to prove the existence of a sharper inequality than the Cauchy
inequality,
Also, we introduce some generalisations of these inequalities.
1991 Mathematics Subject Classijication. Primary 26D15.
Key woms and phrases. Cauchy A-G inequalitYj Arithmetie meanj Geometrie meanj Weighted
means.
485
G. V. Milovanovic (ed.), Recent Progress in lnequalities, 485-487.
© 1998 Kluwer Academic Publishers.
Z. MIJALKOVIC AND M. MIJALKOVIC
486
Lemma 1. For a finite sequence 0/ positive numbers 0: + al, 0: + a2,' .. ,0: + an,
where 0:;::: 0, ai > 0 (i = 1, ... ,n), the inequality
(1)
(0: + a2)'" (0: + an) + ... + (0: + ad'" (0: + an-d _ 1 ;::: 0
n '../«0: + ad(o: + a2)'" (0: + an))n-l
holds.
Proof. Indeed, applying Cauchy inequality we have
n-l
n-l
A
A
'(0: + a2)'" (0: + an )'+ .. · + '(0: + ad'" (0: + an-S
;::: n '../(0: + ad(o: + a2)'" (0: + an)r- 1 ,
which is equivalent to the inequality (1), with equality if and only if 0: + al
0: + a2 = ... = 0: + an. 0
Lemma 2. The function S(o:, a) is monotonically nondecreasing, i.e.,
when 0: :::; ß.
Proof. Finding a derivative offunction S(o:,a) with respect to 0:, we obtain
Then from (1) we conclude that S~(o:,a) ;::: 0 on [m,n]. Thus, Sn(o:,a) is a
monotonically nondecreasing function, and inequality (2) holds. 0
Theorem 3. For a nonnegative sequence a and 0: ;::: 0, we have
Proof. Since Sn(o:,a) is a nondecreasing function of 0: and Gn(a) = Sn(O,a), we
see that Gn(a) :::; Sn (0:, a).
From
lim Sn (0:, a) = lim
a-too
we find also that
a-too
y/(1 + al/o:)··· (1 + an/o:) - 1
1/0:
SHARPENING OF CAUCHY INEQUALITY
487
Let p = (PI,P2, ... ,Pn) be a finite sequence of positive numbers. Then we can
consider a generalised function in a,
In a similar way, we can obtain the inequality
where
and
A n (a,p) -_ Pial + P2 a2 + ... + Pnan
PI +P2+···+Pn
represent the weighted geometrie and arithmetie mean, respectively.
References
1. P. S. Bullen, D. S. Mitrinovic and P. M. Vasic, Means and Their Inequalities, Reidel Publishing Co., Dordrecht - Boston, 1988.
2. D. S. Mitrinovic (with P. M. Vasic), Analytic Inequalities, Springer Verlag, Berlin - Heidelberg - New York, 1970.
A NOTE ON THE LEAST CONSTANT IN LANDAU
INEQUALITY ON A FINITE INTERVAL
A. YU. SHADRIN
Computing Center, Siberian Branch, Russian Academy 01 Sciences,
630090 Novosibirsk, Russia
Abstract. In the Landau inequality on the unit interval
with 11 . 118 := 11 . IIL. [0,1]' 1 :5 p, q, r :5 00, 0 :5 k < n, we find the least value Ao of the
first constant a.
1. We are concerned here with the problem of Burenkov on sharp constants in
Landau-type inequality on the unit interval
(1.1)
with
11·118:= 1I·IIL.[o,I],
1 '5:,p,q,r:5 00,
0'5:, k < n.
Denote by r the set of all pairs (a,ß) for which (1.1) holds for any 1 E W:[O, 1].
The general problem is to find the complete collection G = {(A, B)} of sharp
constants in (1.1) which are defined as
(1.2)
A ~ A o :=
inf
(a,ß)er
a,
B:= B(A):=
inf
(A,ß)er
ß.
Here we define the least value of the first constant
(1.3)
A o = Ao(n, k,p, q, r).
2. The Landau-type inequalities in the additive form (1.1) were firstly studied by
H. Cartan and Gorny for p = q = r = 00. For arbitrary p, q, r E [1,00] they were
obtained by Gabushin [2].
1991 Mathematics Subject Classijication. Primary 41A17, 41A44.
Key woms and phrases. Landau inequality, Markov inequality, Lagrange interpolation.
Supported by agrant from the Alexander von Humboldt - Stiftung
489
G. V. Milovanovic (ed.), Recent Progress in lnequalities, 489-491.
© 1998 Kluwer Academic Publishers.
A. YU. SHADRIN
490
Burenkov [1] was first who was looking for the sharp constant (1.3) and the corresponding constant B o := B(Ao). He proved that
Ao
= Mo,
k
= n - 1,
1 ~ p, q, r ~ 00,
where
(1.4)
._
._
M o ·- Mo(n -l,k,p,q).-
sup
PE'II"n-l
IIP(k) IIq
IIPII '
P
is the best constant in the Markov-type inequality of different metrics for algebraic
polynomials.
In [3-4] it was shown that
(1.5)
A o = Mo,
0 < k < n,
p = q = r = 00.
Moreover, the exact value for B o was also found. (In fact, Eq. (1.5) was proved
much earlier by H. Cartan, though with a poor second constant.)
Here we give an elementary proof of the following result:
Theorem 1. For any n, k,p, q, r
A o = Mo·
3. Notice, that for all n, k,p, q, r the value Mo provides the lower bound for A o,
i.e.
(1.6)
A o ~ Mo.
To see that, one can substitute in (1.1) instead of 1 an algebraic polynomial p. of
degree n - 1 extremal for the Markov inequality (1.4).
Thus, it is enough to prove that (1.1) holds with 0: = Mo and some ß < 00
(the smaller is the better). We do it by finding an appropriate approximation to
1 E W~. Such a method was used by H. Cartan and Gorny, and was given in the
most general form by S. B. Stechkin [5].
Pro%/ Theorem 1. Let 1 E w~, and let P: W~ -t 7rn - l be any projector from
W~ onto the space 1Tn - l of algebraic polynomials of degree n - 1. Then
111(k)lIq ~ IIP(k) (J)lIq + 111(k) - p(k) (J)lIq
~ Mo IIP(J)lIp + 111(k) - p(k) (J)lIq
~ Mo 11111p + Mo 111 - P(J)lIp + 111(k) - p(k) (J)lIq.
Set
A NOTE ON THE LEAST CONSTANT IN LANDAU INEQUALITY
491
and No = infp No{P). It is easy to show that No < 00. For example, one can take
as a P the Lagrange interpolating polynomial.
Hence,
that is
Ao :::; Mo,
Bo :::; No·
With respect to (1.6) this means that
Ao = Mo,
which completes the proof.
References
1. v.!. Burenkov, On sharp constants in inequalities between nonns 01 intennediate derivatives
on a finite interoal, Trudy Mat. Inst. AN SSSR (Proe. Steklov Math. Inst.) 156 (1980), 2229. (Russian)
2. V. N. Gabushin, Inequalities lor the nonns 01 a function and its derivatives in metric L p ,
Mat. Zametki 1 (1967), no. 3, 291-298 (Russian) [Eng!. Trans.: Math. Notes 1 (1967),
194-198].
3. H. Kallioniemi, The Landau problem on compact intervals and optimal numerical differentiation, J. Approx. Theory 63 (1990), 72-91.
4. A. Yu. Shadrin, To the Landau-Kolmogorov problem on a finite interoal, Open Problems in
Approximation Theory (B. Bojanov, ed.), SCT Publishing, Singapore, 1994, pp. 192-204.
5. S. B. Stechkin, Best approximation 01 linear operators, Mat. Zametki 1 (1967), 137-148
(Russian) [Eng!. Trans.: Math. Notes 1 (1967), 91-100J.
SOME INEQUALITIES INVOLVING HARMONIe
NUMBERS
MIOMIR S. STANKOVIC
Faculty 0/ the Occupational Sa/ety, Carnojevica 10a, 18000 Nis, Yugoslavia
BRATISLAV M. DANKOVIC
Faculty 0/ Electronic Engineering, Department 0/ Automatics, P.O. Box 73,
18000 Nis, Yugoslavia
SLOBODAN B. TRICKOVIC
Faculty 0/ Civil Engineering, Beogradska 14, 18000 Nis, Yugoslavia
Abstract. In this paper we consider some inequalities for convex functions and derive
sharper lower and upper bounds for harmonie numbers. Using the Hadamard's integral inequality we get some better estimates. Also, we give a few applieations to some
functions.
1. Introduction
We define the n-th partial sum of the harmonie series as the n-th harmonie number,
1 . In the analysis of algorithms, harmonie numbers frequently occurs.
Hn =
-k
f:
k=l
For example, Knuth [5] dedieates one section to the study of these numbers and
gives some basie identities. Riordan [12], Lafon [5] and Karr [3] also give some
identities with these numbers. Generalised harmonie numbers are defined as
(1.1)
and have been investigated by Kemp [4], Sedgewiek [13], and Spiess [15]. In [16] it
was given a systematic investigation of identities involving harmonie numbers and
generalised harmonie numbers. Some inequalities with harmonie numbers were
investigated in [2-3].
In [11] it was stated a result from [1]. Namely, the following inequality
holds, where C = 0.57721566 ... is the Euler's constant.
1991 Mathematics Subject Classification. Primary 26D15; Seeondary 05A20, llB68.
Key woms and phrases. Harmonie numbers; Hadamard's inequality; Bernoulli numbers; Euler
eonstant; Euler-Maclaurin formula.
493
G.v. Milovanovic (ed.), Recent Progress in lnequalities, 493-498.
© 1998 Kluwer Academic Publishers.
494
M. S. STANKOVIC, B. M. DANKOVIC, S. B. TRICKOVIC
In this paper we consider some inequalities involving lower and upper bounds for
(1.2) S(nl, n2, ... , nk) = ~ H ni lSiSk
~
Hn;nj
+ ... + (_I)k-l Hnln2 ...nlo.
lSi<jSk
2. Inequalities for Harmonie Numbers
The harmonie number H n can be represented as
(2.1)
n I l
H n = ~ k = C + t/J(n + 1) = C + ;;: + t/J(n),
k=l
where C is the Euler's constant and t/J(s) = r'(s)/r(s). In order to get better
estimates than (see [2])
1
- +logn:::; H n :::; 1 +logn,
n
(2.2)
n> 1,
we use Hadamard's inequalities
J e ; b) :::; b ~ alb J(x)dx :::; J(a); J(b),
(2.3)
which hold if f"(x) ~ 0, i.e., if J is a convex function.
Taking the function J(x) = I/x, a = k -1, b = k (2 :::; k :::; n), the right inequality
in (2.3) reduces to
(k = 2, ... ,n).
Summing these inequalities, we get
(2.4)
1
1
1
1
1
-2n + -2 + logn < H n = 1 + -2 + -3 + ... + -n'
which gives a sharper lower bound for H n in comparison to (2.2), because
1
1
1
;;: + logn < 2n + '2 + logn,
n>1.
Also, we can find a sharper estimate for H~ = 1 + ~ + ~ + ... + 2n ~ 1. Namely,
1
1
1
..
1
-2 - - + -log(2n + 1) <
H < 1 + - log n.
4n+2 2
- n 2
Let f" is a nondecreasing function. Then
1
1
j'(x) + '2f"(x) :::; J(x + 1) - J(x) :::; j'(x) + '2f"(x + 1).
SOME INEQUALITIES INVOLVING HARMONIe NUMBERS
495
These inequalities are special case of some more general inequalities proved in [10].
Taking f(x) = log x, we obtain the following estimates
In
(2.5)
1
niin
i
"2 L (k + 1)2 + log(n + 1) ~ L k ~ "2 L k 2 + log(n + 1).
k=l
k=l
k=l
Comparing this result with (2.2) (see [2]), it is easy to see that we have now a
sharper lower bound, because of the inequality
1
1
1
n
;;; + logn ~ "2 {; (k + 1)2 + log(n + 1),
n ~ 1.
Similarly, in the case of the upper bound
1
1
"2 L k 2 + log(n + 1) ~ 1 + logn,.
n
n ~ 3,
k=l
we can conclude that inequalities (2.5), for n ~ 3, give sharper bounds for the
harmonie numbers H n .
In order to illustrate the obtained results, we compute numerieal values for the
lower bounds of H n , given by (2.2), (2.4) and (2.5), as well as the exact values of
H n given by (2.1), for 2 ~ n ~ 10.
n
2
3
4
5
6
7
8
9
10
(2.2)
1.19314718
1.43194562
1.63629436
1.80943791
1.95842614
2.08876729
2.20444154
2.30833569
2.40258509
(2.4)
1.44314718
1.76527896
2.01129436
2.20943791
2.37509280
2.51733872
2.64194154
2.75278013
2.85258509
(2.5)
1.27916784
1.59809992
1.84124347
2.03745391
2.20180868
2.34315257
2.46710844
2.57746896
2.67691137
Hn
1.50000000
1.83333333
2.08333333
2.28333333
2.45000000
2.59285714
2.71785714
2.82896825
2.92896825
As we can see, the lower bound in (2.4), among these bounds, gives the best results.
Consider now the function f(x) = I/x r +1 (r E Z) in order to get an inequality
for the generalised harmonie numbers H~r+1) defined in (1.1). If r > 0 and x > 0
it follows that f"(x) > O. Applying again the Hadamard's inequality, we get an
estimate of the lower bound of H~+1 in the form
~(I-~) + ~ + _1_ < H(r+1).
r
nr
2
2nr+1
n
This gives a sharper upper bound in comparing to [2], because of inequality
I(
I(1 - 1) +"21 + 1
1)
;:- 1 - (n + I)r <;:-
nr
2nr+1 '
n > 1.
This is easy to show, taking s = r + 1 and a = 1 for parameters which appear [2].
All these inequalities can be obtained by (3.3).
496
M. S. STANKOVIC, B. M. DANKOVIC, S. B. TRICKOVIC
3. Inequalities Based on Harmonie N umbers
Theorem 1. The following inequalities
(3.1)
k = 2,3, ... ,
hold, where C is the Euler's constant. In particular, we have
k = 2,3, ....
(3.2)
Proof. Applying Euler-Maclaurin summation formula (see [8])
n
r
j=O
a
1
LJ(j) = in f(x) dx + i (J(a) + f(n))
rn-I
+ '"" B 2s [f(2S-I)(n) _ f(2S-I)(a)] + R (n)
~ (2s)!
rn
to (1.2), we find the following representation of the harmonie numbers
(3.3)
1
~ B2 ·
Hn=C+logn+ 2n - ~2jn;j'
j=l
where B 2j are Bernoulli's numbers. Substituting (3.3) in (1.2), we can get an
asymptotie series for S(nl, ... , nk). Namely,
1(1-II(1--.)
k
1) -1(1-II(1-~)
k
1 ) + ... ,
S(nl, ... ,nk)=C+2
i=l
n,
12
i=l
n,
Le.,
Using partial sums of asymptotie series (see [9]) we obtain the inequalities
(3.5)
k = 2,3, ... ,
from whieh we get directly inequality (3.2). If ni = 1, then, because of (1.2), it
follows S(nl, ... , nk) = 1 (k = 2,3, ... ). If ni > 1 (i = 1,2, ... ), because of (3.4),
we get (3.2). 0
SOME INEQUALITIES INVOLVING HARMONIe NUMBERS
497
Theorem 2. Sharper inequalities also hold
k = 2,3,4,
C < S(nl,'" ,nk) ::; 1,
(3.6)
(3.7)
C< S(nl,'" , n5) < 1.00009,
C < S(nl,'" , n6) < 1.00330,
(3.8)
C < S(nl, ... , n7) < 1.00527,
C < S(nl,' .. , ns) < 1.00784.
Proof. After finding ßniS(nl,'" ,nk), we get
ßniS(nl, ... ,nk)<O,
k=2,3,4,5,6;
ni=2,3, ....
The values of S(nl"" ,nk), k = 2,3,4,5,6, and ni = 2, i = 1,2, ... ,k, are
presented in following table (nine digits after decimal point are correct):
k
2
3
4
5
6
S(nl,n2, ... ,nk)
0.9166666667
0.9678571429
0.9906995782
1.000088325
1.003288228
Thus, we can claim now that C < S(nl,n2) < 0.91667, C < S(nl,n2,n3) <
0.96786, C < S(nl,'" , n4) < 0.99070, C < S(nl,'" , n5) < 1.00009, C <
S(nl,'" ,n6) < 1.00330, and so, we have proved inequalities (3.6) i (3.7).
In the case k 2: 7, it is necessary to examine the sign of ßn,s(nl, ... ,nk) in
order to determine the set ni, i = 1,2, ... ,k, for which the term S(nl,'" ,nk)
has the maximal value. In that way, we get inequalities (3.8). In the same way,
the corresponding inequalities can be determined for k > 8. So, we obtain sharper
inequalities (3.1) i (3.2) for any particular k. 0
Theorem 3. The following limits hold
(3.9)
Proof. A limit process in (3.4) gives relations (3.9).
We consider now the numbers
H* = 1 +
n
0
!3 + !5 + ... + _
l_
2n - 1
and the corresponding sums
S*(nl,n2) = H~l + H~2 - H~ln2'
S*(nl, n2, n3) = H~l + H~2 + H~3 - H~ln2 - H~ln3 - H~2n3 + H~ln2n3'
S*(nl, ... ,nk)=
L H~i- L
l~i~k
l~i<j~k
H~;nj+ ... +(-I)k-1H~ln2 ... nk·
M. S. STANKOVIC, B. M. DANKOVIC, S. B. TRICKOVIC
498
Theorem 4. The following inequalities hold
(3.10)
~C+Iog2 < S*(nl, .. ' ,nk) < ~C+Iog2+ 4~'
(3.11)
~C+Iog2<S*(nl, ... ,nk)<1,
k=2,3,4,5,6,7,8.
Proof. At first,
Further , we have
H* - C
logn I
_1____7--:n - 2 + 2 + og 2 + 48n 2
1920n4 + ... ,
as weIl as
(3.12)
S*(nl,'" ,nk) =
~ +log2+ 418 [1-
g ~r)1
(1-
-_7 [1 - rr (1 - ~)
nt 1+ ....
1920
i=l
From (3.12), it directly foIlows (3.10). After finding ßn,s*(nl,'" , nk), we get
ßniS*(nl,'" ,nk) <0,
k=2,3,4,5,6,7,8,
wherefrom we conclude that inequalities (3.11) hold.
D
References
1. H. F. Sandham, Problem E 819, Amer. Math. Monthly 55 (1948), 317.
2. H. Bateman and A. Erdelyi, Higher Transcendental Ftlnctions, Vols. I, II, McGraw-Hill,
New York, 1953 [Russian Edition: Nauka, Moskow, 1973].
3. M. Karr, Summation in finite terms, J. Assoc. Comput. Mach. 28 (1981), 305-350.
4. R. Kemp, Fundamentals 0/ the Avemge Case Analysis 0/ Particular Algorithms, WileyTeubner Series in Computer Science, Stuttgart, 1984.
5. D. E. Knuth, The Art 0/ Computer Progmmming, Vols. 1-3, Addison-Wesley, Reading,
Mass., 1968.
6. J. C. Lafon, Summation in finite terms, Computing Supp!. 4 (1982), 71-77.
7. B. Martic, On some inequalities, Mat. Vesnik 12 (27) (1975), 95-97.
8. F. Olver, Asymptotic and Special Functions, Academic Press, 1974.
9. G. V. Milovanovic, Numerical Analysis, Part I, 3rd Edition, Naucna Knjiga, Belgrade, 1991.
(Serbian)
10. G. V. Milovanovic and M. S. Stankovic, The genemlization 0/ an inequality /or a /unction
and its derivatives, Univ. Beograd. Pub!. Elektrotehn. Fak. Ser. Mat. Fiz. No. 461 - No.
497 (1974), 253-256.
11. D. S. Mitrinovic (with P. M. Vasic), Analytie Inequalities, Springer Verlag, Berlin - Heidelberg - New York, 1970.
12. J. Riordan, Combinatorial Identities, R. E. Krieger, Huntington, N.Y., 1979.
13. R. Sedgewick, The analysis 0/ quicksort progmms, Acta Inform. 7 (1977), 327-355.
14. D. Slavic, On summation 0/ series, Univ. Beograd. Pub!. Elektrotehn. Fak. Ser. Mat. Fiz.
No. 302 - No. 319 (1970), 53-59.
15. J. Spiess, Mathematische Probleme der Analyse eines Algorithmus, Z. Angew. Math. Mech.
63 (1983), T429-T431.
16. J. Spiess, Some indentities involving harmonie numbers, Math. Comp. 55 (1990), 839-863.
SOME INEQUALITIES FOR POLYNOMIALS
IN L o NORM
E. A. STOROZENKO
Odessa University, IMEM, Petra Velikogo 2, 270000 Odessa, Ukraine
Abstract. Some relations between zeros of a polynomial and zeros of its integrals are
established.
Let
Pn(Z)
= Coz n + C1Zn-1 + ... + cn
be an arbitrary polynomial with complex coefficients and
M(Pn ) = exp
C~
L: Ip
log
n
(e it ) I dt) .
M(Pn ) will be "Lo-norm" of the polynomial Pn . Suppose that Co i 0 and let
a1, a2, ... ,an be zeros of P n (z). By means of the Jensen 's formula
M(Pn ) = ICoI
(1)
n
II max(l, lai!).
i=l
K. Mahler [2] proved inequality
(2)
from which, because of (1), it foHows an important property of zeros of the polynomial P n and its derivative:
n-1
n
i=l
i=l
II max(l, Ißi!) ~ II max(l, lad),
(3)
where ß1, ß2, . .. ,ßn-1 are zeros of the polynomial P~ (z).
In this work we shaH establish some inequalities for the polynomial Pn(z) and its
integral JozP() d(, as weH as for Pn(z) and each of integrals J:z P n () d(, where
J: ih
0< r < 1, and e Pn ( ) d(, where 0 < h< 27r/n. These inequalities, as weH as
(2), will enable to make the corresponding conclusions about connections between
P n () d(,
zeros of the polynomial Pn(z) and each of mentioned polynomials
J:
Jrzz Pn() d(, J: eih Pn() d(.
1991 Mathematics Subject Classijication. Primary 30C15j Secondary 30A10.
Key words and phrases. Inequalitiesj Normj Best constantj Algebraic polynomialsj Composition
of polynomials.
499
G. V. Milovanovic (ed.), Recent Progress in Inequalities, 499-503.
© 1998 Kluwer Academic Publishers.
E. A. STOROZENKO
500
Theorem 1. 1/ Pn- 1(z) is a polynomial 0/ degree n -1, then
(4)
where
. 'lrk
2 sm-.
II
n
n/6<k<5n/6
For some initial values, n = 2,3,4,5,6, we have An = n.
As n ---t 00 we obtain
2 L "3 '
log An "32 log 2 - -;
('Ir)
"-J
where the Lobachevsky function, defined by L(u) = - Jou log cos t dt. An approximate value of An can be done as An ~ (1.4)n.
Since for polynomial Pn- 1(z) = (z + e ia )n-1 the inequality (4) reduces to an
equality, we conclude that the factor A n n- 1 in (4) could not be improved. Comparing to (2), the inequality (4) represents an opposite inequality. In fact, putting
Pn-1 (() d( = Qn(z) in (4), we get the following inequality
J:
M(Q~) ~ A;;1n M(Qn).
Theorem 1'. 1/ 0, 'Y1, ... , 'Yn-1 are zeros 0/ the integral Joz Pn-1 (() d(, then
n-1
n-1
i=1
i=1
II max(l, l'Yi!) :::; An II max(l, lai!) .
The extremal polynomials Pn - 1 and multiplier An are the same as in Theorem 1.
The interest to the integrals Jrz Pn(() d( and Jz
Pn(() d( arose from the theory
of functions in connections with certain investigation of increments of analytical
function in the unit ball along the radius and a circular arc.
ze ih
z
Theorem 2. Let Pn- 1 be a polynomial 0/ degree n - 1 and 0< r < 1. Then
(5)
where
A(n,r) =
II
2 sin 1I"k
n
,
nß/1r<k<n(1r-ß)/1r V(I- r)2 + 4rsin 2;k
. v'i-=T
2
.
ß=arcsm
For a fixed r and n = 2,3,4,6 we have the following values of A(n,r):
2
A(2,r) = -1- ,
+r
A(3,r) = 1
3
2 '
+r+r
4
A(4, r) = (1 + r)(1 + r2) ,
INEQUALITIES FOR POLYNOMIALS IN Lo NORM
A(6,r)
If n -+ 00 (r - fixed), then
n
501
= (1 + r) (1 + r +6r 2 ) (1 - r + r 2 ) .
18r- log
log A(n, r) '" 11" ß
and if r -+ 1
ß
n
dt = - A(r) ,
J(1-r)2 +4rsin2t
11"
2sint
11"
A(r) '" 2" (1 - r).
Inequality (5) reduces to an equality ff Pn - 1 (z) = (z + eia
r-
Theorem 2'. 1/0,81, ... ,8n - 1 are zeros o/thepolynomial
n-1
1.
J:z Pn- 1(() d(, then
n-1
II max(l, 18 l) ~ A(r, n) II max(l, lail) .
i
i=1
i=1
The extremal polynomial Pn - 1 and multiplier An are the same, as in Theorem 2.
Theorem 3. Let Pn- 1(z) be a polynomial 0/ degree n-1 and O<h<211"/n. Then
(6)
where
B(n,h) =
sin 1rk
II
(1/2-h/21r}n<k<n-1
. (d n h)·
sm -+n
2
For h < 211"/n and n = 2,3,4,5 we have:
1
B(2, h) = cos h/2'
va
B(3, h) = 2 sin(1I" /3 - h/2) ,
.j2
B(4, h) = 2cosh /.
2sm (/
11" 4-h / 2) ,
v'5
B(5, h) = 4 sin(211" /5 - h/2) sin(1I" /5 - h/2) .
When n -+ 00 and h is fixed we get
l
n 1r - 1r / n
n
log B(n, h) '" (log sin x - log sin(x + h/2)) dx = - J(n, h)
11" 1r/2-h/4
11"
and
J(n, h) '" - sin ~ ·logsin ~,
h -+ O.
Equality in (6) is attained when Pn(z) = (z + e ia )n-1.
502
E. A. STOROZENKO
Theorem 3'. If O,iI, ... ,in-l are zeros ofthe polynomial f:e. h P n- 1 (() d(, then
n-l
n-l
i=l
i=l
II max(l, liil) ~ B(n, h) II max(l, lail) .
For proofs of Theorems 1-3 it is important to know the representation of each of
.h
the integrals foz , Irzz and fzze in the form of the composition of two polynomials,
one of which is Pn(z). The further reasoning are connected with an application
of inequality of N. J. Bruijn and T. A. Springer for "Lo-norm" of composition. In
more details, let
Then the polynomial
C(z) =
~ (~)akbkzk
is a composition of A(z) and B(z) (see [3]). We shall denote this composition by
ED, Le., C(z) = A(z) ED B(z). Now, we will find the corresponding compositions for
each of the previous integrals.
For
we have
and
r +
(1
10
l +
l +
z
(1
rz
z
ze'h
(1
I: (n - ~
= +
+
= I: (n = +
+
()n-l d( = z
()n-l d(
1)
k
k=O
1) 1 - rk+ 1 zk
k
k 1
z
()n-l d( = z
(1
L
k
1
k=O
n-l (n - 1) 1 - e i (k+1)h
k=O
k
k +1
z)n - 1 ,
n
(1
z)n - (1 + rz)n ,
n
(1 + z)n - (1 + zeih)n
zk = -'----'----'----~n
INEQUALITIES FOR POLYNOMIALS IN Lo NORM
503
Hence
According to the Bruijn-Springer inequality [1, Theorem 7]
M(A(z) E9 B(z)) :::; M(A(z)) . M(B(z)).
In order to complete the proof we must calculate the "Lo-norm" of polynomials
With this purpose we find the zeros of these polynomials and use Jensen's formula
(1). Some technical difficulties arise in connection with the investigation of the
asymptotic behaviour of the factors A(n), A(n, r), and A(n, h).
References
1. N. J. de Bruijn and T. A. Springer, On the zeros of composition-polynomials, Nederl. Akad.
Wetensch. Proc. 50 (1947), 859-903 [= Indag. Math. 9 (1947), 406-414].
2. K. Mahler, On the zeros of the derivative of a polynomial, Proc. Roy. Soc. London 264
(1961), 145-154.
3. G. P6lya und G. Szegö, Aufgaben und Lehrsätze aus der Analysis, Vol. 2, Springer Verlag,
Berlin - Heidelberg - New York, 1971.
SOME INEQUALITIES FOR ALTITUDES AND OTHER
ELEMENTS OF TRIANGLE
MALISA R. ZIZOVIC and MILORAD R. STEVANOVIC
University "S. Markovic" - Kragujevac, Technical Faculty Cacak,
Svetog Save 65, 32000 Cacak, Yugoslavia
Abstract. In this paper we give some improvements of geometrie inequalities from the
reeent monograph [1].
The notation in this paper is taken from the monograph [1].
Theorem 1. We have
'"'
ha ~-<
W
(1)
a
Proof. Since
ha
ß - 'Y
ß 'Y . ß . 'Y
- = cos - - = cos - cos - + sm - sm Wa
2
2
2
2
2'
we have
a = ~
'"'
' " ' cos -ß cos -'Y +
~ -h
Wa
2
2
(2)
L'sm -ß2 sm. -'2Y .
Also,
(3)
ß
(
ß
'"' cos - cos 'J.. < J3 '"' cos 2 - cos 2 'J..
~
2
2~
2
2
) 1/2 = -v'3 [S2 + (4R + r)2]1/2
and
because of ab + bc + ca = 8 2 + 4Rr + r 2 •
1991 Mathematics Subject Classification. Primary 51M16.
Key woms and phmses. Inequalities for triangle.
505
G.v. Milovanovic (ed.), Recent Progress in Inequalities, 505-510.
© 1998 Kluwer Academic Publishers.
4R
M. R. ZIZOVIC AND M. R. STEVANOVIC
506
From (2), (3) and (4), we get
L Wha:s 2..
{v'3[8 2 + (4R+r)2)1/2 + (8 2 +4Rr+r 2)1/2}
4R
a
= 4~ {3[(8 2 + (4R + r)2)j3] 1/2 + (8 2 + 4Rr + r 2)1/2}
1 {
}1/2
:s 4R
4[3(8 2 + (4R + r)2)j3 + 82 + 4Rr + r 2]
,
i.e.,
(5)
Applying Gerretsen's inequality (see [1, p. 45))
to inequality (5), we obtain (1).
D
Corollary 2. The following inequalities hold:
(6)
L ~: :s ~ (6R 2 + 5Rr + 2r2)1/2 :s
v:
[R + (v'6 - 2)r]
1
:s 2R (5R + 2r) :s 3.
Proof. At first the inequality
is true, because of its reduced form r(12v'6 - 29)(R - 2r) 2: O.
Since (5 - 2V6) (R - 2r) 2: 0, we also have
v'6[ R+ (v'6 - 2)r] :s 2R(5R
1
R
+ 2r).
Finally, the inequality (5R + 2r)j(2R) :s 3 is obviously true.
D
The inequalities given in (6) are improvements of the inequality
(see [1, p. 219)).
SOME INEQUALITIES FOR TRIANGLE
507
Corollary 3. The inequality
L
(7)
h a < ...!... (48R 3 + 16R2r - 7Rr2 - 2r 3 ) 1/2
2R-r
wa - 2R
holds.
Proof. Inequality (7) follows from (5) and the inequality
82
< R(4R + r)2
-
2(2R-r)
(see [1, p. 166]).
0
Inequality (7) is stronger than (1).
Corollary 4. We have
LWa>
9R
.
h a - ../6R2 + 5Rr + 2r 2
Proof. The proof follows from (1) and inequality
(8)
and from inequality (1).
Using (6), (7) and (8), we can obtain some other inequalities.
Remarks. (i) In dass 6.1 of triangles with a constant side c, O! -+ 0, ß -+ 0, 'Y -+ 71",
S -+ c, r -+ 0, R -+ 00, we have
.
hm
aEa l
[2
] 1/2 =2.
rn s +(4R+r)(2R+r)
1
R-too Rv2
Thus, in the dass 6. 1, the bound in inequality (5) is 2. (ii) In the dass 6.2 of triangles
with r -+ 0, the corresponding bound in inequality (I) is v'6.
Theorem 5. We have
L
(9)
Wa
< 4 {R . ../13R2 + 14Rr - 8r 2
ha -
Y-:;:
9R-2r
Proof. Since
Wa = 2 (R
ha
Y-:;:
Va (8 - a) , II(28 - a) = 28(8 + 2Rr + r 2),
28-a
2
and
L w = 2 (R L va(8 - a) = 2 (RE va(8 - a)(28 - b)(2s - c) ,
a
ha
Y-:;:
28-a
Y-:;:
I1(28-a)
M. R. ZIZOVIC AND M. R. STEVANOVIC
508
we find
L -h = -I~-r s + 2R1 r+r 'L..J" ya(s - a)(2s - b)(2s - c),
Wa
a
2
S
2
L Va(s - a)(2s - b)(2s - c) = L Va(s - a)(2as + bc)
= 2s Lava(s -a) + abc LJ~ -I,
Using the identities
a3 + b3 + c3 = 2s(s2 - 6Rr - 3r2)
a4 + b4 + c4 = 2s4 - 4(4Rr + 3r2)s2 + 2(4Rr + r 2)2
we obtain
Since
we conclude that
and then
L wha :$ 2 Y-;;:
{ii.
2~
{[Rr(s2 + 4Rr + r 2)] 1/2
s + r+r
a
2
2
+ Va [(4Rr + 6r 2)s2 - 2(4Rr + r 2)2]1/2}.
After some calculations, we find
[Rr(s2 + 4Rr + r 2)] 1/2 +
va [(4Rr + 6r2)s2 _ 2(4Rr + r2)2] 1/2
= [Rr(s2 + 4Rr + r2)] 1/2 + Va [(4Rr + 6r 2)s2 _ 2(4Rr + r2)2] 1/2
:$ 2 [Rr(s2 + 4Rr + r 2) + 3(4Rr + 6r 2)s2 - 2(4Rr + r 2)2)/3]1/2
= 2 [(5Rr + 6r 2)s2 - (4Rr + r 2)(7Rr + 2r2)] 1/2,
509
SOME INEQUALITIES FOR TRIANGLE
and then
(10)
4
W
[
] 1/2
~ h: ~ S2 + 2Rr + r 2 (5R 2 + 6Rr)s2 - (4R 2 + Rr)(7Rr + 2r2)
.
Since, the right hand side of inequality (10) is a decreasing function in S2, we can
apply Gerretsen's inequality
(see [1, p. 45]).
Finally, after some calculations, we get inequality (9). 0
According to (9) and (8) we obtain the following inequality
Remark. Inequality (9) is stronger than inequality of D. M. Milosevic (see [1, p. 219])
(11)
Indeed, it is sufficient to show that
4
fE v'13R2 + 14Rr - 8r < J3 R + r
V-;;:
which is equivalent to
2
9R- 2r
-
r'
(35R 2 - 19Rr - 6r 2 )(R - 2r) ~ o.
The last inequality is evidently true.
Theorem 6. The inequality
9R - 14Rr + 4r
~ -ma < ----=---2
(12)
ha -
2
2Rr
holds.
Proof. Using m a ~ sV3 - Wb - W c (see [1, p. 221]) we have
and
Further,
(14)
~~-~
h - r'
(15)
...,...Rr_+_r_2
~ w a >- ~ h a -- 2rs ~ ~a -_ _s2_+_42R
a
M. R. ZIZOVIC AND M. R. STEVANOVIC
510
Since
from (11), we get
'L...J-<2+-.
"' W a
R
ha 2r
(16)
Now, from (13)-(16) we obtain
(17)
L -ma
< 3- h
r
a -
y
r,:; 8
82
+ 4Rr + r 2 + 2 + -R = -1yr,:;38 - -182 + -R - -r .
2Rr
2r
r
2Rr
2r
2R
va
Finally, using Hadwiger-Finsler inequality
8 :S 4R+r and Gerretsen's inequality
8 2 ~ 16Rr - 8r 2 , (17) reduces to inequality (12).
0
Corollary 7. The inequalities
(18)
'"' m a
9R - 12r
L...J-<---ha 2r
and
(19)
6r
L -mha>3R
---4r
a -
hold.
Proof. In order to prove (18) it is enough to show that
9R2 - 14Rr + 4r 2 < 9R - 12r
2Rr
2r
which is true, because of 2r(R - 2r) ~ O.
Inequality (19) follows directly from (18) and (8).
0
References
1. D. s. Mitrinovic, J. E. Pecaric and v. Volenec, Reeent Advanees in Geometrie Ineqv.alities,
Kluwer, Dordrecht, 1989.
Author Index
A
Abramowitz, M., 321
Adamovic, D. D., 13, 20, 22
Adams, R. A., 100, 104, 106, 108, 109, 121
Agarwal, A. K., 71, 75
Agarwal, R. P., 290, 291, 294, 304, 307, 397,
398,403,404,405,406,412,421,422
Ahlfors, L. V., 39, 53
Ahmed, S., 321
Akhiezer, N. 1., 166, 174
Akrivis, G., 195, 201
Albert, M., 449,455
Alefeld, G., 325, 337, 339
Aleksandrov, A. D., 117,121,341,364,377,
378
Alexandrov, A. D., (see: Aleksandrov, A.
D.)
Alt, H. W., 100, 104, 106, 108, 112, 116, 121
Alzer, H., 203, 292, 307
Amos, R. J., 458,459,462,463
Anderssen, R. S., 13
Andrews, G. E., 71, 75
Andrianov, A. V., 92, 93, 94
Andrica, D., 425,427,431
Andrievskii, V., 31, 33, 37, 42, 53
Appell, J., 273,286,288
Arestov, V. V., 55, 57, 59, 60, 62, 80, 81, 94
Arsenault, M., 252, 253, 266
Artin, E., 310, 321, 481, 484
Askey, R., 5, 63, 66, 68, 74, 75, 201, 216,
222, 238, 312, 321
Astor, P. H., 397, 421
Atanassova, L., 340
Aubin, T., 101, 102, 106, 108, 112, 121
Aviny6, A., 121
Aziz, A., 259, 260, 266
B
Babenko, V. F., 77, 78, 81, 90, 94, 95
Baernstein, A., 75, 119, 121
Bailey, P. B., 463
Bailey, W. N., 70, 71, 73, 75
Bajsanski, B., 247
Baker, Ch. T. H., 194, 201
Baker, G. A., 449, 455
Ballieu, R., 393
Bandic, 1., 5
Bandie, C., 97, 101, 110, 114, 117, 119, 120,
121, 122, 288
Bao, P. G., 335, 337, 339
Bari, N. K., 81, 94
Bari, R., 443
Barnes, E. S., 6, 12, 13
Barnett, S., 387, 395
Barta, J., 120, 122
Bateman, H., 493, 494, 495, 498
Bauer, F. L., 387, 395
Beckenbach, E. F., xi, 105, 122, 289, 307
Beckman, F. S., 364, 366, 369, 374, 378
Beesack, P. R., 132, 158
Bell, F. K., 433
Bellman, R., xi, 105, 122, 289, 307
Benammar, M., 127, 129, 147, 159
Benard, P., 118, 122
Bennewitz, C., 127, 132, 145, 159
Benz, W., 374, 376, 377, 378
Berdyshev, S. V., 80
Berens, H., 374, 378
Bergh, J., 274,288
Berman, D. L., 254, 266
Bernstein, S. N., 55, 58, 61,62, 110, 250,
255, 266
Bertolino, M., 4, 10
Besov, O. V., 80, 90, 94, 95
Besson, G., 118, 122
Beynon, M. J., 127, 129, 151, 156, 157, 158,
159
Bharucha-Reid, A. T., 394, 395
Bhatt, S. S., 395
Bilchev, S. J., 26
Binev, P., 476, 479
Birkhoff, G., 460,463
Bishop, R., 366,378
Bjelica, M., 445, 448
Blaschke, W., 289,307,333,334,339
Block, H. D., 292, 307
Boas, R. P., 65, 249, 259, 266
Bochner, S., 122
Bohr, H., 310, 321
Bojanov, B. D., 90, 95, 96, 161, 163, 168,
174,201,259,261,266,491
Bol, G., 117, 122
Bonnesen, T., 117
Börsken, N. C., 332, 339
Borwein, P. B., 250, 253, 254, 255, 259, 261,
267
511
512
AUTHORINDEX
Bosse, Yu. G., 79, 95
Bottema,O., 6, 11
Bradley, J. S., 151, 153, 159
de Branges, L., 74
Brass, H., 175, 176, 186, 195, 196, 198, 199,
200, 201, 202
Bressoud, D. M., 71, 72, 75
Brezinski, C., 322
BfI!zis, H., 111, 115, 122
Brodlie, K. W., 133, 159
Bronstein, I. N., 122
Brothers, J. E., 105, 119, 122
Browder, A., 264, 267
Brown, B. M., 127, 129, 133, 138, 139, 140,
144, 145, 147, 148, 149, 150, 151, 156, 157,
158, 159
Brown, G., 74, 75
Brudnyi, Ju. A., 476, 479
de Bruijn, N. G., 61,62,256,257,259,260,
265, 267, 502, 503
Brun, V., 119
Bullen, P. S., xi, 7, 12, 203, 205, 206, 210,
211,487
Burago, Y. D., 117,119,122
Burenkov, V. 1., 78, 80, 95, 277, 288, 490,
491
Burton, G. R., 122
Bushell, P. J., 243, 245, 246, 247
Buslaev, A. P., 80, 95
c
Calderon, A. P., 100, 112
Cäpräu, C., 465,467,469
Cartan, H., 91, 95, 490
Cauchy, A. L., 100, 381, 383, 386
Cavalieri, B., 118
Cavaretta, A., 80, 87,96
Cengiz, B., 363
Chan, T. N., 259,267
Cheeger, J., 121, 122
Chen, J., 25,441,442
Chen, W., 304, 307
Cheney, E. W., 8
Cheng, S. S., 294,307
Chihara, T. S., 382, 392, 395
Cholewa, P. W., 449, 455
Chui, C. K., 75, 80
Ciesielski, K., 375, 378
Ciesielski, Z., 174
Coifman, R., 109, 122
Copson, E. T., 132, 148, 159
Cordes, H., 112
C6rdova, A., 31, 33, 36, 37, 46, 48, 53, 54
van der Corput, J. G., 255, 264, 267
Courant, R., 120, 122
Craven, A. M., 243,245, 247
Criscuolo, G., 218, 238
Crstici, B., xii, 4, 5, 8, 10, 12
Cvetkovic, D., 433, 434, 435, 442, 443
Czerwik, S., 449, 455
D
Damelin, S. B., 217, 224, 235, 236, 238
Dankovic, B. M., 493
Das, M., 395
Datt, B., 259, 260, 267
Davis, Ph. J., 176,201,321
Dawood, Q. M., 259, 266
Daykin, D. E., 243,244, 245, 247
Della Vecchia, B. M., 218, 238
Descartes, R., 381
Deutsch, F., 164
Devide, V., 12
DeVore, R. A., 194, 198, 201
Dewan, K. K., 260, 267
Diananda, P. H., 243,244, 245, 247
Dias, N. G. J., 127,129, 151, 152, 153, 154,
155, 159
Dimik, P., 12
Dimitrovski, D., 4, 10
Dirichlet, P. G. L., 113
Ditzian, Z., 80, 90, 95
Djokovic, D. Z., 5, 12, 13, 19, 20, 21, 22, 27,
243, 244, 246, 248
Djordjevic, R. Z., 4, 5, 6, 9, 10, 11
Doncker-Kapenga, E., 200, 202
Doob, M., 433, 434, 442
Dörfler, W., 121
Doronin, V. G., 78, 95
DoS!;i, Z., 313, 321
Dnibek, P., 272, 277, 288
Drasin, D., 75
Drimbe, M.I., 431
Drinfel'd, V. G., 244,247,248
Duffin, R. J., 254, 267, 89, 90, 95
Durand, A., 250, 267
Durand, L., 186, 201, 320, 321
DureIl, C. V., 248
Duren, P., 75
Duris, C. S., 397, 421
Dzyadyk, V. K., 38, 54, 90, 95, 250, 267
AUTHORINDEX
E
Edelman, A., 394, 395
Egervary, E., 41, 54, 304, 307
Ehrich, S., 200, 201
Elbert, A., 314, 315, 316, 317, 320, 321, 322
Erdelyi, A., 322, 493, 494, 495, 498
Erdelyi, T., 8, 250, 253, 254, 255, 259, 261,
267
Erdös, P., 250, 252, 259, 267
Espelid, T. 0., 202
Evans, W. D., 127, 129, 131, 132, 133, 134,
135, 136, 137, 138, 139, 140, 143, 144, 145,
146, 147, 148, 149, 150, 151, 156, 157, 158,
159, 160, 277, 288
Everitt, W. N., xi, 6, 127, 129, 131, 132,
133, 134, 135, 136, 137, 138, 139, 140, 141,
142, 143, 144, 145, 146, 147, 151, 153, 156,
157, 159, 160, 288, 457, 458, 459, 460, 461,
462,463
F
Fan, K., 289,290,292,304,307,445,447
Favard, J., 383, 395
Fefferman, C., 99, 109, 112, 122
Fejer, L., 43, 63, 64, 66, 68, 75, 250, 267,
304,307
Fekete, M., 250, 267
Feldheim, E., 74, 75
Feiten, M., 201
Fiedler, H., 176,201
Fike, C. T., 387, 395
Fink, A. M., xi, xii, 5, 8, 12, 128, 152, 155,
160, 241, 243, 245, 248, 290, 308, 428, 431
Fitch, J., 68,75
Fleming, R. J., 363,378
Flucher, M., 97, 106, 107, 115, 119, 120, 122
Förster, K. -J., 175, 184, 186, 187, 193, 194,
195, 199, 200, 201
Forti, G. L., 449, 455
Fourier, J. B., 381
Fransen, A., 484
Frappier, C., 257,258,260,267
Frehse, J., 105, 122
Freidkin, E. S., 243, 248
Freidkin, S. A., 243, 248
Fucik, S., 97, 104, 113, 123
G
Gabushin, V. N., 78,80,95,489,491
Gagliardo, E., 106, 122
Gallot, S., 118, 122
Gamelin, T. W., 354, 378
Gardner, R., 259, 261, 267, 268
513
Gargantini, 1., 325, 326, 339, 340
Garnett, J. B., 238
Gasper, G., 74, 75, 311, 322
Gatteschi, L., 317, 321
Gävru\ä, P., 465, 467, 469
Gehring, F. W., 39, 42, 54
Geisberg, S. P., 81, 95
Genz, A., 202
Ger, R., 455
Ghizetti, A., 176,201
Giaquinta, M., 97,105, 112, 113, 114, 115,
116, 122
Gilbarg, D., 97, 101, 104, 109, 110, 112, 113,
114, 122
Giordano, C., 322
Giroux, A., 260, 268, 382, 389, 395
Girshovich, J., 199, 201
Godunova, E. K, 243, 244, 245, 246, 247,
248
Gol'berg, E. M., 382, 383, 388, 395
Goldberg, D., 99, 123
Goldberg, K., 243
Gonchar, A. A., 264
Gonska, H. H., 196, 201
Goodearl, K. R., 354,378
Gori, L. N.-A., 316,322
Gorny, A., 490
Goulden, I. E., 71, 75
Govedarica, V., 471
Govil, N. K., 249, 256, 257, 259, 260, 261,
263, 267, 268, 8
Greiner, R., 33, 40, 54
Grisvard, P., 111, 123, 274, 288
Gronwall, T. H., 64, 75
Grosche, G., 122
Guc, A., 374, 378
Gunning, R. C., 122
Günther, C., 48, 54
Günttner, R., 196, 201
Gusakov, V. A., 80, 95
Guy, R., 434, 443
H
Hadamard, J., 79,95
Hadwiger, H., 117,119,123
Hakopian, H., 168, 174
Hämmerlin, G., 201,202
Hanany, H., 434,443
Harary, F., 433,443
Hardy, G. H., xi, 5, 79, 95, 102, 103, 118,
119, 123, 128, 132, 141, 151, 160, 289, 308
Harnack, A., 114
Harris, L. A., 250, 264, 265, 268
Hartman, P., 313, 322
514
AUTHORINDEX
Hayman, W. K., 135, 136, 160, 320, 322
Heilbronner, E., 434
Heining, H., 271,272,273,277,280,288
Heinz, E., 109, 123
Henrici, P., 325, 326, 340
Hermite, Ch., 381
Hersch, J., 119, 120, 123
Herschern, M., 248
Herzberger, J., 325,337,339,340
Hewitt, E., 74, 75, 104, 123
Hilbert, D., 120, 122
Hirschfeld, R. A., 359, 378
Hölder, 0., 100, 114
Holland, F., 44, 54
Hong, Y., 441, 443
Hörmander, L., 81, 95, 264, 268
Hossu, M., 465, 467, 469
Huber, A., 117,123
Hutson, V., 463
Hyers, D. H., 449, 465, 469
I
Il'in, V. P., 80, 95
Ismail, M. E. H., 318, 322
Ito, K., 102, 103, 123
Ivanov, K., 476, 479
lvanov, V. 1., 250, 268, 56, 62
J
Jackson, D., 64, 75
Jain, V. K., 260, 268
Jakovlev, G. N., 274,288
Jaminson, J. E., 363,378
Janie, R. R., 5, 6, 9, 10, 11, 13, 14
Janous, W., 24
Jarosz, K., 353, 356, 359, 378
Jarvis, R. J., 160
Jensen, J. L. W. V., 101
Jha, S. W., 238
Jia, Rong-Qing, 250, 266, 268
Jiang, D., 90, 95
John,F., 310,322,341,351,352,378
John, 0., 97, 104, 113, 123
Johnson, B. E., 353
Jones, D. S., 134, 135, 136, 160
Joung, H., 223, 238
Jovanovie, M., 471
Jung, S. M., 466, 469
K
Kac, A. M., 394
Kahaner, D. K., 200,202
Kairies, H.-H., 322
Kalajdzie, G., 24
Kallioniemi, H., 490, 491
Karlin, S., 80
Karr, M., 493, 498
Kato, T., 101, 108, 115, 123, 132, 160
Kawohl, B., 119, 123
Kedlaya, K., 203,209,211
Keedwell, A. D., 442, 443
Kemp, R., 493,498
Kerimov, M. K., 317,322
Keckie, J. D., 4, 5, 9, 10, 12, 13, 14, 23, 24
Kirby, V. G., 127, 129, 138, 139, 140, 144,
159, 160
Klambauer, G., 473
Knopfmacher, A., 217,220,238
Knuth, D. E., 493, 498
Kocie, V., 433, 442, 443
Kofanov, V. A., 81, 94
Köhler, P., 182, 183, 196, 201
Kokologiannaki, C. G., 317,322
Kolmogoroff, A. N., (see: Kolmogorov, A.
N.)
Kolmogorov, A. N., 79, 95, 116
König, H., 217,220,221,224,225,235,236,
238
Konovaiov, V. N., 80, 85, 95
Korneichuk, N. P., 78,87,90,95
Kostlan, E., 394, 395
Koumandos, S., 74
Kovalenko, L. G., 476, 479
Kraus, L., 435,442
Kristiansen, G. K., 243
Krotov, V. G., 56, 62
Krull, W., 310,322
Kryakin, Yu. V., 162, 174,475,476,479
Kuczma, M., 322
Kufner, A., 97, 103, 104, 106, 107, 113, 120,
123, 124, 271, 272, 273, 274, 275, 277, 280,
286,288
Kuijlaars, A. B. J., 332,334,340
Kuptsov, N. P., 80,96
Kuz'minyh, A. V., 374,378
Kwong, M. K., 80, 96
L
Labelle, G., 259, 260, 268
Lachance, M. A., 48, 54
Lackovie, I. B., 9, 23, 24
Lafon, J. C., 493,498
AUTHORINDEX
Laforgia, A., 314,315,316,317,321,322,
323
Laguerre, E., 381
Lalli, B. S., 398, 412, 421
Landau, E., 64,79,80,87,96,250
Lc\szI6, L., 382, 389, 395
Lavrentiev, M., 40,54
Lax, P. D., 259, 268
Lazer, A. C., 110, 123
Lazov, P. R., 5
Lebesgue, H. L., 104
Lee, C. M., 307,308
Lester, J., 377, 378
Leviatan, D., 90, 95
Levin, A. L., 218, 219, 220, 226, 227, 228,
238, 239, 250, 261, 268
Levin, M., 199, 201
Levin, V. 1., 243, 244, 245, 246, 247, 248
Lewis, J. T., 40,54,314,323
Li, P., 121, 123
Li, Xin, 250, 254, 256, 261, 262, 263, 268
Lieb, E., 111, 115, 122
LighthilI, M. J., 242,243,244
Ligun, A. A., 78, 80, 81, 90, 94, 95, 96, 183,
201
Lions, P. L., 109, 122
Littlejohn, L. L., 127, 147, 148, 149, 150,
159
Littlewood, J. E., xi, 5, 79, 95, 102, 103,
118, 119, 123, 128, 132, 141, 151, 160, 289,
308
Ljubic, Ju. 1., (see: Ljubich, Yu. 1.)
Ljubich, Yu. 1., 80, 96, 151, 160
Löfström, J., 274,288
Longinetti, M., 473
Lorch, L., 313, 314, 316, 318, 323
Lorentz, G. G., 183, 201, 422
Losonczi, L., 288,303,308
Lovblom, G., 363, 378
Lubinsky, D. S., 213,217,218,219,220,
221, 222, 224, 226, 227, 228, 229, 230, 233,
235,236,238,239,250,261,268,269
Lunter, G., 304, 307, 308
Lyche, T., 397, 421
M
Macke, D. H., 201
Magaril-Il'jaev, G. G, 80, 81, 96
Mahajan, A., 318, 323
Mahler, K., 499,503
Makai, E., 313, 314, 323, 394, 395
Malcolm, M. A., 243, 244, 245, 248
Malik, M. A., 257, 259, 265, 267, 269
Malozemov, V. N., 382,383,388,395
515
Mangasarian, O. L., 397, 421, 422
Marcinkiewicz, J., 216,221,239
Marcus, M., 110, 122
Marden, A., 75
Marden, M., 60, 62, 385, 393, 395
Mare, L., 425, 427, 431
Marinkovic, L. Z., 307, 308
Markoff, A. A., (see: Markov, A. A.)
Markoff, V. A., (see: Markov, V. A.)
Markov, A. A., 249, 251, 269
Markov, V. A., 89, 96, 251, 252, 269
Marsh, D. C. B., 6, 12, 14
Marshali, A., 481, 483, 484
Martio, 0., 42,54
Martic, B., 498
Martos, B., 471, 473
Mastroianni, G., 218, 238, 239, 240
Matano, H., 115, 123
Mate, A., 217,222,224,239
Matjila, D. M., 217, 220, 221, 239
Matorin, A. P., 80, 87, 96
Matsuda, T., 203,209,211
Maz'ja, V. G., 104, 107, 110, 111, 123
McCoy, T. L., 332, 334, 340
McKenna, P. J., 110, 123
McLaughlin, H. W., 6
Mehler, F. G., 190, 191, 192
Meier, J., 196,201
Meinardus, G., 32, 54, 250, 269
Mendeleev, D., 249, 269
Merkle, M., 10, 481, 484
Metcalf, F. T., 6
Meyer, Y., 109, 122
Meyers, N. G., 112, 123
Mhaskar, H. N., 218, 227, 239
Micchelli, C. A., 168,174
Michael, J. H., 13
Mielnik, B., 369, 378
Mihailovic, D., 12
Mihajlovic, M. D., 377
Mijalkovic, M., 485
Mijalkovic, Z., 485
Milman, M., 109, 123
Milojkovic, S., 4
Milovanovic, G. V., xii, 1, 4, 8, 9, 10, 12, 25,
26,64,75,90,96,158,239,250,256,269,
289, 290, 292, 294, 295, 297, 298, 303, 306,
307,308,377,381,385,395,495,496,498
Milovanovic, 1. Z., 9, 289, 290, 292, 294, 295,
298, 303, 307, 308
Mil08evic-Rakocevic, K, 5
Mil08evic, D. M., 509
Minkowski, H., 101, 118
Miranda, C., 110, 123
516
AUTHORINDEX
Mitrinovic, D. S., xi, xii, 1, 2, 3, 4, 5, 6, 7,
8, 9, 10, 11, 63, 64, 65, 75, 96, 127, 128, 152,
155, 160, 203, 205, 206, 210, 211, 239, 243,
245, 248, 250, 256, 269, 289, 290, 295, 308,
318, 323, 339, 381, 385, 395, 428, 429, 431,
445, 447, 448, 472, 473, 481, 484, 485, 487,
493,498,505,506,507,509,510
Mitrinovic, R. S., 1, 19, 20, 21, 22, 23, 27
Mitrovic, Z. M., 448
Moak, D. S., 312, 323
Modenov, P. S., 366, 378
Mohammad, Q. G., 259, 266
Mohapatra, R. N., 249, 250, 253, 254, 256,
257, 261, 262, 263, 268, 269
Mollerup, J., 310, 321
Montel, P., 393
Montgomery, H. L., 224, 239
Moore, R. E., 325,340
Mora, X., 121
MordelI, L. J., 242,243,244,245,247,248
Morrey, B. M., 97, 104, 105, 113, 123
Moser, J., 108, 112, 123
Mthembu, T. Z., 217,218,239,250,261,
269
Muckenhoupt, B., 233, 239, 273, 281, 288
Muldoon, M. E., 309, 310, 313, 314, 315,
316,317,318, 320, 321, 322, 323
Mulholland, H. P., 46, 54
Müller, M. M., 201
Müller, S., 99, 107, 109, 114, 121, 122, 123
N
Nachbin, L., 268
Nagasawa, M., 352, 378
Naimark, M. A., 458,459, 461, 463
Nanjundiah, T. S., 203, 204, 206, 207, 210,
211
NaselI, 1., 318, 323
Natanson, I. P., 161, 174
Nesbitt, A. M., 241, 243, 244, 248
Neumaier, A., 438, 441, 443
Nevai, P., 216, 217, 219, 222, 223, 224, 239,
240, 250, 261, 269
Nevanlinna, F., 352
Newton, 1., 381
Nielsen, N. J., 217, 221, 224, 236, 238
Nikolskii, S. M., 80, 95
Nirenberg, L., 106, 112, 113, 124
Novotna, J., 293,307,308
Nowosad, P., 243, 244, 245, 246, 247, 248
o
O'Hara, P. J., 250,253,256,257,260,261,
269
Obreschkoff, N., 394, 395
Olkin, 1., 481, 483, 484
Olver, F., 496, 498
Opic, B., 103, 106, 120, 124, 271, 288
Orlicz, W., 108, 109
Ortiz, E. L., 320, 322
Osserman, R., 117,124
Ossicini, A., 176, 201
Ostrowski, A. M., 445
Osval'd, P., 56, 62
p
Pacella, F., 117, 124
Parkhomenko, A. S., 366, 378
Passow, E., 166,174
Payne, L. E., 110, 120, 121, 124
Pecaric, J. E., xi, xii, 6, 7, 8, 12, 14, 24, 25,
26, 128, 152, 155, 160, 243, 245, 248, 290,
308, 428, 431, 505, 506, 507, 509, 510
Peck, J. E. L., 248
Pekarskil, A. A., 250,264,269
Peller, V. V., 217,237,239
Persson, L. E., 25, 271, 273, 275, 277, 280,
288
Percinkova, D., 5
Petkovic, Lj. D., 325, 327, 328, 329, 330,
332, 333, 340
Petkovic, M. S., 325, 327, 329, 330, 332, 340
Petras, K., 176, 184, 193, 194, 197, 198, 201,
202
Petrovic, Lj., 481, 484
Petrovic, M., 1, 2
Petrusev, P., (see: Petrushev, P. P.)
Petrushev, P. P, 250,264,269
Pfeffer, A. M., 306,308
Phelps, D., 243, 244
Philippin, G. A., 110, 124
Ph6ng, VÜ Quöc, 132, 160
Pichugov, S. A., 81, 94
Piessens, R., 200, 202
Pinkus, A., 240, 80, 87, 96
Piperevski, B., 5
Plamenevskir, B. A., 111, 123
Plum, M., 129, 159
Poincare, H., 104, 106, 108, 120
Pollard, H., 74, 75
P6lya, G., xi, 5, 60, 62, 66, 79, 95, 97, 118,
119, 120, 123, 124, 128, 132, 141, 151, 160,
289, 308, 502, 503
Pommerenke, eh., 42,54
Ponomarenko, A., 188, 189, 202
AUTHORINDEX
Pop-Stojanovic, Z. R., 21
Popadie, M. S., 9, 12
Popescu, D., 465, 467, 469
Popov, V. A., 250, 264, 269, 476, 480
Popov, B. S., 5, 27
Potts, R. B., 13
Powers, L. D., 441, 443
PreSic, S. B., 20, 21
Protter, M. H., 109, 110, 120, 124
Pryce, J. D., 129, 159
Q
Quarles, D. A., 364, 366, 369, 374, 378
R
Rabinowitz, Ph., 176, 195, 201, 202
Rademacher, H., 342, 378
Radok, J. R. M., 6,12,13
Radon, J., 176,202
Rahman, M., 311, 322
Rahman, Q. 1., 250, 252, 253, 256, 257, 258,
259, 260, 261, 267, 268, 269, 385, 386, 387,
388, 390, 392, 393, 395
Rahmanov, E. A., 227, 240
Rankin, R. A., 243, 244, 248
Ra§a, 1., 431
Rassias, Th. M., xii, 8, 12, 25, 26, 64, 75,
239, 240, 250, 256, 269, 295, 308, 341, 369,
373, 374, 375, 378, 379, 381, 385, 395, 449,
455, 465, 469
Ratschek, H., 325, 330, 335, 336, 340
Ratz, J., 455
Raynal, J., 70,72,75
Reed, M., 110, 124
Richtmeyer, R. D., 326
Riesz, F., 250, 269, 118, 177, 202
Riesz, M., 55, 62, 75, 250, 269
Rigler, D. A. R., 141, 160
Riordan, J., 493, 498
Rochberg, R., 353, 359, 363, 379
Rodriguez, R. S., 250, 253, 254, 256, 257,
260, 261, 262, 263, 268, 269
Rogosinski, W. W., 250,269
Rokne, J., 325,330,335,336,337,339,340
Rota, G.-C., 460, 463
Rowlinson, P., 433, 442, 443
Rubinov, A. M., 471, 473
Rumpf, M., 120, 122
Rusak, V. N., 264
Ruscheweyh, St., 31, 33, 36, 37, 40, 41, 46,
47,48,53,54,135,160,257,258,267,269
RusselI, A., 152, 153, 154, 160
Russo, M. G., 240
517
s
Sachs, H., 433,434,442
Saff, E. B., 48, 54, 76, 218, 227, 239, 250,
260, 261, 269
Sahakian, A., 168,174
Salinas, L., 48, 54
Sambandham, M., 394, 395
Sandor, J., xii, 8, 12
Sandham, H. F., 493,498
Sapkarev,1., 5, 12
Sarantopoulos, Y., 264,270
Sard, A., 176, 202
Sarvas, J., 42, 54
Sato, M., 80
Sauer, N., 434, 443
Sawyer, E., 273, 281, 288
Schaake, G., 255,264,267
Schaeffer, A. C., 89, 90, 95, 250, 254, 267,
270
Scheick, J. T., 250, 270
Schempp, W., 174
Schiffer, M., 120, 124
Schmeisser, G., 195, 200, 201, 250, 252, 259,
260, 261, 268, 269, 381, 385, 386, 387, 388,
390, 392, 393, 394, 395, 396
Schmidt, E., 118
Schoenberg,1. J., 80,87,96,307,308
Schönheim, J., 434,443
Schumaker, L. L., 75, 397, 421, 422
Schwarz, H. A., 100, 118
Schwenk, A. J., 434,443
Scott, L. R., 194, 198, 201
Scraton, R. E., 247,248
Searcy, J. L., 248
Secrest, D. H., 190, 202
Sedgewick, R., 493,498
Seidel, J. J., 438,443
Semendjajew, K. A., 122
Semmes, S., 99, 109, 122, 124
Semrl, P., 369, 373, 378, 449, 455
Sendov, BI., 162, 172, 173, 174, 475, 476,
477, 479, 480
Shadrin, A. Yu., 80, 89, 90, 96, 489, 490, 491
Shapiro, H. S., 241, 242, 243, 244, 248
Sharma, A., 250, 270
Sharma, C. S., 374, 378
Sheil-Small, T., 260, 269
Shen, X. C., 228,240
Shilov, G. E., 79, 95
Shisha, 0., 294,308
Siafarikas, P. D., 317,322
Sierpinski, W., 431
Simic, S. K., 433, 435, 442, 443
Simon, B., 110, 124
518
AUTHORINDEX
Simon, Lo, 114, 124
Singh, Vo, 250, 270
Skorokhodov, So Lo, 317,322
Slavic, Do, 498
Sleeman, Bo Do, 160
Slipieevic, K., 19
Smale, So, 36, 54
Smith, Jo Ho, 434, 443
Smith, Po Wo, 80
Sobolev, So Lo, 102, 103, 106, 107, 108, 109
Soljar, Vo Go, 80, 96
Specht, Wo, 381, 382, 383, 386, 387, 388, 396
Sperb, Ro, 124
Spiess, Jo, 493, 498
Springer, To Ao, 61, 62,502, 503
Srivastava, Ho Mo, 8, 26, 308, 378
Stamate, 1o, 5
Stancu; Do Do, 194, 202
Stankovic, Ljo Ro, 9
Stankovic, Mo So, 23, 493, 495, 498
Stanojevic, Co, 27
Stechkin, So Bo, 78, 96, 490, 491
Stefanovic, Lo Vo, 325, 340
Stegun, I. Ao, 321
Stein, Eo Mo, 80, 96, 99, 102, 109, 112, 113,
122, 124
Steinig, Jo, 66, 74, 75
Stevanovic, Mo Ro, 505
Storozenko, Eo Ao, 56, 62, 476, 480, 499
Stout, Eo Lo, 352, 355, 379
Stromberg, K., 104, 123
Stroud, Ao Ho, 190, 202
Struwe, Mo, 115, 116, 124
Stuart, Jo, 243, 248
Sturm, Cho, 313, 323
Suffridge, To Jo, 37, 48, 54
Sweers, Go, 113
SZo-Nagy, Bo, 177, 202
Szabados, Jo, 80,217,240
SZ3sZ, 0o, 304, 307, 316
Szegö, Go, 46, 54, 55, 57, 58, 60, 62, 66, 67,
68, 69, 74, 75, 97, 118, 119, 120, 121, 124,
186, 202, 255, 256, 259, 265, 270, 304, 308,
320, 323, 384, 392, 396, 502, 503
Szego, Po, 313, 314, 316, 318, 323
T
Tabor, Jo, 449,455
Taikov, Lo Vo, 80, 96
Takev, Mo Do, 476, 477, 479, 480
Talenti, Go, 105, 117, 118, 119, 124
Tanasescu, Co, 24
Tashev, So, 164, 174
Taussky, 0o, 289, 290, 292, 304, 307
Telyakovskii, So Ao, 250,270
Thomas, Do Go So, 248
Thorin, Go 0o, 102
Tihomirov, Vo Mo, 80, 81, 95, 96
Timofeev, Vo Go, 80, 96
Timoshin, 0o Ao, 80, 85, 96
Titchmarsh, Eo Co, 128, 160, 461, 463
Todd,Jo, 289,290,292,304,307,445,447
Torchinsky, Ao, 109, 124
ToSic, Do Djo, 13
Totik, Vo, 90, 95, 240
Trajkovic, Mo, 327,328,332,335,340
Tricarico, Mo, 117, 124
Triebei, Ho, 274, 288
Triekovic, So Bo, 493
Troesch, Bo Ao, 243,244,245,247,248
Trudinger, No So, 97, 101, 104, 108, 109, 110,
112, 113, 114, 122
Turajlic, So So, 23
Turan, Po, 250, 259, 270, 316, 381, 382, 392,
394, 395, 396
u
Überhuber, Co Wo, 200, 202
Ubhaya, Vo Ao, 471, 473
Ulam, So Mo, 449
Ulear, Jo, 12, 13
Usmani, Ro Ao, 403, 422
v
Vakarchuk, Mo Bo, 81, 94
de la ValIee Poussin, Co, 250, 270
Varga, Ro So, 47, 48, 54, 76
Varma, Ao K., 250, 267
Vasic, Po Mo, xi, 4, 5, 6, 7, 10, 11, 12, 13, 14,
21, 22, 23, 27, 128, 160, 203, 205, 206, 210,
211, 289, 290, 308, 445, 447, 448, 472, 473,
487,493,498
Velikova, Eo Ao, 26
Vermes, Ro, 382, 396
Vertesi, Po, 217,239,240
Vetterlen, Do Ho, 260, 268
Vidav,l., 17
Vietoris, Lo, 64, 65, 66, 70, 76
van Vleck, Eo Bo, 393
Volenec, Vo, xi, 6, 12, 24, 25, 505, 506, 507,
509, 510
Vorob'eva, Ao 1., 471, 473
Voronovskaja, Eo Vo, 252, 254, 270
Vosmansky, Jo, 323
AUTHORINDEX
w
Walter, W., 159, 288, 308
Wang, K. Y., 74
Wang, X., 325, 340
Ward, J. D., 75
Watson, G. N., 312,313,314,315,317,318,
319, 323
Weinberger, H., 109, 110, 111, 118, 120, 121,
124
Weiss, G., 102, 124
Wente, H. C., 109, 125
Whippie, F. J. W., 71
Whitney, H., 110,162,174,475,476,477,
480
Wilkinson, J. H., 325
Wilson, J., 70, 72, 73, 76
Wirths, K. J., 48,54
Wirtinger, W., 289,290
Wong, J. S. W., 307,308
Wong, P. J. Y., 397, 404, 405, 406, 421, 422
Wrigge, S., 484
Wu, T., 325, 335, 336, 337, 340
x
Xu, Y., 217,230,231,238,240
y
Yang, G. S., 307, 308
519
Yanushauskas, A., 26,308
Yau, S. T., 121, 123, 125
Yin, X. R., 290, 304, 308
You, C. D., 307, 308
Young, W. H., 101
z
Zahar, R. V. M., 54, 322
Zalgaller, V. A., 117, 119, 122
Zang, T., 261
Zaric, B., 5
Zeidler, E., 106, 125
Zelazko, W., 359,378
Zeller, K., 174
Zettl, A., 80, 96, 132, 133, 156, 157, 160, 463
Zheng, S., 325, 340
Zhong, L., 217,225,228,237,240,267
Zhu,L., 217,225,228,237,240
Ziegler, D., 122
Ziegler, V., 122
Ziemer, W. P., 97, 99, 100, 101, 102, 104,
105, 119, 122, 125
Zizovic, M. R., 505
Zulauf, A., 242, 243, 244, 245, 248
Zvengrowski, P., 366
Zvyagintsev, A. 1., 80
Zygmund, A., 56, 57, 62, 112, 214, 216, 221,
229, 239, 240, 250, 260, 270
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