Extremal properties characterizing weakly l-valent principal functions
by Dennis Evo Garoutte
A thesis submitted to the Graduate Faculty in partial fulfillment of the requirements for the degree
DOCTOR OF PHILOSOPHY in Mathematics of
Montana State University
© Copyright by Dennis Evo Garoutte (1966)
Abstract:
Extremal properties, characterizing weakly λ-valent principal functions F0^ and F1^ defined on planar
bordered Riemann surfaces, are developed. The λ-valent principal functions are constructed by using
Sario’s linear operator method. The extremal properties derived are analogous, in a rather natural way,
to those of the univalent principal functions F0 and F1 of Sario. Such extremal properties characterize
the λ-valent principal functions in a class much larger than the class of weakly λ-valent functions. The
class consists of functions λ-valent near the border and near a fixed interior point, with the valence
being arbitrary elsewhere.
The principal function F0^λ is shown to be the λ-th power of the univalent function F0. The function
F0^λ then has the geometric property of being a weakly λ-valent radial slit disk mapping. The
functional extremized by F1^λ is slightly different than the expected one. It does reduce to the expected
one, however, when the class of competing functions is taken to be a smaller class consisting only of
properly normalized λ-th powers of univalent functions. ,
I1
1
j:
EXTREMAL PROPERTIES CHARACTERIZING WEAKLY
X-VALENT PRINCIPAL FUNCTIONS
DENNIS EVO GAROUTTE
■!
A thesis submitted to the Graduate Ihculty in partial
fulfillment of the requirements for the degree
i
i '
DOCTOR OF PHILOSOPHY!
Mathematics
Approved:
ad, Major Department
< ? - 9 u
Chairman, Examining Committee
Ii
^ G r a d u a t e Dean
”
I
. i
I
MONTANA STATE! UNIVERSITY
Bozeman, Montana I
!
i ;
December, 19 6 6 ■ I
/
.ii
1' 1
I
I
iii
ACKNOWLEDGMENT
The author wishes to thank his thesis advisor, Dr. Paul Nickel,
;
1
for his proposal of the thesis topic, a nd for his assistance during its
solution.
'
i
'!
i I
i:
■I
I -
■
, I
! I
;I
sI
Mi
I
TABLE O F ,COETEHTS
Chapter
Page
INTRODUCTION
I.
'PRELIMINARIES
§
I.
S . 2.
§
II.
3«
§
3*
g
6.
Basic Properties; of Rieinann !surfaces . .
Normal Linear Operators,' on a Riemann
Surface
. . . . . .
The Reduction Theorem
i. . . .
L- . .j . . ; . . . .
§
Approximating Surfaces
. . .
.
Extremal Properties of:Harmbnic Functions
on Approximating Surfaces
. . . . . .
Extremal Properties of Analytic Functions
on Approximating Surfaces . . . . . . .
J.
Extremal Properties
. Principal Harmonic
8 . Extremal Properties
Principal Analytic
9*
§10.
X-th
. . .
X-th
. . .
The Extremal Property,of the Second X-th
Principal Harmonic Function . . . . . .
Extremal Properties of the;Second X-th
Principal Analytic' F u n c t i o n ...........■
LITERATURE CITED
' W
of; the'First
Function
.
o f the First
Function
.
EXTREMAL PROPERTIES OF THE SECOND X - T H PRINCIPAL
HARMONIC A N D ANALYTIC FUNCTIONS 1 . . . . . . .
g
4
4
6
9
11
Construction of Harmonic Functions on
EXTREMAL PROPERTIES OF THE FIRST X-TH PRINCIPAL
HARMONIC A N D ANALYTIC FUNCTIONS ..............
§
I V .1
. j. . .j...........
EXTREMAL PROPERTIES OF 'HARMONIC A N D ANALYTIC
FUNCTIONS ON APPROXIMATING SURFACES . . . . .
• I
'
::
'
1
.
§ '4.
III.
. . .
. . . . . . . . . ...........
11
12
16
19
19
23
28
28
34
36
V
ABSTRACT
Extremal pr operties , characterizing weakly A-valent principal
functions F 0 a n d F^
defined on planar Bordered Riemann surfaces,
are developed.
The X-vulent principal functions are constructed by
using S a r i o 1s linear operator method.
The extremal properties derived
are a n a logous, in a rather natural way, to those of the univalent prin c i ­
pal functions F 0 and F-] of S a r i o . Such extremal properties characterize
the X-valent principal functions in a class much larger than the class of
weakly X-valent functions. . The class consists of functions X-valent near
the border and near a fixed interior point, with the valence being arbit­
rary elsewhere.
The principal function F 0 is shown to b e the X-th power of the
univalent function F 0 . The function Fg then has the geometric property
of b eing a w eakly X-valent radial slit disk mapping.
The functional
extremized b y F^
is slightly different than the expected one.
It does
reduce to the expected one, however, when the class of competing functions
is taken to be a smaller' class consisting only of properly normalized X-th
powers of univalent f u n ctions.
INTRODUCTION
L. Sario has used linear operators to establish the existence of
certain univalent canonical mappings of planar bordered Riemann surfaces
W
onto slit disks [ 6 ] .
These mappings
F q and F^-, called principal
analytic functions, are foraied from principal harmonic functions, which
themselves are constructed by applying the linear operator method of [ 5 / 2 •
The singularity function
s, wh i ch plays a central role in the linear
operator method, is defined near the border y- as the flux function s ^ ,
2 rc
which is constant on y w i t h flux
point
surface
I it is equal to
log
\z
there, while near a fixed interior
- £ |'.
By exhausting the planar bordered
W, one constructs the mappings
Fq
and
F
of
W
onto a slit
plane disk, w i t h radial or circular slits, possibly degenerate, removed.
In this investigation, we study the extremal properties of k-th
principal functions
s
is taken to be
Fq
S^
and
near
F^
T
to be k-th
X
Fq
x
and
powers by
F^
2jtX
log
.
jz
- g |
near £ .
Here s^
In a manner similar to [ 6 ] the
are constructed.
These maps were first shown
Nickel ['4,] and hence their geometric properties
are completely understood.
mappings.
X
and
is constant on T with a flux of
canonical maps
which result when the singularity function
Namely, they are weakly k-valent slit disk
In a manner different from [ 4 ], it will be shown b y using
derived extr'emal properties that
similar property for
F^
F^
is the
k-th power of •F q .
The
is not proved here because it is needed to
I
i I
derive a n extremal property for
F
i ,*
Chapter I is introductory b y !nature,! and !contains important theorems
and definitions used in the sequel. ;
. j
: ' : I
■:
■ ' _
In Claapter II, we consider a 'planar bordered Riemann surface W
of infinite connectivity.
border c o m p m e n t
sion.
T
It is assumed,that
The surface W is approximated by an!increasing sequence of compact
containing T
and
are constructed.
, F
W
£
.
F
W
n
!
, each of;finite connectivity and each
:
:
:
x
the analytic functions
F
;
; j
:|
! ,
on
'
x
and
Among a class of normalized analytic functions
Fin
F
of
maximizes the functional
on
log r(F)
+ ' Aj (F) :
minimizes
in
2jt X
Here
f
nj
On
: 2rtX
and
I
i- j
£ e Int Wj to be held fixed in our discus­
and a point
bordered Riemann surfaces
W
has one isolated, compact,
log r(F)
-
A (Fy
I
2L
_
i
V
log IF
■ I
;
jj
represents the radius of the image :of T
r(F)
is the complementary logarithmic area.
!i
d arg F.
under
F and
A(F)
;
:
ij
jIn Chapter III it is then shown :b y using the, Reduction Theorem
Ls in the limit for the limit
(Sario [ 8 ]) that the extremal property holds
function
F
X
o
,X
that F “ is the
A deviation formula is. derived, from w h i c h it is established
;
U'
X-th
po w e r of the univalent, ,function
geometric properties of
F
is a w eakly
X
Fq
'
i!
m a y be inferred from those of
X -valent radial slit, disk mapping.
:r.
ties of F
o
Fq
are also derived,
:i. -i .
I I
iiI
I
I
:
.1
of [ 6 ].
Fq .
The
In fact,
Other extremal proper-
-3.I
It is shown in Chapter IV that the extremal property holds in
}y.
the limit for the limit function
.
'
S a r i o 1s Reduction Theorem could
not be applied and an argument based on the convergence of certain
Dirichlet integrals is used.
established.
One additional ,extremal property is also
:
,
::
Such extremal properties characterize the
1-valent principal
I
functions in a m u c h larger class of functions. Namely, this class of
functions consists of functions w h i c h are
I
X-valent near T and
;
are no further restrictions.
I
;
:
i .
There
CHAPTER I
PRELIMINARIES
Important definitions and theorems used throughout this discourse
are listed in this chapter.
g
I.
Basic properties of Riemann surfaces.
Def i n i t i o n .
A Riemann surface (bordered Riemann surface) is a
connected Hausdorff topological space
open sets
O
and a family
W
together with a covering
of mappings cp
V
by
satisfying the following
pr o p e r t i e s :
I.
Each
cp e $
of the complex plane
2.
Every
3.
If
is a homeomorphism of an
onto an open s e t '
(closed upper half plane).
O e V
O
O e V
is the domain for some
and Og
are the domains of
is a directly conformal mapping of cp
(o
cp e $ .
Cp^
and cp^
then Cp^ ° cp_J
fl O g ) .
We w i l l denote a bordered Riemann surface by
W
and
W
will be its
interior, which is itself a Riemann surface.
D e f inition.
The border of a bordered Riemann surface
set of points belonging to
W
W
is that
wh ich is mapped into the real axis by some
q> e f .
A bordered Riemann surface
W
is known to be orientable, and hence
the border has a positive as well as a negative direction.
Definition.
Surface W.
C
C
be
an arbitrary Jordan curve on a Riemann
The surface is called planar if whenever each
a neighborhood
then
Let
W - C
N
such that if
R - C
Def i ni t i o n .
called harmonic
W
has
is composed of two components,
is composed of at least two components-.
locally separates
p on C
then it also separates
That is to say, if
W.
A real-valued (complex-valued) function
f ° <p 1
(a n a l y t i c ) if the functions
f
on W
is
a r e , where they are
defined, harmonic (analytic) in the usual sense, for all 9 e j|.
Definition.
"{^nj"
A n exhaustion
of a Riemann surface
W
is a sequence
comPac^ bordered Riemann surfaces satisfying:
2.
The boundary of
W
consists of a finite number of disjoint
analytic Jordan c u r v e s .
3.
E a c h 'component of
4.
W.
=
The sets
Let
surface W
u
U
w
W - W^
is relatively non-compact.
n
W^ w i l l be referred to as "approximating surfaces."
and v
be harmonic functions on a compact bordered Riemann
Then Green's formula m ay be expressed as
(I)
W
where
•
B(W) is the border of .W
and
dux-
/
is the' conjugate differential.
-6The left side of (l) is the mi x e d Dirichlet integral for the surface
and is denoted by
D^( u, v ) .
G r e e n ’s formula m ay also be expressed as
B(W) W
For functions
B ( W )}
pdp*
W
vdu *
*
p, harmonic on
is understood as
0.
=
Int W, but possibly not defined on
Iim
the border of an approximating surface
^
^ pdp*.
Here
denotes
and is homologous to
.
We shall be concerned w i t h an important'solution of the Dirichlet
boundary value problem for a Eiemann su r f a c e .
Such is called a flux
function a n d ■is defined in the following manner.
component of a compact bordered Riemann surface
o'
let
be a 'curve contained in-the
homologous to
In
w
W (a
=
0
)
on
a
let
a ,
§
of
be a border
In addition,
W ' whi c h
o and o'
is
Denote the region bounded b y
w
solve the following boundary value problem; namely,
w
=
I
s^
in
W .
a
.
on o' ,
function for a neighborhood of
Evidently,
interior
Let
=
has flux
a
and w
is harmonic in
by
'U(a ).
W( a ).
A flux
is defined as:
(2 rtX /
dw*)
w.
2 % X , is constant on o', 0 on
a
, and harmonic
W( a ) . ’
2.
Formal linear operators on a Riemann surface.
Definition.
if (l)
G
is a regular region of a n open Riemann surface
G is relatively compact,
(2)
G and its exterior have a common
boundary which is a I -dimensional analytic
submanifold, a nd (5 )
all
W
components of
W - G
-
7 -:
.
I
•
are non-compact,.
I
N o w let
ji
.
K
be a n y compact set! contained in W and let
u be a
i
'j
function harmonic on its complement. 'It is possible to enclose K .with
a regular region
1 .1
1 '
with a positively oriented boundary P .
G
'
r
u
w i l l be denoted b y
of
G
i
'
/R d u *.
The flux of
ii
That this is independent of the choice
■
i
. :l
:
f
is a consequence of G r e e n ’s f ormula. !As described above,
P
will
I .
1I
I:i
be said to represent the ideal boundary of W,i. The border of a compact
•
'
—
bordered Riemann surface
of
W, while
W
1
I :.
I!
trivially represents the ideal boundary
Uo , where P .
■
I
a r e 'the border components of a n exhausting
ii
1
'
■
W n , represents the ideal boundary:of the R i e m a n n !surface
Definition.
W
,
•J
I
,
W
.
M
I 1. '
:
"
j ,: f
j
:
set of functions harmonic on
I ;
W
=
f
on a.
'
■
1
lK
:
W
• with
fx +
= S f2I =
5-
LI
=. I.
4.
If
f Ss O , then
5*.
L
d(Lf)*
-
Lf
0. '
!
a a nd
. and continuous ,on
L has the followihg properties:
2.
I
'
i!
i:
i ,
We denote b y L a n operator whose domain consists of
I
'
■!
I
I '
the set of continuous real-valued functions; on
L(f)
'
:
' ;
1I
i i'
, ,
be the boundary of a Regularly embedded open set
:
compact complement.
I.
are distinguishable
and its !exterior have the same boundary and
this boundary is a, I -dimensional submanifold;
O
.
if the interior and exterior o f • W.
in the following s e n s e :
Let
Wi
is a regular Iy embedded open set of the open
■
Riemann surface
P
°x;Lfx T
:
;
I '
^ i0.
I/
11'
:
j
i
whose range is the
W
In addition,
: ’
I
-8Here p represents; the ideal boundary of
1
;•
a normal linear o p e rator.
I
W. , Such an operator is called
,1
1 ; . 1•
;
,
;
.
Some important examples of normal linear operators are the principal
r
'
I
'
'
.
:
operators
Lq
of C 2 ].
and
boundary value problem in
To define L
we let
W*
Por a giyon
f j on
Oi ,
L Qf
solves the
w i t h k vanishing normal derivative on
P (W).
I
I ! .
i
be the solution o f /the boundary value problem
'
j
-I
,
Uf
1
w i t h boundary values f on Oi and O on P . ■ Let ; "w be the harmonic
/
i
function w h i c h has boundary values
O on. a! a n d '.I on P
Then L^ f is defined b y the equation
L xf
Observe that
L f
=
i.
Uf
equals
- Up
f
oh
;
;
<a:(uf)* /Vpdw-* )
a
w. ' ■
and is constant on P w i t h vanishing
1
i
:•I
flux t h e r e .
I
|
i'
" ' ' ■
7'
!
'
A third example is the operator L*, referred to as the operator
. i
associated vrith a removable
centered at a point
B of W
I
defined on the boundary
A, a n d equal to
Oi of
A, then
W 1
In general,
1 '
; ';
need not be connected.
separating-
W '
a.
.
f. b e 'the restrictions of
L.
I
is defined on each
'i!
1
'
b y setting L f
=
the operators
L..
I*
I
W
.
1
.
;
I
-We can therefore consider
i
W*.
'
.f''t o ' a n d
with boundary
x
suppose that an operator
Then we can define a n ew operator ■ L on W 1
:
i
' :
L ^ f Such an operator
11
continuous on
■
,I
into a finite number of disjoint sets
:
Let
:
:■
,
L*f, is harmonic on A,
■ ■
on 0! .
f
i
singularity.
If A i is a parametric disk
;
!
.
and f ' is a continuous real-valued function
IL
I
I
,
i
i
I
is called the direct sum of
' .■
— C) H
The following linear operator theorem (Ahlfors and Sario [ 2 ])
is used to construct harmonic functions which will play a key role in
this investigation.
Theorem.
Let
W
he an open Riemann surface and
L
a normal linear
operator defined w i t h respect to a regularly embedded open set
where W 1
—
on
W
W
has compact complement.
If s
W1
is harmonic on
Wt c
w
and continuous
«
w i t h vanishing flux, then there exists a function
satisfying on
W
p - s
.=
L(p - s ) .
p
harmonic on
The function p is uniquely
determined to within a constant.
That is to say, p modulo
§
p.
s
has the properties of
L
on W 1
The Reduction Theorem.
he a n exhaustion of the open surface
Let
he a sequence of functions such that .f
Definition.
compact set.
W,
N
i f
is liarmonic (analytic) on
S fn j* converges on compact sets of
K c W , there exists an
and
such that
W
-j^ fn ; n
W .
if for every
-
R
converges uniformly on K.
T n fis a normal family on
Definition.
of
W
when every sequence
contains a subsequence which converges on compact sets of
W.
A w e l l known sufficient condition for a family of analytic functions
fn f to he normal is the f o l lowing.
exists a constant
M
and an integer
I f for eyery compact set
R
such that
'If ( z)I < M
K, there
for all
-10n> N
and a ll
z g K,
then I f
^
is a normal family.
T l '
Let -) W f he an exhaustion of
L nj
he classes of functions w i t h domains
let
m
and
m
W
W
and let
and
he functionals defined on
n
Reduction Theorem (Sario C 8 ]).
and
Hin
and
C :
n
C
and C
n
respectively.
W
and
W
n
for each
In addition,
.
Suppose that the functionals
satisfy the following conditions w i t h respect to the classes
I.
m(f)
2.
If
he true if
3.
W
=
m
Iim
n
c W
n
is replaced h y
fe C
n
W
and
f e
n
among a l l
°
k<_n
and
C
f jTT e
Wm .
f e
m n (fk )
f
C .
m
The same must
h y C.
then
There exists a function
functional m
then
is a sequence of functions in
If
m
(f).
and if
n
converges uniformly to
4.
m
n
n
e C
n
C
n
an d if i f,
I k
converges to
such that
f
n
m^(f).
minimizes the
C .
■ n
5«
Por
f e
m^(f)
^
m_^(f).
6.
The family of minimizing functions mentioned in (4) is a normal
family w i t h the limiting functions belonging to
Then any limit function
a n d the value of the minimum is
f
=
m(f)
Iim f
n
n
C.
minimizes
llm Bn (I11) •
m
among all f e C,
I
(■
I
CHAPTER II
EXTREMAL PROPERTIES OF HARMONIC:A E D ANALYTIC
FUNCTIONS ON APPROXIMATING SURFACES
Consider a planar bordered Riemann surface
W
of infinite connecti­
vity a nd having one compact border component' Y
•. In order to describe
I
'I
the remaining part of the boundary of W, we-: recall that such a surface can
be embedded in a Riemann sphere S
2
(Ahlfors and Sario [2l ).
to this embedding, it is assumed that no point of
•
Y
1
•.
With respect
is a limit point of
points belonging to any other boundary components. 'Such a border compoi
_
nent is called "isolated". Also, it; is assumed that the surface W - Y
2
is open in S .
■
:
:
—
S ' belonging to the interior of W - Y
We fix a point
—
Operations in W
following.
.
.
of compact bordered surfaces
-j W
^
Y
Y
and
£
.
In this chapter
of finite connectivity and each
, _
.!
CO
Furthermore,; W jnc W ^ +1 and W = ^ W^.
.
n
by P1, P2 , ..., Pk ^
W ca'n be exhausted b y a sequence
r',' each
11
is one border component of W
Evidently,
Denote the remaining border components
1
i
and for convenience 'set
we propose
in the
I
It is known that such a surface
containing
2
I
such as I n t , boundary, e t c . , are referred to S
^
'i
to develop extremal
:
'
:;
i ^i*
properties of certain
harmonic and analytic functions defined on 'the approximating surfaces W .
: I'
g 4
.
1 : ^
Construction of h a r m o n i c .functions:on approximating surfaces.
!I
I
Using S a r i o 1s linear operator method; described in [ 5 ] , ;the har-
i
monic functions p ^
, on
and
p.
m
I
j
will be constructed on each of the ,
I*,
,
-12-
approximating regions W
k log 12 -£
.
The singularity function s will he taken to be
j in a neighborhood of £ , identically zero in neighborhoods of
the Pi , and equal to the singularity function s
k
function s
near y
.
^ ■
is that function w h i c h has flux equal to SjrX0
The singularity
Uote that ■the
total flux o f s is zero*
L1 near y, Lq near each Pi ,
The direct sum of the linear operators
and L ' near S , w i t h an application of the linear operator method, yields
the X-th harmonic function p Q^ up to a constant.
malized at £ so as to get property
The function
ii) of H ( X) satisfied.
is n o r ­
Here, L* is
the operator associated w i t h the class of harmonic functions having a
removable singularity at
£ .
The X-th harmonic function p ^ is derived
from the same direct sum of operators with the exception that
near each
is used
Pi *
The functions p ^
and p ^
possess the necessary properties required
to belong to a class of harmonic functions HfiC X) which will be defined in
the next section.
In addition, the function p
Pp ^
0-Q
=
O on each P., while
.on
X
on
has its normal derivative
is constant on each P i w i t h flux ■
1
/ p . dpin*
=
°*
These functions will be seen to satisfy certain extremal
pr o p e r t i e s .
S 5
.
Extremal properties of harmonic functions on approximating s u r f a c e s .
The extremal properties that the X-th harmonic functions are -to
enjoy are w i t h respect to classes of functions defined as follows:
-13r
D e f inition.
H (^ )
n
is the set of functions p ( z ) which are har-
;
.
;
:
monic on W n -I while satisfying i ; ) p ( z ) = const. = c(p) for zey and
J
^.dp* = 2 jcX , and ii.) the function!
harmonic continuation to £
with
h(z) = :p(z)
- Xlog
|z-£| has a
'
',
h(;i) = O.;
X
The harmonjLc functions p Q^ and p ^ "belong to the class
( X )•
A
lemma that is essential in obtaining their extremal properties will first
i
be stated and proved.
Lemma T_. iIf
5 is the border of a parametric disk about
orientation induced b y the surface Wn , t h e n 1for every p and
Pdq-X-
have
Proof.
(I)
J b P<3a*
q.e
i
, with
Hn ( X ) we
Cjdp-X-
Adding and subtracting the same term we have
- qdp* = j^(p-q) dq-x- - q f(P" q )> • '
!
!
■
Since the function p-q_ has a removable singularity at
, i t has a single
valued conjugate and, upon'using an! integration'by parts, the right hand
i
.
I
:
side of (I) m a y be written as
'
■ i
|
'J^5Cp -q ) dq-x- + ; (p-q
)* d q
.
B u f this is equal to I m ^ s (P-Q) dQ - where P = p, + ip*
and Q =
q + iq * .
The functions P and Q are multiple valued but the function P-Q is single
I
valued.
'I
:
Using property ii.) of H (:X )'together w i t h Cauchy's Integral
11 :
I
I
:
Formula this can be seen to be zero.The extremal property that
theorem.
"X
$A
■ i
possesses is .stated in the following
I
II
"
-I l}.-
Theorem I .
---------functional
Among all pe H f X )
nx '
(p) = 2jrXc(p)
-
JJ Pn
I'
x> ^
the function
pdp* .
-on
maximizes the
The value of the maximum is
. ; ■ ■ ■ ■ • ■
The deviation of this: functional from its maximum is B 7 (p-p
w V1x^ -irOn'
,,I
2jrX c'(p ^).
on'
and the maximizing function is u n i q u e .
Proof.
y
By Green's formula we have
X,
\
X
But since p and p
fe-pon\ =JyuPn (I1-pOn) 4 (p-pon>* '
P '■ X ■
X
SEtn ( X ) , J r (P-Po n ) 'd (P-Po n ) * =
O .
:
■
.
i
j
Gr e e n ’s formula then Becomes
(2)
(P-Pon) = / 5n pap* + J p n Pon a P0J » - :/pn P0J ap, + p
•on 7
The second term of the right hand expression-vanishes Because the normal
dp ^
derivative
O on
’ ■ For the same reason J g^pdp^i* = 0 .
Therefore we change its sign and upon applying Green's formula obtain for
the last t e r m of (2 ) the expression
V y u B F 0h
j
■ F ^ P on*..*
Here 5 is the Border of a parametric ^disk A about s w i t h .orientation
induced b y H - A .
By lemma I .) the contribution of 5 to this integral is 0.
Using property i . ) of H ( X )
n
the part due to iy is 2jtX c (p ^ ) - 2 rtX c(p).
J
: T on
Collecting all of these contributions we obtain the formula
(3)
X
X
:
-r
■
2jrxC(Pon) - Hfn(P-Pon) = SjtX CjXp ) - Jpii pap* .
Since Ijjy (p-p
\
on
'
I
) -O
we have the desired extremal property.
is clearly Dgn (P-P o n ) 1
h
'
The deviation
I
-15 -
As for the uniqueness of the maximizing function, suppose another
function
p
maximizes the functional.
and hence p 1 - P q^ = C (constant).
and p
(p ~Po n ) = 0
;But h y the normalization at
£,
C = O
•on
The function p
p
Then the deviation
X
xn
possesses a n extremal property similar to that of
X
, hut the functional that it extremizes differs from the one originally
on
anticipated, wh i c h was 2 jcX c(p) +
pdp-x-.l
,
Theorem 2.
----------
The function
this m i n imum is
+; J p n W
'- sjpn
The minimum value is I 2 m X ;c ( p ^ ) T h e
(p - p
wn
Proof.
minimizes .'the functional
;
:
atx o(p)
1L ( P )
among all p e H ( X ) .
p ^
jrXn
ap» •
deviation from
1
i
:
) and the minimizing function is u n i q u e .
ln
::
X
We have h y using G r e e n ’s Theorem
\
W j p - pl ) a P
(P "
- pJ p *•
As in Theorem I the contribution of- T to- the integral is 0, and we write
(4)
Dgn ( p - P l n )
=
X
pdPil" + J p n P xb d P ln*
• 2JpnpIPap>i•
Since
p
;
.
;
r
J
4 V
0
on each
Upon applying G r e e n ’s Theorem again,
'
the third t e r m m a y he written as
p d p in* " p I dpit
■I
is constant on e a c h P. an d the Ifltuc
the second term in (4) vanishes.
-
x :
pUS p d p Init
■
-
I
:i
i
;
-X
p I n d p *-
By lemma I . the contribution of 5 to the jibove integral is 0 and, using
property i.) of H ^ ( X ) , this reduces to. 2jtX-c(p) - 2rtX c (p i^) •
'
I•
Hence,
(4)
-16-
becomes
27tX c ( p ^ )
(5)
Since D g
Bg
,+
(p-p
-
(P-P1^)
P j a c(p)
=
Jg
+
pdp*
0 we have the desired!result.
I'
;
-
2jg p Indp*'
The uniqueness of the
■ ■
minimizing functional follows exactly as the argument used in Theorem I .
,
- ■
I 6. . Extremal
i
properties of analytic functions■on approximating .surfaces.
.
;
. !
1
The classes of functions in which the X-th analytic functions are to
!
i
I
enjoy their extremal properties are !defined! as f ollows: ,
I
'
■
;
Definition.
A (X) is the class of functions
----------n
:
. i
::
such that the following properties are satisfied1:
i.)
ii.)
If (z )
I
=
r(f)
=
F
analytic on
n
.
Iim
F(Z)z/
(z-G )"X =
!
I
I
Some interesting connections between t h e 1:classes
i
:
are expressed in the following theorem.
I
Theorem 3-
If
F e A
X-th analytic functions
A
*
0
- G .
number of zeros
=
and
A (X)
n
i
l o g : lpi e H
(x).
F urthermore, the
(pon t 1 pOniti 'ani ' A n = exptp^
c
The function
on W
A
F
(X) then
H (X)
n
i pin*)
(X).
Proof.
F
2 aX.
const, !for z e y w i t h d arg F =
F has a X-th order zerb at , s
iii.)
belong to
W
Zq
log
If I
will be harmonic on
W
- £ provided
:
:
A n application of, the argument principle gives the
on
- s r JB(Hn)
Wn
as
a arE F
Hence, the only place where
=:
F = O
~k
on
Ir
W
d arg, F
is at I.'
+
d arg F
X.
i
!
-17-
Wow F has a X-th order zero at £
G(z) with G ( ^ )
4=
0.
so -that near £ , F(z.)
=
I.
!
Hence, the function h(z) mentioned in property
ii.) O f H n ( X ) may he written as h(z!) = log ■IF I I- Xlogjz- £ | =
loglG(z) J
A sufficient condition for h ( z ) to ;he harmonically extendable is that
^iijP (z- £) h(z). =
0.
This is satisfied and furthermore, by continuity
'
i
!
'
:
considerations,' the extension takes the proper vdlue O since ^i^ h(z)
log IG( £ ) j = 0.
H
Property ii.) of
property i.) is readily checked.
i
j
i
!
!
:
j
I
We prove the second part of Theorem 3
\
as a function.
=
(X ) is therefore satisfied and
:n
I
gous proof for F
' ■
I
X
'for F
-
and omit the analo-
'
'
I
X:,
We must "see that F a = ‘exp (p A 4- i p A *) is defined
on
■on
on
The multiple valued conjugate function p a^ x- has periods
I
2jrX
'rd P on*
and
k
has a period of 2 jtX
*
for each i . 1 That is to say, it
O
d v
and we have F
on
: defined as.a function.
. /
•;
■
Here, F
' on
;
is normalized b y setting h*, the conjugate ;of the continuation for
X
•on
, _
X log
I
„I
.
,.
jz - £ j, equal to zero at
zed in a similar manner.
for z e Y", and
of H n (X
’
)}
/
/
in
We have
X
on
' :
£
^ o n ^
_X
The mapping F ■ is normali­
X
' = ■ exP (p ^L(z ) )
■’o n '
-
const.
I
T
.X
d !Pon-X- 7 2 jrX, .
properties i i , ) a n d iii.) of
Theorem 3
and F
d arg F
JT .
X
By the normalization condition iii.) of Afi( X) it
,
follows that G( I )
(%-£ )
=
i
Using property ii.)
) may'be obtained.
I
■
X
leads us to the following extremal properties for F
on
stated below in Theorems 4. and 3.1
:
i.
-1 8 -
Theorem 4.
The function F
2JtI
among F e A ^ X ).
log r(F)
/jjj log
jF I d arg F
The maximum is 2jti log r ( F ^ ) :and the deviation from
the maximum is D - (log
Proof.
uniquely maximizes the functional
on
Jf / f
).
on
We have log .|f
' X1 ■
p
and according to Theorem I . this
-tOn
on
maximizes the functional 2 jtX c(p)
pdp*.
-
And "by Theorem 3 •, if
Pn
F eAn (X ), then log
2 Jt^ log r(F)
-
Ls
|f | e
log
H e n c e , IrQn maximizes the functional
( I).
|F j d argF among all F'e A (X ).
x
'
- -
% '
D^Ciog If I - log| FonI ) - ,Ilt4i (log If / Fon I).
I
Uniqueness is argued as in Theorem I.
II
ii
The deviation is
*'
;
'
1
In a n analogous manner the following theorem expressing the extremal
: ,
property for F
may he proved.
Theorem 5.
--------—
; :
The function F
2 jtX log r(F)
among F e A ( X ) .
n.
minimum is D^n (log
+
in
Jg
uniquely minimizes the functional
log
Jf |id
argF
los
X
'
''
/
fF in ^darg F
The minimum is 2 jtX log r (F
) and the deviation frcm the
■ x
,
. in
’
If / F i n I )•
:
i
CHAPTER III
'
I
EXTREMAL PROPERTIES OF THE FIR S T 1X-TKl PRINCIPAL
HARMONIC A N D ANALYTIC -FUNCTIONS :
Extremal properties for surfaces of infinite connectivity which
!
1
/
;
generalize the results of Chapter II will he derived in this chapter, with
i
M
the main tool u s e d being S a r i o ’s Reduction Theorem [8 ] i
I
T
■
Extremal properties of the first ^-th principal harmonic function.
Let ¥
he a planar bordered Riemann !surface and -jw. f
;
set as described in Chapter II.
rx V
Lemma T_.
Proof.
family
-^F“
of a family
The family
Because p
is normal.
jPo n j
X
log If
on
I
;
I
I1
■ !'
I nJ
'
an exhausting
.
I -
a normal family on the surface W - 5.
on
it suffices to show that the
A w e l l known sufficient condition for normality
f >■ of analytic functions on a region is that the functions.
for sufficiently large n, be uniformly bounded, .on every compact subset of
'I
the region.
I
;:
!
Let K be an arbitrary compact set contained in ¥
and choose N
large- e nough such that
K c ¥^.
log If JJ1I d arg F ^
on
on
Denote b y A ( F ^ , W y ) the quantity
for n - N . ' This is a ,non-negative quantity) for
consider the Dirichlet integral over the surface
!
tion is induced b y ¥.
(,)
0 - %
•
W^
■
W y , whose orienta'f
Then
- W j 1= S lL n l1 =
J f e 1=S ^ o n
P g
+
A<-*L’V
■
':
-
20
-
X
The first integral of (l) is zero and
(2)
2^X log r(F h
^
2z(X log r(F^)
-0.'
Hence we have
+ A(?.^ W j : 6
on'
,
-N
X,
2*X log r(P^).
:
The last inequality results from the extremal theorem for the function
on W jj.
It must be noted that the 'expression in the middle is the
functional given in the extremal theorem (Theorem 4. of Chapter II).
the
r(F
X
. '
' '
' ::
) are bounded.for all n ^ N. An application of the argument
on
principle, yields
is a z
Thus
on
(z) | ^
r(F
) for z e W'!.
o n ,.
xi
To see this, suppose there
.
w i t h W = F ^(zn ) such that l"W„ j > r(F
. Then, there is no
o
on °
°
on
' X, . :•
I
change in the argument of the function jF
(z) - w
around each of the cycles
o
e W
IF
n
i
T and P. for every i.
F
'
i.
Therefore, b y the argument principle, the function
X
I
'
X
(z) - w
has no zeros on W , i . e .,ithe value w
is'assumed b y F
for
on
o
. _
n'
.
o
.
on
no z e W .
n
Hence,-
'FonJ
^or n - ^ is uniformly bounded on every compact
subset and is therefore a normal f a m i l y on
W.
i
We let p ^ be some limit of the.' normal family
:
X
wi l l see that there is exactly one such p .
o
ip
.
Later, we
i
■
X
The function p
is referred
o
to as the "first" X-th principal harmonic function to distinguish it from
x
;
the function p. wh i c h has an analogous developments
1
;;
:
The Reduction Theorem w i l l be used to develop an extremal property
for p ^ in the class of functions defined as follows:
0
.
Definition.
H(X)
.
I!
;
is the class of functions jp
: ;
;;
on W - I and satisfy the properties: :
;
which are harmonic
-21
i.)
p(z)
c(p)
ii.)
h(z')
P(z)
"with
const., for z e r with
-
2jtX.
hasia harmonic continuation
0.
h(" ■£)
The function
X log jjz - §;
dp*
pO
'
insure the existence of certain integrals is stated and,proved "below.
Lemma 2.
If
m<n
the inequality 7 5
pdp*'
J
all p e H (X).
Proof.
•
• I
Consider the surface j W
The Dirichlet integral of
p
!!
^ 7 5 pdp*
v ^n
'
holds for
Wm , with orientation induced "by
on this surface is non-negative and we
have
0
i
\
-
H1 (p>
'Pn - :Pm papi!
In the following,
Hn
'
pdp* is to he understood as
pdp*.
Iim
pdp*,
the existence of the limit being guaranteed b y the monotonicity of such
integrals as shown in lemma 2.
f
4
haustion
I
i
r as can be seen b y applying. G r e e n ’s Theorem.
Theorem 1_.
<j) (p)
=
2 jcX c(pQ )
This definition is independent of the ex-
2 tcX c(p)
The function
-
/ppap*
X:
p^
i
uniquely maximizes the functional
among all p e E ( X ) .
and the deviation from this maximum is given b y
,
Proof.
-----
0
:
D ^ (p - P q )*
,
^
It w i l l be shown via the Reduction Theorem that p
the functional
p
The maximum is
maximizes the functional
;,
-2xX c(p)
+
L
pdp*.
0
minimizes
Then it will follow that
—
The functional - §
22—
satisfies
- (| (p)
Iim
(In Cp) L the existence
n
of the limit "being insured by the monotonicity of the integrals
as proved in lemma 2.
W
m
a pdp*,
Pn
The restrictions of functions defined on surfaces
belong to the class H ( X )
n'
defined on W.
J
for
n £
The same m ay be said of functions
Condition 3«) is satisfied because of standard theorems on
interchange of limits.
ments of 4.).
The extremal theorem for p ^ fulfills the requireon
Lemma 2.) provides us with the monotone property of the
while the requirements of 6 .) are met b y lemma I .
functionals
the functional
Gj)
is maximized by p^"
o
Thus
with the maximum being Iim 0 (p ^) =
n
n on
2 rtX c(p^) .
As for the deviation formula, consider the function p
e Cp-P 0 ) which belongs to H ( X ) for arbitrary e.
functional
(3 )
ffl
In (pe)
n
=
Po +
The value it gives the
is
2k
X . X
c(p^)
-
1P p O dbo* +
n °
X
% =
-
=
X,
a(p-pj*
/p.(p-po )
~ n
wh i c h is a quadratic polynomial in e with a ^ the coefficient of the e term.
The integral in the last term is precisely IH (p-p^).
X
N o w suppose L^C p -Pq )
=
X
'°
Iim Ey (P“P 0 ) is finite.
X
n
of EyCp-P 0 ) determines the finiteness of the integrals
^ P 0 d P^" *
P
/ppdp* and
as can be seen b y using the triangle inequality for Dirichlet
integrals on surfaces W - A 7 where A
setting e
The finiteness
=1,
is a parametric d i s k about
consider taking limits on both sides of (3-) •"
i.
Upon
In view of
the preceding rema r k s , each term with the possible exception of an possesses
a finite limit.
Therefore so does a^.
Denote this limit b y a.
Taking
■
limits of (3 ) we obtain
(^)
I
(Pg )
=
:
I (Pq )
+
a .e
,!
:
- e 2 D_(p - p ^ ) . ■
But according to the first part of this theorem <j) (p
;
e
=
0.
=
) has a maximum when
6
;
Therefore its derivative with respect to e evaluated at e
must vanish; i.e., a
e
■
-
0.
=
0
The desired deviation formula results when
I is substituted into (4).
.
..
■
I
If Dg(p-pJ":) is infinite, then the triangle inequality for Dirichlet
integrals on surfaces W - A
can be used to show that
Ij (p)
is
infinite as well. ' The deviation formula holds in' the sense that both sides
. J
:
are i n f inite.
'
■
The uniqueness of the maximizing function .'follows'exactly as in the
i
pr o o f of the extrbmal theorem in the; finite case.
I
;
.
■
;
!
uniqueness of the maximizing function is the fact
I
limit of the normal
T
x
family |Pon
An analytic function
that of
on
i.e., F^
" 0
0
that p^
is the only
':
'
This completes the proof of Theorem I.
will be
exp (p^
referred to as the "first”
i
:
A consequence of the
d e f i n e d .in an analogous manner to
+ i p^ i ).
0 * •
The function F^
:
0
will be
1 -th principal analytic function to distin­
guish it from another principal function F^; .
§
8 .
Extremal Properties of the First 1-th Principal Analytic Function
The extremal property of the'function p
I.;
'' leads immediately to the
extremal property for the 1 -th principal analytic function
A class of competing analytic functions is defined on W as fo l l o w s :
Definition.
A (l)
is the class of !functions F analytic on
W and
-24satisfying the p r o p e r t i e s :
i.)
ii.)
iii.)
I F(z
)I =
I
r(F)
c o n s t . for !zey with /
F has a X-th order zero at §.
/
X
( z -g)A
Iim F( z ) /
.Z^S
The function
F^ ,
i °i
■'
■
:
=
darg F = O .
I
1
4
:'
'
already defined a b o v e ,,helongs to the class A ( X )
r
i
■;
The quantity - L l o g IFj darg, F^ denoted b y A(F),:iis understood as
: Iim - /n
n
log
Jf |
d arg F.
■
Theorem 2.
:
J
:i
:
X
uniquely maximized by the function F ” .
the deviation from the maximum is
■
If F e
A(X
(log
) then log
-5- A(F)
is
The maximum is 2 jrX log r(F^^) and
!
Proof.
,
Among F e A( X ) the functional G ^ X log r(F)
I FIZ^Fq I ).
;■
i
If J e H :( X ).
:
The argument used in
Theorem 4 of the preceding chapter yields the result. .
:
:X
An important corollary characterizing F q
univalent function F
of [6] , i.e.;, the principal function corresponding
°
to
X=
' i
as a 'I^th power of the
,■
I ^ w i l l now be drawn from Theorem 2.
Corollary 1_.
function F •
o
Proof.
X
The function F q
1
'
is the X-th power of the univalent
4
■;
:
'
:
It is first observed that Cf
( z )^
;
;
.
belongs to A( X •)-.
Because of the uniqueness of the maximizing function in Theorem
remains to be seen that the deviation
2,
it only
Dfl(log .I (F0(z )X/ Fq (z ) * ) =
For convenience, denote b y f ( z ) the analytic; function (F ( z ) ) / F
;
-I
O
' O
w i t h a removable singularity at B .; By G r e e n ’s Theorem
O(z )
i
:
I
;
-25D- (log
j fI ) '
'yUp
' I f I darg f .
log
The contihution of T .to the integral is 0.
( 5)
(log I f ! )
O - D -
=
L
N ow
l o g i f J da'rg f
1
Let f
131
( z)
=
( F ^ .( z )
OIil
w i t h a limit "being f.
)
F
( z-).
Oul
=
I
log If Idarg f .
!
The family K f > i s a normal family
,•
• 331J
Therefore, interchanging limits and. integration, the'
right hand side of (5 ) m a y he written, in the sense of iterated limits, as
y™ AJft Jpn 10S I V
(6)
D0 (IogIfI)
-ve
1»^
darB
=
lfi'U m J p n
log |fm l, d ar g fm
Considering the Dirichlet integral of log |f_|
we obtain
0.
over;the surface
Wn^
■'
.
W
a
=
0 "
i:
= /@ml08lIiB ld arg:fm " / % losliB l d ars fH
:
The
first integral is 0 a n d we have
(8)
r
s
log | f j d arg f
^
1
;
’ '
:i .
for every im ^
0
■
J
*.
- :
n.
;
Thus, taking iterated limits of (8),jwe obtain
p:
„
(9) 1Jjl A^JPnlog lfB l a ars
Combining (6)
and (9)
ij
fm' s
'
'
0J
' .
J
’
the desired result D ^ '(logl f Ij
= 0.
This
completes the pr o o f of Corollary I
,1
A consequence of this corollary is the fact 'that F q
—
is a weakly
I:
X -valent radial slit mapping of W into a disc in the following s e n s e :
Definition: ' The m a p ping F( z ]) is called weakly
I
X -valent if
for
each w e F ( W ) , the set F ”1(w) consists of at most
X
points z e W, and for
some w e F ( W ) the set F - (w) consist^ o f .exactly
X
points.
A weakly.
X-
valent mapping F(z) of W into the point set Si is called a radial slit
-
'I
f
• •
,
mapping of W into S if e a c h component1 of the set i v e 8 ; F ~
(w) contains
- I
i
I
1
at most k-1 points z e W r is a radial slit of point.
:
J
i '•
i
x
Ttro corollaries stating other extremal properties of
F
will he
°
dratm.
Define a class of functions
differing from
A(l) in that
5' (l) which is a' subclass of
A(F) ' ^i
0
for each F e
A(X) and
^J-(X).
: I
X1■
F q1 has the property
Corollary 2. 'The function
fiSgl)
Proof.
Let F e
PmXlogr(F)
=;
3(x) ^
TheA(F)
'I
PmXl o g r(F); +
&
band;
'I
A(F)
. '
::
The last inequality is the extremal property, of
:
Define a class of functions
Jtj'-(x)
1
:
g)- (X) only in thd condition |f (z ) I '=
Corollary 3«
I
Proof;
r > r(F ).
: j
■
I
| '
-N
^ 2 m X :log r( F ).
-X
F
• ■
- 0
. :Hence the r e s u l t .
!°
defined on W and differing from
;;
i.
:
c o n s t . on y is dropped.
.
The function F | has the extremal property
oI
,
,Sgfo
W
1
g
P ? 1 " :
i
Suppose there is a function G e .%|:(x) with
mi^. |c(z)|
Let W jhe the simply connected region !enclosed hy
=
G(y).
Choose a function cp belonging to the' class'of univalent functions A(l)
‘
I'
i;
:• ;
defined for W $ and consider the disk of radius ^ 1Whicli is the image of
W 1 under ©.
Let C be tbie circle, contained entirely in W s , having radius
:
;
: .
r and center at 6. Let M be the maximum value of j® ( z ) | for z e C.
i L
.
:
By Cauchy's estimate for the derivative of ah analytic function we have
-27-:
ICp 5(o)j
cp!(o)
I= M / r .
The normalization condition' iii.) of A(j_) implies that
3, and we have r A M.
function cp ° G Belongs to
Because M ^ p
we have
p > r(
).
The
3 ( X ) and we have- a contradiction to Corollary •I
:
M
I
:
■
CHAPTER! IV
. !
I
!
EXTREMAL PROPERTIES OF TEE SECOND X -TH PRINCIPAL
HARMONIC A N D ANALYTIC FUNCTIONS :
I
1
;
I'
In this chapter extremal properties of certain X-th principal functions on W w i l l he derived.
§
9 •
.
,
The extremal property of the second X-th principal harmonic function.
,
First, a few properties of the' functions p Q
X
. are developed which are
needed to obtain the normality of the. family
Lemma I .. If
Proof.
m-n,
then
Jg
p Qm
X
d P om*
0.
Dy lemma 2 of Chapter :3
11
'p
p O m d p Om*
Lemma 2.
X
X I
'P ^ o m ^ o m * ' !'
•m ■ i,
i;
-
X
1c (p o^ ) f
The sequence
is! a decreasing sequence.
'
Proof.
Let m - n .
T
:i
i
X
Using the'extremal property for p
we have
:
on
2>a =Cp 0^ ) SI 2«x o(po^ )
Q
pO m d J o m * ! P.
:
i
2 * X 6 (p « ) -
The last inequality follows from lemma I.
Lemma j5*
Proof.
log
( Xi
F
I .
The family
X
^p "j" -is a normal family on W - L
It suffices to show the family
;I
'
large enough such that K c W ™
F
on
,: • •
For an arbitrary compact set ; K
!
we write for
Xl ■
X
F_._ f" ;normal because p_q
L I^ n j
in
■
;
!
contained in
W
Using the 'extremal property on
n - N
I
chose
N
!
W
for
-292rt)v.log r(F ^
_ in
^
2jtX log r ( F X )
on
^
2jtX log r ( F X
Oh
).
The last inequality follows from lemma 2. upon noting that log r ( F ^ )
=
c CPyn )-
Thus the
r Clni n ^
are uniformly hounded for
n ^ N.
An a p p l i ­
cation of the argument principle as in lemma I . of Chapter III provides
us w i t h
If
^ ( z)
I^
r(F ^ )
for z e K.
Therefore,
is uniformly hounded on the compact set K.
-j? H
'for n ^ M
We infer that
r is a
normal family.
We denote h y p ^
a limit of the 'family'
w i l l he referred to as the "second"
was not a h Ie to he applied.
.
Such a function
1-th principal harmonic function.
Due to the extra term of the functional
orem
-jp^^j-
^
of Chapter II, the Reduction The
Namely, it is not known whether the
I
functional is an increasing one or not.
\
A sequence of lemmas involving
convergence of Dirichlet integrals is needed to obtain the extremal theorem
_
X
for P i
Lemma 4.
--------
For the functions
Sr
Proof.
X
n
According to C 4 ]
■exp (Pi
+
i p^*)
X log
|f
I=
X p
.
^
in
K n "
we
2% )
have the relation
=
the function
and F i
[ 6] , corresponding to the case
p
X=
( F 1Cz) ) ^
F i (z)
where
is the univalent principal function of
I.
X
Thus we have p -"-
L i k ewise, for the functions
p^
=
we have
log
p^
I Fi I =
= X n
The concern for these properties is manifested in the following equation:
(I)
Ds
n
(p^
- 'P^
)
X 2
D;
(Pj
Pm)'
,
'i
-3q~
The restriction
because
/„
d p *
Pi
J
of
p
to
W ■ belongs to the class
d arg
i
F^;
consequence of the univalence of F1
:0 j for,"each
.
p>
of [ 6 ]
This is a
B y 'ldmma 2. of [6] , i.e., the
extremal theorem for p . , w e have
in
(2)
(l>
where
- P,-)
is. the functional
and the family "j P i n T
The functional
9(p)
Theorem
Iim cP (p
Iim
(P1
Tl
%
n
j!i Hl
-
)
P111)
cPvi (p' ) ,
<? J p 1 )
2^ c(p)
+
pdpx- .
satisfies the conditions:of the Reduction Theorem.
is defined as'lim^p (p)
= . cP (p ) .
ILa
0.
'and we have via the Reduction
Taking limits of
,
(2)
we obtain
Equation (l) then yields the r e s u l t .
-
Lemma 5 .
If p e H(l)
and-
. (p)
, then
lira
_
n
Proof.
is finite, where A
'ij\.' X
( p ! - p ^ p)
1■
.
!1
j
parametric disk about
=
is a
0.
According to the Cauchy-Schwarz inequality for the mixed
• I
. :'
]!
Dirichlet integral we have
(DW - A (pin - pi ' p> )2
n
:
%
n
has the limit 0.
'
i
- A (pin "
Us i n g lemma 4. and noting the finiteness of
co m p l e t e .
The functional
^(p)
'A
• .
n
_ A (p) '
the right hand side
Therefore, the left side must also, and the proof is
■
.
.
':
;
I
;!
;;
-31
If
A
is a parametric disk about
1
:
I ' we have the following l e m m a .
'
:
■i
:'
1 :
,
is finite, :then .Iim Dr '
An
.
■1
In
.W ■ - A
in
;
1
.; ,
n
exists for each pi e H ( X ) , and is equal to
j(p .
p).
.Lemma 6.
If
--------W - A
B-
„ (p)
.'
Proof.
:
' I
~
1
i
. n)
.
The linearity of the inixed Pirichlet integral furnishes us
wi t h
(5)
X-A
(p“
n
’ p)
"
dH n - A
(i,x i
> p)
“ I
: X ni;
By lemma 5-) the right hand side of ;(3) tends to i O .
complete if the sequence
jD^ _ ^ ( p ^
b
)•
The proof will be
p)j-possesses a l i m i t .
It will be
n
,
' :
I
established that this sequence is Cauchy.
The Cauchy-Schwarz inequality
i
supplies us w i t h
I
M
X
m
-W
n
A
< P)
-
dS
m
- W (Px )-IdB -W (P)
n
!
m ■
:n
The existence of
Iim B g (p)
for each p ei H(X); implies that each of the
n
.
terms on the right hand side of (4) n an be made ,small for sufficiently
i I
large m and n.
:!
Thus the desired sequence is Cadchy, a n d in view of previous
remarks the result is established.
i
I
' 1
A corollary of this lemma is Ilater Used in establishing a deviation
I
fo r m u l a .
Corollary 1_.
Xx
D1^(P-P1 ) •
:
If
Dg _ ^
(p)
' Ii
is finite,Hthen
Iim D 5 ( p - p ^ )
W
"in'
n.
-32- Proof.
disk
A
It suffices to prove 'the stj.ted .relation-with a parametric
about
§' removed from each; of the surfaces W
and W.
i
I
■ i!
n
the quadratic character of the Dirichlet integral, we m a y write
(P -
Vm )
=
D5 _ & (p)
2 I DW - A (P' ?3.n?
n
.;.
;
<5) X - A
;i
We consider taking limits of (5 )•
side is
limit
:
Due to
nM - A
n
'
■
The limit of the first term on the right
D g _ ^ ( p ) , while according to ;lemma 6.; t h e :middle term has the
/ 1 \
(p, P 1 ) .
-2 D t-t
a
W - A
:
:I
The triangle inequality for Dirichlet integrals
is used to write
(G)
1%
n
- A
DW - A
n
X jI
-
%
X n X l-
- A
dW
Xn- X
The last inequality follows from the fact that the Dirichlet integral is an
■
increasing function of the region. " %• lemma
the right hand side of (6)
approaches 0 w i t h increasing n. . H e n c e , the term DT-T 'A (p
possesses
,
w - A
in
I
'
I
n ■
the limit D,-^
A (p ). After taking limits of (5 ) and using the quadratic
IrV ~
ZX
I
;
X
I
character of D g Cp-Pi ), the result is established.
H o w it can be seen that the f u n c t i o n a l '
Cf)
has a limit on n.
=■
2jtXc(p)
+jfp
pdp*:: -
P i^ dp*
The second term'of the functional possesses a limit
because of the monotone property for buch integrals as proved in lemma 2
of Chapter III.
If
Dr.
A (p)
is: finite, then lemma 6. furnishes us w i t h
■!
-33-:
the existence of
Iim B g •_ ^ ( p ^ / p ) ,
and this implies the existence
■ n“
;
\
of the limit of the last term of the functional.
,Lemma 6. also furnishes
X
p^ dp* .
us with the value of the limit, that being
VJhen D g
A (p)
is infinite, the triangle inequality,:as in equation’(6) of Corollary I.
with
p
in the role of
p^
i
* n (p )
and we have that
Iim
n
^
2 jcX c(p 3
1Ir (p)
n
by
^(p)
xKp)
Iim
p
from this value is
'
uniquely ,minimizes the functional
-f
L
-
.
p^. dp-x-
2 jcX c(p^)
' '
I
and the deviation
2-
;
1
:
p
X
furnishes us with
i
(p)
i (p).
it follows that
p
'
1
2j^
pdp* j:-
.
X
Dr. (p - p ).
¥
I
Therefore we define a
Li
The m i n imum value is
I
2JtX c(p^)
1
The extremal property!of
^
dH
(P - P 1^)
■ n
X
'
2jtX c(pi n )
Is
, V (p).
11.
2aX c(p)
among all p e H(X),.
+
is infinite as'well.
i
Theorem 1_. The function
Proof.
d i m DT
-=(p ^ - p)
■n
Vy in
-By equation (5 ) of Chapter, II
infinite as well.
n e w functional
t h e r e , implies that
■
.
for e a c h p G iH(X). . Passing to the limit we obtain
Using the factb that ' 'p^
V (p )
=
2jtX c.(p ) „
Hence
=
Xp^
and.^
dp^x- = 0,
the minimum property of
As for the deviation recall the equation (5 ). from Chapter II, namely
tl/ n (p)
n
==
■
Dg
X
(p-Pi n )
n ■
in
+
X
2 jcX c(p i^).
• in
By iCorollary ■I . to lemma 6. the
: :
; . . ■
.
)
-34limit of the term
is f i n i t e .
%
Hence, if
(p - p ^ )
^_^(p )
is
(p - p^)
providing'
D jj _ A
(p) '
is finite, we obtain the desired deviation
formula after passage to the limit of the above expression for ^n (p)•
^W-A
Dg.
i® infinite, then so'is
(p - p ^ ) .
If
The use of the triangle
inequality for' Dirichlet integrals obtains this fact.
Therefore, the devia­
tion formula holds in the.sense that both sides are infinite.
i 10.
Extremal properties of the second X-th principal analytic function.
We define the function .
normalized so as to get
h-x-( £)
nuation of p ^ - log Jz - I |.
and for e a c h
Fe
A(X)
as
=
exp (p^ + i p^-x-)
0.
Here,
h-x-( I )
The function
we have
IogjFj
where
p \
is the harmonic conti
belongs to the class
belonging to
is
H(x).
A(l)
These remarks
together w i t h Theorem I establish, in a manner analogous to the proof of
Theorem 2 of Chapter'III, the following theorem.
Theorem 2.
The
X-th principal function
F^
uniquely minimizes the
A(X).
The minimum' value is
functional
23tX log r(F)
+
f IogjFj
among a l l functions F
2 jtX log r(F^)
d argF
belonging to the class
and the deviation from this value is
It should be remarked that
disk mapping.
- 2^
F^
This is a consequence of
univalent function
F
(IogjFzZF^I)'.
is a weakly X-valent circular slit
F ^ being a X-th power of the
w h i c h is proved in
[ 4]
.
This could not be proved
X
here since the fact was used to derive the extremal property of F^ .
_35_.
A corollary exhibiting another extremal property of
drawn from Theorem 2.
Denote hy
B(F)
will he
the quantity
I
%
Define a class o f •functions
(l) which is to he a subclass of
w i t h the additional restriction that
Corollary T_.
Proof.
2aX
log
Let
r(F)
The function
min
r(F)
3(x)
-
F e
F
=
B(F)
X
^
A n (X)
0.
has the extremal property
r(F^).
i
3 (x).
Then
2itX log r(F)
+. B(F)
^
2 jtX log r ( F ^ ) .
The last inequality is the extremal property of F^ .
It is a n open question whether
F^
Hence the result.
has a min-max extremal property
similar to that of Corollary 2. of Chapter III.
A proof similar to the
pr o o f there wo u l d he complete if it could he established that the function
<P° G
in
£}
belongs to the class
(X)
with
m|x
| G(z)
3 (X).
|
=
Here
G
r < r(F^)
is a function assumed to he
.
I
I;
! I
I
I
LITERATURE CITED
I
!
;
I
.1
: ;
: i
I'
LITERATURE CITED
[ I ]
L. A l i l f o r s Complex A n a lysis', McGraw-Hiil Book- Company, Inc.,
(1952).
|
: j
' I
-I
C 2 ]
L. A h l f o r s , and L. S a r i o , Riemann Surf a c e s , Princeton University ■
Press, (i9 6 0 ).
;
;
[ 3 ]
P. E i c k e I, On extremal properties for annular r a d i a l ,and circular
slit mappings of bordered Riemann surfaces, Pacific J. M a t h . ,
1 1 , 1487-1503.
,
i
[ 4 ]
P. Eic k e I, A note on principal functions and multipI y -valent
canonical m a p p i n g s . Pacific J. M a t h . , (to a p p e a r ) .
L 5 ]
■
;i
.
L. S a r i o , A linear operator -method on arbitrary Riemann surfaces,
Trans. A m e r . Math. Soc., J2 (l952), 281-295.
[ 6 ] ,L. S a r i o , Strong and weak boundary components, J. Analyse M a t h . ,
5, (1956-57), 389-398.;
"
I
-' ! '
[ 7 3 E. S a r i o , Capacity of a boundary and a 'boundary component, Ann.
of Math., 52, TT95 ii-T, 135-144.
[ 8 ]
L. S a r i o , Extremal probl e ms'and' harmonic interpolation on open
Riemann surfaces, T r a n s . A m e r . Math. Soc., 79, (3955),
362-377.
i
I
i
I,'
I
MONTANA STATE UNIVERSITY LIBRARIES
CO
111I 11(111IIIIII
7(32 10C 1C>40*
O
■ cop.2 ^
Garoutte, D. E.
Extremal properties character: .Z
ing weakly I -valent principal
functions
"
AND ADONtCOa
q n
(2 67/9
'
^