Journal of Inequalities in Pure and
Applied Mathematics
ON A CONVOLUTION CONJECTURE OF BOUNDED FUNCTIONS
1
KRZYSZTOF PIEJKO, 1 JANUSZ SOKÓŁ AND 2 JAN STANKIEWICZ
1 Department
of Mathematics
Rzeszów University of Technology
ul. W. Pola 2, 35-959 Rzeszów, Poland.
EMail: piejko@prz.rzeszow.pl
volume 6, issue 2, article 34,
2005.
Received 30 November, 2004;
accepted 03 March, 2005.
Communicated by: K. Nikodem
EMail: jsokol@prz.rzeszow.pl
2 Institute
of Mathematics
University of Rzeszów
ul. Rejtana 16A
35-959 Rzeszów, Poland.
EMail: jan.stankiewicz@prz.rzeszow.pl
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
229-04
Quit
Abstract
We consider the convolution P (A, B) ? P (C, D) of the classes of analytic func1+Cz
1+Az
tions subordinated to the homographies 1−Bz
and 1−Dz
respectively, where
A, B, C, D are some complex numbers. In 1988 J. Stankiewicz and Z. Stankiewicz
[11] showed that for certain A, B, C, D there exist X, Y such that P (A, B) ?
P (C, D) ⊂ P (X, Y ). In this paper we verify the conjecture that P (X, Y ) ⊂
(A, B) ? P (C, D) for some A, B, C, D, X, Y.
2000 Mathematics Subject Classification: Primary 30C45; Secondary 30C55.
Key words: Hadamard product, Convolution, Subordination, Bounded functions.
Krzysztof Piejko, Janusz Sokół
and Jan Stankiewicz
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
On a Convolution Conjecture of
Bounded Functions
Title Page
3
6
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 16
J. Ineq. Pure and Appl. Math. 6(2) Art. 34, 2005
http://jipam.vu.edu.au
1.
Introduction
Let ∆ = {z ∈ C : |z| < 1} denote the open unit disc and let H be the
class of functions regular in ∆. We will denote by N the class of functions
f ∈ H normalized by f (0) = 1. The class of Schwarz functions Ω is the
class of functions ω ∈ H, such that ω(0) = 0 and |ω(z)| < 1 for z ∈ ∆. We
say that a function f is subordinate to a function g in ∆ (and write f ≺ g or
f (z) ≺ g(z)) if there exists a function ω ∈ Ω such that f (z) = g (ω(z)); z ∈ ∆.
If the function g is univalent in ∆, then f ≺ g if and only if f (0) = g(0) and
f (∆) ⊂ g(∆). In this case we have ω(z) = g −1 (f (z)).
Let the functions f and g be of the form
f (z) =
∞
X
n=0
n
an z ,
g(z) =
∞
X
bn z n ,
Title Page
We say that the Hadamard product of f and g is the function f ? g if
(f ? g)(z) = f (z) ? g(z) =
Krzysztof Piejko, Janusz Sokół
and Jan Stankiewicz
z ∈ ∆.
n=0
∞
X
On a Convolution Conjecture of
Bounded Functions
n
an b n z .
n=0
Contents
JJ
J
II
I
Go Back
J. Hadamard [2] proved, that the radius of convergence of f ? g is the product of
the radii of convergence of the corresponding series f and g. The function f ? g
is also called the convolution of the functions f and g.
For the classes Q1 ⊂ H and Q2 ⊂ H the convolution Q1 ? Q2 is defined as
Q1 ? Q2 = {h ∈ H; h = f ? g, f ∈ Q1 , g ∈ Q2 }.
Close
Quit
Page 3 of 16
J. Ineq. Pure and Appl. Math. 6(2) Art. 34, 2005
http://jipam.vu.edu.au
The problem of connections between functions f, g and their convolution f ?
g or between Q1 , Q2 and Q1 ? Q2 has often been investigated. Many conjectures
have been given, however, many of them have still not been verified.
In 1958, G. Pólya and I.J. Schoenberg [7] conjectured that the Hadamard
product of two convex mappings is a convex mapping. In 1961 H.S. Wilf [12]
gave the more general supposition that if F and G are convex mappings in ∆
and f is subordinate to F , then the convolution f ? G is subordinate to F ? G.
In 1973, S. Rusheweyh and T. Sheil-Small [9] proved both conjectures and
more results of this type. Their very important results we may write as:
Theorem A. If f ∈ S c and g ∈ S c , then f ? g ∈ S c , where S c is the class of
univalent and convex functions. Moreover S c ? S c = S c .
On a Convolution Conjecture of
Bounded Functions
Krzysztof Piejko, Janusz Sokół
and Jan Stankiewicz
Theorem B. If F ∈ S c , G ∈ S c and f ≺ F, then f ? G ≺ F ? G.
Title Page
In 1985, S. Ruscheweyh and J. Stankiewicz [10] proved some generalizations of Theorems A and B:
Theorem C. If the functions F and G are univalent and convex in ∆, then for
all functions f and g, if f ≺ F and g ≺ G then f ? g ≺ F ? G.
Contents
JJ
J
II
I
For the given complex numbers A, B such that A + B 6= 0 and |B| ≤ 1 let
us denote
1 + Az
P (A, B) = f ∈ N : f (z) ≺
.
1 − Bz
Go Back
W. Janowski introduced the class P (A, B) in [3] and considered it for some
real A and B. If A = B = 1 then the class P (1, 1) is the class of functions
with positive real part (Carathéodory functions). Note, that for |B| < 1 the
Page 4 of 16
Close
Quit
J. Ineq. Pure and Appl. Math. 6(2) Art. 34, 2005
http://jipam.vu.edu.au
class P (A, B) is the class of bounded functions. In 1988 J. Stankiewicz and Z.
Stankiewicz [11] investigated the convolution properties of the class P (A, B)
and proved the following theorem:
Theorem D. If A, B, C, D are some complex numbers such that A + B 6= 0,
C + D 6= 0, |B| ≤ 1, |D| ≤ 1, then
P (A, B) ? P (C, D) ⊂ P (AC + AD + BC, BD),
moreover, if |B| = 1 or |D| = 1, then
P (A, B) ? P (C, D) = P (AC + AD + BC, BD).
The equality between the class P (A, B) ? P (C, D) and the class P (AC +
AD+BC, BD) for |B| < 1 and |D| < 1 was an open problem. In [5] K. Piejko,
J. Sokół and J. Stankiewicz proved that the above mentioned classes are different. In this paper we give an extension of this result. First we need two
theorems.
Theorem E (G. Eneström [1], S. Kakeya [4]). If a0 > a1 > · · · > an > 0,
where n ∈ N, then the polynomial p(z) = a0 + a1 z + a2 z 2 + · · · + an z n has no
roots in ∆ = {z ∈ C : |z| ≤ 1}.
P∞
n
Theorem F (W. Rogosinski [8]).
P∞ If the nfunction f (z)P=∞ n=0 2αn z Pis∞subordinated to the function F (z) = n=0 βn z in ∆, then n=0 |αn | ≤ n=0 |βn |2 .
On a Convolution Conjecture of
Bounded Functions
Krzysztof Piejko, Janusz Sokół
and Jan Stankiewicz
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 5 of 16
J. Ineq. Pure and Appl. Math. 6(2) Art. 34, 2005
http://jipam.vu.edu.au
2.
Main Result
We prove the following theorem.
Theorem 2.1. Let A, B, C, D be some complex numbers such that B + A 6= 0,
C + D 6= 0, |B| < 1, |D| < 1, then there are not complex numbers X, Y ,
X + Y 6= 0, |Y | ≤ 1 such that P (X, Y ) ⊂ P (A, B) ? P (C, D).
Proof. As in [5], the proof will be divided into three steps. First we give a
family of bounded functions ω ν ; ν = 1, 2, 3 . . . having special properties of
coefficients. Afterwards we construct, using ω ν , a function belonging to the
class P (X, Y ) and finally we will show that such a function is not in the class
P (A, B) ? P (C, D).
Now we use a method of E. Landau [6] and find some functions ω ν , ν = 1, 2,
3, . . ., which are basic in this proof. We observe that
1
=
1−z
1
√
1−z
2
=
∞
X
pk z
2
3
= 1 + z + z + z + ··· ,
k=0
where
(2.1)
1 · 3 · · · · · (2k − 1)
; k = 1, 2, 3, . . .
p0 = 1 and pk =
2 · 4 · · · · · 2k
Kν (z) =
k=0
Title Page
JJ
J
II
I
Go Back
Close
Quit
For some ν ∈ N we set
ν
X
Krzysztof Piejko, Janusz Sokół
and Jan Stankiewicz
Contents
!2
k
On a Convolution Conjecture of
Bounded Functions
Page 6 of 16
pk z k = 1 + p1 z + p2 z 2 + p3 z 3 + · · · + pν z ν
J. Ineq. Pure and Appl. Math. 6(2) Art. 34, 2005
http://jipam.vu.edu.au
and note that
Kν2 (z) = 1 + z + z 2 + · · · + z ν + bν+1 z ν+1 + · · · + b2ν z 2ν ,
where bν+1 , . . . , b2ν ∈ C. Let ω ν be given by
z ν+1 Kν z1
z ν + p1 z ν−1 + p2 z ν−2 + · · · + pν
ν
(2.2)
ω (z) =
=z
.
Kν (z)
1 + p1 z + p2 z 2 + · · · + pν z ν
On a Convolution Conjecture of
Bounded Functions
Since
pk > p k
2k + 1
= pk+1
2k + 2
where k = 0, 1, 2, 3, . . . ,
then for ν ∈ N we have 1 > p1 > p2 > p3 > · · · > pν > 0.
Applying Theorem E to the polynomial Kν we obtain Kν (z) 6= 0 for |z| ≤ 1,
hence the function ω ν is regular in ∆. Moreover ω ν (0) = 0 and on the circle
|z| = 1 we have
K (eit )
(ν+1)it
−it
ν
ν it e
Kν (e ) ω e = = |Kν (eit )| = 1; t ∈ R.
Kν (eit )
In this way we conclude that for ν ∈ {1, 2, 3, . . .}
(2.3)
ω ν ∈ Ω.
Krzysztof Piejko, Janusz Sokół
and Jan Stankiewicz
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 7 of 16
ν
Let, for a certain ν ∈ N, the function ω be represented by following power
series expansions: ω ν (z) = γ1ν z + γ2ν z 2 + γ3ν z 3 + · · · and let sνn (z) denote the
J. Ineq. Pure and Appl. Math. 6(2) Art. 34, 2005
http://jipam.vu.edu.au
partial sum
sνn (z)
=
n
X
γkν z k = γ1ν z + γ2ν z 2 + γ3ν z 3 + · · · + γnν z n
k=1
for all n ∈ {1, 2, 3, . . . , ν + 1}.
Now we will estimate sνn (1) = γ1ν + γ2ν + γ3ν + · · · + γnν . If we integrate on
a circle C : |z| = r in a counterclockwise direction with 0 < r < 1, then we
obtain
Z
Z
a
m
(2.4)
az dz = 0 and
dz = 2πai,
z
C
C
On a Convolution Conjecture of
Bounded Functions
Krzysztof Piejko, Janusz Sokół
and Jan Stankiewicz
for all integers m 6= −1 and a ∈ C. Hence
sνn (1) = γ1ν + γ2ν + γ3ν + · · · + γnν
Z
1
1
1
1
1
ν
ω (z)
+
+
+ · · · + n+1 dz
=
2πi
z2 z3 z4
z
C
Z ν
ω (z)
1
=
1 + z + z 2 + z 3 + · · · + z n−1 dz
n+1
2πi
z
C
Z ν
1
ω (z)
=
Q(z)dz,
2πi
z n+1
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
C
Page 8 of 16
where
Q(z) = 1 + z + z 2 + · · · + z n−1 + dn zn + dn+1 z n+1 + · · · + ds z s
J. Ineq. Pure and Appl. Math. 6(2) Art. 34, 2005
http://jipam.vu.edu.au
is any polynomial whose first n terms are equal to 1 + z + z 2 + · · · + z n−1 .
If for n ≤ ν + 1 we set
Q(z) = Kν2 (z) = 1+z +z 2 +· · ·+z n−1 +z n +· · ·+z ν +bν+1 z ν+1 +· · ·+b2ν z 2ν ,
then we obtain
sνn (1)
1
=
2πi
Z
ω ν (z) 2
K (z)dz.
z n+1 ν
C
From (2.2) it follows that
sνn (1) =
1
2πi
Z
z ν+1 Kν ( z1 )
Kν (z)
Kν2 (z)dz
z n+1
On a Convolution Conjecture of
Bounded Functions
Krzysztof Piejko, Janusz Sokół
and Jan Stankiewicz
C
1
z Kν
Kν (z)dz
z
C
Z
1
1
1
1
ν−n
=
z
1 + p1 + p2 2 + · · · + pν ν
2πi
z
z
z
C
× 1 + p1 z + p2 z 2 + · · · + pν z ν dz.
1
=
2πi
Z
ν−n
Contents
JJ
J
II
I
Go Back
Using (2.4) we get
sνn (1) = pν−n+1 + pν−n+2 p1 + pν−n+3 p2 + · · · + pν pn−1 ,
where (pk ) are given by (2.1), and so
(2.5)
Title Page
sνn (1) =
n
X
k=1
γkν =
n
X
k=1
Close
Quit
Page 9 of 16
pν−n+k pk−1 .
J. Ineq. Pure and Appl. Math. 6(2) Art. 34, 2005
http://jipam.vu.edu.au
By the above we have γ1ν = sν1 (1) = pν and for all ν ∈ {1, 2, 3, . . .} and
n ∈ {2, 3, 4, . . . , ν + 1}
γnν = sνn (1) − sνn−1 (1) =
n
X
pν−n+k pk−1 −
k=1
n−1
X
pν−n+1+k pk−1 .
k=1
Therefore
γnν
(2.6)
=
n−1
X
(pν−n+k − pν−n+1+k ) pk−1 + pν pn−1 .
On a Convolution Conjecture of
Bounded Functions
k=1
The sequence (pn ) is positive and decreasing, then from (2.6) it follows that
γnν > 0 for all ν ∈ N and n ∈ {1, 2, 3, . . . , ν + 1}.
(2.7)
Title Page
We conclude from (2.5) that for n = ν + 1
ν
sνν+1 (1) = γ1ν + γ2ν + · · · + γνν + γν+1
=
ν+1
X
pk−1 pk−1 = 1 +
k=1
Since
(2.8)
Krzysztof Piejko, Janusz Sokół
and Jan Stankiewicz
P∞
k=1
ν
X
Contents
p2k .
k=1
p2k = ∞ then
II
I
Go Back
lim sνν+1 (1) = lim
ν→∞
JJ
J
ν→∞
1+
ν
X
!
p2k
= +∞.
k=1
Now using the properties of ω ν we construct the function belonging to the
class P (X, Y ) which is not in the class P (A, B) ? P (C, D), where |B| < 1 and
|D| < 1.
Close
Quit
Page 10 of 16
J. Ineq. Pure and Appl. Math. 6(2) Art. 34, 2005
http://jipam.vu.edu.au
Since for |x| = 1 we have P (A, B) = P (Ax, Bx), we can assume without
loss of generality that B ∈ [0; 1), D ∈ [0; 1) and Y ∈ [0; 1].
For fixed ν ∈ N let hν be given by
hν (z) =
(2.9)
1 + Xω ν (z)
,
1 − Y ω ν (z)
where wν ∈ Ω is given by (2.2). It is clearly that hν ∈ P (X, Y ). Suppose that
there exist the functions f ∈ P (A, B), g ∈ P (C, D) such that
f (z) ? g(z) = hν (z).
(2.10)
Let the functions f and g have the following form:
f (z) =
and
1 + Aω1 (z)
ω1 (z)
= 1 + (A + B)
1 − Bω1 (z)
1 − Bω1 (z)
ω2 (z)
1 + Cω2 (z)
g(z) =
= 1 + (C + D)
,
1 − Dω2 (z)
1 − Dω2 (z)
where ω1 , ω2 ∈ Ω. For simplicity of notation we write
f (z) = 1 + (A + B)f˜(z),
g(z) = 1 + (C + D)g̃(z),
On a Convolution Conjecture of
Bounded Functions
Krzysztof Piejko, Janusz Sokół
and Jan Stankiewicz
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
where
(2.11)
Page 11 of 16
f˜(z) =
ω1 (z)
1 − Bω1 (z)
and
g̃(z) =
ω2 (z)
.
1 − Dω2 (z)
J. Ineq. Pure and Appl. Math. 6(2) Art. 34, 2005
http://jipam.vu.edu.au
Using these notations we can rewrite (2.10) as
h
i
1 + Xω ν (z)
˜
1 + (A + B)f (z) ? [1 + (C + D)g̃(z)] =
1 − Y ω ν (z)
and so
f˜(z) ? g̃(z) =
(2.12)
where h̃ν (z) =
X +Y
h̃ν (z),
(A + B)(C + D)
ω ν (z)
.
1−BDω ν (z)
ν ˜
On a Convolution Conjecture of
Bounded Functions
Let the functions h̃ , f and g̃ have the following expansions in ∆:
(2.13)
h̃ν (z) =
∞
X
f˜(z) =
cνn z n ,
n=1
∞
X
an z n ,
g̃(z) =
n=1
∞
X
bn z n .
n=1
z
z
From (2.11) it follows, that f˜(z) ≺ 1−Bz
and g̃(z) ≺ 1−Dz
, and hence by
Theorem F (and since 0 ≤ B < 1 and 0 ≤ D < 1) we obtain
∞
X
1
|an |2 ≤
1 − B2
n=1
and
∞
X
1
|bn |2 ≤
.
2
1
−
D
n=1
Let us note, that
∞
X
n=1
Krzysztof Piejko, Janusz Sokół
and Jan Stankiewicz
Title Page
Contents
JJ
J
II
I
Go Back
Close
|an |2 + |bn |2 =
∞
X
(|an | − |bn |)2 + 2|an ||bn |
n=1
1
1
≤
+
.
1 − |B|2 1 − |D|2
Quit
Page 12 of 16
J. Ineq. Pure and Appl. Math. 6(2) Art. 34, 2005
http://jipam.vu.edu.au
From (2.12) and (2.13) we obtain
X +Y
cνn ,
(A + B)(C + D)
an b n =
for
n = 1, 2, 3, . . . ,
therefore
∞
X
∞
X ν
1
2(X
+
Y
)
1
(2.14)
+
−
|cn |.
(|an | − |bn |) ≤
(A + B)(C + D) 2
2
1
−
B
1
−
D
n=1
n=1
2
On a Convolution Conjecture of
Bounded Functions
Now we observe that
∞
X
cνn z n
n=1
Krzysztof Piejko, Janusz Sokół
and Jan Stankiewicz
ω ν (z)
= h̃ (z) =
1 − Y ω ν (z)
ν
= ω ν (z) + Y [ω ν (z)]2 + Y 2 [ω ν (z)]3 + · · ·
2
= γ1ν z + γ2ν z 2 + · · · + Y γ1ν z + γ2ν z 2 + · · ·
3
+ Y 2 γ1ν z + γ2ν z 2 + · · · + · · ·
= γ1ν z + γ2ν + Y (γ1ν )2 z 2 + γ3ν + 2Y γ1ν γ2ν + Y 2 (γ1ν )3 z 3 + · · · .
Contents
JJ
J
II
I
Go Back
Since Y ∈ [0; 1] and since (2.7) we have
∞
X
Title Page
Close
|cνn |
=
|γ1ν |
+ γ2ν + Y (γ1ν )2 + γ3ν + 2Y γ1ν γ2ν + Y 2 (γ1ν )3 + · · ·
Quit
n=1
≥
ν+1
X
n=1
Page 13 of 16
|cνn |
≥
γ1ν
+
γ2ν
+
γ3ν
+ ... +
ν
γν+1
=
sνν+1 (1).
J. Ineq. Pure and Appl. Math. 6(2) Art. 34, 2005
http://jipam.vu.edu.au
From the above we have for all ν ∈ {1, 2, 3, . . .}
∞
X
(2.15)
|cνn | ≥ sνν+1 (1).
n=1
Combining (2.14) and (2.15) we obtain
∞
X
ν
1
2(X
+
Y
)
1
(2.16)
(|an | − |bn |) ≤
+
−
(A + B)(C + D) sν+1 (1).
2
2
1
−
B
1
−
D
n=1
2
From (2.8) it follows that we are able to choose a suitable ν such that the
right side of (2.16) is negative. In this way (2.16) follows the contradiction and
the proof is complete.
From Theorem 2.1 and Theorem D we immediately have the following
Corollary 2.2. The classes P (A, B) ? P (C, D) and P (AD + AC + BC, BD)
are equal if and only if |B| = 1 or |D| = 1.
On a Convolution Conjecture of
Bounded Functions
Krzysztof Piejko, Janusz Sokół
and Jan Stankiewicz
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 14 of 16
J. Ineq. Pure and Appl. Math. 6(2) Art. 34, 2005
http://jipam.vu.edu.au
References
[1] G. ENESTRÖM, Härledning af en allmä formel för antalet pensionärer
som vid en godtycklig tidpunkt förefinnas inom en sluten pensionskassa,
Öfversigt af Kongl. Vetenskaps – Akademiens Förhandlingar, 50 (1893),
405–415.
[2] J. HADAMARD, Théoréme sur les séries entiéres, Acta Mathematica, 22
(1898), 55–63.
[3] W. JANOWSKI, Some extremal problems for certain families of analytic
functions, Ann. Polon. Math., 28 (1957), 297–326.
[4] S. KAKEYA, On the limits of the roots of an algebraic equation with positive coefficients, Tôhoku Math. J., 2 (1912), 140–142.
On a Convolution Conjecture of
Bounded Functions
Krzysztof Piejko, Janusz Sokół
and Jan Stankiewicz
Title Page
[5] K. PIEJKO, J. SOKÓŁ AND J. STANKIEWICZ, On some problem of the
convolution of bounded functions, North-Holland Mathematics Studies,
197 (2004), 229–238.
[6] E. LANDAU, Darstellung und Begrundung einiger neurer Ergebnise der
Funktionentheorie, Chelsea Publishing Co., New York, N.Y. (1946).
[7] G. PÓLYA AND I.J. SCHOENBERG, Remarks on de la Vallée Pousin
means and convex conformal maps of the circle, Pacific J. Math., 8 (1958),
295–334.
[8] W. ROGOSINSKI, On the coefficients of subordinate functions, Proc.
London Math. Soc., 48(2) (1943), 48–82.
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 15 of 16
J. Ineq. Pure and Appl. Math. 6(2) Art. 34, 2005
http://jipam.vu.edu.au
[9] S. RUSCHEWEYH AND T. SHEIL-SMALL, Hadamard product of
schlicht functions and the Pólya-Schoenberg conjecture, Comtnt. Math.
Helv., 48 (1973), 119–135.
[10] S. RUSCHEWEYH AND J. STANKIEWICZ, Subordination under convex
univalent functions, Bull. Polon. Acad. Sci. Math., 33 (1985), 499–502.
[11] J. STANKIEWICZ AND Z. STANKIEWICZ, Convolution of some classes
of function, Folia Sci. Univ. Techn. Resov., 48 (1988), 93–101.
[12] H.S. WILF, Subordintating factor sequences for convex maps of the unit
circle, Proc. Amer. Math. Soc., 12 (1961), 689–693.
On a Convolution Conjecture of
Bounded Functions
Krzysztof Piejko, Janusz Sokół
and Jan Stankiewicz
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 16 of 16
J. Ineq. Pure and Appl. Math. 6(2) Art. 34, 2005
http://jipam.vu.edu.au