Convolutions in Geometric Function Theory (Séminaire de mathématiques supérieures)

CONVOLUTIONS IN GEOMETRIC FUNCTION THEORY

STEPHAN RUSCHEWEYH

SEMINAIRE DE MATHEMATIQUES SUPERIEURES SEMINAIRE SCIENTIFIQUE OTAN (NATO ADVANCED STUDY INSTITUTE) DEPARTEMENT DE MATHEMATIQUES ET DE STATISTIQUE - UNIVERSITE DE MONTREAL

CONVOLUTIONS IN GEOMETRIC FUNCTION THEORY

STEPHAN RUSCHEWEYH Universitat Wiirzburg

1982 LES PRESSES DE L'UNIVERSITE DE MONTREAL c.P. 6128, succ. «A», Montreal (Quebec) Canada H3C 317

ISBN 2-7606-0600-7 DEP6T LEGAL - 3" TRIMESTRE 1982 - BIBLIOTHEQUE NATIONALE DU QUEBEC

Tous droits de reproductIOn, d'adaptation ou de traductIOn reserves © Les Presses de }'Universite de Montreal, 1982

To my Friends and Colleagues in Afghanistan

CONTENTS

INTRODUCTION . . .

11

Chapter 1 DUALITY

15

1.1

The duality principle.

15

1.2

Test sets . .

19

1.3

Special cases (1) .

22

1.4

Special cases (2)

28

1.5

Convolution invariance

35

1.6

Additional information

41

Chapter 2 APPLICATIONS TO GEOMETRIC FUNCTION THEORY

45

2.1

Introductory remarks

45

2.2

Prestarlike functions .

48

2.3

Application to close-to-convex and related functions

63

2.4

Related criteria for univalence . .

70

2.5

M and related classes of univalent functions

75

2.6

Convex subordination

84

2.7

Univa1ence criteria via convolution and applications

94

2.8

Additional information

97

. . . . ..

.••....

10

Chapter 3 LINEAR TRANSFORMATIONS BETWEEN DUAL SETS . 3.1

Some more duality theory.

105 105

3.2 Special cases

112

3.3

118

Additional information

Chapter 4 CONVOLUTION AND POLYNOMIALS

121

4.1

Bound and hull preserving operators

121

4.2

Application to univalent functions.

130

4.3

Polynomials nonvanishing in the unit disc

136

4.4

An extension of Szego's theorem

140

4.5

Additional information . . . . .

141

Chapter 5 APPLICATIONS TO CERTAIN ELLIPTIC POE'S

145

5.1

Connection with convolutions

145

5.2

Univalent solutions

148

5.3

Extension of Schwarz' Lemma

154

5.4

Additional remarks .

156

REFERENCES

157

SUBJECT INDEX

167

LIST OF SYMBOLS AND ABBREVIATIONS .

168

INTRODUCTION

For two functions

f

analytic in

Izl




= y(f)

y

E R such that

0, z E U.

and note that

A is a domain

Assume (1.11) to be false.

Then there is a

19

straight line origin.

Let

xo'yo E U.

p (f

through the origin which intersects

* g)(x O)' (f * h) (yO)

Then there exists

A on both sides of the

be such points on

to E (0,1)

p,

where

g,h E U,

such that

(1.12) But (1.12) is imnossible since 2)

From (1.11) we deduce g E V**.

by Theorem 1.1, for Re e iy (f * (tg + (1

V**

U is complete and

is convex.

If

t)h))(z) > 0

Re e iy (f

*

f E V*.

g) (z) > 0, z E U,

g,h E V**, t E [0,1], in

U,

for

g E V and,

we therefore have

and this shows

tg + (1 - t)h E V**:

From (1.4) we obtain

co U =

co V = co V** = (/**

and thus the assertion. Note that Corollary 1.1 applies to the situation described in Theorem 1.3.

It implies that in complete and compact convex sets in

the form

l. 2.

functionals of

A1 /A 2 , Al ,A 2 E A, are extremized by convex linear combinations of at

most two extreme ?oints and their rotations. Stieltjes

AO'

inte~rals

A similar result for analytic

in [5lJ has found many apnlications (for instance [87J, [79J).

Test sets The most difficult part in the application of the duality princiole is

the determination of the second dual for a given interesting set be applied, i.e.,

U c AO to find a (small) set U c V**.

V c AO or, in turn, for a given V c U,

such that Theorem 1.1 can

This latter problem is somewhat easier to handle and

leads to the introduction of test sets. DEFINITION:

Let

U c AO'

Then T c AO

is called a

tv..t !.Jet for

U

20

(written

T

~>

U)

if

(1.13)

T cUe T** .

The following simple observation will be useful:

a set

Uc

AO is a

dual set if and only if (1.14)

U

= U**

.

In particular,

V*

(1.15 ) for arbitrary

=

V***

v cA. o

(1.18) (1.19) (1. 20) (1.2l) ('rJ k:

(1. 22) PROOF:

Tk

~>

U) k

>

(1.16) follows from the definition of dual sets.

(1.17) we have

T1 c T2 c Ti*

of (1.15) gives (1.17).

T*1 :) T*2 :) T*** l ' An application obtain Ti:) Ti :) T3, while (1.17) gives

and (1.16) implies

From (1.16)

~e

From the assumption in

21

T*1 = T3· Thus T*1 = T*2 and T** and = T** = T** 1 2 3

= T** T2 c T** 2 1

T** T3 c T** 3 = 2 .

and

T3 c T** 2

the result follows.

and this implies

( U T*) * = k

( U Tk) **

U T** k

:J

= (n

and thus

T** c T**** = T**. 2 1 1

T3 c T** 1 . Since

(1. 20) is immediate from the definition of duality,

n T** k

( U Tk)**:J U Tk, serve that

To nrove (1.19) we have to show

T2 c T** 1

From the assumption we have

(1.18) follows from

An application of (1.16) gives

:J

which is (1.21).

Tk ) *

:J

U Tk*

For the proof of (1.22) we ob-

by (1. 20) J (1. 21) •

(1. 22) follows from

U Uk' For

U,V

C

Ao let U· V be the direct product U • V = {f

THEORE~1

1.5:

Let

I

f = g • h,g E U,h E V} •

Tk , 11 k , V

C

AO' Tk

Qomplete..

The.n

(1. 23)

(1. 24)

Tk

~> Uk' k

= 1,2

=>

T1

T2

•

~> Ul

• U2

In the proof of this theorem and on other occasions we use the notation etc. if the convolution is to be performed w.r. to the variable

z, x,

etc.

Note that

convolution involving various variables is associative:

f(x) * (F(z,x) * g(z)) = (f(x) * F(z,x)) * z g(z) . x z x PROOF of Theorem 1.5:

To prove (1.23) we may assume that

just one element, say

g.

f E T l , h E (T l • V)*

such that for any

V contains

The general case then follows by applying (1.22).

Ixl

~ 1

(completeness!)

Let

22

o 1:

(1.25)

F : z

fixed) is in

x,z E U.

Ti.

1

~ h(z) *z l-xz x ~ h(z) * g(z)

For arbitrary

f E Ul

Thus (1.25) holds with the new

• V)**

Ti*

C

f,

we obtain

Ul • V C (r l • V)**.

and finally

Fz (x) * x f(x)

too, and the limit

(h * (f • g)) (z) 1: 0, z

with Hurwitz' theorem gives fg E (T

*/(x), z E U .

U ~ C,

F: z

This shows that the function

(z

!~~~

h(z) *z(f(xz)g(z)) = h(z) *z

f

U.

(1. 24)

x

~

1

~

0,

together

This implies is an iterated appli-

Special cases (1)

1.3.

In the next two sections we shall determine a fairly big class of sets in AO

to which the above concepts apply. THEOREM 1.6:

V**

=

Let

V

= {(I

A simple but crucial result is:

+ xz)/(l + yz)

H. f E AO

H denotes the class of functions Re eiYf(z) for a certain PROOF:

Y E

>

0,

x

I

Z

EU ,

R.

We write l+xz l+yz

(1. 26)

f E AO

I Ixl = IYI = l}.

is in

V*

if and only if

=

(1 -

y) l+yz

x

+

Y

such that

Thel1

23

for

Ix I

z E UI

=

Iy I

l+xz ltyz

*

f

a

(;

f(-yz) # £-1 • £ = x/y .

fixed and varying

y

straigth line f(O)

x x - y)f(-yz) t y ¢

or

= 1,

(1.27) For

= (1

Re w

= 1: Re fez)

=~.

> ~,

x,

the right hand side of (1.27) represents the

Thus

feU)

z E U.

cannot intersect that line and because of

This condition is also sufficient for

f E V*.

Any such (and no other) function has a Herglotz representation

(1.28)

f(z)

=J

d~

c;)

l-1';z

ClU where

~

is a probability measure on

(1. 29)

(f

*

g) (z) =

ClU.

J

Now if

g E H we have

g(r;z)d~CI';)

ClU such that the range impli es

of

(f

(f * g)(U)

find a two-point measure satisfies

0 E (fO

* g)(U)

is contained in the interior of

and thus: ~o

g E V**.

If

g E

g

f V**.

This

AO is not in H, one can

such that the corresponding function

* g) eU) and this shows

co g(U).

fO E V*

We omjt the details.

The following result which generalizes Theorem 1.6 will be refined in the next section.

Therefore we state it as a lemma.

Note that for

a,S> 0

we have

For

a > 0 we use the notation

24

and that l+xz I +yz E H, x,y E U

LEMMA 1.1:

Fo~

a

~

1 we

h~ve

(1.30)

[CX]

PROOF: 1.6.

IT VI and VI complete with VI ~> H by Theorem k=1 An iterated application of (1.23) gives l' cx ~> Vcx-[cx] H[cx] . If F E Ha - l ,

Vcx

We have

= Va_[cx]·

we have ItxZJ I - CX +[CX]F E ( Ityz

H[a]

' x,y

EU,

and thus

V c V • Ha- 1 c V cx

V ~> V • Ha - 1

(1.18) implies

cx

a-[a]

1

H[a] VI • Ha-I

and (1.23) gives

1

~>

follows now from (1.19).

THEORE~1

1. 7:

Fo~

aI' ... ,cx n E

n V = { IT

(1 + x. z) J j=l

Then

OM

aVlY

f

~

V** , x E U,

ak

E

C te:t

Ao

x.

J

E

V,

j = 1, ... ,n}

we. have

(1 + XZ)CXf(Z+X ~ If(x) E V** l+XZ) n

wheJte

a::

L

j= 1

PROOF:

For

a .. J

x E U let a (T) =

T+X

I+XT

, b (1") =

T+X T+XT

J

'" Hu;.

The result

2S

be automorphisms of

U.

These functions are correlated by

(l.31)

1 + a(T)z = l+xz(l + b(Z)T), T,Z E U • l+xT

For

x. E IT, w E U, J

n (1.32)

II

put

y. = a (wx.) J

such that with (1.31)

J

n

01..

01. (1 + y.z) J = e(l + xz)

J

j=l

01..

e1 + b (z)wx.) J

II

J

j=l

with

e=

n IT

01..

(1

+ n.w) J J

j=l Note that

C

is independent of

z.

Thus for arbitrary

g E

V* we obtain from

(l. 32) :

o 'I

g

*z

n II

(1 + xz) 01.

01..

(1

+ b(z)x.w) J J

j=l n

01.

(l+xz) = g *z 1-b(z)w *w

Since this is true for arbitrary

z,w

f

01..

(1

IT

j=l

+ x.w) J J

U we deduce that the function

Fzew) E AO

with g (z) F (w) =

z

is in

V*,

Now let

o 'F

f

f

V**

g(z) *

few) *w FzCw) =

f 'F 0

in

z

el+xz)OI.

such that

g(z)

We note that

(l+xz)OI.

* Z 1-b(z)w

* z (l+XZ) 01.

, z,w E U •

U by the duality nrinciple, in particular

f(x) 'I O.

26

Thus we may apply Hurwitz' theorem g(z)

(1. 33)

+

1)

to deduce

(1 + xz)af(b(z)) ~ 0, z E U ,

*

because the function (1.33) is g E V*

(w

~

0

z

in

= o.

(1.33) holds for arbitrary

and this implies (1 + xz)af(b(z))/f(x) E 1/** .

(1.34)

This result has a number of useful applications.

A very important

special case is contained in the next corollary. COROLLARY 1.2: k = l, ... ,m,

Le:t II be.

-tn The.altern 1. 7.

M

m

= {rr

(1 + YkZ)

f\

Yk E 0, k

k=l

+ xz)a-S

{(I

PROOF:

Let

E C,

M.6ume. U

Then

Fait c.e.Jttun Sk

I

fEU.

= l, ... ,m}

c V** .

m

x E IT} • U c V**,

wheJte. S =

I Sk'

k=l

Using (1.31) we can write the function (1.34) as

B

m (l+a(Yk)Z) k . + xz)a rr

(1

Since

a

k=l

is an automorphism of

l+xz

U the proof is complete.

An imDressive demonstration of the power of Corollary 1.2 is the proof of our next theorem which, in fact, is equivalent to Szego's theorem (0.1). denote the set of polynomials P E Pn , p

nonvanishing in THEOREM 1.8:

p, deg p

~

n,

and

p

is in

....

p

n

if and only if

U.

Fait n E N let Vn

= {(I

+ xz)n

I

x E

Let

U1. Then

Pn

27

PROOF:

1)

Using the functionals

principle we see:

V** c n

p

n

Pn

~>

and the duality E U,

can be

~

V** c p n A. n n 0

V** cannot vanish in U: n

It

n AO'

The next step is to prove

2)

in

Vn

n,

>

n A0 • The functional A(f) = f(zO)' Zo

used to show that the functions in remains to show:

Ak(f) = f(k)(O) E A, k

In fact 1 if

f E

V*n'

we have

U:

o f.

(1. 35)

(1

+ z)n

*

n

f = II (1 + ZkZ) . k=l and

(1.35) implies

fl (0)

1

n

=- L

n k=l

Thus

zk'

Ifl (0) I

~ 1

and (1 + xz) * f = 1 + Xfl(O)Z f. 0, z E U , which gives

f E

Now we proceed by mathematical induction.

3)

trivial. therefore

Assume it holds for (1

n - 1.

IT. + yz) E V** n ' Y E

Vn _ l ~>

Pn - l n ~O

For

n = 1 the claim is

V** 1

From 2) we know that

Corollary 1.2 applied to

lin ~> {(I + xz)n-1 C1 + yz)

(1.36)

Since

Vi.

I

1/

c

V** n

J

and

= V , U = V** gives n 1 •

x.y E U} = VI • Vn _1 •

we can apply (1.23) to obtain

(1. 37)

The result follows from (1.19). The idea to this proof as well as Corollary 1.2 are due to Sheil-Small [74J.

Note that a polynomial

V*n

if and only if

28

p (z)

=

I

k=a

(~] a k zk

":f;

~

Thus Theorem 1.8 shows that for every This is Szego's Theorem 0.1.

q E Pn

a,

z EU .

n Ao

p *

we have

q ":f;

0, z E U.

Of course, Theorem 1.7 carries more information

since the duality principle applies to this situation.

1.4.

Special cases (2) For

a,S

~

a

let

rca,S)

IS

T(a,S)

The final result (Theorem 1.9) is due to Sheil-Small [74J

are test sets.

to determine fairly large sets

K(a,G)

The aim of this section

and slightly weaker formulations are in [58J.

for which

In both previous approaches a geo-

metric property of functions J "starlike of order

a" J

was a crucial ingredient.

The proof presented in this section makes no use of that result. We start with a preliminary observation.

LEMMA 1.2:

Fo~

S

~

1 we have

T(l,S - 1)*

J

T(I,S)*.

In the proof we need a method which recently found many applications and reflects, in fact, a special case of the JUlia-Wolff Theorem.

It is known as

Jack's Lemma [26J:

LEMMA 1.3: e~n 6ottow~

Let w

be m~omo~ph~e {n

Zo E U the ~nequalLty

that zaw'(zo)/w(za)

~ 1.

U, w(o) = o.

Iw(z) I ~ Iw(zo)1

holdo 6o~

Then

{~ 6o~

a

Izl ~ IZa l , ~

29

PROOF of Lemma 1.2: of the functions

1)

f E T(l,y)*.

First we give an alternative characterization A slight modification of the definition is

f * (l-z)-y ---'--""""""'I-y ¢: f * (l-z) -

(1.38)

x

X-T '

IT, z

x E

E U ,

which is equivalent to the statement that the left hand side of (1.38) has real part

>

1

in

V.

Let f

Jf

::

(1.39)

* ( 1 - z) l-y, y 1: 1

If

Y

,

y :: 1 •

Then the identity (1.40)

_1__ :::

_....:1:...---::-

(l-z)Y

(l_z)y-l

*

[y - 2

-L + _1_

y-l 1-z

1

]

y-1 (1_z)2

leads to the relation (1.41)

A combination of (1.38) and (1.40) shows:

f E T(l,y)*

z

~

if and only if

V, Y 1: 1 ,

(1. 42)

Re f Y > 12., Z E U, Y = 1 . Note that this holds for 2)

Now let

T(l,l)** = H ~ T(I,O) exists an

f E

S~

y 1.

~

O. If

and thus

$

=1

we conclude from Theorem 1.6 that

T(l,l)* c T(I,O)*.

T(I,B)*\T(l,B - 1)*.

For

S>

1

assume there

30

If we write

zf

(8=2

(1. 43)

S_l

fS-l

w(z) , B ¢ = l-w(z)

fS-l

1 = l-w (z) , S = 2

then if follows from our assumptions that that there exists

Zo

E U such that

From Lemma 1.3 we get

x

,

w is meromorphic in

Iw(zo) I

= zow' (zO)/w(ZO)

2 ,

~

1.

= 1,

Iw(z)1 :'" 1

U, w(o)

= 0,

and

Iz I :'" Iz 0 I .

for

Taking the logarithmic derivative

of (1.43) and using (1. 41) we obtain after some manipulation w(z) (1 _ l-zw'(Z)/w(z)) l-w(z) B-1

(1. 44)

Since

f E T(l,B)*

This shows that

fS ¢ 0

it is clear that

in

U,

in particular,

fS(zO) ¢ O.

w(zO) ¢ 1 and thus

(1.45 )

(1.45) contradicts (1.42) with For y

Y

+ xkz)

Yk

k=l

LEMMA 1. 4:

PROOF:

The proof is complete.

we define

> 0

m

V = {IT (1

y = B.

m

E AO I mEN, Yk ~ 0, xk E U, k

Let

a., 8

~ 1.

= 1, ... ,m, I

k= 1

Yk = Y} .

The.VI.

The proof consists of a large number of test set operations as described

in Theorems 1.4, 1.5.

We start with the case

T(l,B - 1)** c T(l,S)**,

a. = 1.

and thus, by Corollary 1.2,

From Lemma 1.3 we have

31

T(l,S) N> {

l+xz 1 (l+yz) C1+UZ)S-

= T(l,l) • TCO,S H'. TCO,S -

~>

(compare Theorem 1.6).

T(l,S) -> {

contains

l+xz 1 q(l+uz)S-

I

q

E VI' x,u E IT}

q E VI

we obtain

= V(O,l) • T(l,a -

This is the desired result for a = 1. [aJ-1 ~ ((1 + xz)(l + vz) E p[aJ n AO)

The latter set

TCa - raJ,S) •

1) •

H ~ Vel,S)

Now let

a> 1.

{(I + xz)[aJ

I

From Theorem 1.7 we get

x E IT}

[aJ-1 a-raJ {(l+xz) (l+vz) (l+yz) l x,v,y,u E UUJ ' S C1+uz)

~>

= T(l,S) •

T(a - 1,0)

Vel,S) • T(a - 1,0)

->

The latter set contains

V(l,O) • T(a - 1,S),

and an inductive argument gives

TCa,S) -> V([aJ,O) • T(a - raJ,S) This set contains

1) .

and thus

T(l,S) -> V(O,S -

=

1)

Tel,S) -> V(O,[SJ) • T(l,S - [SJ).

V(O,S - 1) • T(l,l),

TCa,S)

1)

(1 + xz)/q E H for

Since

An inductive argument gives

I x,y,u E IT}

P(a - 1,0) • T(l,S),

and thus

T(a,S) -> VCa - 1,0) • Vel,S) COROLLARY 1.3:

Let 1

~

a

~

S. Then

= V(a,S)

.

32

{

C1. 47)

TCa,S) ""> Ha • V(O,S - a) T(S,a) ""'> Ha • V ((3 - a. 0)

This is an obvious consequence of Lemmas 1.1. 1.4

T(a,S)

C

Va • V(O,S - a)

C

since (in the first case):

V(a,S) c Ha • D(O.S - a).

The second case is similar. Since second duals are closed we may improve Corollary 1.3 by taking the closures of the right hand sides of (1.47).

=

q (z)

m

II

(1 + xkz)

-Yk

Ha

is already closed.

E DeO,y), y:::

Let

a .

k=l Then we have Re zg I (z) q (z)

It is well known that the functions

are dense in the set of functions Thus

V(O,y)

f E A with

is dense in the (closed) set

Re(zg' (z)jg(z))

> -

L2

frO) = 0 and

K(O,y)

of functions

Now let

(1.48)

and (1.49)

K(a,S)

=

{t I

f E

KCS,a)},

0

Re f

~ S~ a .

>

-

~

g E AO

in with

U.

33

Thus we get

Fon a

THEOREM 1.9:

~

1, S

1 we have

~

T(a,~) "'>

(1.50) K(a,S) the fact that

Kapla~ elao~~

are called the K(1,3)

K(a,S) . of type

(a,S).

This is due to

is the class of derivatives of the so-called close-to-

convex functions, first introduced by Kaplan [29J (see Chapter 2). Kaplan used an intrinsic definition of this class, namely if and only if it is nonvanishing in arg f(re This extends to

(1.51)

i& 2 i&l ) - arg f(re )

~

is in

K(1,3)

&1 < &2 < &1 + 2n, 0 < r < I, -n + &1 - &2 .

K(a,B):

60n &1

arg f(re

a~d ~o~va~~~g ~~

f E AO

THEOREM 1.10: ~n a~d o~y ~n

U and for

f E AD

In his work,

U

~ ~~

K(a,S), a,S

~

0,

< &2 < &1 + 2n,

i&2

) - arg f(re

i&l

) ~ -an - l(a - 8)(&1 - &2) .

For a proof, using Kaplan's original idea, see Sheil-Small [74J. seems to be a weakness in Theorem 1.9.

Is it perhaps true that

T(a,S)

There can be

replaced by the sets

(1. 52)

The answer is not known but a hint in this direction is contained in the following theorem. THEOREM 1.11:

pJtObab..i1.Uy

meMuJte

]J

Let a

O~

~

(au) 2

1, B ~ 1,

.6ueh that

a~d

f E K(a,S).

The~

th0te

~

a

34

fez) ::

a. (l+xz) 8 d~ ( ), z EU • x,y (l+yz)

J (au)2

PROOF:

It follows from Theorem 1.9 and the duality principle (compare Theorem

1.2) that every extreme point

f E co K(a.,S)

has the form

fez)

(1.53)

where we may assume

0 < Y :: a. - [a.] < 1.

We have to prove

from Theorems 1.2, 1.0 that there is a probability measure

(l+XZ)l-Y(l+YZ)Y:: l+uz

J

It is clear

x:: y.

on

~

such that

au

l+l;;z d l+nz

J..l

(aU) 2

Thus

fez) ::

J (au) 2

Thus

f

is represented as a convex combination of members in

assumption shows that

(1.54)

J..l

K(a.,S)

and the

~O:

is concentrated at one point, say

fez) ::

Comparing (1.53) and (1.54) we immediately deduce

no:: u,

~o

:: x, the assertion.

This proof is due to Clunie (see [74]) and a similar approach has been used in [58].

For an alternate argument see section 6 of this chapter.

certain values of the parameters (for instance,

a.:: S

~

1,

CI. ::

S- 2

~

For 1)

35

Theorem 1.11 has previously been obtained by pure convexity theory methods (compare [70J, chapters 1,2).

As an application, we mention

Let

COROLLARY 1.4:

f E

K(a,S), a,S

~

Th~n

1.

(1. 55)

Here we use the symbol fez)

= L akz k ,

g(z) =

I

for coefficient majorization:

~

bkz k , b k ~ 0

for

k ~ 0,

we have

for

f ~ g

if and only if

~

~

lakl ~ b k , k ~ O.

PROOF:

Brannan, Clunie and Kirwan [9J proved that for

Ixl

1, a

1,

(l+xz)a ~ (ltz)a l-z l-z Thus we obtain for (ltxz)

x,y E

au

a

(l+yz)B The result follows from Theorem 1.11.

1.5.

Convolution invariance U c AD

In this section we study sets

which are invariant under convo-

lution: f,gEU

(1. 56)

>f*gEU.

We make use of the following simple criterion. LEMMA 1.5:

(1. 57)

Fait

V

C

AD

a..6.6Um~

f E V*, h E

(II

:that V,..,.> Wand

-> f

*

hEW •

36

Then (1.56) hold6 PROOF: (f

*

Let

g)

*

V~*

If we use (1.57) twice, we obtain

Since this function is nonvanishing in

f * g E V***

EXAMPLE: relation

U = V*.

f,g E V*. hEW.

hEW.

we conclude

6o~

.....

*

(f E V*,g

*

(g

*

=

h)

h E V**),

= V*.

From Theorem 1.8 we have for

= Pn n AO'

U

f

Vn

= {(l + xz)n

I

x E IT}

the

Thus it follows already from the definition of duality

Vn has property (1.57) and we conclude that V*n is closed under convo-

that

lution.

Note that this is exactly Szego's Theorem 0.1.

T(a,S)*, a.S

Next we study the sets

~

1.

To show that these sets are

invariant under convolution we need some preliminary results which will be useful also in other situations. THEOREM 1.12:

g E AO

let

v = {l+xz

g(z)

Fo~

l+yz

Then

nO~ eV~1j

f E

V* and

PROOF:

f

We have

V "'> H • {g}

for

HE

H.

f * g ¢ 0

and thus For

*

(Fg) (U) g

*

in

U.

y E U fixed and 1

co(F (U)) .

C

Since from Theorems 1.5. 1.6 we obtain

f E (H • {g})*,

H

is in

F E A we have

f

(1. 58)

x,y E IT} •

we conclude that

a ER

= (l-yz -

!

(f

z

(y)

(Hg))j(f * g) ¢ 0

the function

+ ia)/n + ia)

H and inserting this into the above inequality we get

F

*

Re Fz (y)

>

!

for

37

Herglotz' formula implies the existence of a measure

f (1

=

Fz(Y)

~z

on

au

such that

Sy)-ld~z(~)

-

au and thus for

F EA

= F(y) *

fez) * F(yz)g(z) fez) * g(z)

(1.58) is the limiting case

y

(1.58) hold~

60n F E A.

PROOF:

x.y E IT we have

For

1.

~

Let a.a

COROLLARY 1.5:

=

F (y) z

y

~ 1, f

E T(a.a)*, g E K(a - 1.S - 1).

Then

l+xz l+yz g E K(a,S) . Theorem 1.9 gives

= K(a.S)*

T(a.S)*

and thus Theorem 1.12 applies.

The next two theorems are generalizations and refinements of Corollary 1.5. THEOREM 1.13:

and let

f

a,a.y,o.~,v

Let

a

~

y

a

~ ~ ~

~

~u~h th~

a

~

0

~

a-I

y, a

~

v

~

S - 0 •

a-I. a -

E R be

E T(a,S)*, g E K(y,o), F E

K(~,v).

Then

f * gF E Hmax{~.u} •

(1.59)

f

*

g

Special cases of Theorem 1.13 are in Sheil-Small [74J and in [58]. PROOF:

First assume

~ ~

v.

If

~ ~

1

the assertion is a special case of

38

Corollary 1.5.

Now let

>

R E K(~

There are functions and

~

1 and without loss of generality assume

- v,O),

S E

K(v,v)

=R

f

• S.

Let

~

1.

m = [~]

Q = R11m such that

Q E KCC~ - v)/m,O) For

with

v

k

= O,l, ... ,m

H(~-v)/m c H .

C

- 1 we have

gQk E Key + k(~ - v)/m,o)

K(y + ~ - 1.0)

C

C

KCa - I,B - 1)

and thus by Corollary 1.5 f

f

* (gQkQ) E H(~-v)/m, k = O,l." .• m - 1.

* (gQk)

Multiplication of all these functions yields f f

(1. 60)

v-I Now let

n = [v]

and

P

= Snv

E

** gR g

E

~-v

H

•

v-I

K(V~l

Vn-l) =

Hn

c

H.

For

k = O•.••• n

we

have (1.61)

gRpk E K(y + ~ - v + ~Cv

- 1).0

+

*Cv - 1))

c K(a -

1.B - 1)

and by Corollary 1.5 V-I f

* (gRpkp)

f

* (gRpk)

Multiplication of these functions for

k

E Hn

= O,l, ...• n

- 1

gives

V-I f

(1.62)

* gRS v f

* gR

E

v 1 H- .

v-I Finally. (1.61) for

k

=n

shows

gRS v

E K(a - 1.S -1)

and since

Sl/v E H we

39

conclude from Corollary 1.5 that

(1.63)

A multiplication of (1.60). (1.62) and (1.63) gives the result.

The case

~


f * h E TCa.B)*, h E K(a,S)

~

f

* h E K(a,B),

h E T(a,B)** ==> f

* h E T(a,S)**.

Without loss of generality we assume

1

~

B~

a.

According to Lemma 1.5

40

relation i) follows from ii). g E

K(a - B,O), F E KeB,S) f

FO E KeS,B).

with and

f E T(a,S)*

T(a,S)*. Since

C

Let

with

*

= gF.

h

h = f

*

be such that there are functions

From Theorem 1.13 we obtain

= (f *

(gF)

g) • F 0

ii) follows from Theorem 1.14 since T(a - B + 1.1)*.

From i) we obtain

fa

h E K(a,S)

To prove iii) let

fO * f E T(a.S)*

is arbitrary in T(a,S)*

g E K(a - S,O) • K(O,O)

fO

fO * f * h # 0

and thus

we conclude

f

* h

be a second function in

f

in

U.

T(a,S)**.

Theorem 1.15 has first been proved in [58J and by Sheil-Small [74J.

T(n,O), n E N,

that the example given above states that Theorem 1.15 holds for as well.

For

T(O,n),

however, it fails.

Note

The exact range of the parameters

a,S

for which Theorem 1.15 is valid is unknown. To conlude this section we prove that under certain circumstances, convolution invariance of a set transfers to larger sets. THEOREM 1.16:

LeA:

V c AO

be. c.omp.fe-te. aYl.d c.ompact.

UYl.de!L c.OYl.VO.fu.U.OM the .6ame ..L6 br.ue Let

PROOF:

h * f

Now choose

Al E A with

1..1 (q)

c..fMe.d

A2 (q)

the duality principle shows that are left with the proof of h * fO

= (h *

go

f * q) (z),

h * f * gO"# 0

in

U.

* q)(z). Another application of

A2 (f)

= A2 (f O)

* go

'I: 0 in

finition of duality and the assumption

= (h *

go E V such that

and the result follows if we can show that

A2 E A such that

.u

V**.

From the duality principle we obtain a function

= A1 (gO)

V

It will be sufficient to prove that

* g "# 0 in U. For z fixed let

q E AO' A1 (g)

f,g E V**, h E V*.

nOll.

16

for a certain U.

fO E V and we

But this follows from the de-

fO * go E V.

A similar statement deals with convex sets in

Ao.

Although this result

41

is not directly related to duality we prefer to mention it at this stage. THEOREM 1.17:

6unc.tlon h E Aa

LeX.

~uc.h

V

6o~

that

Aa

C

W-Lth

ate

W:::

co V

c.ompac..t.

AMume theJr..e

.v.. a

f,g E V we have

(1.64)

Then (1.64) holM PROOF: in

Let

V,

Vc

ate

nO~

f,g E W.

denote the set of finite convex linear combinations of functions

VC ::: W.

such that

holds for arbitrary

f,g E

W is convex we first conclude that

Since

(1.64)

Vc . Since W is compact, (1.64) holds for f,g

E

Vc

as well.

1.6.

Additional information 1)

We wish to mention two more structural properties of duality. n (1 + z) )

have seen (Theorem 1.8) that a single function (namely set for a large set. such properties.

We

can be a test

It would be very interesting to determine all functions with

A negative result in this direction is contained in the next

theorem. THEOREM 1. 18 :

V ::: {O,

LeX.

f E

whe~e

A and f(-l)

a

ex..wu.

Then.

V** ::: {f(xz) I Ixl ~ I} .

Here we denote by

f

f

(1. 65)

Clearly, if

only if i)

(-1 )

,,00

f::: La akz

k

E Aa

a k # O. k ~ 0,

the solution of

*

f(-l):::

1

1- z

the equation (1.65) can be solved in

and ii)

la k / l / k

+

I, k

+

00.

if and

42

PROOF:

Under the assumptions we have

V* = h E V**

Now let U.

* f(-l) I g E Ao' g

{g

(h * f ( -1 )) * g

such that

The functions

g E T(l ,S).

*

h

B ~ 1,

'¢

'¢

0 in

U} .

0 for arbitrary

g E AO' g '¢ 0

in

have this property and thus

f ( -1) E

n (T (1 , S) *)

•

~1

In Chapter 2 (Theorem 2.3) we shall prove that the latter set consists of the functions

(1 -

xz) -1 ,x E -U.

THEOREM 1.19:

L~

Thus we have

T1,T2

Ao

C

h

=f *

(1 -

xz) -1 ,

the result.

be ~omptete and ~ompact.

Fo~

y E R tet

Then

vY ~> PROOF:

Let

g

= yg1

yT** + (1 - y)T 2** . I

If

+ (1 - y)gz' gj E

duality principle the existence of

A E~, we conclude from the

such that for

f.

J

we have

In particular, for h E V* y

is in

~

and thus

(h

and

z E U fixed, the functional

* g)(z) = (h * f)(z)

'¢

O.

This implies

A(q) :: (h

g E V** Y

*

q) (z)

which is

the result. 2)

The following corollary to the duality principle has a number of sur-

prising applications since it permits to transfer certain extremal problems for second duals to different extremal problems for not related test sets.

43

Let Tj E AO'

THEOREM 1.20:

j

= 1,2, be

g E Ao'

compact and complete,

Then we have (1.66)

(1.67) PROOF:

First we prove the theorem with (1.67) replaced by g * h E Ti

(1.68) f E Ti*'

In fact, assume

h E

for arbitrary (1.68).

T2 .

h E T** 1

h E Ti* .

Then (1.66) shows that

g

= Ti.

g * f E T2**

T2* and thus

*

h

*

f

= (g *

f)

*

h

¢

0

Therefore (1.66) implies

The other direction follows by interchanging the subscripts

Obviously, (1.68) implies (1.67). f E

for all

1,2.

To prove (1.67) ==> (1.68) choose an arbitrary

From the duality principle it is clear that h E Tl .

if the same is true for all

g * f * h ¢ 0

for all

The proof is complete.

Some applications will be given in Chapter 2, section 8; compare [50J. We return to Theorem 1.11 and give an alternate proof which, however,

3)

works only for

a

~

In fact, consider

2.

= {(l+xz) (l+yz) a-I I

x,y,u E IT}

(l+uz)S

Writing

Tl (y,O)

K(a,S)

~

= {(I

TO(a,S)

This implies for

a,S

+ xz)Y

I

= T(l,S) •

~

1

x E

U}

we have

Tla - 1,0)

~>

K(l,S) •

T{a - 1,0)

~

T(a,S) .

44

(1.69) According to Theorem 1.1 the extreme points of sets,

TO(a,s)

and

T(a,S).

For

a

~

2,

co K(a,S)

are contained in both

however, the intersection consists of

the functions Cl+xz)a Q

,

x,y E U •

(I+yz)P Since functions with

x E U or

y E

U cannot be extreme points, the conclUSIon

follows. Comparison of (1.69) with (1.50) leads to the following problem:

TI , T2 are test sets

Is it true that if the compact and complete sets for the same set

U,

the intersection

The answer is unknown.

TI

n T2

is also a test set for

If it is affirmative, we would have a proof for

the problem mentioned after Theorem 1.10, at least for (1.50).

U?

a

~

2,

using (1.69) and

Chapter 2

APPLICATIONS TO GEOMETRIC FUNCTION THEORY

2.1.

Introductory remarks In this chapter we shall apply the duality theory to concrete situations

in geometric function theory. in particular to (classes of) univalent functions, Most of the functions

f E A of interest in this context are normalized by the

conditions f(O) = 0. fl (0) = 1 , and the collection of these functions is denoted by with f E Al

AO

AI'

Since duality is dealing

a direct application of the previous results is not possible, if and only if

f/z E AO f

*

and for g # 0, 0
f/z E K(O.2 - 2a) .

A function f E Al exists

~

a

It is clear from (2.1) that

AI'

(2.2)

a

~

S .

is said to be in the class

g ES • ~ E

a

R,

Ca , a

~

1,

if and only if there

such that (z) 0 z E U Re e i~ z f' g(z) > , ,

(2.3)

which is equivalent to

f' E K(1.3 -

(2.4)

The functions in subclass of

S

Co

are called

2a) •

cto~e-to-convex

and they form an important

SO).

(larger than

Another even larger subset of S is formed by the Baz~ev~Q 6un~o~ B(a,S), a> 0. S E R. such that for a certain

(2.5)

where

Re (f(z)/z)a+iS-l

f E B(a,S)

Here ~

e iRe


Wte

1.1

on.

au

Le:t

fE_C;,O'.~1.

a

.6uc.h :tha;t

(2.27)

fez)

= J au

z 2-20'. d1.1es)

(l-~z)

fez) a, z

(2.53)

For

0 S a < 1

these functions are convex univalent and the obvious relation f E K zfl E S , a a

(2.54)

holds.

U .

f

Many extremal problems in

K

~ S

1

J

are solved by the function

a

z

J (1

ha(z) :::

- t)2a-2dt

o which satisfies zh"

a

'il'" +

(2.55)

=

1

a

1+(1-2a)z 1- z

The following problem has been studied several times: number

S::: Sea)

such that

Ka

C

Clearly

SS'

8(0):::~.

find the largest

The following nice

result due to MacGregor [37J gives an indirect solution of the above problem

(-
0, z E U .

i"'2 holds. This

using the charac-

terization of Bazi1evic functions by Sheil-Small [71J will show that in

B(a,S)

for any

a

>

0, S E R.

functions which are not in

M.

pez)

is not

On the other hand, there are Bazilevic

A geometrical description of the members in

M

not yet known. The following two theorems are merely reformulations of Theorems 1.15, iii) and 1.11 (which holds for WEOREM 2.29:

Le;t

T(a,S)**

f E

M,

as well as for

00

WEOREM 2.30:

Lu

f (z) =

L

k=l meMWte

]J

ovt

(aU) 2

.6 u.c.h

that

Thevt

g E KO' akz

k

E

f

*

K(a,S)).

gEM.

M. Thevt theJr.e -U a.

p!to ba.b,uuy

is

79

z + _x;_y z2 (2.83)

f

(z)

=

--~2""'--

dll(X,Y) .

(ltxz)

rYL paJr.ti.c..u1.aJt,

lan I ~ n, n ~

(2.84)

Theorem 2.29 implies that for

2 .

£ E M we have

z

= J f~t)

h(z)

(2. 85)

dt EM,

o since

h(z)

=-

10g(1 - z) * f.

z(l - z)-l-i E S

For the spiral-like function

it is known (Krzyz, Lewandowski [31J) that

z

=

assumes the value

0

fo

fO (t)

infinitely often in

have another proof of

t

dt

U.

This shows that

M ~ S.

The next theorem implies an extension of (2.85). THEOREM 2.31:

Let

f E

M, Zo E U.

Th~n

£(z)-£(zo) (2.86)

-----"'- E T(l,2)** . z-zO

Since

fO(z) =

T(l,2)** c T(1,3)**

COROLLARY 2.5:

Fo~

f

E

we get in particular:

M, Zo E U, we have

fO E S\M

and we

80

f(t)-f(zO)

(2.87)

PROOF:

----"-- dt

EM.

t-z o

Let

g E T(1,2)*.

zg(xz)/(l - zOz) E SO'

Then for

x E IT we have

zg(xz) E R1

2:

= S12:

From the definition of M ((2.78)) we get

such that

f~r

O a 1 (t) -+ 1, t -+ to'

(a 2 (t)

-

- 1)

1) / (a 1 (t)

-+

4, t

-+

to'

F ,v., e. g •. To prove this result we shall need two theorems due to Korovkin (compare

[15J) (Lemma 2.3) and Brickman [10] (Lemma 2.4). LEMMA 2.3:

(ak(t) - 1)/(a 1 (t) - 1) LEMMA 2.4:

60Jt

n -+

on

UndeJt the M.6umpti.OY1..6 -+

Let

TheM.em 2.42 we have

2

k , kEN. be -i..n

G(Z,T)

n

noJt a .6equenee

A

co.

AMume theJte ex-i...6:t6 a g E A wUh

i)

G(Z,T ) n

-
rOo THEOREM 2.44:

2.8.

Izl < r

In fact,

f E Al ~

U if and only if there is a probability measure

on

is typically real [0 J nJ

'IT

fez) :::

f

o (l-e

i~

z -i~ z) (l-e z)

which shows that the set of these functions equals

V ::: {f~

I

f~ (z) :::

Z

--I"""'~:-------:-i~-::---

(l-e

z)(l-e

t)

dflC~) co

V where

, 0 5 ~ 5 'IT}.

such that

98

We have to prove

h

*

f

*

~

g

co V for

h

=-

log(l - z)

cording to Theorem 1.17 it suffices to do so for

f,g E

V.

and

f,g E co V.

Ac-

V c So and thus

But

f E V implies z

h * f =

J (f(t)/t)dt

E KO .

o This gives

(h

*

f)

*

g E So

for

f,g E

V and since h * f * g has real

Maclaurin coefficients it is typically real:

* f * g E co V.

h

This completes

the proof. This result naturally leads to the question whether the MandelbrojtSchiffer conjecture (see (2.73)) holds at least for univalent typically real functions.

However, even this weaker conjecture is false.

that the coefficient body

(a 2 ,a 3 )

Bshouty [13J has shown

of typically real univalent functions (previ-

ously determined by Jenkins) fails to have the necessary invariance property. Robertson's Theorem 0.2 implies in particular that real if

f,g

are typically real and, in addition,

g

f

*

g

is typically

is convex univalent.

It is

not known whether the corresponding conclusion holds if in all cases typically real is replaced by typically real univalent.

In particular, it is still possible

that any typically real univalent function is in 2)

M (see section 2.5).

It is an old conjecture due to Robinson [47J that for any

f ES

function ~[f(z)

(2.129) is univalent in where

Izl

C

Co

M

C

from the fact that

This observation is due to Barnard and Kellog

[5] and they were able to draw the same conclusion for any spiral-like function

f.

We give a slightly simplified version of their proof. Let

f

be spiral-like such that for a certain zf'

lui

=1

we have

ltuz l+z

f This implies that

zf'> '3' 1 Re f

Furthermore, for

F

= (1

Iz I
z-z o

It is worth mentioning that if (3.31) holds for a normalized function and any

Zo

E V, we can conclude

fER y .

For a proof of this fact see [54J.

f

An

application of Corollary 3.4 will be given in Chapter 4.

Additional information

3.3.

It is of interest to characterize functions

a.,S

:5

f E Al

such that for

I,

Denote the class of these functions by Chapter 2 that

Rea.,S).

It is obvious from the results in

119

The first result in this direction has been obtained in [63J. L~

THEOREM 3.9:

Th~~

a 5 S 5 1.

~n a~d o~y ~6

f E R(a,S)

f E Al

a~d

(3.32)

*

f

PROOF:

Let

Sy

z/(l - z)2-2y

=

cessity of (3.32) is clear.

z ES . (l_z)2-2a S

and

Now let

as before.

Sy(-1) g

E Ss' f

Since

= s~-l) *

g.

E S the nea a Then for any h E S a 5

we have

*

f

Smce s

(-1 )

h =

(S~-l) * 5 S *

h)

* (S~-l) *

g) .

s~-l) * hERa C RS we get s~-l) * Ss * hESS'

Furthermore,

* g E RS and from (2.23) we get f * hESS' One can show that Theorem 3.9 is false if

Silverman and Silvia [76J. characterization of

PROOF:

l( f * z

S

a

In the same paper, however, we find the following

L~

a,S 5 1.

* Ss(- 1) * z

~ ~~

f E Al

1+~Z2B)

g ER a

* f * sa *

(-1)

Ss

E RS

or, equivalently, if and only if f * h

(-1)

* Ss

R(a,S)

E T(1,3 - 2a)**,

(1-z) -

(3.33) is obviously fulfilled i f and only if g

for each

see Sheil-Small,

R(a,S).

THEOREM 3.10:

(3.33)

a > S,

E RS

~6 a~d o~y ~6

Ixl

51.

120

for each

h E S(). •

This is equivalent to

f E

R(a,S).

Using this characterization it has been shown in [76J that

RcLo)

i f and only if

~(zf'(z) + it) E T(l,2)**,

(3.34)

z

f (z)

(3.34) is fulfilled if there exists

(3.35)

holds in

U.

A E [O,lJ

t

E

R.

such that

f E Al

is in

Chapter 4 CONVOLUTION AND POLYNOMIALS

4.1.

Bound and hull preserving operators We recall de Bruijn's theorem (extension of Szego's theorem 0.1, compare

Theorem 2.51 with

n

= 0):

(4.1)

a polynomial (p

*

p E Pn

n AO

q)(U) c q(U), q E P

n

has the property ,

if and on ly i f

(4.2)

p

* (1 + z) n # 0, z E U .

Thus the polynomials (4.2) form the class of range preserving convolution operators (w. r. t .

Pn (f E B) n (4.3)

U)

on

P. n

A function

f E

A is said to be bound preserving on

if and only if

IIf *

qll .5 Ilqll, q E Pn

where

I

Ilhll = sup h (z) zW

I

The following theorem, due to Sheil-Small [73], characterizes the class

Bn .

122

on

au wah

'('6 and only

f E Bn

THEOREM 4.1:

,(,~

theAe ex"u,a a c.omplex me.MLULe

]J

1 and a 6unc.tion F analytic.,(,n U .6uc.h that,(,n u,

111111 ==

(4.4)

f 1-1;z dll + zn+l F(z). 1

fez) =

au PROOF:

If

f

satisfies (4.4), then for

*

f

q

=

q E Pn we have

J q(1;z)dll

au and hence

Ilf * qll

==

Ilqll •

111111 ~

Ilqll

which shows

fEB . n

Now let

fEB. n

On

the space

we define a linear functional

= (f * which satisfies

IIAII

Hahn-Banach theorem

q)(l),

q E Pn

Pn Pn is a subspace of cO (aU) and thus by the A extends to COcaU) without increasing the norm. The

==

1 on

Riesz representation theorem then yields a complex (Borel) measure 111111 ~

1 such that A(H)

=

J H(1;)dll(1;) , H E CO(au)

.

au Choose

z E u, q(1;)

= (1

- (z1;)n+l)/(l - 1;z) E P

n

such that

11

with norm

123

=

A(eD n+1 f 1- (zl;;) 1-zZ;

d (r-)

j1.."

au =

1 f 1-ZI;; dj1

n+1

+ Z

au Thus (4.4) holds with

F

= R2

- R1

R2 (Z).

U.

which is obviously analytic in

Note that Theorem 4.1 implies:

fEB

= U Bn

i f and only if there exists

n

a complex measure

j1

with

(4.5)

1Ij111:r; 1 such that

fez)

=

f l:ZZ; dj1(Z;),

z EU .

au If

fEB

n

and

f(O)

= 1,

be a probability measure.

(4.6)

and

(f

f

prese:', fEB. n

j]

g on

Pn

j1

in (4.4) can clearly be assumed to

In this case, however,

q)(z)

= J q(l';z)dj1 au

is even convex hull preserving.

E

co q(U),

z EU,

If, on the other hand,

it needs to fulfil both

f(O) = 1

(since

q

f

is convex hUll

=1

E P) n

and

Thus we obtain THEOREM 4.2:

th~~ ~x~u

(4.7)

*

then the measure

f E

a pnobab-iLUy

A

~ Qonv~x

M~a,6uJl.~

fez)

=

on

hull

au

~e4~nv~n9

and

F

analyuQ

1 n+1 J 1-zl'; dj1 + z F(z).

au

on

Pn

~6

and only

~n

~

U

.6uQh thctt

124

From the well-known Herglotz representation theorem it is clear that the set of functions (compare (1.28))

= J au

fez)

equals the set

d~, ~

l=ZS

probability measure,

R: f E AO' Re

fez)

> ~, z

EU.

Thus we have

The. 6oUoW- 1

:: 1

= aOb O + whenev~

-6(1.cJl

tha.t nOlL antj .6~qt1.~nc.e.

n alblz ...... + a n bn z #- 0, f

e (z) x

-L6 no,t one 06 the

z = l-xz , Ixl = 1

N. Th~VI. th~~

2.

Iz I

ak l-Izl fez) - T+TzT

AO

valid for functions in

'

with real part positive in

U (see(4.29)!).

Thus

a

is finite and we obtain (4.31)

h (z)

= .!. 0'.

f (z) -0'.

z-l

=1

7T.

I

+

(1 -

j=l

J-

1(1)

. )zJ

0'.

= I

j=O

Hence

11 _

(4.32)

But

o = q(l)/O'.

1(1)

7T.

J- 0'.

I

= 1 lB. JO

.

1, j E r~

is a convex linear combination of

froM (4.32) we deduce the existence of shows that

1-
~f. Theo~y ~n ~~odeJt~

TSUJI, t1.,

[93J

d~e (JJWtze1..~

TOEPLITZ, 0., V~e ~ne~en votlkommenen Rawme deJt Funk~onentheoJUe.

[89J

[92J

tlbeJt

(1922), 28-55.

typ~c.ally ~eal nU~~o~

Comment. Math. He1v. 23

[91]

G~ac.e

(1931), 163-166.

Belg. Bull. Cl. Sci. (5) 66 [88J

Indiana Univ.

(1979), 429-443.

algeb~~QheJt

[86J

Q~~on no~ ~~ke 6un~0~.

Me:tho~

~o~ Solv~ng E~pUc. EquaUo~.

New York, Wiley

&Sons,

1968.

SuboJtd-