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

<

RI , g

analytic in

Izl

<

R2

and represented by their power series expansions

00

(0. I)

fez)

= I

akz k , g(z)

k=O

let

f * g

=

denote the function 00

I

(f * g)(z) =

(0.2)

akbkz k

k=O A simple calculation shows that

f

the Hadamand

g

p~oduct

of

f

and

*

g

is analytic in

(0.3)

(f

It is called

in honor of J. Hadamard's famous theorem con-

cerning the location of singularities of the "factors".

Izi < RIR2'

f * g

in terms of the singularities of

He used the alternative representation as a convolution integral:

* g)(z)

=

I

2'lTi

I

f(z/r;)g(r;)dr;/r;, Izl/RI < P < R2 .

Ir;/=p For this reason

f * g

is also called the

~onvoiution

of

f

and

g.

The constant theme in this book is to study properties of operators

12

where

u

f

= {Izl

A of functions analytic in the unit disc

is an element of the set <

I}

B c A.

and

In particular, we try to characterize operators which

B c A into itself. An important classical example of such

send a certain given

a result is the following (unit disc version of a) theorem due to Szego [8SJ (a corollary to the famous theorem of Grace [23J). THEOREM 0.1:

hM the

~a.me

U.

N

l~

=

fez)

be nonvan-i..tJung -i..n

n E

F04

Then

pJ!.ope.4:ty.

Note that it is possible to state Theorem 0.1 as a convolution theorem characterizing functions which preserve the class of nonvanishing polynomials of fixed degree.

Other operators preserve the range, the univalence, geometric

properties of the image domains, certain norms, etc.

A stimulating result in this

direction was Robertson's convolution theorem for typically real functions (i.e., functions

f E

A

with

f(O)

THEOREM 0.2 [45J:

= 0,

f'(O)

= 1,

1m fez) • 1m z

16 00

f (z) =

L

k=l

akz k • g(z)

=

~

a

in

U).

13

h (z)

=

haJ.:, :the. .6ame. pfLOpeJt:ty. Since this theorem has many applications, among them a simple solution of the coefficient problem for these functions, it was hoped that the class A or certain subclasses of S

normalized univalent functions in

invariance property (Mandelhrojt-Schiffer, POlya-Schoenberg).

S of

share this

The attempts to

solve these problems produced a number of important general insights into properties of Hadamard products which are discussed in the following chapters. The following notion turned out to be useful: functions

f E A with

V*

(0.4)

and

V**

= (V*)*,

= {g

f(O)

= 1.

E AO

I

Then for

V f E V:

the second dual.

let

AO

consist of the

V C AO define the dual

(f * g)(z) # 0

in

.6e.:t

U} ,

For instance, Theorem 0.1 has the equivalent

formulation (0.5) {(I + z)n}**

= {p

E AO

I

P

polynomial of degree ~ n, P(z) # 0

The "duality principle" states that under fairly weak conditions on linear (and other) extremal problems in

V**

information since in many cases of interest

are solved in

V**

V.

in

V,

U} . many

This is a useful

is much larger than

V (compare

(0.5)), and various classical theorems from different fields can be obtained by a unified approach. Most of the results in this book are no more than ten years old (a considerable number of them have not even been published before) and many parts of

14

the theory are still developing and have not yet found a final form.

Although it

was impossible to include every result in the field, I have tried to give a fairly complete survey of the available material. These notes are an enlarged version of a series of lectures delivered at the Seminaire de mathematiques superieures, Universite de Montreal, August 1981. I should like to thank the organizers of this conference, Prof. Q.I. Rahman and Prof. G. Sabidussi, for the opportunity to present this part of convolution theory.

Wurzburg, January 1982

St. Ruscheweyh

Chapter 1

DUALITY

1.1.

The duality principle We are using dual sets as defined in 0.4.

convergence in space.

U

The space

A

the space

With the topology of compact

is a locally convex separated topological vector

A of continuous linear functionals on A is described in the

following basic theorem of Toeplitz [88J. THEOREM A:

Iz I

~ 1

I.l u~h

A E

that fl O~

A

~h a~d

only

~~ th~~~ ~

=

* f)(l) .

A(f)

(g

The correspondence (1.1) is denoted by eompiet~

~u~~~o~

g

a~aiy~~ ~~

f E A

( 1.1)

to be

a

A

=g.

A subset

Vc

Ao

is said

if it has the following property:

(1. 2)

f E V

Here we used the notation

f

x

>V

txl

~ 1:

f

(z) = f(xz). z E U.

x

E V •

Note that any dual set is com-

plete (and closed). THEOREM 1.1 (Duality principle, [50J): piet~.

Th~Vl

Let V c AO

b~ ~ompaet a~d ~om­

16

"d A E A;

(1. 3)

co(V) :: co (V**)

(1. 4)

(co

A(V)::::: A(V**) •

stands for the closed convex hull of a set.)

PROOF:

Since

V**

V c

we have

clusion we need to show that

\(V**). \ E A.

A(V) c

f

a

\(V)

implies

f

a

and clearly we can restrict ourselves to the case A

=g.

with

f

0

compactness of

g

A(V).

*

g

is a compact set of analytic functions in uCl)

~

0

u E U.

for

Xo

<:

we obtain

Let

R.

(g

Ixl < 1 and trary

Thus consider

R for a certain

<:

*

f) (1)

::

f E V:

(g

(g

*

f) (x O) f.

* f) ex) :: we get

I

\ E A.

R > 1.

The

f E V}

Izl

<:

R.

Our assumption on

g is analytic in

Then

g(z) ::: g(xOz).

z E U, f E V**.

f E V**.

a:: O.

A and

(g

o.

Since

* f x )(1)

cg * f) (z) f. O.

This proves (1.3).

with

in particular in a point

1.

ACf) :: (g * f) (1) :: for

a E C. \ E A.

for

By compactness we conclude that the same is

true in a certain neighborhood of 1 <

A(V**)

V then shows that U :: {f

CI.l) gives

Izl

is analytic in

To prove the inverse in-

cg

Now assume

V

f.

I zl ~ 1

and for

is comolete we have for

o.

Thus

g

The choice

E

V*

and for arbi-

z:: l/xO

then gives

* f) (I/xOJ f. 0 co(V) f.

co(V**).

Then by a sepa-

ration theorem in locally convex separated topological vector spaces (compare section 20. 7. (1); section 16, 3. (l)J) there exists ments of

V**

from

V.

f EV

[3~,

A E A which separates ele-

This is impossible by (1.3).

Although (1.4) indicates a certain connection of duality and convexity it turns out that the second dual

V**

is in general more closely related to a set

17

V than co(V).

The following corollary is false with

Let V be

COROLLARY 1.1:

o ~ A2 (V). Then 60n any

f E

eompa~

V**

and eomplete.

replaced by

co(V).

Fon Al ,A 2 E A

aA~ume

V** th~e e~~ an fo E V ~ueh that Al (f)

(1. 5)

PROOF:

A2 (f)

First note that

A = Al - (AI (f)/A 2 (f)) fO E V with

A(fO)

A2 (f) 0

~

0 by (1.3).

A2 E A we have

0

~

For the functional

A(V**).

Again using (1.3) we find

= O.

The following two theorems deal with the relation between second duals and convex hulls.

In this discussion we use some elementary convexity theory as

t(o)

we denote the set of extreme points of

Note that for compact sets

V C AO we have (Krein-Milman Theorem)

discussed in [70, Appendix AJ. a given set.

By

co(E(co V)) = co(V)

(1. 6)

E(co(V))

(1. 7)

C

V .

and

Then 60n any

f E

V** thene f(x)

ex,-Wu

=

f

a pnobaUWy m~u/te

II

on

CaU)n

~ueh that

g(~lz'~2z"",snz)dll.

(aU)n

PROOF:

V is compact and complete.

it suffices to show that

In view of (1.4), (1.5) and Choquet's Theorem

18

g(xlz •...• xnz) E V with

Consider

Wi thout loss of generality assume

IXkl < 1 for a certain

= 1.

k

la that there is a probability measure

For

It is a consequence of Herglotz' formu-

au

on

V

1: au

-:-l--~-l-Z = J

(1. 9)

k E {1 •...• n}.

Z;;z

such that

dv (z;)

z2" ..• zn E U fixed we obtain by convolution with (1.9) the relation g(x l z.Z 2 ·····z n )

=J

g(l;z.z2····· zn)dV(I;)

au The choice

Z.

J

= x.z. j = 2 •. ".n. J

then gives

~(xlz.X2z •...• xnz) = J

(1.10)

g(l;z.x 2z, ... ,xnz)dV(I;)

au Since v not in

is not concentrated in one point we conclude that the function (1.10) is E(co(V))

first variable).

(excluding the trivial case when

g

does not depend on the

(1.7) now gives (1.8).

THEOREM 1.3:

The.11

co II

= V** ,

whVte.

v = {tg PROOF:

1)

Let

~

(1.11)

Let (1

f E V*.

A be the union of E (f

*

g)(U)

for any

+ (1 - t)h

I

g.h E U. 0 ~ t ~ I} .

We wish to show that there is a g E U:

(f g

Re eiYCf

* g)(z)

* p,)(U), g E U. E U).

>

= 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

~

<

v

can be proved by exactly the same method.

Then

f

PROOP:

*

g E

a.B

Fo~

THEOREM 1.14:

~

1 let

F E AO

A function

in

U,

K(a - 1,0) • K(O.S - 1)

is in

FI , F2 with

Re F2 ~ ~(1 - a)

= F2(0) = 0,

FICO)

= gl

if artd only if there

Re PI ~ lCl - S),

such that

zF' --P-zg'/g

K(a - 1,0)-KCO,S - 1).

KCa - 1.0) • K(O,S - 1).

exist functions

Now let

f E T(a.S)*, g E

- gz

where

= Fl

- F2 .

gl' gz

satisfy the conditions mentioned above.

The identity zef * g)' f

*

g

= f * zg' f * g

*

=f f

ggl

*

gives the result once we have shown that

g

But this follows from the assumptions for

g E

KCa - I.B - 1).

i)

ii) iii)

PROOF:

Let a.B

~

* gg2 f * g

Re hI ~ 1(1 - B), Re h2 ~ 1(1 - a)

U.

THEOREM 1.15:

f

1 and

gl' gz

f E

in

and Corollary 1.5 since

T(a.B)*.

Then

h E T(a.B)* ==> 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,g E Al

Izl

However.

we have

< I •

if and only i f

iQl z * ~ z #

0, z E U ,

and so there is an obvious transformation of duality to

A function

f E Al

is called

¢~~e

06

ond~

AI' a

<

I

if and only if

46

(2.1)

zf' (z) fez) ~ a, z E U •

Re

The set of these functions is denoted by usual notation

S*). a

S

(for obvious reasons we avoid the

a

In particular, z E S (l_z)2-2a a

and these functions play an important role in extremal problems for

Sa

well known that

S if and only if 0

C

univalent functions in

1,

where

It is

S is the set of all

f E Sa < >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 i
=1

E

if and only if there exists

R

zf' (z) (f(Z))a+iS-1 g (z) z J > 0, z E U , at

z

= O.

An equivalent condition is

g E Sl-a

47

1

f(Z) a+iB-I

f'(z) (-z--J

(2.6)

E K(I,2a + 1) .

Another frequently studied extension of the close-to-convex functions are the c..lM e.-to-c..orr.vex nU.rr.c..:tiOn.6

on

only i f there exist

R. such that

g E SO' cp E

(2.7)

larg e

olLdeA

icp zf' (z) g (z)

r

B.

is such a function if and

f E Al

B71' , z E U ,

<-

2

which is equivalent to (2.8)

f' E

A function k ~ 2,

of feU)

f E Al

K(B,B +

is said to be

2) •

on

bou.rr.dalty

1L0t~rr.

at

mo~t

k71',

if in a limiting sense the variation of the tangent angle at the boundary is at most

k71',

see [70,p.23J.

These functions are characterized by

the representation (2.9)

f'(z)

= g(z)/h(z),

g E S2_k' h E S6-k -4-

-4-

and thus

k k k k f' E K(2 - I,D) • K(D, 2 + 1) c K(2 - I, 2 + 1) .

(2.10)

We see that the notion of Kaplan classes describing various geometrical situations.

K(a,B)

Since the duality theory applies to

Some other sets of functions, directly related to

will be discussed as well.

06

unifies all these definitions

we can expect to obtain some valuable information regarding the above-

mentioned functions.

t(Qe.

K(a.B)

olLden a

~

1

In particular, a function

if and only if

f E Al

is called

K(a.B), plL~taA­

48

f/z E TCl.3 - 2a)* .

(2.11)

Although it is not immediately clear from the definition what the particula interest of these classes may be. our results will show that they playa cen r role in some situations. Finally we wish to mention the class (2.12)

f E Al

M of functions

such

f' E T(1.3)** •

As we shall see, M contains only univalent functions and seems to be a fairly large subset of S.

2.2.

Prestarlike functions Let

(2.11).

Ra

be the class of prestarlike functions of order

A simple calculation using (1.39). (1.42) shows that

a

as defin d

fER,a:::l.

a

and only i f f*

z22 ES.a
(2.13)

Re fez) > Z

Note that the "factor" name "prestarlike".

zl (1 - z) 2-2a

1

2•

z E U. a. ::: 1 .

is itself in

If we introduce the operator

(2.13) justifies th

S. a.

oy:

Al

-+

(2.14)

we deduce from (1.38) the equivalent condition for

(2.15)

fER :

a

Al

with

49

Since = ~,(zn-lf)(n), n = 0 , 1 n.

(2.16 )

,...

the relation (2.15) takes a particularly simple form if cases

a = o,~

f E

(2.18)

f E RO

members map

2 - 2a

f

N.

The special

give

(2.17)

Thus we have

,

R~

=

S~

and

R1

<-~>

RO

>Re

<

zf' 1 ~ > 2,

z EU ,

zfll Re(--y;- + 1) > 0, z E U •

= KO'

KO

where

is the subclass of

S whose

U onto convex domains.

The following theorem is basic for the theory of prestarlike functions. THEOREM 2. 1:

PROOF:

i)

Let a

1 and

~

f,g E R. a

Then f * g E Ra .

i) is a reformulation of Theorem 1.15, i) using definition (2.11).

prove ii), note that

K(1,3 - 2a)

K(1,3 - 26)

~

To

and thus by Theorem 1.9 and

(1.17),

T(1,3 - 2a)* = K(1.3 - 2a)* c K(1,3 - 26)*

= T(1,3

- 26)* .

?art i) of Theorem 2.1 has three cases of particular interest (2.19)

f,g E AI' Re

f z>

1

2.

, Re -g > 2 Z

>Re f*g>l 2 Z

(2.20)

f,g E S,

> f * g E S, ,

(2.21)

f, g E KO

> f

2

2

*

g E KO

(in U) ,

(a=l,~,O):

50

Although (2.19) can easily be obtained from the Herglotz integral representation for such functions, (2.20) and (2.21) are much stronger. conjecture of Palya and Schoenberg [42J is valid.

(2.21) states that the

Theorem 2.1, ii) implies that

KO c S~ c R1 '

an old result due to Strohhacker [81J. For

a

~

o,l.

!heoreD 2.1, i) was first

~roved

in [62J.

plete proof of Theorem 2.1 has been given by Suffridge [83J.

The first com-

He proved a deep and

much stronger theorem on the composition of polynomials with certain restrictions on their zeros (see Chapter 4, Theorems 4.14-4.17), and showed that the following relations - equivalent to Theorem 2.1 - are a limiting case of his result:

Let

00

z

-----= L (l-z) 2-2a

ck(a)zk, a s 1 .

kd

Then if

(2.22)

we have

(2.23)

Furthermore. if

a <

SS

1,

then

(2.24)

Compare (2.22), (2.23) with Szego's theorem (O.l)! 2.1 is due to Lewis [33J.

Yet another proof of Theorem

51

Since we have a Herg10tz formula for

f E R1 ,

the following corollary

is a consequence of Theorem 2.1, ii). COROLLARY 2.1:

me.a.6Wte

1.1

on.

au

fER,O'.~1.

Le:t

a

.6uc.h :tha;t

(2. 25)

fez)

l~~Z

J au

=

f

(2.26)

~

d1.1es)

_z_ 1-z

Using the characterization (2.13) we obtain COROLLARY 2. 2:

me.aJ.>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) <{

(2.28)

z e1_z)2-2O'.

This representation has first been proved (by a different method) by Brickman. Hallenbeck, MacGregor and Wilken [12J.

It is remarkable, however, that the sets

of measures

a

1.1

in (2.27) decrease for

decreasing.

This is a consequence of

Theorem 2.1, ii) and (2.25). The functions in

R. a ~ a

l,

can be re~resented with the aid of a

certain convolution integral and functions in for a,B < 1 we have

To this end we note that

52

2-2S g E Sa < >z(g/z)2-2a E Ss .

(2.29) Since

(1 -

have for

z)-a ~ 2FI(a,I,I,z),

where

2FI

is the hypergeometric function, we

a > 1 (

(2.30)

)(-1)

1

~

(I_z)a

For the notation

1

~

i l (1, I,a,z)

(a - 1)

I

(1 - t)a-2 l:tz dt

o fe-I)

see (1.65).

A combination of these formulas and (2.13)

gives

THEOREM 2.2:

Let a < ~.

Then we have fER a

~n

and only

~6 the~e

.6uc.h that

exM.t6 agE R1 2:

fez)

(2.31)

~

I 1

(1 -

2a)z

(1 -

t)-2a[g~~Z)r-2adt

.

o As a result of Theorem 2.1, ii) we see that the classes decreasing

a.

Ra

become narrower for

The next result shows that their intersection is very small [49J.

THEOREM 2.3:

R , a a

~

1,

ziti - xz), x E IT. PROOF:

It will be enough to study the intersection of

n EN.

Let 00

f (z) =

I

j=I We wish to show that for (2.32)

k

~

Ia k

n

a. zj E R J I-n 2

the estimate

- y(n,k)a k-I/ 2

~

1 - y(n,k)

Ra

with

a = (1 - n)/2,

S3

holds with

Let gez) =

r

j=l be the function appearing in the representation of

a 2 = b 2,

lation shows that

z(g/z) nT I

where

q

The

k-th coefficient

by (2.31),

f

A simple ca1cu-

dk of the expansion of

has the form

is a polynomial with non-negative coefficients in its variables.

We

obtain

Idk since

(ntl} k-l bk-l, 2

(~~i]

q(I, ... ,l) t

(2.31) we deduce

-

dk

_ [ntl] -c q(l,." ,I) -- (ntk-l] k-I k-l

is the

= (n;~~I]ak

k-th coefficient of

z/(l - z)n+l.

But from

and thus

(n+ 1]

(ntk-IJ _ k-I k-l which is (2.32). Now assume that

f E R1_n

for every

n E N.

Then (2.32) holds for

k

-2-

fixed and

n::: k,

Since

lim y(n,k)

=1

we obtain

n~

or 00

(2.33)

fez)

=

\' k-l k a2 z k=l L

Iak

- a k-ll 2

=0

for this

f

54

which is a function of the form mentioned in the assertion. ia21 ~ 1 belongs to

(2.33) with

Ra' a

~ 1,

That any function

is an immediate consequence of the

definition of prestar1ike functions. 00

REMARK:

For

fez)

= L

akz k E Ra

one can use the standard techniques

k=l to obtain the sharp inequality

(2.34)

which, in fact, describes the exact coefficient body

(a 2 ,a 3 )

Another consequence of Theorem 2.1, ii) is that This is no longer true for

THEOREM 2.3:

1 2.

< et

~

a

c S

Ret . for

et

~

1 2. •

1.

ha.tdJ., A.-I and only

R c S et

P-

in

A.-~

a :s

1 2..

This result has first been published by Silverman and Silvia [77J. give a proof in the context of duality. g E T(1,3 - 2a)**

g(z)

=

In fact, the duality principle shows that

implies

~ 2(1 - /zl)2et-3, z E U •

Ig(z)/

Thus

(1 - z)-2 ~ T(1,3 - 2et)**, ~ < a ~ 1,

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

We

such that

fl(z)

= (1

- z)2

and we find

f E:

r.a.

* (f/z) has a zero in U.

(f/z E This

f

is not even locally univalent.

The following convolution result for prestarlike and starlike functions is a reformulation of a special case of Corollary 1.5.

THEOREM 2.4:

Fo~

a

~

1 .tet

f E P. , g E

a

Sa. F E A. I

Then

55

* gF(V) c co(F(V)) . * g

f f

(2.35 )

(2.35) is fundamental for prestarlike functions and cOlltains Theorem 2.1.

Another

(equivalent) statement is even more useful for certain applications, in particular because it clearly points out the advantage compared with convexity theory. THEOREM 2.5:

Fo~

a

1 let

<

fER, g E S , F E a a

A,

and

Then (2.36 )

PROOF:

Let

be the solution of Pa

F

* g

f

*

g

g

=

( -1)

* pa.

( -1)

g * Pa

From our assumptions we have

*

F

* f g

* Pa

Then we have

z).

= z/(I -

F* * pC-I)) * [(f* Pa)f * Pa P J a

(g

a

=

* Pa.

*

* Pa( -1)

(g

(-1)

* Pa.

pC-I) E Rand a. a

f

* Cf * Pa )

)

*

pES. a. a

The result

follows from (2.35). Note that (2.36) with the choice

f

z

==

is equivalent to (2.27).

(2.36)

S.

Extremal a problems of various fields are covered by Theorem 2.5, and a number of appliemphasizes the role of

Pa

as an extremal element in the class

cations will be given within the next sections.

At present we confine ourselves

to two examples. THEOREM 2.6:

(2.37)

PROOF:

Let x,y E V, 0 Re

~

t

~

1,

and

g

E R,. .2

Then

xt+y (l-t) g(x)-g(y) g(xt+y (l-t)) x-y

In Theorem 2.5 we choose

a =

,.2,

F (z)

z

= (l-xz) (I-yz)

and

56

fez) = z(l- (xt + y(l - t))z)-l. (p

ex

Note that

f E R, .

Then we obtain

i

= z/(l - z)): xHy (I-t) g(x)-g(y) * g, = g(xt+y(I-t)) x-y g z=I f *

E CO[(l-exttYCl-t))z] (U)]

F

The latter set is in the halfplane

Re w :::

Theorem 2.6 is due to V. Singh.

(l-xz) (l-yz)

,i

and this gives (2.37).

The special case

The next result deals with the convolution of functions in

t = 0 S , ex

is in [62J.

1~

Note

ex < 1.

that in this range

sex

(2.38)

Let

THEOREM 2.7: -6t.aJz..UkeYlel>-6

PROOF:

Of

n

f

*

g

L6

~ ~ ex ~

at

c

S, 2

S<

c R

0.

1, f E

iea.ot ~e OhdeJL

SS'

on

g E

So.'

Then

~he ohd~

.6~aJtUkeYlel>-6 06

pa

* nS

06 =

We apply (2.36) twice to obtain

c co (

ZP'

a

Pa

* PB * Ps

cU)

]

* p )' -- z Cpa s CU) ] Pa * Ps

= co (

V. Singh (unpublished) used the continued fractions expansion of obtain the following lower bound for the order of starlikeness of

Pa * PS'

2Fl

to

57

1S

a

S

f3

<:

1: 8 + (1-8) (2a-l) a+28(1-a)

We omit the details.

Note that the same method permits one to study the equivaS,~sa<1.

lent problem for convolutions of finitely many functions in

a

The following application of Theorem 2.4 has a flavor similar to that of Theorem 2.7.

a

It is valid, however, for

S

1.

For

f E Ra

let

C(f)

such that

C(f)

L~

THEOREM 2.8:

f,g E R , a S 1. a C(f * g) c C(f)

(2.39)

PROOF:

8+ 1

D8 (f

DSg E S. a

*

(f

*

Iw of

11 < 1

KO

(2.40)

n C(g)

.

8+ I

g) (U) =

(U) c co ( D

8

g(U)

J

og

g)

Interchanging the roles of

If, for example.

The~

(8 = 2 - 2a)

Theorem 2.4 yields

D

since

Re w ~ ~.

lies in the halfplane

a = 0 and

we conclude that

C(f), C(g)

C(f * g)

consisting of all functions

f,g

gives the result. are contained in the disc

lies in the same disc:

f E Al

with

zf"(z) Iff(z) 1< 1, z E U,

the subclass

T

58

is closed under convolution. f E Al

(0 =~)

The same holds

for the class of functions

with

Izf'(z) f (z)

(2.41)

11

_

'

< ZJ

Z

E U •

It is clear how Theorem 2.8 can be applied in other similar situations. It is well known that if have

gez) ::

statement:

IZ

f(/2) E $,.

f

is an odd starlike function of order

0

we

This can easily be extended to the following

z

if 00

(2.42)

L

p(z) :: Z

b znk E $ k

k=O

I -zn '

we have 00

(2.43)

L

q(z) :: Z

k=O

bkz

k

E $, 2

This fact can be used to obtain the following interesting structural formula for

R, o

0 ::

THEOREM 2.9:

1 - n/2. L~

n

E Nand 00

fez) ::

Z

L k=O

akz

k

E Rl ,

-zn

.

Then 00

g(z) = z

PROOF:

The function

5(Z):::

z/O

Theorem 2.1, i) and (2.13) we see that The conclusion follows from (2.43).

L

k=O

aknz k E R, Z

is in s * f

$1

1

-2 n

and by an application of

is a function of the form (2.42).

59

We conclude this section with the determination of a number of special starlike and prestarlike functions.

An important subclass of

Rl

is described

in the following classical result of Fejer [21J:

THEOREM 2.10:

-L6 a conv ex.

{a k }

A~~ume

al

= 1, and that

de.c~e.~ing ~ e.que.nce I A....

e..

ak ~ 0

nO~

k

~

2,

~uch

GCz,l)(-l)

*

G(z,x)

I

Then

Let

A>

-1,

-1 ::: x ::: 1,

and G CZ J x)

z = ---~:--::---r2 1.+1

Cl-2xz+Z)

= Pl-A

such that

Clearly

G(o,x) E Sl-A

E R, ,.

This latter statement is equivalent to

and

GC·,l)

2

2,-11.

p~AJA)cx)

(Xl

where

I

z

(2.44 )

k=O p(A,A)Cl) k

are the Jacobi polynomials

I-x

-) 2

Lewis [34J has given a far-reaching extension of C2.44):

THEOREH 2. 11 :

lsI

Fo~

(Xl

(2.45 )

z

I

s a, -1 ::: x ::: 1,

pea,S) (x) k

k=O p£a,s) (1)

z

k

we. have.

E R,.2 ( 1-0.-8 )

.

that

60

His proof uses a careful study of the function (2.45) written in terms of the hypergeometric function. x ::: -1

It is too complicated to be reproduced here.

of (2.45) is of particular interest since it shows (a ~

lSi)

The case that

(2.46) The study of hypergeometric functions is very natural in this context, compare Theorem 2.7.

The following extension of (2.46) is useful.

z 2FI(1,1 +

(2.47) PROOF:

B+

iy,l + a + iy,z) E R!(l-a-S) .

z 2Fl(1 + a + 6,1 + S + iy,l + a + iy,z) E S!(1-a-6) .

We follow the argument in [34J. is trivial, we may assume F(z)

Since

w(z), wen) ::: 0, zF' (z) F (z)

1 in

U.

Iw(z)1 s Iw(zo)1 ::: 1, Izi

>

O.

~

Let

We have to show that

(2.49)

is bounded by

1 + a + 6

a::: 1 + a + 6

= 2F1 (a,b,c,z).

the meromorphic function

F(z)

Th~n

We shall prove the equivalent statement

(2.48)

and

+ 1 ~ Is + 11.

Let a,S,y E R, a

THEOREM 2.12:

Izol.

b

and the case

=1

+

1 + a + 6 ::: 0

S + iy, c ::: 1

Re(zF' (z)/F(z))

>

-a/2

+

a + iy

or that

defined by

w(z) ::: a 1-w(z)

If this is false we find S

0

Zo E U with

According to Lemma 1.3, x ::: Zow' (zO)/w(zO) ~ 1.

satisfies the differential equation

(2. SO)

z (1 - z)F" + [c - (a + b + 1) zJF' - abF ::: 0 .

Taking the derivative of (2.49) and eliminating to the relation

F"

leads after some manipulation

61

(2.51)

zA(z) = B(z)

with A(z)

= a[b

+ (a - b +

aw(z)[z:~~)) + c - 1

B(z) =

Inserting the values of

a,b,c

we find

A(zO) ~ 0 under the restrictions for

z:~~)))W(Z)J + (a + 1 - c)w(z)J .

IA(zo)1 = IB(Zo)l. a,S.

Furthermore,

Thus (2.51) gives

Izol

= 1,

a con-

tradiction. (2.48) gives no information if

a <

-1.

The following result partially

fills this gap. THEOREM 2.13:

Fo~

-1

<

S~ a

W~ hav~

(2.52) PROOF:

The function

z(l _ z)-l-S

z 2Fl(1,1 + S,l + a,z)

is in

Thus

=(___z~__](-1)* (l-z) l+a

Note that for -~ <

0 <

S < min{O,a}, however,

B<

a,

z . (l-z) 1+f.,J E R-2' (1-"') u. Q

(2.47) is a better result.

(2.52) is better.

For

This shows that both results are

not sharp for certain nontrivial values of the parameters.

(2.52) in a "starlike"

version has first been proved by Merkes and Scott [40J using continued fractions. We wish to mention the following simple application of the results of this section. only if

A function

is called

Qonv~x

06

o~den

a (f E K ) a

if and

62

zf"(z) Re( fl(z) + 1) > 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

(-<

subordination): THEOREM 2.14:

Fo~

0 S a < 1 and

h EK a

we have

zh h _z_-< ~ I

(2.56)

h

This implies that to calculate

I

h

a

Sea) one only has to consider h

:::

h a

.

Later Wilken and Feng [ 93J proved

8 (a)

:::

h' (-1) a , 0 h (-1) a

S 0.<1

We give a proof of Theorem 2.14 in terms of prestarlike functions. fact, let

H

a

:::

zh '/h . a a

Then

H a

In

satisfies the differential equation (see 2.55)

63

= 1+ (1- 20.) z

(2.57)

l-z

Writing this in terms of l/H a

l/H

satisfies a hypergeometric differential equation and we obtain 1

=I

H ( z) a

But

and differentiating (2.57) we easily deduce that

a

z 2Fl(l,2a,3,z) E

R_1

C

RO

= KO

is a convex univalent mapping of In Theorem 2.5 we choose f

= z/(l - z) E R. a

F

II (1 , 20. , 3 , z)

+ (a - 1) z

by Theorem 2.13 and we conclude that

U.

= -log(l

.

Now let - z)

h EK a

such that

such that F * g

= zh'

g

l/Ha

ES . a

= hand

Then we obtain

From the above-mentioned properties of

l/H

a

we conclude h /zh'

-<

l/H

a

and thus

(2.56). For related results see Eenigenburg, Miller, Mocanu, Reade [19J and [64J, [66].

2.3.

Application to close-to-convex and related functions In section 2.1 we found the relation of certain function classes with

K(o.,S)

for certain

a,S.

We restate some of the results of Chapter 1 for these

functions. Th~n

THEOREM 2.15: PROOF:

We have

f/z E T(l,3 - 2a)*, g' E K(l,3 - 20.),

*

g)'

= (f / z) *

* gEe a ,

By Theorem 1.15, ii) we

obtain (f

f

g' E K(1 J 3 - 20.)

64

which is the result.

An independent proof is due to Lewis [33J. Fo~

THEOREM 2.16: kn

k

and g/z E T(~ , ~ + 2)*.

PROOF:

*

g/z E T(S,S + 2)*.

obtain

~

Fo~

Then

f

f

K(l,3)

Let we have

h * (g/z) E K(O.2).

(f

*

g

bound~y ~otation

at m06t

~ 06 boundaAy ~otation at mOht kn. and thus by Theorem 1.14:

k

k

(g/z) E K(2 - 1,0) • K(0, 2 + 1) .

B~ 1

* g ,u

let

f

be

~lOhe-to-convex

c.lOhe-to-~onvex

06

o~d~

06

o~d~

Sand

S.

S if and only if

f' = hF, h E K(O.2), F E K(S.S).

T(S,S + 2)* c T(l,3)*,

Since

and from Theorem 1.14 we

An application of (1.59) gives

(f * g) 1 = fl * (g/z) h * (g/z) h * (g/z)

and thus

be 06

f E Al

is close-to-convex of order

f' E K(S,S) • K(O,2).

K(S.B + 2)

Then

= f' *

g)'

THEOREM 2. 17 :

f E Al

2 let

k k f' E K(2 -1,0) • KeO, 2 + 1)

We have (f

PROOF:

~

_ hF * (g/z) - h * (g/z)

* g)' :: G • (h * (g/z)) E K(S,S) •

=G

E

K(S,S)

K(O,2) •

Note that Theorem 1.11 (representation of the convex hulls) and Corollary 1.4 (coefficient majorization) apply to the three classes mentioned in Theorems 2.15-2.17.

These results have previously been obtained by pure convexity methods,

compare Brickman, MacGregor and Wilken [llJ and Brannan, Clunie and Kirwan [9J. We omit the details. We now turn to the Bazilevic functions.

T ( z)

a

A simple calculation shows

= lr (a

a'-

-

1)

z (1-z) +

f E B(a,S), a = a +

Let

z (1- z)

2

J

is

and

6S

* CzCf/z)a) = 2f'Cf/z)a-1 .

T

a

Thus with 00

=z

a zk h Cz) =T(-l)Cz) = L a a k=l a+k-1 we

see that

~

z (f z) ) a

(2.58)

and

2F1 (l,a,a + l,z)

every

for a

f E Al

> 0

=ha *

z F, F E K(1 , 1 +

2a) ,

with a representation (2.58) is in

B(a,S).

Since

BCa,S) c S

(compare Bazi1evic [6]) we conclude that the functions (2.58) are non-

vanishing in

and this in turn implies

ha /z E K(l,1 + 2a)*.

(2.59)

h a E R1 -a .

Compare (2.59) with Theorem 2. 12!

From Theorem 1.15, ii) we obtain ([56]):

THEOREM 2.18:

Let a z (~

>

or

0, S E Rand f E Sea,S), g E R1 -a • Then

fJa+iS) 1/(a+iB) * (z

E

B(a,S) .

Another consequence of (2.58). (2.59) is (f/z)a+i B E K(l,l + 2a)

(2.60)

for f E BCa,S).

For a

THEOREM 2.19:

~

1, S = 0 we can improve this result to

Let a

~

1, S = O. f/z E

Then

K(1,2)

6o~

f

E S(a,S)

we have

•

Theorem 2.19 and some similar - even stronger - results depending on a

66

are due to Sheil-Small [75J.

PROOF: H E

H.

We write Let

p E K(O,a. + 1).

h

a.

*

a.

=h *

= g (a.+l)/2a. ,

p

Fa E K(a.,a.).

(f/z)

We give a slightly different proof.

q ::: g

(a.-1) /2a. •

Theorem 1.13 gives

It remains to prove

zp E Sl-a./2'

But

= h a. /z

h

where

gH

h

E T( 1, 1 + 2a.) * , g E

= Po.

Thus

g

=h *

pF ::: (h

h

*

gH

*

p

E K(O,a.)

and

*

F

= qH

p)F o'

K(0, 2a.),

E K(l,a.),

where

or equivalently,

or

zp E S(1-a.)/2

* zp E R(I-a.)/2

c

R1-a./2 .

This implies

The function in the parentheses is For

a.::: I (B(l,O)

h

a.

is the class of close-to-convex functions) Theorem

2.19 is well known (Pommerenke [43J). result 2.4. for

f = gH,

g E RO' H E

where

and the proof is complE,te.

H,

In fact, in this case we have the stronger which is easily established using Theorem

Theorem 2.19, however, is not strong enough to give Bieberbach's conjecture

B(a.,O), a.

>

1.

a.::: lin, n E N,

For

this conjecture follows from (2.58):

Fak a::: lin, n E N,

THEOREM 2.20:

f~z(1-z)

PROOF: obtain

and

f E

=

(£/z)a

[iJ Z

a.

G;

we have

-2

(h /z) * FJ F E K(1, I + 20.). a F ~ (1 + z)/(1 - z)1+2a. and thus

We have

B(a,O)

(h / z) * 1+z ::: a. (1-z) 1+2a.

1

From COH,llary 1.4 we

67

Raising this to the n-th power and letting a:: lin

gives the result.

Theorem 2.20 has first been proved by Zamorski [94J.

Our proof is in

Sheil-Small [75J. A combination of (2.59) and the various convolution properties of prestarlike functions give the following integral transform invariances: THEOREM 2.21:

i)

Let a < 1 attd

,in any 06 .the c..iM.6U

f

R, S , C • a a a

Then z

hI_a * f :: (1 - a)za

(2.61)

f t-I-af(t)dt o

ii)

In

f

,i,6,in eLtheJr. 06 .the cW.6e-6

nunct(on (2.61) ,i,6,in S(1+a)/2' C(1+a)/2'

Sa/2

on Ca/2 '

.then, .the

ItU pec.tiv ely.

Sa/2' is a reformulation of the result in the second part of the proof of Theorem 2.19. If zft :: gF E Ca / 2 with

PROOF:

g E Sa/2

i) is

and

c1e[~r

by (2.59).

ii), for

F E H we obtain from Theorem 2.4 that z (hI

* f)'

hI

---=_-..;;,;a:....-__:: -a hI -a * g hI -a

* *

gF

E H .'

g

the conclusion follows.

Since

Results like those in Theorem 2.21 are numerous in the literature. well-known case is

a::-1 z

L(z)

=f f

o

f(t)dt ,

A

68

the so called Libera transform [35J. if

f

has the same property.

Libera proved that

L(z)

is in

So

or

KO

Theorem 2.21 contains this result and part ii), in

fact, improves it:

This is of interest because it shows that the Libera transform sends the

S_!'

class

containing non-univalent functions, into

2.12 (with

= B+

a

So c S.

Note that Theorem

admits an extension of Theorem 2.21, i) to integral trans-

1)

forms of the form z

(1 - c)zC

I

t-I-Cf(t)dt

o with

Re c

~

a,

compare [48J.

We omit the details.

Let us return to the c1ose-to-convex functions. only if

f' E K(1,3)

l/f' E K(3,1).

or

We know

f E Co

Thus Theorem 1.11 and Corollary 1.4

yield:

THEOREM 2.22: me.aJ.,Wte

]J

on

(aU) 2

Fo~

E Co

f

~nd

n

EN

3n

=

J

(l+xz) n (1+yz)

(au) 2

In particular (1+ z) 3n

(2.62)

(l-z) For

n = 1,

th~e ex~~ ~ pnob~b~y

~uc.h tha:t

(~,r

(2.62) reads

if and

n

d]J ( x,y )

69

~~

(2.63)

£1

=1

0+z)3 l-z

so that the coefficients of to study the same problem for

00

+ 4z + 7z2 + 8

L

k=3

l/f', f'

S.

CO'

f

are bounded by

It is of interest

8.

f E Sand

If

(2.64 )

one can show that the coefficients

aI' a 2 , a 3

satisfy exactly the same inequali-

ties as in (2.63), namely

The complete set of estimates contained in (2.63), however, is not valid in In fact, there is an example (see [58J) for a function ficients

in (2.64) such that

lim lakl

=

00.

£ ES

S.

with real coef-

It would be interesting to de-

k~

termine the maximal rate of growth of these coefficients,

f E S.

Finally, we determine the subordinating convolution operators for

L e.., f

(2.65) ~6

and only

PROOF:

i)

in

f E

Assume

* g

-< g,

g E Co '

T(2,2)*. f

satisfies (2.65).

The function 1

g(z)

=z

x+y + 2 z

(l+yz)

2

CO'

70

is in

x,y E V.

Co for arbitrary

For

x # y

we have

(2y - 2x)

-1 .I:

l g(U)

and

it follows from (2.65) that (f

*

g) (z) # (2y - 2x)

-1

, z E U .

This condition can be rewritten as

o#

[f

*

(g - (2y - 2x)

and thus

f E T(2,2)*.

ii)

g E CO'

Let

-1

1 [f )](z):: 2(x-y)

w E C\g(U)

Then with every

which also belongs to

C\g(U)

*

J

( 1+xz 2 l+yz J, z E U ,

there is a ray from

w to

00

and this proves g(z)-w E H2 :: K(2 2) g(O)-w ,.

If

f E T(2,2)*,

we obtain from Theorem 1.9 that f

or

f * g # w in

we conclude

U.

*

(g (z) - w) # 0, z E U •

(f * g)(U) c g(U)

This shows

and since

g

is univalent

f * g ~ g.

A result reflecting part ii) of the above proof is due to Sheil-Small [74J.

Note that Theorem 2.23 holds as well for the larger class of linear ac-

cessible functions w E C\g(U)

2.4.

g ES

there is a ray

which are defined by the property that with any [w,oo)

C

C\g(U).

Related criteria for univalence The main results of this section are due to Jankovic [28J.

2.2 we observed that the conditions

In section

71

(2.66 )

a ~ ~,

with

B =~.

imply univalence (prestarlikeness of order

a)

of

a

=l

This result, however, is not best possible as seen from the case

f E AI' where

(2.66) reads

Re zf' (z) f (z)

and for univalence of

sufficient.

f

f E Al

THEOREM 2.24:

Let a

Re

.-L6 uMvafe.rU:.

v-Lolrung the. Let

Ta

r

U

0:-

)

1

B(a)

~

1 and

03-2a f I > 02-2af - 2

0

will be

is the minimal number in

U.

f E Al

•

replaced by

2:

S = Sea)

to be univalent in

(2.67)

f

z

this same condition with

So the question arises which

(2.66) forcing

Then

1

>"2'

1-2a_ 2-2a-

~uQh

that

e(a)

•

Qannot be. Jte.p.tac.e.d by any

~ma.Ue.Jt

numbe.Jt wUhaut

QanQf~-lon.

denote the class of functions in

Al

satisfying (2.67).

Using the re-

lation (compare (1.40))

we see that

It is convenient to introduce the class functions

f

E=

f\

Ea consisting of those

such that there exists agE R a

and

~

E R with

72

D2- 2a f 2-20. > 0, z E U .

i
(2.68)

Re e

Note that

= CO'

Eo

D

g

the class of close-to-convex functions.

Related classes

have also been considered by Al-Amiri [2J.

PROOF:

Let

f E

To.

D2 - 2a f E

such that

S

There is agE

a.-:z1·

0,

and thus holds.

fEE

l'

a.-:z The function

Now let

fEE

and

1

a.-:z

z

h(z)

= (1_z)3-28

is in

R 1 a-:z

*

z

E

Ra_~

with

E U ,

R

g E R 1, a-z


(z

J(-l)

such that (2.68)

(1_z)3-2a

and we obtain

D3- 2a f 3-2a)J = Re _ _ _ _ _-:--:::--_D_...........&<--h * D3- 2a g h

*

3 2

.

[0 - ag(e1
:;: 0 by Theorem 2.4 and (2.68).

COROLLARY 2.3: Qo~ve.xl ~~

U.

Since

Fan a.

~

we conclude

0

a~y nu~ctia~

f E Ea.

f E

Eo1J-2l '

~ u~~val~nt (Qlo~~-to­

73

PROOF of Theorem 2.24: convex) in show that

That

f E

Ta. • a.

c:;

-

1

2.

is univalent (close-to-

U is a consequence of Lemma 2.1 and Corollary 2.3.

It remains to

Sea) cannot be renlaced by a smaller number. We consider the

functions

=

f (z)

which satisfy

l-2a.-E 2-2a. The coefficients

bk and thus

bk/k

-+

bk

are given by

= r(2-2a.) r(2-2a.+k+E) rC3-2a+s)

00

~

rCl-2a.+k)

rC2-2a.) kl+s r(3-2a.+s)

which is impossible if

Note that Theorem 2.24 with

a

=0

(k -+ 00)

f E S. gives the univalence criterion

which has been known for a long time (Umezawa [90J). It follows from (2.68) that (2.69 )

Since

Ca.

is invariant under convolution with THEOREH 2. 2S:

Ra. we conclude

a. < I, fEE , g E R , a a.

then

f

* gEEa .

We wish to mention an interesting set of functions belonging to

74

EC1 - A)/2' A ~ O. n E N, A ~ 0, the

Fo~

THEOREM 2.26:

(2.70)

Pn (z)

n! ~ (l+A)n_k ktl (1+A)n k~O --::-Cn--"""k-:-)-:-!- Z

==

Let A~ 0,

COROLLARY 2.4:

polyno~

00

E N.

n

Let

g(z)

=z

r akz k E SCl-A)!2' k=O

(2.71)

r

n

(ltA)k CltA)n_k zktl k! (n-k)! ~ E C(1-A)/2 .

n! (ltA)n k=O

(2. 72)

(2.72) is due to Lewis [34J, and from (2.69) and Theorem 2.25 we conclude that (2.72) contains both Theorem 2.26 and (2.71).

The proof of (2.72) rests on

the fact that the polynomial q (z)

n

has all its zeros on zeros satisfy

~

-

~ >

Izl

qn E K(l,A + 2) [34J.

\' (ltA)k (ltA)n_k k (n-k) ! z (ltA)n k=O k!

=I

l..

and that the arguments

2n/Cn +

ditions of Theorem 1.10 with

n

n! = ..,..,....;~-

I

t

A).

a = I,

of two subsequent

From this, one can conclude that the con-

B=At

which is the assertion.

~ > ~

2 are fulfilled, and thus

For the complete argument we refer to

75

2.5.

M and related classes of univalent functions We recall the Mandelbrojt-Schiffer conjecture: 00

fez)

= L

akz k E S, g(z)

=

k=l

implies (2.73)

Clearly (2.73) implies (2.74)

(f

*

g)(z) ~ 0, 0 < Izl < 1

J

and thus

z

lz

(2.75)

f i1!l dt * g t

'4 ~

0, z E U •

o To disprove this conjecture (even the weaker form (2.75)) we just observe that g E Al

with

g' E T(1,3)

are members of S.

z

lz for any

f E S.

f il!l dt o

t

z

E T(l,3)*

or

J f~t)

dt E RO

o

By (2.13) this says

(2.74) can hold for arbitrary

So (2.75) implies

S c So which is false.

g E S only if

f E SO'

This shows that

However, not every

f E So

has this property which follows from the next theorem. Let g E S.

ZeAl

be the class of functions

From the relation

f

such that (2.74) holds for any

70

z (l-xz)(l-yz)

(2.76)

we deduce for

*

V**

= {g/z I

Then

f'({lzl < r})

(2.77)

= z(l

- z)

-2

~

g E S}

that

,

= {f/z I fEZ}

Let fEZ.

THEOREM 2.27:

k (z)

x,y E uJ

V = {[(I - xz)(l - yz)]-l

V*

wh~e

g(xz) -g(yz) , x,y E V x-y

g ~

the Koebe

C

~o~

0

<

r

<

1

we have

k'({lzl
nun~~on.

(2.77) is the so-called Marx conjecture originally stated for arbitrary f E SO'

In [17J Duren and McLaughlin showed that (2.77) holds for the functions

fez) = z[(l - xz)(l - yz)] Z.

For the whole class

Note that this implies

-1

-

• x,y E V,

SO'

and thus - by the duality principle - for

however, (2.77) is false as Hummel [25] has shown.

Z # SO'

as claimed above.

We note that a careful study of tremal problems in

S.

Z would be of great importance for ex-

For example, the Bieberbach conjecture is equivalent to

the statement n

z +

!.... n

E

Z. n

:::0: 2

•

The above considerations suggest the study of the class such that (2.78) for arbitrary

f* g# 0,0<

f E SO'

Since

f E So

Izi

< 1,

if and only if

~4 C

Al

of functions

g

77

z

.!.

z

we immediately deduce for

THEOREM 2.28:

Co

C

M.

"

AI:

gEM <=> g' E T(1,3)**

so that our theory applies to

PROOF:

dt E T(l 3)*

o

g E

(2.79)

f fet) t

M.

We have

Co c M C S an.d

two oil the6 e -6 W Me.. equ.al..

g' E K(1,3)

is close-to-convex if and only if

g

Vl.O

T(1,3)**

C

On the other hand, the first factor in (2.76) is in

we conclude

M C S.

To show

Clearly

M# S

since otherwise

Z

So

which shows

and from (2.79)

So which is false.

~

Co # M is harder since - according to the duality principle

many properties valid in

K(1,3)

remain true for

T(1,3)**.

Our claim will

follow from the next result.

LEMMA 2.2:

Let

g(z)

=z

+

az 2

+

Sz 3 E S,

~JS

E R.

Then. gEM.

The proof of Lemma 2.2 - contained in [67J - is somewhat involved. just outline the steps. in S

It is known (Brannan [8J) that

g

if and only if

1;3S , _ 1/3 ~ S ~ 1/5 , (2.80)

I~I ~ {

2(S - S2)!, 1/5 < S ~ 1/3

Furthermore, the exact coefficient body g(z)

(a 2 ,a 3)

with

We

as in the theorem is

78

is given by (see Pesch1 [41J and (2.34)) (2.81) To prove Lemma 2.2 we have to show

(a,S)

whenever

satisfies (2.80) and

(a 2 ,a 3)

satisfies (2.81).

For the details

see [67J. To complete the proof of Theorem 2.28 consider the polynomial (2.82) which belongs to

M according to Lemma 2.2.

A result of Suffridge [83J states

that a polynomial like (2.82) can be close-to-convex only if for the zeros the relation is obviously not the case.

In fact, a careful study of

(j)

1 -

pez)

(j)

7T

2 >"'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<:lzl<:l o # zg(xz) * l-ZOZ

fez) Z

= z[g(x)

= z[g(x) * x

f(xz)-f(zoz) xz-zoz ]

Thus f(xz)-f(zoz) x

for

z E U.

r+- - - - - - -

xz-zoz

From the compactness of

E T(l,2)**

T(1,2)**

the result follows, letting

z .... 1. Theorem 2.31 is in close relation to properties of the class of linear accessible functions in

S

(compare the remark after Theorem 2.23).

any close-to-convex function is linear accessible. that

f E Al

Note that

Sheil-Small [72J has shown

is linear accessible if and only if for any

Zo

E U

f(z)-f(zO) - - - - E /«(1,2) . z-zO Since

/«(1,2) c T(I,2)**

functions are in

M.

it is natural to ask whether all linear accessible

This would be the case if the following inverse to Theorem

81

2.31 were true:

f E Al

for

f' E T(I,3)**

we have

if

f(z)-f(zO) - - - - E T(l,2)**

z-zO

holds for any

Zo E U.

We are not able to decide this question.

Note that

Bieberbach's conjecture is open for linear accessible functions. M -L6 Un.eaJt -tn.VaJU..a.M.

THEOREM 2.32:

f E M and arbitrary

This means that for

x E U we have

f(x+z )-f(x) ll+xzJ

(2.88)

----""""":::2:-- EM.

f'(x)(l-Ixl ) PROOF:

From Theorem 1.7 and

g'(z)

f' E T(l,3)**

= (1

+

we obtain

XZ)-2f'(X+~J/f'(X)

E T(l,3)**

l+xz and

g

is the function in (2.88). Next we prove a refinement of (2.84).

This result ([sOJ) is apparently

difficult to prove without duality and has not been known even for starlike functions.

Of course, it is not known whether it holds for

THEOREM 2.33:

fez)

Folt

= n

(2.89)

1T

n

(z, f)

S.

= L

k

akz , n EN.

k=l

Then We have -tn. U: 1T

(2.90)

11 -

(z,f)

n fez) I -< (n + l)lzln + nlzln+l, n EN.

82

holdc 60n the Koebe 6unetion.

Equ~y

PROOF:

For

f E M we have

f' E T(1,3)**.

0

have to prove (2.90) for fixed

< Izl <

fl E T(1,3)).

A2 (f') = fez) # 0 for

Thus by the duality principle we

1 only for

fl E T(1,3)

(note that

This reduces (2.90) to a straight-forward

calculation.

M is invariant under convolutions with convex uni-

We have seen that

valent functions (Theorem 2.29).

T

C

KO of functions

with (compare (2.40))

f E Al

(2.91)

If we restrict ourselves to the smaller class

t fll

(z) I

tf

5

1

(z) I, z E U ,

we get an even stronger result.

Let

THEOREM 2.34: PROOF:

f E

T,

Re

*

g E So'

zef * g)' (f/z) * g' Al (g') - - = Re --""--''--''--..-...-'---- = Re -:--:--"e'""'f * g z 1.2 (g 1 )

(1:.Z

0

f

We have to show

o <:

for

Then

gEM.

<:

Izi

<:

1.

J iJ!l dt) o

t

From Theorem 2.29 we see that

ty principle applies:

*

g'

A2(gl) # 0

we only need to prove the theorem for

such that the duali-

g' E T(I,3)

(using a suitable parametrization for the functions in T(1,3))

g (z)

a

= ---11 +a

[

z

(l-z)

for all

2 + all J, Re a ~ 0 • -z

Thus we are left with the conditions

(2.92)

zf' * ga Re ---;:;;--..;;.;. f * g a

= Re

z 2f"+(a+l)zf' zf'+af

~

0, z E U, Re a

~

0 .

or g

a

with

83

But the expression under the real part is a Moebius transform in

a

and using

this property we obtain the equivalent inequality

[Im(TVTTWT v w) J2

(2.93)

where v Now

f E T implies

implies

Re(l/w) >

= f(z)/zf' (z),

f E

l

KO

C S~

w

=1

+ zf"(z)/f' ez)

such that

Re(l/v) >

1.

Furthermore, (2.91)

such that the left hand side of (2.93) is

>

1.

This com-

pletes the proof. For the sake of completeness we mention that for

z (ff * ........... g)'I~~

(2.94)

If

*

gEM,

fulfilled.

g

f E T, g E So

we have

- 11 < 1, z E U •

however, this stronger condition for starlikeness is not necessarily Theorem 2.34 suggests a new conjecture for

S.

We shall discuss this

matter in Chapter 4. The duality principle can prove only such properties for valid and sharp for method.

M which are

Co as well. This shows clearly the limitations of the

Some known results for close-to-convex functions are not obtainable by

dualitYi for instance, the simple estimate larg f~Z)1 < ~TI

(2.95)

true for

f E

M.

z EU ,

Co is not known to hold in the whole class M simply because du-

ality just permits to compare plane sets. in

,

In fact, we doubt that (2.95) is true

84

2.6.

Convex subordination In this section we study subordination in connection with convex uni-

valent functions in functions.

U,

mainly consequences of subordination under such

The historical roots of these considerations are the following re-

suIts of P6lya and Schoenberg [42J on the de la Vallee-Poussin polynomials (means): n

v

(2.96)

Let

n

J

L

(z)

2n zk , n EN. ( n+k

k=O

K denote the class of convex univalent functions in

the origin). (2.97)

POlya and Schoenberg proved: f EK<

(2.98)

A (not normalized at

>(V

n

fEK=>V

* f E K for

n

n E N) ,

*f-
(2.99) Since

fez) = (1 - z)-l E K,

(2.97), (2.98) imply in particular

Vn E K,

(2.100)

n

EN,

V ~_l_ EN n 'l-Z ' n •

(2.101)

While (2.97) lead to the so-called POlya-Schoenberg conjecture proved in Theorem 2.1, (2.98) (in connection with (2.101)) is a special case of the following conjecture later made by Wilf [92J (in fact, he conjectured a slightly weaker resuIt): (2.102)

for

f.h E K. g E A we have g-

g * f-
A proof of (2.102) is given in Theorem 2.36 below.

However, it seems well worth

85

to mention that Wilf's conjecture is stronger than the Palya-Schoenberg conjecture.

To see this we refer to a result which has been suggested by (2.98) and

will be a consequence of Theorem 2.42: (2.103)

(V

n

* f

-< f

for

for

Now let (2.102), (2.103) be true and

A, f'(O)

~

0,

we have

n E N) => f E K • f,g E

Vn

f E

K.

Then, for

n E

N,

* f-
by (2.98) and thus

by (2.102).

Then (2.103) applied to

f

*

g

f * g E

shows

K.

the Palya-

Schoenberg conjecture. (2.103) is an example for "convexity generating" sequences which will be characterized below. nation chain.

(2.99) shows that the

form a (discrete) convex subordi-

V n

Continuous convexity generating convex subordination chains are

important for the study of global mapping properties of solutions of certain POE's. For details see Chapter 5. THEOREM 2.35 [62J: z E U,

Ixl

<:

zfl (z) Re f(z)-g(x) First assume that (2.104) holds.

such that function

g(x O) h(z)

real part on

A..n and onllj A..n noJt.

g -< f

Then we have

Iz I,

(2.104)

PROOF:

f E K.

LeX.

f

f({lzl

<:

r})

= Zfl (z)/(f(z) Izl = r

>

If

for a certain

0 • g

~

r

f.

with

- g(x O)) is analytic in

and thus in

Izl ~ r.

then there exists IXol Izl

<:

<: T <:

r

1.

Xo E U Then the

and has positive

But this is impossible since

86

h(O) = O. f

On the other hand,

(restricted to

/z/ < r)

g-{f

Xo

implies that for

is starlike with respect to

EU

and

g(x O)

IXol


which is ana-

lytically expressed by (2.104). We note that for

f E K we have in particular

Ix I < Iz I <

zf' (z) > 0, Re f(z)-f(x)

(2.105)

1 .

Brickman [lOJ proved that (2.105) is also sufficient for in

U

to be in

f E

A with f' (0)

# 0

K.

Theorem 2.35 is, beside duality, the crucial part in the proof of Wilf's conjecture [92J: THEOREM 2.36: PROOF:

Foll.

We clearlY can assume

f,g,h E AI'

F (z) = (f * h) (z) - (f

z (f

g -< h we. ha.ve. f * g

f,h E K a.n.d

Then for f

-< f

x E U

* h'( )[h(z)-g(xz)J f * zh'

= _ _z_::,"z_~z~h;...'..l:.(;;:"Z)'--_

* g) (xz)

* h)' (z)

By assumption,the function in square brackets has positive real part. more,

f E KO = RO

clude

Re F(z) > 0 in

since

f

and

zh' E SO'

U.

* h.

Thus, by Theorem 2.4 (with a

Further-

= 0) we con-

Another application of Theorem 2.35 gives the result

* h E Ko by Theorem 2.1. Theorem 2.36 has found applications in various fields.

ample of a fairly general consequence [55J. THEOREM 2.37:

= 1,

Let G E K, G(O)

z F (z)

=z

exp CI

o

a.n.d

_G""",(~=)_-_l dx) .

We give an ex-

87

Then

6o~

f

E Al

we have zf' (z) fez)

s, t E

U,

tf(sz) sf(tz)

(2.106)

-< G(z)

-< tF(sz)

sF(tz)

Many special cases of (2.106) have previously been known. if f E SO' then Theorem 2.37 with

G(z) = ~~~

gives

tf(sz) -«1-tz}2 sf(tz) I-szJ

(2.107)

which for

s = I, t

+

0

is the classical result

fez)

z of Marx [39J.

-<

, f E So .

1

(1_z)2

For other applications of (2.106) see [65J.

PROOF of Theorem 2.37: "only if" part.

We just give a proof for the (more important)

The function z

p (z) =

J (l=SX o

is in

For example,

K if s,t

E UJ s

t.

fez)

For

l~tXJ dx

Thus

(zf' (z) _ 1) * p

(2.108) by Theorem 2.36.

~

-

h E A with

-<

(G(z) - 1) * p

h(O) = 0 we have tz

(h

*

p) (z) =

J sz

and therefore (2.108) 1S equivalent to

h(x)dx x

88

tz

Q(z) =

I

1J

(x) dx (fl f(x) x

tz

-<

sz This implies

exp(Q)

-<

We now turn to

J

(G(x) - l)dx = R(z) x

sz

exp(R)

which is the assertion.

~onvex hubo~~nation ~hain~

(abbreviated:

c.s.c.).

Theo-

rem 2.36 in combination with (2.98) shows that (2.109) for

f E K.

A C.S.c. is

(2.109) was conjectured by P61ya and Schoenberg [42J.

defined as a function f:

where

I

is a certain set in i)

R,

U x I -+ C ,

such that

f(o.t) E K. t E I.

(2.109) is a c.s.c. over

I

= N.

Our previous results admit some insights into

the structure of such chains. THEOREM 2.38: t

E I.

Then

f

~

a

Let f(z.t): ~.h.c. ov~

I

U x I -+ C be ~n

and only

~n

~6 6o~

A

with f' (O.t) # 0

any

Ixl

<

Izl

<

nO~

1 and

(2.110)

This is an immediate consequence of Theorem 2.35 and Brickman's observation concerning (2.105).

89

Let fl(z,t), f 2 (z,t)

THEOREM 2.39:

oven a

~et

PROOF:

I.

Clearly

be convex

Then f(z,t) = fl(z,t) * f 2 (z,t) f(·,t) E K for

t E I.

~

Now let

~ubond~nation chai~

albo a

c.~.c.

oven I.

t l ,t 2 E I, tl < t 2 •

Then

applying Theorem 2.36 twice we obtain

Note that Theorem 2.39 applies in particular if

f 2 (·,t) = f E K for

t E I.

Next we give a sufficient criterion for c.s.c.'s which is applicable in particular in combination with Hopf's maximum principle for elliptic PDE's.

Let f:

THEOREM 2.40: f(O,t) = const.

60n

0 < t < 1.

w (z )

(2.111)

PROOF:

o<

y

16 60n any

fezl,t l )

1 l-

Then there exists

the latter set being closed and convex. fCe

icp

,t l )

and

We conclude that

f(U,t 2)

wy(z)

y E

co~nuo~.

Let f(·,t) E K,

R,

J

Assume that for two such

f(U,t 2)·

C be

= Re e i Yf (e icp t), z = t e icp

It will be sufficient to show that

t1 < t2 < 1.

~

IT x [0,1)

f(U,t 1) c f(U,t 2)

for

t l ,t 2

zl E U such that

there is a

cp E R such that

fee ~ ,t 1) 1 l f(U,t 2L

There is a straight line separating

and by a suitable rotation we obtain

has a local maximum in

Izl

<

t2

and thus in

U.

This

90

contradicts our assumption. Note that a similar argument shows that under the assumptions of Theorem 2.40 we can deduce that w(z)

(2.112) is injective in

= fee iq> ,t),

U.

THEOREM 2.41:

Let f E K.

a

Th~~

I_t 2

= - - tzfl (tz) + f(tz)

(2.113)

~

z

IH2

~o~v~x ~ubo~dlnatio~ ~hai~ ov~

sf(z,t)

(O,lJ.

will be a standard example for a number of purposes in the

sequel.

To prove our claim we observe that the corresponding functions

(2.114)

w = w (z) = Re e iYC1-lzl2 ""2 zf'(z) + f(z)J y

II

1+ z

(compare (2.111)) are solutions of the POE

2 4 (zw + zw-) = 0 . w-+ zz l-Izl z z

(2.115)

To this elliptic equation Hopfls maximum principle applies and gives the second assumption in Theorem 2.40. f=f

this for

o

=_1_ l-z

From the definition of Sf (z.t)

a

dition

*

zg' :F 0

in

To prove

sf(z,t) E K it will be sufficient to do

For the general

f E

K it then follows by convolution.

M (see section 2.5) it is clear that we just have to show 0

<

Izl

<

1

for arbitrary

gEM.

This leads to the con-

91

2 l_t 2 - - tzg"(tz) + - - g' (tz) :f:. 0 l+t2 1+t 2 for

z E U, gEM,

which is equivalent to

tzg"(tz) g' (t z)

But this follows from the general estimate

zg"(z) - 21z1221 I,() g z l-Izl valid for any function a c.s.c. over

(0,1).

g

_<

_4.......I.zl-:::. . . 2,zEU, l-Izl

in the larger class

S.

TItis proves that

sf(z,t)

is

By Caratheodory's theorem on the kernel of a sequence of

domains it clearly extends to the limiting case

t

= 1.

In particular we deduce from Theorem 2.41 that l_t 2 - - ztf'(zt) + f(zt) 1+t2

(2.116)

whenever

f

Let function t

E I,

F:

0 < t < 1 ,

E K. I c R such that Ux I

~

to

C is called

and if for arbitrary

f E f

(2. 11 7)

implies

-< f,

*

=

sup I

is an accumulation point of

eonv~y

genenating (e.g.) if

F(., t

I. )

E Ao'

A with f'(O):f:. 0 the condition

F (. , t)

-<

A

f, t E I ,

f E K. The following theorem applies to most of the known c.g. functions.

92

Let

THEOREM 2.42:

co

l

F(z,t) = 1 +

ak(t)zk E

A, t E l ,

k=l be .6 ueh tha.t

i) ii) iii) iv) Then

ak(t) E R, kEN, t E I, 1

Re F(z,t)

z E U, t E I,

> 2,

1 > 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

-<

q(z)

= lim

n-+co

a

.6uc..h that

g(z), n E N, G(Z,T )-g(z)

ii)

g' (0) "F

n > 0,

T

~

e~:t6 -i..n

U.

n

Then

Re

(2.118)

-i..n

U\{z

I

q(z)

zg'(z)

~

a

g'(z) = a}.

PROOF of Theorem 2.42:

Let

tn E I, tn

-+

to

such that

w-Lth

T

n

-+

0

93

G(Z,T) = f * F(z,t ), g(z) = fez). n n Then by Lemma 2.3 and in the notation of Lemma 2.4,

Let

q (z)

00

= -f *

L

k=l

Here we used the inequality (see [51J)

00

for any function form convergence.

= I

akz k E AD k=O Thus by (2.118)

h(z)

with

Re h(z)

>! in U, to ensure uni-

zf" (z) Re(f' (z) + 1) ~ 0 whenever

f' # O.

This shows

f E K.

We give two applications of Theorem 2.42. F(z,n)

= Vn (z),

n E N.

From (2.101) condition ii) of Theorem 2.42 follows and for

the Maclaurin coefficients

ak(n)

a1( n ) --

of

Vn

n

is e.g., compare (2.103).

ak(n) E Rand

----+1

4n+3 n+2

V

we have

2n J n (n2n)-1(n+1 - n+l

---+

Thus

First consider

4 .

'

94 1

= -l-z

For the chain in (2.113) with Theorem 2.42 is a consequence of (2.116).

the second condition of

The Maclaurin coefficients are

and using Theorems 2.42. 2.41 we find that 1_t 2 zt 1 --+ --1+t2 (1-tz)2 l-tz

(2.119)

is a convexity generating convex subordination chain.

2.7.

Vnivalence criteria via convolution and applications For a number of applications it turned out to be useful to represent

certain subclasses of starlike univalent

S

essentially as dual sets.

(E SO)

f E Al

is

if and only if

1 + x-I z 2-"'2'--" #; 0. f/z * _ _

(2.120)

For instance.

z E VJ Ixl

= 1 .

(1-z)

This is most easily verified by a direct computation of the Hadamard product in (2.120) (compare (1.38)).

For reference we list a number of similar conditions

for other classes: 1)

f E S , 0: < 1, a

(2.121)

f/z

2)

f E Al

*

if and only if 1+ x+2a-1 _ _2_---.;20:"--_z #; 0, z E O-z)2

is spiral-like of type

v. Ix I = 1

A E (- ;

,%-J

.

(defined by

9S

Re[e

iA

zf' (z)/f(z)J > 0, z E U)

if and only i f

-2iA

x-e 1+ -2'1.. z l+e l

z E U, Ixl

f/z * --:"--=-2- 1: 0, (l-z)

(2.122)

3)

I zf' (z)/f(z)

f E Al fulfills

f/z

(2.123)

* l+(x-l~Z I- 0,

11

-

<

= 1 .

if and only if

1

z E U, Ixl

= 1 •

(1-z) 1), 2) have been pointed out by Silverman, Silvia and Telage [79J, while 3)

has been used by Fournier [22J.

Note that prestarlike functions obey a similar

relation. We wish to mention two applications.

The first one deals with neighbor00

hoods of univalent functions, see [6lJ.

For

f =

L akz

k

E Al

we say

1

is in the

a-neighborhood

N8 (f)

of

f

if

00

(2.124)

It is a well-known result that

Nl (fa)

C

SO'

where

fO(z) - z.

terization (2.120) we obtain the following related result:

PROOF:

Let

x-I

h (z) X

Then since

hx(z) E Co

=z

1+ -2- z ---::-(1_z)2

we obtain

=

Using the charac-

96

(2.125)

I c k (x) I ::: k J k ::: 2 J Ix I

hx * f E Co

and

C

S

which by Koebe's

I

(2.126)

Now assume

g E N,(f). 4

*

(h

x

f) (z)

z

=1

,

a-theorem implies

I

Then using (2.125), (2.126) we obtain for

z E U J Ixl = 1:

1 > - -

4

l_ - 4

00

o.

>

g E SO'

An application of (2.120) shows

A similar method was used by Fournier [22J to obtain the following resuIt.

Let

f ET

(see (2.91)).

Izg' (z)/g(z) - 11 < 1 coefficients

a 2 , ... ,a n

U.

in of

Then any

g E N8 (f)

with

8

=

l/e

fulfills

Both of these estimates can be improved if the f

are zero.

For details see [6lJ, [22J.

The next application is dealing with the determination of the radius of starlikeness for the functions in

co SO'

This problem was solved by Hamilton and

Tuan [24J using Theorem 1.20 and the following results due to Robertson: rO

= .4035 .. ,

for any (2.127)

be the positive root of the equation

f E S there exists

a E R such that

r6 + Sr 4 + 79r 2 = 13.

let Then

97

and there exists any

fO E

co So

such that

fO

is not univalent in

Let

g E

co SO.

Then

if there is a probability measure

~

gez) :::

au

on

Since

-l-Cg(roz) rOz

hx E S,

*

.6:t.oJc.Li.ke. uni.va1.e.1'l.t -i.n

g E

In fact,

co So

if and only

such that

( z dfl (y) . J (l-yz) 2 dU

According to (2.120) the theorem holds if for

(2.128)

.u.

g

We give a quick alternative argument.

h (z)) ::: x

Ixl = 1, z E U,

f h~(yroz)dfl(Y) ~

we have

0 .

au

(2.127) applies and gives (2.128).

That

by any larger number follows from Robertson's example

rO

cannot be replaced

f O'

Additional information 1)

We show that Robertson's Theorem 0.2 on the convolution of typically

real functions can be proved using duality. in

for

r > 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

<~.

+ zf' (z)J

Since the function (2.129) can be written as

f

*

h,

the

99

h (z)

Z = i1 [ l-z

Z

+

]

(l-z) 2

So

Robinson's conjecture follows for the classes is convex univalent in

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 <:

+ xz)(1 + yz), Ixl

zF' Re -F-

= Re

1 2

= IYI = 1,

we have

xz yz 4 I I l+xz - Re l+yz > - '3' z

1

<: 2

•

This shows 1 + z£'(z) #- _ zF'(z), Izl < fez) F(z)

1

2

,

or equivalently (2.130)

h(z/2) * l+xz fez) 1+yz

¢

0, 0

<

Izi < 1 .

After a simple modification, Theorem 1.12 applies to (2.130) and yields for any h(z/2)): (2.131)

h* h*

Ff(U) f

C

co(F(U)) •

h

100

With

F = zf'/f

we obtain

~

which proves that

z(h~

*

f)' (U) c -zf' co(~(U))

h

*

f

(h * f)(z)

= (h *

f)(z/2)

is spiral-like and thus univalent.

In their original proof Barnard and Kellog established the deeper result f E S.

that (2.130) (and thus (2.131)) holds for any

However, if

f

is not

spiral-like, (2.131) seems not to be strong enough to establish Robinson's conj ecture.

3) functions.

We wish to mention a few more results concerning prestarlike The following estimate is due to Silverman and Silvia [77J. 00

THEOREM 2.45:

Let fez) =

I

k=l

akz k E R , a ~ 1. a

Then

, k :::. 3 .

These bounds are not sharp in general. [84J we obtain better bounds if

a2

Using a deep

= 0:

k-l ~ II 2j-1 k 2J'+1-2a' j=l

(2.132)

result of Suffridge

= 2 , 3 , ...

,

and k-l k-l

(2.133)

~-k

2j-l

II 2' -a . 1 J+1-2a J=

(2.132) is sharp for any

k:::. 2,

J

k

= 2,3, ....

while (2.133) is sharp only for

The following result is in [60J.

k

= 2.

101 00

THEOREM 2.46:

In

= I

f

akz

k=l

6 olr..d eJt a

all. e. pJt eo .:to.Jt.V..k e. 0

k

ERa' a ~ 1,

Izl

i~

<

~he.~ ali

patr..tial ~~

1/(4-2a).

The proof is a simple technical affair since it suffices to give it for the extremal function of R. a

= z/(l-z)

fez)

and then to apply the convolution invariance

Theorem 2.46 can be used to establish - via convolution - similar results

for functions in

Sa' Ca ,

and even for certain

p-valent functions.

This unifies

a number of previous theorems due to Szego, Goel, Padmanabhan and others.

For

details see [60J. Al-Amiri [3J has studied the inverse question to the general relation

lliEOREM 2. 47:

Ld

fER,a~1.

The.n

a

fer z)/r E R 1, a a a-2

whe.tr..e.

= [1- 2a+ 214=2a] ~

r

a

5-2a+2/4-2a]

In the proof he employs the well-known method of Zmorovic.

We omit the details.

Several formal generalizations and special subclasses of prestarlike functions have been studied by a number of authors, among them Al-Amiri [lJ, [2J,

S. Singh and R. Singh [80J.

Many important questions, however, in particular the

description of geometrical properties of prestarlike functions and their convolutions,are still open.

One important step in this direction is due to Suffridge

[83J. originally stated for functions in THEOREM 2.48:

rg

O~

au 06

le.~g~h

Ld

f,g E R , a a

Y f' Y g'

sa . ~

1.

For the notation see (2.15). Ld

f,g

be.

a~a.1.ljtic. O~ Mc...6

lr..e6pe.c.tive.llj, a~d I.lUPP0I.le. ~h~

102

Re

D3-2af ~--=-~ 2 2 D - af

r

Then thVte .,w an Me.

c

='2: I

au

3-2a Z

Er f'R e D2- 2ag D g

= 2, 1

z Er

g

06 length Yf + Yg - 2n 011. wrue.h

f

*

g

.,w ana-

lyuc. and ~ a.t.-W 6-iu D3 - 2a (f 2 2 D - a(f

Re

*

g)

*

g)

= ~,

z E

r .

The proof is too complicated to be reproduced here. 4)

We refer to [23J.

We give a simple example for the application of "dual" statements in

the sense of Theorem 1.20.

To do so, we first establish a suitable theorem and

then look for the "dual" theorem. THEOREM 2.49:

c.onvex un-ivaleYt.t 60n (2.134)

Let

p

Iz I

< r

r

=

n

n

be a pohJnom-<.a.e. w-<..th

P/z E

Pn

Then

P M

w-<..th 1

2(n+I)2

(2 + 3n - ~n2 + 4n) .

This is an old result due to Dieudonne [16].

Using duality and Theorem

1.8 we easily see that in order to minimize

Re

zP"(z)

P' (z)

+

1

over all polynomials under consideration, it suffices to deal with the polynomial PO(z) = z(l - z) n .

We immediately obtain

zP"(z)

Re P' (z) for

/zl

<

l/(n + 1).

+ I

~

I -

fu:1lhl J::TZI

A simple calculation completes the proof.

103

Now let - in the notation of Theorem 1.20 -

Tl = {(I - xz)n and

g (z) :::: 1/(1 - r z).

I Ixl = I},

T2 : : T(1,3) ,

Then Theorem 2.49 is equivalent to

n

and Theorem 1.20 gives (2.135)

T** is known, we can reformulate (2.135) into the dual theorem (we recall

Since f E

2

M <--> f' E T(1,3)**

f E AI)'

for

00

I

LeX fez) : : z

THEOREM 2.50:

akz

k

The.n nolt

E M.

n

E N,

k::::O

<::

r

~

n

the.

nwmb~ ~n

The numbers 5)

r

r

n

(2.134).

are best possible in both theorems.

n

a,S E N,

Theorem 1.9 for

has an interesting application to rational

functions. THEOREM 2.51:

LeX m,n E N, m ~ n,

(2.136)

Let

have.

R(z) :::: P(z)/Q(z) E AO

f

*

and f E Ao

~ueh

that

m (1+ xz) (1+z) n

be. a

1 0, z, x E U •

It~o»at 6uYle~oYl w~h

The.n we. PEP, m Q E Pn ,

104

(2.137)

(f

PROOF:

Assume

K(m,n)

and thus For

w

n

f

R(U).

f

*

= 0,

RO

Then ~

0

in

* R)(U)

C

R(U) .

ROez) = (P(z) - wQ(z))/[(l U.

This implies

f

*

R

~

w)Q(z)J

is in

w which is (2.137).

Theorem 2.51 is a refined version of Szego's Theorem 0.1 (in

this form due to de Brujn [7J) which will be used in Chapter 4.

It is worthwhile

to point out that the duality princjl'le, in particular Corollary 1.1, applies to any rational function in

K(m,n), m,n E N.

For example, it is possible to extend

Theorem 2.49 to functions of the form

P,Q E

AO

R(z)

=z

nonvanishing polynomials in

U.

P (z)

Q(z) ,

Chapter 3

LINEAR TRANSFORMATIONS BETWEEN DUAL SETS

3.1.

Some more duality theory

U C AO is a dual set if and only if there exists

We recall that a set

WC AO with U = W*.

Any dual set is closed and cOMplete and is characterized

by the property (3.1)

U

= U**

•

This relation is a special case of the following simple but important observation.

TIIEOREM 3.1: ta-i..n..i..ng

FOJt

V

C

Ao

:the

M . c.ond

duai..

A.....6

:the .6ma...Ue6:t dual... .6 e:t c.on-

V.

The justification of the term "smallest" comes from the obvious fact that any finite or infinite intersection of dual sets is a dual set.

The content of

Theorem 3.1 can also be stated in the following perhaps more suggestive form:

V**

is the dual hull of

Note that for compact and complete sets

V C Ao •

V the dual hull is contained in the

closed convex hUll.

PROOF of Theorem 3.1:

Assume

V C W*.

Then obviously

V*

~

W** and

106

V** c W*** = W*.

V contains V**.

Thus any dual set containing

Since

V** is

V the conclusion follows.

itself a dual set containing

Another useful criterion to identify members of

V**

is given in the

next theorem. THEOREM 3.2:

then

f E

Let V C Ao and f E Ao'

A(f) E A(V)

6o~

any A E A,

V**.

Note that for compact and complete f E V**

necessary for

Zo

g E V*,

If

in A.

Thus there exists

* fO)(zO)

# O.

V

C

AO'

the condition of Theorem 3.2 is also

(duality principle).

PROOF:

(g

16

EU

the functional

J

fa E

This shows

g

v

A defined by

such that

* f

in

# 0

(g

A(h) = (g

* h)(zo) is

* f)(zO) = ACf) = A(fo) =

U for any

g E V*

which is the

assertion. Let

n be the set of continuous linear operators

q:

A + A.

To

q En

we assign its kernel function

which is analytic in

z E UJ q(f)(z)

~

E U.

Using this kernel we have

= HCz,~) *

f(~)I~=I' f E

A

By nO

we denote the set of origin-preserving operators

q(f)(O)

= f(O).

q E n,

i.e.,

This latter condition is equivalent to H(O,~)

- 1,

s

E IT

We wish to establish some simple results concerning the transformation of dual sets under operators in

n and related operators.

107

Let V be compact and complete, and let q E n be

THEOREM 3.3:

that q(f)(O) # 0,

E V.

f

(3.2)

Then

~o~

Q(f)

q(f) = q (f) (0)

~uch

op~o~

the

. V**

.

-+

A

0

we have

O(V**) c O(V)**

(3.3) Equatit~ hotd~ ~~

PROOF:

and only

~~

dual.

~

Q(V**)

First note that according to the duality principle the functional

V**

is nonvanishing on

Aoq E A and for

= Q(g),

f

Q is defined.

so that

g E V**,

there exists

For any

q(f)(O)

A E A we have

go E V (Corollary 1.1) such

that

_ ( q(g) 1 _ (Aoq) (g) _ (Aoq)(gO) _ ACf) - A q(g)(O)j - q(g)(O) - q(go) (0) - A(Q(gO)) E A(Q(V)) . f E Q(V)**.

Theorem 3.2 gives it contains

q(V)**

If

O(V**)

is dual, then Theorem 3.1 shows that

and equality holds in (3.3).

The other directlon is trivial.

Of course, it is of interest to characterize cases where the condition on equality in the above theorem is fulfilled.

A fairly general result is obtained

in the next theorem.

Let q E no be ~veJtt~ble wUh q-l E no'

TflEOREM 3.4: ~un~on

H(z,s)

o~

q

ah~ume

=1

H(z,O)

~o~

and complete. and (3.4)

q(V**)

~

complete,

the.n. q(V**)

= q(V)** .

z E U.

1~

V

C

Fo~ the k.~nel

Ao

~

compact

108

PROOF: -1

H

By Theorem 3.3 it suffices to show that be the kernel function of

(z,r;)

q

-1

is a dual set.

q(V**)

A simple calculation using

Let f

o -= 1

gives = q

-1

-1

(q(fO))(z) = H (z,O)

or (3.5)

H

-1

(z, 0) - 1, z E U •

Let 01 = {f E V*

I

f

analytic in

IT}

~

and let

01

consist of all functions

~

with

fEW.

From (3.5) we deduce

01

C

AO'

We wish to prove

~

(3.6)

0)* = q (V**) •

It is a consequence of Hurwitz' theorem that

V** = (11*

and thus (3.6) is

equivalent to ~

(3.7)

01*

= q(01*)

•

h (z)

h(pz) E 01*

~

Let

h E 01*

~

which is equivalent to

(3.8) for (3.9)

p

~

for any

hp(r;) *r;H-I(Z,Z;) *z f(z)lz=1 ¢ 0 p,r; E U, fEW.

Since

01

is complete, (3.8) is equivalent to

P E U and to

109

for

P,s,z E U,

fEW.

The freedom in choosing the parameter

p

shows that

(3.9) is equivalent to (3.10) for

p,z E U, fEW.

Thus (3.10) is equivalent to

h

(3.11)

E q(W*),

P

EU,

p

and thus to (3.12) since

h E q (01*) ,

q(W*)

and that

= q(V**)

q(V**)

is compact and complete by assumption.

This proves (3.6)

is a dual set.

We remark that in Theorem 3.4 we can replace (3.4) by the condition: there exists a set

U, V cUe V**,

(3.13)

for which

q(U)

is complete.

In fact, in the notation of the proof above, we have h E W* <

(3. 14)

If (3.13) holds we conclude

>h

P

q(U) c W*

Ipl

E q(V**),

~ 1 •

and thus by Theorem 3.1:

q(U)**

C

W*.

Using (3.14) we get

q(V)**

~

C

q(U)**

C

W*

C

q(V**)

which together with (3.3) proves the assertion of Theorem 3.4. The most simple examples for operators satisfying the conditions of Theorem 3.4 are given by

110

where

g E AO

with existing

g(-l) E AO

H(z,s)

= g(zs),

as solution of the equation

* gC-l) =

g

In this case

(f * g)(z)

~

q(f)

(3.15)

1

l-z

and the assertion of Theorem 3.4 yields in a sym-

bolic notation g

*

= (g *

V**

V)

**

Note that in this case (3.4) is automatically fulfilled. It is easy, however, to construct examples of operators (and sets satisfying the assumptions of Theorem 3.4 which are not of type (3.15).

V)

In gener-

al it will not be possible to relax completely conditions like the existence of the inverse operator

-1 Q

in Theorem 3.4. q(f)

which is not invertible. V** = V,

For

V

= (1 ~

- z)n

{I/Cl -

For instance, choose

* xz)

n E Nand

f E nO

I Ixl ~ I}

we have (Theorem 1.18)

and q(V**)

q (V)

= {(l

* * = {( 1

-

xz)n I Ixl

- x z) n

I Ix I

~ l} ,

~ I} * * .

These sets are different by Theorem 1.8. Finally we mention that under the

assum~tions

have (3.16)

Q(V**)**

= Q(V)**

.

of Theorem 3.3 we obviously

111

In order to exploit the relevance to duality of linear operators we prefer to use a more symmetric approach. U2 .

Then for any

Let

H(z,s)

be analytic in the polydisc

analyti~ in U let

f

and

If

q(f)(O) t 0 we write

(3.17)

Q(f) = q(f)/q(f)(O) . ~

The operator

Q is defined analogously. ~ = {f(pz) I Ipl

For any complete set <:

V C AO we define

1, f E V}

In this notation we obtain the following general result for operators sending one dual set into another. THEOREM 3.5: atM

Q

M

i)

ii)

Let V,DJ

-tVl. (3.17) th e.

noUor~ng

Q -L6 de.MVl.e.d on

1I~ *

'"

W~*

Q -L6 de. n-tne.d on

~

iii)

AO

C

Q -L6 de. n-tn e.d on OJ

be. c.ornpac..t aVl.d c.ompi.ete..

The.VI.

nO~

aVl.Y

op~-

J.dateme.n:to Me. e.qu-tva..te.nt: and Q(t1**) c (JJ*; aVl.d Q(DY**)

C

V*;

and Q(Do,) c V*.

Note that this is a far-reaching extension of Theorem 1.20. PROOF:

Clearly ii)

>

iii).

the function associated with

Now assume that iii) is valid.

q

we obtain for

f E 01, g E V~*

H(z,s) *z f(z)l z=l g(~) *~ H(z,O) *z f(z)l z=l ¢ 0, ~ E U .

Then if

H(z,s)

is

112

From the completeness of (3.18)

W we

get

g(l;;) *1;; H(z,1;;) g E V~*

Since

*z

fez) # 0, z,l;; E IT .

implies the existence of a number

Ipi > 1

see that (3.18) holds for

s =1

(3.19)

q(g)(z) * fez) # 0, z E U .

This proves in particular

q(g)(O) # 0

holds for any rem.

o

fEW

g(ps) E V~*

with

we

as well, and thus

and

Q is defined on

we deduce the same inequality for

fEW

V~*.

Since (3.19)

by Hurwitz' theo-

Thus Q(g) = q(g)/q(g) (0) E 00*

and i) follows.

We have shown:

V, W we get i)

and

ii) ==> i).

Interchanginf, the roles of

Q,

Q

> ii), and this completes the argument.

An application of Theorem 3.5 is given in the next section.

3.2.

SpeCial cases We start with the characterization of operators of type (3.17) which send

o

T(a,S)*

into

T(a' ,B')*.

THEOREM 3.6:

the 60nm (3.17) (3.20)

(3.21)

Then

Theorem 3.6 is essentially due to Sheil-Small [74J.

Let 1

~

c=

min{a' .S'}

~

min{a.S}.

a6~ume

Q((l - I;;z)-l) E T(a',S')*, I;; E U,

Fo~

an

op~ato~

Q

06

113

Q(TCa~B)*) c TCa' ,8')* •

(3.22) PROOF:

We wish to apply Theorem 3.5

v = TCa,S)*,

with

W = TCa' ,S').

Thus to

prove (3.22) we only have to show e

~

QCTCa',S')) c TCa,S)** .

(3 23) First we observe that for

f E T Ca' J3' ) **

= H(z,s) *

(3.24)

H(O.s)

(3.24) extends to q(f)(O)

we have

t 0:

Q

z

=1

and thus

q(f)(s)

fez) t 0,

t O. s E U.

s.z E U •

which gives in particular

T(a':S')** and in particular on T(aP.S').

is defined on

Next

let f (z) =

( I_XZ)Ca'J(I_yz)a'-Ca'J -

-

E

(l-vz)8' such that for any

Ipi

1,f(pz) E T(a,e. S').

<:

Then we have for a certain

£

largCe

i£ e

Assume for the moment

= £(x,y.v) i£

(1 - pvz)

An obvious refinement of the case

Iarg

T(a',S')

II = \)

S' - a' f(pz)JI

= a'

<:

a'TI

z E U •

2

in Theorem 1. 13 gives

Q((l-z s )-l) * f(pz) z -----:-,-~S::-:-, I ----=-l--Q((l- z s)- ) * (l_pvz)a

a'TI

<:--

2

z

for

z,s E U.

valid for

g

This again extends to

analytic in

U, we get

z

=1

and using the relation

<5

= a'

::: S'.

114

Q(f(pz))

(3.25) where

H

p

K(a' ,a')

E

Ha '.

and

But the right hand side of (3.25) is a product of functions in

K(a - at,S - a')

proof for the case

a'

>

S'

K(a,S) c T(a,B)**.

and is therefore in

The

is similar and we omit the details.

The following Corollary 3.1 is an immediate consequence of (3.23) and Theorem 3.5.

the

Und~

COROLLARY 3.1:

a~~umptionA

06 Theonem 3.6 we have

~(T(a' ,S')**) c T(a,S)** .

(3.26)

Although (3.20) is clearly necessary for the conclusion of Theorem 3.6 this is not true in general for (3.21).

This problem needs further investigation.

One special case in which (3.21) is empty is described in the following

Let a

COROLLARY 3.2:

Q(T(a,a)*) ~6

and only

~

1 and

q E ~O'

~

C

Then

0

T(a,a)*, Q(T(a,a)**)

C

T(a,a)**

~6

q((l - sz)-l) E T(a,a)*, s E U In another special case Sheil-Small [74J has given the following supp1ement to Corollary 3.1. THEOREM 3.7: 6~~

Let 1

~

a

~

S. 16 an

op~aton

Q

a = a' = 0, S = S', then 60n any

(3.20), (3.21) with ~

0

Q(K(a-E,S-E))

C

K(a - E,S -

£) .

06 type

(3.17)

0 ~ E ~ a

~~-

we have

115

PROOF:

We start with the case

see that

E

= a.

From the assumptions and Corollary 3.1 we

o

Q is defined on T(a,S)**.

Now assume

o

f E T(I,I+S-a)*

such that by

an obvious modification of (2.13),

This shows that the operator given by P (f) (z)

(3.27) is defined on

o

T(l,l+S-a)

and (3.21) we get for

~

=

((1 - z)a-S) (-1) * 'O(f * CI _ z)a-S) and we observe that

P

is of type (3.17).

From (3.27)

E U

Furthermore,

Q((l - z~)-I) E T(a,S)*

since

C

T(I,I + S - a)*.

Thus Theorem 3.6 applies to

and gives o

P(T(l,ItB-a)*) ~rom

C

T(I,l t B - a)* .

(3.27) we obtain 0

~

(3.28)

Q(K(O,S-a)) c K(O,S - a) .

No',. let

0 =: E < a

and

f

g E K(O,S - a), F E K(a - E,a - E). Q(f(pz)) and QCC1 - pz)-I) E TCa,S)*

C

E

K(a - E,S - E).

Since for

= Q(CI

Ipi

<

Then

f =

~F

where

1.

pz)-l) * g(z)F(z)

TCa - E,(:3 - E)*, we get from Theorem 1.13

P

116

Q(f(pz)) with

H E Ha - E

= K(a -

E,a - E).

= H(z)Q(g(pz))

Since

Q(g(pz)) E K(O,S - a)

by (3.28), we

obtain Q(f(pz)) E K(a - E,a - E) • K(O,S - a)

= K(a

- E,S - E) ,

the assertion. To simplify later avpIications we restate Theorems 3.6, 3.7 and Corollary 3.1 for the most important case

a

Let S.S'

COROLLARY 3.3:

= a' = 1. ~

1.

Fo~ a~ op~ato~

Q 06

th~

6orom

(3.17)

Q((l - zs)-I) E T(l,S')*, s E U •

The~

Q(T(I:B)*) c T(l,S')* , Q(T(l.~')**) c T(l.S)** , a~d ~6

S = B',

0 ~

E

~ 1, ~

the~ 0

Q(K(I-E,B-E))

C

K(l - E,S - E) .

It is obvious from this theorem how to characterize operators sending classes of prestarlike functions into other such classes or (choosing E

= 0) close-to-convex functions into close-to-convex functions.

statements of this type see Sheil-Small [74J.

B

= S' = 3,

For explicit

We give a general example of such

117

operators. Then

In fact, let

~

0

~

I

and

g E

Ro' h E Sl+y_o' h analytic in IT.

tEO, where

(3.29)

For

y

t(f)(z) =

f E Ry

g(r;z)h~?;) * f(r;)Ir;=l .

we have (t(f))'(O) = her;) * f(r;)Ir;=l"l 0

RY C Rl+y-o' and thus h * f E Sl +y-us.

since

T(f)

is defined on

let

We have

t (f)

(t (f)) , (0)

Ry '

THEOREM 3.8 [56J: PROOF:

=

This shows that the operator

fER y

q(b) = t(zb)/z

T(R ) cR. y

0

if and only if

such that

f/z E T(1,3 - 2y)*.

q(b) (0) = (t(zb))'(O)

For

and thus

q(b)(z) = ! T(zb) Q(b) = q(b) (0) z A calculation yields Q((l - r;z)-l) = g(r;z) E T(I.3 - 20)* , r;z

Since h(z)/z E K(O.20 - 2y), g(r;z) *~ (1 _ r;)20-2 E K(O,2 - 28) r;z '"

b E T(1.3 - 2y)

118

we get '" QC(1 -

~z)

28-2 ) E

2y)

K(0,2 -

and Corollary 3.3 gives o

Q(T(1,3-2y)*) c T(I,3 - 28)* .

(3.30)

T is continuous and

The assertion follows from the fact that

Zo

For an application of Theorem 3.8 let g(z)

= z/(l

- z), 0

=y

+

Ry,R o are compact.

E V, h(z) E z/(l - zOz) E S1'

l.

COROLLARY 3.4 [56J:

Fo~

fER, y
we have

f(z)-f(zO) -----ER

(3.31)

y+zl '

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-<-l'I.g -6tcLte.me.n..t6 Me. equ-<-valen.t:

COROLLARY 4.1:

n Ao'

i)

f E Bn

ii)

co[ (f

* q) (U)]

iii)

thete

e.x~t

C

co(q(U)),

q E Pn ,

hER, F E A,

The limiting case

n

-+

¢ueh that f

=h

n+1 + z F.

of the equivalence ii) <=> iii) has been known

00

for a long time but was first explicitely stated by Wi If [92J. In a famous theorem Caratheodory and Toeplitz (see Tsuji [89J) characterized the body of the first

n Maclaurin coefficients of functions in

R. A combi-

nation of their result and Corollary 4.1 gives 00

THEOREM 4.3:

fez)

=1 + I

akz k E Bn

and only -<-6 the. H~n

,to

k=l

Hn (f) =

1

al

a1

1

an

a

a

a1

a

n n-l

•

~

PMWVe.

.6 emLde.n,tnde.

n-l

1

125

As an application we deduce an improvement of Bernstein's famous inequality Ilq'll ~ nllqll, q E Pn THEOREM 4.4:

Folt

(4.8)

we. ha.ve.

~

~~2Iq(o)1

Ilq'll

The. numbe.lt PROOF:

q E P , n ::: 2, n

For

,v., be6t po.6.6-i.ble..

2n/ (n + 2) n::: 2,

£

nllqll -

E C,

we have

QE (z)

= L -

n-l n-k k=O

We wish to determine

E

Toeplitz theory that

MnCQ o )

such that

z

n

n

n

E

n AO

£)

are positive.

MnCQE)

5

n

•

It follows from the Caratheodory-

is positive definite since

Hence the principal minors of order

det(Mn CQ E )) ::: 0

+ ~ zn E Pn

Q EB .

R.

Thus

k

of

M (Q ) n

E

Qo

is a polynomial in

(which do not depend on

is positive semidefinite if and only if

and we are left with

A straight-forward calculation yields

We conclude:

QE E Bn

if

IE[ ~ 2n(n + 2)

and there is no larger disc with

center at the origin for which the same result is true. For

q EP

n

we define

q

EP

n

by the equation

126

(4.9)

Since for any

p,q E Pn

we have

* qll = lip *

lip

qll

it is clear that the relatl n

(4.10)

holds.

for

If we apply this to

fEf -< ~ n+2

QE

we get

QE (z)

= n

E

+

k

n

I -

k=l n

k z

E Bn

and for no larger such disc.

To complete the proof let

a(n)

be the largest number such that

Ilq' (z)11 ~ nllqll - a(n) /qe O) f . This is equivalent to each of the following four statements: a(n) Iq(D) I

+ I zq' (z) I =:: nllqll, zEIT

/Eq(D) + zq' (z)1 ~ nllqll, zEIT, IEf =:: a(n) ,

Together with the previous results this shows

a(n) = 2n/(n + 2)

which is the

assertion. We remark that it is not yet clear whether there exist polynomials different from

q(z) ::: pzn, Ipi = 1,

for which equality holds in (4.8).

Another general idea to construct new bound preserving polynomials from

127

old ones is given in the next theorem.

Let q E Pn

THEOREM 4.5:

n R.

Th~n 6o~

any

I€I ~ 1 and

m ~ 1 W~

ha.v~

q + €z !Ir"q E B . n+m PROOF:

Let

= q(z)+€Z m(q(z)-z n ) ~

F (z)

1-e:z n+m

= q(z) For

1£1

<

1 the function

Re F (z)

is analytic in

F

= Re

0 .

~

Izi ~ 1 and on

FER

for

1£1

< 1.

Fon

Izi

q EP

n

= 12

=1

we obtain

•

Since

I€I ~ 1 which by Corollary 4.1 shows

COROLLARY 4.2: (4.11)

), Z

Re q(z) - 1) - Re q(z) + 1

Izl ~ 1 and thus

conclusion holds for

n+m

2 Re q(z)-l _ Re q(z) + 1 1-£z n+m

~ ~(2

This extends to

~

+ e:z q(z) + o(z

f

R is compact the

=q

nr-

+ e:z q

E

Bn+m .

W~ hav~

flzq' - (q(z) - q(O))1I + IIq(z) - q(O)11

~

nllqll .

This result has first been proved by Frappier (unpublished) who used a different method.

Note that (4.11) is another improvement of Bernstein's inequality since

the left hand side is never less than PROOF:

IIq' II.

As we saw above, 2 Q(z) = ni n-k-1 zk E Pn - 1

k=O

n-1

nR.

128

For

Q(z) + EZ

m = n - 1 we obtain

n-l~

Q(z) E B2n - 2 and if we take the

partial sum of this polynomial we clearly obtain lEI ~ 1.

n-th

Q(z) + ~l zn E B n P , nn n

Using (4.10) we get n \' L

k-1l zn ER, n=l + Q(z) =-1+ nk=2 nn E

IE I

~1,

q E Pn by convolution with these functions,

and for

1

n-1 IIEq(O) + zq' - (q(z) - q(O))1I ~ IIqll, lEI ~ 1 , which implies IIzq'(z) - (q(z) - qeO))1I ~ (n - l)lIqll - Iq(O)1 . The result follows using the obvious inequality en - l)llqll - Iq(O)1 ~ nllqll - Ilq(z) - q(O)1I

REMARK: the class

Using the

~-operator one has to keep in mind that it depends on

Pn in which a polynomial is considered to be (even if its degree is

< n).

Let Q E Pn ' Q(O) = 0,

COROLLARY 4.3: q E

~u~h

that

Q E R.

Then

nO~

P , n

(4.12)

I (Q

PROOF:

IE I s

By Theorem 4.5,

*

q)(z) I + I (Q m = 1,

*

we get

q)(z) I ~ IIqll, z E U .

Q+ EQ

E Bn' lEI ~ 1,

1,

I(Q * q)(z) + E(Q which implies (4.12).

*

q)(z) I ~ Ilqll

and thus for

129

n

The choice

Q(z) =

L

k=l

k k - z

leads to the well-known result

n

Iq' (z) I + Iq' (z) I ::: nllqll

(4.13)

(see Malik [38J).

In fact, we have [69J max (lql(Z)1 + Iq'(z)l) = nllqll . Izl=l

(4.14)

If, in particular, a polynomial for a certain

q(z) = Eq(Z)

q

is self-inverse

lEI = 1,

(q E Pn )

which means

we obtain under the assumptions of Corol-

1ary 4.3, (4. IS)

and by (4.14), n

IIqlll = 2" IIqll .

(4.16)

The class

Bn

has an interesting invariance property with respect to

subord ination: THEOREM 4.6: PROOF: f

Let

w(z)

Let

fEB, g -< f. n

The..YL

be a Schwarz function such that

g E Bn . g(z) = f(w(z)).

Assume that

has the representation

I l:~Z d~ (~)

fez) =

+ zn+l F

au (see Theorem 4.1).

For

q E Pn

q(z)

*

g(z)

we have

=

Iau

(q(z)

n+l w (z)F(w(z))

* q(z) -

0

and thus

130

This proves (4.17)

Ilq

but since

* gil :: sup Iq(z) * 1- Z;;w1 (z) z,Z;;EU

1/(1 - Z;;w(z)) E R for any

1

Z;; E U. the right hand side of (4.17) is

Theorem 4.6 and some similar more general results are due to Sheil-Small [?3] .

4.2.

Application to univalent functions In [73J Sheil-Small showed that there is some evidence for the truth of

the following conjecture. CONJECTURE

S:

Ilf * qll :: nllqll .

(4.18) Note that (4.18) with Bernstein's inequality.

f

f ES n

q (z) = z

CONJECTURE

the Koebe function

is exactly

> fIn E 13 , n EN. n

we see that Conjecture B:

z/(l - z)2

Clearly, (4.18) is equivalent to the statement

(4.19) Setting

a.Y!.d q E Pn we. ha.ve.

f ES

Faft

Faft

fez)

S

is stronger than Bieberbach's

=

However, in view of Theorem 4.6 and (4.19) we deduce that Conjecture also contains Rogosinski's 00

CONJECTURE

R:

Lu

g(z) =

I

k=l n :': 2.

akz k

~ f E s. The.Y!. we. have.

lan 1:: n,

S

131

Sheil-Small has given a proof for (4.18) when spiral-like.

f

is close-to-convex or

We give an alternative proof for the close-to-convex case.

THEOREM 4.7: REMARK:

Le.t

Since

g

-< f,

wheJte.

f EM.

The.~

gin E B

n

nO~

n

E N.

Bn is a convex family, Theorem 4.7 holds also for

co M

and thus in particular for any (even non-univalent) typically real function, see [73J.

PROOF of Theorem 4.7:

Let

q E Pn

and

q (z) :: q (z) - q (0) + Eznllqll . E

We wish to show that fact, for

h (z) = q (z)/q' (0) E: E E

is in

So

(starlike univalent).

In

I z I = I, zh' (z) Ih (z) E

- 11

Here we used inequality (4.11). Theorem 2.29 we obtain

This proves

fin E B

n

=

This estimate clearly extends to

* f)(z)

0

¢

(qE

for

l(q z

*

f)(z) 1: Enllqll, z E U, lEI = 1.

Izi < 1.

By

0 < Izl < 1 and thus

and the final conclusion follows from Theorem 4.6.

Our proof together with Theorem 2.34 admits the following nice extension of Theorem 4.7: THEOREM 4.8:

Le.t

f l '£2 EM,

132 z

f (z)

(4.20)

fI(t) * f 2 (t) = J - -t d t

o

The.n..to PROOF:

g

-< f

I

we. have. gin E Bn , n E N.

According to the preceding proof we have

ZJ

hE:t(t) - - dt E T

o and thus

1h<~t) fl *

Z

dt

=

0

This implies for

o < Izl

f

* hc

E

So .

0

< 1

z f 1 (t)

f2

fICt) dt t

*

f0

t

dt

*

h

C

::: f

* hE:

~

0 .

The result follows as above. Note that (4.20) is the Mandelbrojt-Schiffer operator and However. Theorem 4.8 may extend to

univalent in general.

h

is not

S.

In view of the extremal property of the Koebe function in

S one may be

lead to the conjecture (4.21)

IIf * qll ::" z

(l-z)

for let

f E S.

2

* qll ::: liqlll, q

E Pn '

This is clearly stronger than (4.18) but false in general.

In fact,

133

q(z)

(4.22)

=

1 - z _ z2 + z3

and 1 = l+iO'.

f

f ES

Then

[" 10'.

z

z

]

l-z + (l-z)2 '

08 0'.=.

•

(close-to-convex) and a simple numerical calculation shows

Ilf * qll ::: 1.00211q ' ll . Thus (4.21) fails to hold even for close-to-convex functions.

It is somewhat

surprising that it turns out to be true for close-to-convex functions (or members of

M)

and certain self-inverse polynomials. THEOREH 4.9:

f E M

(q E

Ilf * qll

~

IY/. 6a.c..;t, (4.23) ho.td.6 e.ve.Y/. 60lt ntmc..b:oY1..6 Note that the assumption

E

q(O)

= O.

The.Y/. )..6

I[q'll . f

06 ;the. 60Jrm (4.20).

cannot be relaxed since (4.22) is in

P3 "

Let

= q E/q E (0) . I

Then for

h

E

zh' (z) Ih

It is readily seen that we get

=0

q(O)

q (z) E h

w~h

we. ha.ve.

(4.23)

PROOF:

P) n

~Z) E

=

q (z) + Ellq'llz, lEI

and

Iz r

=

,

1 we obtain

1 - 11 = I Z9 - 9 1 qE

q/z E '"Pn-2

=1

=

I (g/ z 2I I qE

and applying (4.16) twice (to

q

and

q/z)

134

1

Again this extends to

(q/z) , qE

Izl < 1

1

~ n-22 IIq/zll IIq'[[- q[[

=1

hE E SO'

and we found:

The same arguments as in

the proofs of Theorems 4.7, 4.8 complete this proof. It should be observed that (4.18) and (4.16) imply the weaker conjecture: Ilf * qll

(4.24 )

5

211q'll. q E 15n • f E S .

It is not difficult to see that the factor

2

in (4.24) can be replaced by the

(probably not best possible) numbers

f

if

is in

M.

In section 2.5 we introduced the class

Z of functions

g E Al

such

that f * g # 0, 0 We have seen that

Z

~

SO'

<

Izl

<

In view of the proofs of Theorems 4.7, 4.9 and Theorem

2.34 one is lead to the question whether for 1 I zf'(z) fez) - 1

It

1S

I, f E S .

<

f

f

Al

we have

fEZ

if

I, z E U .

clear from our results that an affirmative answer implies the conjectures

B, R, S as well as an extension of Theorem 4.9 to

S.

We conclude this section with a description of Sheil-Small's method to obtain his results mentioned above.

He uses Littlewood's subordination theorem

for integral means and the Cauchy-Schwarz inequality.

135

Let

THEOREM 4.10:

f,g E AO

b~ ~uQh

that

f<_l_ g<_l_ l-z ' l-z

(4.24)

Ld P I a,

T

~uyl.c;uoyL.6.

be. SQhtvaJtz

(4.25)

h(z)

The.n oaf!. the. nunQuon

= p(z)f(a(z))g(T(z))

we. have. h/n E Bn' n E N. PROOF:

Let

f n , gn

q E Pn

we have

be the

q

* h

n-th oartial sums of f

= q

and

g, respectively.

For

* (p(z)fn (a(z))g n ('r(z))

and thus, using the inequalities mentioned above and (0.3),

I (q

* h)(rei~)1 ~ Ilqll ;TI

2TI

f

Ifn(a(re it ))I fgn(T(reit))ldt

o

~

Ilqll

(~TI

2TI

J

Ifn(a(reit))12dt)~(;TI

o

~

J o

J o

2TI Ilqll (;TI

2TI

If

n

(reit)12dt)~(l2TI

2TI

J o

:: Ilqll III . In . In the last step we used (4.24) and Parseval's formula. One can show that any spiral-like function and any extreme point of the set of close-to-convex functions has a representation (4.25), so that we have a

136

proof of Conjecture single

for these cases.

n - the conjecture

functions in Conjecture

4.3.

S

S

A similar idea shows that - for each

is contained in Robertson's conjecture on odd

S

for the same

n.

Since the latter conjecture is true for

S holds also at least for

n

~

4.

n:5 4)

For the details we refer to [73].

Polynomials nonvanishing in the unit disc

A considerable amount of research in classical function theory has been devoted to the study of the location of zeros of polynomials, in particlllar if they are partial sums of the

~aclaurin

functions in the unit disc.

Convolution theory, in particular the theory of pre-

expansion of certain nonvanishing analytic

starlike functions, can be useful in this context. Enestrom-Kakeya theorem [20]:

We recall the classical

let ~

(4.26 )

a

n

~

0 •

Then the polynomial

is in

Pn

(nonvanishing in

si~ple

special case of a general result concerninp, partial sums of starlike

U).

The following theorem shO\'ls that this is a

1

functions of order

2. •

oa

Let fez)

THEOREM 4.11 [57]:

= z

L

bkZ k E Sl, n E

k=O ~w.t6

(t

(4.27)

U1e have

nt1.l'l1beJt

p

= pef,n)

q(z) pef,n) > 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 <:

~ u. VI. c.t-Lo VI..6

p

-6a.;(:"U~tj-LVl.rJ (4.26),

137

For the proof we need the following well-known fact for functions in

R:

let 00

I

h(z) =

akz k E R .

k=O Then we have

kO E

If there exists

N

with

la k

I = 1,

o

then for

mEN J k = 0 J 1, ... , kO - 1 J

(4.28) In particular, the sequence

la k _ l - akl

PROOF of Theorem 4.11:

is periodic with period

kO'

Let

~k=O bk k

7T. (z) =

Z

J

We observe that the set of polynomials closed convex hull of the functions

J. J

=

O,I •... ,n.

considered in (4.27) is exactly the

q

7T., j = 0,1, ... ,no J

Therefore, in order to

nrove the theoreM it will be sufficient to show

Re(z7T.(z)/f(z)) > O. z E U ,

(4.29)

J

q(z) 'I- 0,

(4.30)

whenever

f

t

Izl = 1 ,

e , Ixl = 1. x

To prove (4.29) we employ Corollary 3.4.

Since

f

f

S, 2

w E U h (z) w

where

=

w f(z)-f(w) = few) z-w

00

1 +

I

j=l

A.(w)zj E R, J

we have for

136

proof of Conjecture single

n - the conjecture

functions in Conjecture

4.3.

S for these cases.

S

A similar idea shows that - for each

S is contained in Robertson's conjecture on odd

for the same

n.

Since the latter conj ecture is true for

S holds also at least for

n

~

4.

n =: 4,

For the details we refer to [73J.

Polynomials nonvanishing in the unit disc

A considerable amount of research in classical function theory has been devoted to the study of the location of zeros of polynomials, in particlllar if they are partial sums of the

~1aclaurin

functions in the unit disc.

Convolution theory, in particular the theory of pre-

expansion of certain nonvanishing analytic

starlike functions, can be useful in this context. Enestrom-Kakeya theorem [20J:

We recall the classical

let a

~

(4.26)

n

~

0 .

Then the polynomial

1S

in

Pn

(nonvanishing in

U).

The following theorem shows that this is a

siMple special case of a general result concerninp, partial sums of starlike

,

functions of order

;1 •

L~ fez)

THEOREM 4.11 [57J: ~X~~

a

n~~b~

p

= p(f,n)

00

=

L

z

bkzk E S1' n E N.

~

~~eh

1

that

~o~

any

~~q~~ne~

(4.27)

Ole.

hav~

p (f ,n) > 1

Th~n th~~

k=O

whe.n~v~

f

~

not

on~

on

the.

~~net,(.oYl.6

ak

~~6y~nn (4.26),

137

For the proof we need the following well-known fact for functions in

R:

let 00

h(z) ~

L

akz k E R .

k~O

Then we have

Ia k I

~ 1, k E

fL

If there exists

kO E

N with

la k

o

I

=

1, then for

mEN, k = 0,1, ... ,k O - I,

(4.28)

ak

In particular, the sequence

k = ak(a k )m . +m 0 0

la k _ l - akl

PROOF of Theorem 4.11:

is periodic with period

kO'

Let

i

7T. (z) =

J

k=O

b k Z k ' J'

We observe that the set of polynomials closed convex hull of the functions

J

O,l, ... ,n.

considered in (4.27) is exactly the

q

7T.,

:::

j

= 0,1, ... ,no

Therefore, in order to

urove the theorem it will be sufficient to show

Re(Z7T.(z)/f(z)) > 0, z E U ,

(4.29)

J

(4.30)

whenever

q(z) 1:- 0,

f -;:. e , x

Ixl

Izl = 1

J

= 1.

To prove (4.29) we employ Corollary 3.4.

Since

f f Sl

2

wE U h (z)

w

where

=

w

f(z)-f(w)

few)

z-w

=1

00

+ j

I1 ~

A.(w)zj E R, J

we have for

138 lew) Jfew) ) J j

W7T.

A. (w) = '!"""(l J

!A.(w)! ~ 1

Since

J

for

wJ

E

j

N, w E U, we immediately deduce (4.29).

+

1,

such that for i)

ii)

f(w k ) h

wk

k

= O.

q(l)

without loss of generality that wk

EN.

We choose a sequence

wk E (0,1),

+ ~ :

a,

tends to a finite or infinite limit

(z)

Now assume

is compact convergent to a function

First we rule out the possibility that

a

=

hER.

In fact, this would imply

00.

which contradicts the estimate Izq(z)! > 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-<

7T.(l)/O'., j J

jo E N such that

7T.

J -

o

=

1 (1)

and that the remark after (4.28) applies: 00

fez) =

0'.

L j= 1

(B. 1 - B.) z JJ

k

O,I, ... ,n,

= O.

for

and

This

139

lB.J- 1

we see that

-

B·I J

1S

periodic.

On the other hand, it is known

(Eenigenburg and Keogh [18J) that each function in the Hardy space sequence.

HI

f E S"

2

f ¢ e

for

x

Ix I

= 1,

is

and this implies that its coefficients form a zero

We arrived at a contradiction and the proof is complete. For an application of this result we use Theorem 2.11. THEOREM 4.12:

k = O, ... ,n,

Let

all.6u.r1e (4.26).

lsi

~ a, -1 < x < 1, n E

N.

Fo~

the

~umb~

ak,

TheV!.l'Je ha.ve

(4.33)

For the proof just observe that under the

assu~pti0ns

we have

R(1-a-S)/2 c

R~ =

S, . 2

(4.33) for [86J.

For

a =

S>

a =

-I

,

S = -2 and

a k = 1 J k = 0,1, ... , n J

it has been conjectured by Askey [4J.

is due to Szego A related theorem,

resulting from Theorems 2.11, 2.15 and Corollary 2.4, is essentially due to Lewis [34J: THEOREM 4.13:

The~ 6o~

(4.34)

n E

N,

Let

-1 < x < 1,

° ~ A ~ a + S, a ~ lsi,

a.~d

140 (l+A)k (I+A)n_k p(a,B)(x) k k z ;f. 0, z E U • L k! (n-k)! pea,S) (1) k::O n

(4.35)

k

Note that due to our method we cannot claim that this latter inequality Izl :: 1.

holds for

Inequalities of type (4.35), restricted to the interval

z E (-1,1), have been thoroughly studied by Askey, Gasper and others. number of important applications in several fields.

They have a

It seems to be worthwile to

try convolution methods in other related problems.

4.4.

An extension of Szego's

theore~

In this section we give a brief account of the deep results of Suffridge [83J which estahlish a beautiful and rather surprisinp link between convolution

properties for certain polynomials and the corresponding theory for starlike functions of various orders.

The proofs for these results are rather complicated

and we are not in the position to reproduce them here.

We strongly endorse the

opinion - expressed in [83J - that one should look for another proof. perhaps for an even more general theorem. In an earlier paper ([82J) Suffridge built up a theory how the functions in

S

can be characterized as limits of certain

polynomials which characterize the functions in ia.

n

p ({j) :: { II n

Here we put

a

(1 + ze

J)

j::l

n+l

= a,1

+ 27T.

I a.J +1

polyno~ials.

S , a

- a. ::: J

2{) ,

Let

j=l

(1 + ze

Tr

< -

n

are the polynomials

i (2j -n-l){j

n

nEN,05{j

j ::: l, ... ,n} .

Specia 1 members of

II

There is a class of

)

and

141

Concerning this class Suffridge established the following results. THEOREM 4.14:

Ld p,q E Pn .6uc.h:tha.t 6011. .6ame. fJ E (0,

Qn (z,fJ) * pEPn (fJ), 0~ (z,&)

*

i),

n (fJ) •

q f P

The.n

Compare this with Theorem 0.1 ! THEOREM 4.15:

Ld pEP, n

THEOREM 4.16:

co(P (fJ)) n

fJ E (O,~)

n

.6uc.h tha.t Qn (z,fJ) * pEPn (fJ).

= co({Qn (ze 2TIi (k-l)/n,fJ)

The link with the theory of the functions in

Sa

l~k~n}).

is given in the follow-

ing theorem. L6 -tn S, a a

THEOREM 4.17:

the.Jte. L6 a .6e.que.nc.e.

nk

E N, n k

al.o

'Pk E Pnk(TI/(n k + 2 - 2a))

with

l~

f.

~

00

6011. k

~

00,

<:

1,

.6uc.h that 6011. c.e.Jtta-tn

polyno~­

the..6e.que.nc.e. zP k L6 c.ompac.tc.onve.JtgeY/.t-tn U

FoJt a g-tve.n f E Sa one. c.an c.hoo.6e nk

= 2k

- 1, kEN.

Theorem 2.1 (in the version (2.22) - (2.24) as well as Corollary 2.2 is now a simple consequence of these results and the fact that

z

22

,ZEU.

(1-z) - a

4.5.

Additional information 1)

In connection with section 4.3 it is of interest to determine special

142

range preserving functions on

Let a,B

THEOREM 4.18:

see (4.1).

Pn •

-1, n E N,

>

n

a~d

(a+B+n)n_k (a+S+n)

= I

p (z)

We give a non-trivial example.

n

k=O The~

(p PROOF:

We have to show

*

q)(U)

C

q(U), q E Pn n

t (z)

=p *

t(x)

= ~~n~ P Ca, (a+B+n) n n

(1 + z)

# 0, z E U.

Ca)

pea,S) n

where the

are as above.

B)

(1 + 2x)

From the well-known properties of these poly-

nomials we see that their zeros are in the interval for

Iz I :::

1.

But

(-1,1)

t (z)

and thus

'f:. 0

This proves the assertion.

Consider the case

a = O.

Then Theorem 4.18 is equivalent to the

statement that the Cesaro means of order

y::: n - 1

are range preserving on

Pn :

n

(4.36 )

(q

* L

-'-r-----r.,J...

zk) (U)

C

q(U)

k=O

(4.36) is also valid for

J

q E Pn

y = 1,2, ... ,n - 2 but not for arbitrary y

>

O.

This

result is related to the work of Bustoz [14] who studied the invariance of univalence under these means.

We note that the coefficients of the polynomial

n

(4.37)

L

k=O

........,.-............. zk

143

satisfy (4.26) so that Theorem 4.11 applies to these means. Let

2)

S be a set of non-negative integers,

0 E S.

Let

AS

be the

A which have an expansion

set of those functions in

Clearly fS (z) =

z

1:

k

kES

AS'

represents the identity operator (w.r.t. *) in

Let

Vs

= {fS }'

VS* which we call the "shadow of f S ".

ested in the elements of the set

name is justified by the fact that the elements of the shadow of important properties with THEOREH 4.19:

fS '

LeX

f

E VS*, g E AS'

(4.40)

* g)(U)

(f

zg g

fS

This

share many

We give a few examples.

(4.38) (4.39)

We are inter-

uI1-<-va1.e.iu: -<-n

> z (f

U

cmwex uMvaie.iU: -<-n

U -> f

The.n we. have. g(U)

C

* g) uMvaie.iU: -<-11

U;

* g c.onve.x uvuvaie.iU: -<-n

U.

In (4.39) we can replace "univalent" by any of the properties starlike, spiral-like, close-to-convex and many others. apply the duality principle to the set equals

Us (xz)

For the proof of Theorem 4.19 just

I Ixl

~ l}

whose second dual

** . VS It seems to be a difficult task to determine the shadow of the identity For

S

= Sn = {O,l, ... ,n},

however, we clearly have

144

V** sn

= {p E

Pn n A0

To see this we just note that

Vs

I

p

* (1 + z)n 1 a,z E u}

equals n

0.1.

Pn n Aa

and apply Szego's Theorem

Thus the shadow may contain a fairly large number of members which is rather

surprising in view of the strong properties in Theorem 4.19.

If

S = N U {a},

however, the shadow consists just of the functions 1

l-xz which follows from Theorem 1.18. sets beside

S

, Ixl

~ 1 ,

It is not known whether there exist infinite

such that the shadow is non-trivial, i.e.

J

contains other functions

fS(xz), Ixl ~ 1. We conclude with an explicit application of Theorem 4.19,

THEOREM 4.20:

y E {l, ...

,n - 2}

Olt

AM ume. that fi aIt

Pn the.

Y ~ n - 1 the. polynomial

n

z (q

*

I

k=a

hM :the. .6 arne. pita pe/1.:ty .

q E

nun~:tion

zq

S = S . n

ha.f.J one. atl th e.

Chapter 5

APPLICATIONS TO CERTAIN ELLIPTIC POE'S

5.1.

Connection with convolutions We are dealing with equations of the form

(5.1) where

= wzz-

E(w) is analytic in

a

+ a(zz)(zw

z + zw-) z

U and real-valued on

=0

,

[-I,IJ,

It is a consequence of

Vekua's theory ([91J, compare [52J) that there exists a function

. In

k(t,r)

analytic

-u2 such that the system

= e ik~ c I k I (r),

(5.2)

k E

Z, z = re

i~

with r

(S ,3)

ck(r)

= r k[exp(-2 I

I

ta(t 2)dt) -

o

I

tkdk(t,r)J , k

~

0 ,

o

is a complete system of solutions of (5.1).

The functions

ck(r)

are solutions

(analytic at the origin) of the Fuehsian equation

2 2 - 4r a(r ))c e"k + l(l r

(S. 4)

Co (r) -

1.

k-

k2 - 2 ek r

= 0,

kEN

,

From the maximum principle for (S. 4) and ckeO) = 0, kEN,

we deduce

146

that

ck(r)

are monotone functions of

dk(r) = ck(r)/ck(l), k the function

a

~

O.

(but not on

r,

Then by (5.3) we obtain a constant k,r)

Now let

C depending on

such that 1

(5.5)

ek(l) ¢ O.

in particular

Idk(r) - rk exp (2

J

k

ta(t 2)dt) I ~ ~C k

r

Thus we have THEOREM 5.1: :theJte ex.-U :t .6 eq u. en.c. e.6

w(z)

.-U a J.JO iuJ/Lo n. o~

E(w) = 0

-t11

U -tn an.d on.iy -tfl

{a k } , {h k } /)J,l:th

r1ml a k rl/k k-tOO

~

1, lim!bkl l / k k-)OO

~

1

(5.6)

If

a - 0

we have complex-valued harmonic functions in

and since we are just interested in the extension of analytic which corresponds to the case

b k = 0, kEN,

restriction in the general case:

U

fun~tion

it is natural to impose the same

we shall deal with the class

A(a)

of (5.1) having a representation (5.7)

Theorem 5.1 allows the definition of the following invertible operator T:

a

A ~ A(a)

by the relation:

for

theory

f E A with

of solutions

147 00

fez) =

I

akz k

k=O we set (5.8)

As an example we use (5.9)

l-(s

2s2 222 ' 0 ~ s < 1 .

Izl )

In this case 2

= r

(5.10)

1i-k l-(sr) k 1+(sr)2

--"-'::"-""-::-2-' k ::: 0 .

l+k

Note that the limiting case Now let ate with

w

= Ta (f)

s

~

1

E A(a)

l:.L. 1+5 2

is (2.115). be given by (5.8).

For

0

<

t

<

1 we associ-

w the analytic functions 00

L

(5.11)

akdk(t)zk, zEIT

k=O We obviously have (5.12)

with the associated functions damental relation holds: (5.13)

for

f E A, 0 < t

<

1,

-ft'

we have

then the following fun-

148

This is the point where convolution theory becomes applicable to (5.1).

ft

Clearly. the determination of properties of this discussion.

(or

;)

is the crucial part in

The following result gives an important general information.

For a proof we refer to [52J. THEOREM 5.2:

Let w

= Ta (f),

feU)

(5.14)

C

w(U)

C

co(f(U))

This is immediate from (5.14) since

J.

Re z ~

Then

z E U.

Re w(z)

COROLLARY 5.1:

f E A.

co(f(U)),

is the half-plane

That equality cannot hold in Corollary 5.1 follows from Hopf's maximum

principle applied to the solution

- Re

w of

(5.1).

As a consequence of Corol-

lary 5.1 we deduce

f t E R. a

(5.15)

5.2.

< t ~

1 .

Univalent solutions In this section we study the classes

Sea)

of normalized univalent so-

lutions

in

A(a),

compare [53]. THEOREM 5.3:

a

$

0 theJr.e e~th If

w

curve)

0 < t < 1.

E Sea)

Thus

= Ta (f)

E Sea). then

~u~h that

w = TaCf)

w

f ES

= Ta(f)

PROOF:

In

ft

we see that

ftCz)

is univalent in

f E

f

S.

On the otheJr. ha.nd, 1..6

Sea). maps

au

U and since

onto a closed Jordan ft

is compact

149

convergent to univalent:

f

for

t

we conclude that

f

(which is non-constant) is

f E S.

Now assume that functions

1,

+

ft

T

a

for each

(f) E Sea)

f E S.

Then for

form a differentiable subordination chain for

sense of Pommerenke [44J.

a

at

f t (z)

Re --;:-:-:;--::--

zf~(z)

0 < t

f ES

the

1

in the

<::

Hence

a

* at

f

= Re

f

~

f t (z)

* zf~(Z)

> 0, 0

<:: t

<:: 1,

z E U ,

t

a

R, z

E U ,

or f * for arbitrary constant, is in

[iazf~

f E S.

For

+

~t ftJ ~ a

~

0

0

0 <::

<:: 1,

E

this shows that the second factor, up to a

Z (as defined in section 2.5) and thus in SO'

The limit

t

gives 00

(5.16)

1 '( ) [ iaz 2 + L\ dk'(l)Zk J E 0' S aE R\{O} . iatd l 1 (l-z) k=l Now it is well known that for any function 00

g(z)

=Z

L

t

bkz k E So

k=2

exactly one of the following statements holds: (5.17)

(5.18)

g(z)

= z/(l

- xz)2, Ixl

bk lim -

k-+oo k

If we apply this to (5.16) with varying

=0 a

=1

,

.

it is readily seen that (5.18) is

+

1

150

impossible in this case and we are left with the conditions (5.19)

Now let 1

(5.20)

q(r)

= exp(2

I ta(t 2)dt) r

and (5.21) An application of (5.3) yields for

(~k(r))(n)

(5.22)

r, n

= (Q(r))(n) +

fixed:

0(1), k ~

00

,

and (5.4) is transformed into " ~k

(5.23)

2k+l n'J' 2 0' + [ - r- + 2 ~ Q ~ k + -r ~ Q ~k =

o.

From (5.19), (5.22) we get (5.24 )

k~i(l)

This can be true only i f

= ~k(l) = Q'(l)

~Ji (1)

= 0

+ 0(1), k ~

00

and thus

~k(l)

= 0,

Q' (1)

=0

kEN,

(5.25)

Since Q" (1) above n E N,

Q(l)

= O.

= 1,

(5.25) yields

~" (1)

k

.

= 0,

kEN,

and from (5.22) we get

Upon differentiation of (5.23) we deduce

Q'" (1)

= O.

and since

~k' (1)

= 0,

A repetition of this process finally gives Q is analytic we obtain

Q

= 1.

kEN,

Q(n) (1)

and as

= 0,

This immediately leads to

IS 1

a - 0 which is the assertion. THEOREM 5.4: PROOF:

In

w

= TaCf)

E sea),

In the preceding proof we saw that

pact convergent to ft--
for

f

S.

~

0 < t < 1

th~n

ft

w(U)

= feU).

forms a subordination chain com-

Thus by Caratheodory's kernel theorem we deduce

and because of (5.12), weU) c feU)

follows.

(5.14) com-

pletes the proof. Now we are looking for sufficient conditions for univalence. consequence of Hopf's maximum principle and Theorem 2.40 that valent if all

ft

are convex univalent in

IT, 0

< t < 1.

the assumption that the tangent vector of the curves and

zft(z)

~

O. z E U.

. a d4>

g::

zf'

is uni-

This is equivalent to

ftCei~)

turns monotonically

f t ( e i4> ) :: Lag ( ) ( te i4> )

,~

E R ,

we easily obtain

THEOREH 5.5: i)

w:: TAf

Using the obvious relation

-~

with

It is a

Le;t

1.z T a (zf')

f E Al

.6uc.h tha;t

b~

~oJt

z EU

f. 0,

T (z2 f ,,) ii)

Th~n

T (f) a

Re

a a

T (zf'

E Sea)

)

>-l.

and map.6

U onto

th~ c.onv~x do~n

We denote the class of these functions by implies

o<

ft

convex univalent in

K(a).

feU). Since

w=

T

a

(f) f K(a)

IT, it is clear that for g E K we have for

t < 1, using Theorem 2.1,

convex univalent in U.

Thus

TaCf * g) E K(a)

by the above discussion.

We

152

have thus proved the following extension of the Palya-Schoenberg conjecture:

Folt w = TaCf)

THEOREH 5.6: 111 pa.J!.tic..ul.aJt.,'(6

wl

= TaCf),

E K(a), g E K,

we. ha.ve. TaCf * g)

w2 = TaCg) E K(a), the.VI.

w = w10w2

E K(a).

= TaCf *

g) E

K(a). In the following theorem we determine a specific class of functions in KCa).

a - O.

This result is well known for 00

THEOREM 5.7:

Let fez) =

r akz k E Al k= 1

2 r k Iakl k=2 00

The.VI. PROOF:

T (fJ

a

be ~u~h that

~ 1 .

E KCa).

Without loss of generality we may assume

fez)

f

z.

A look into the

second order differential equations satisfied by the functions

kEN,

dk(r)/dl(r),

shows that

Hence, for

0 <

Izl

< I

we get

This estimate contains both conditions of Theorem 5.5.

The assertion follows.

153

REMARK:

It is well known that in the analytic case the condition

00

cS

I

(5.27)

k lakl

<

1

k=2

with

=1

cS

implies univalence of the corresponding function.

ample (5.9) with

s

can be used to show that there is no exponent

-+ 1

(5.27 ) such that for arbitrary corresponding functions a (zz)

we have

T

a

Our standard ex-

(f) E S (a) :

f

cS <

2

in

and each admissible

Theorem 5.7 is best possible in this sense.

In view of Theorems 5.5, 5.6 there is a case of particular interest: a (zz)

assume

TA(Z/(l - z))

has the property that

convex univalent for

0 < t < 1).

Then

T

a

f

K(a)

(f)

f

(which is equivalent

K(a)

f E K.

for any

believe that t~is property holds for a large variety of functions

We

a(zz).

To conclude this section we mention a result for the limiting case where aCt)

has a singularity at

t

= 1.

One can show that if this singularity is a

simple pole the representation theory in section 1 and the previous results in this section remain valid.

For the proof of the following theorem which is an

application of Fuchs' theory for second order differential equations and Theorem

2.42 we refer to [53J. THEOREM 5.B:

and tet the

t

1

=

dom~n

w(U)

be a ~

Let aCt) ~hmpte

pote

be analytiQ

on

a

~n

t

E u,

a

w~h ~~~duum

~eal-valued nO~ ~eal

~

1/4.

Then

~6

t,

w E Sea)

Qonvex.

A special case is our standard example with

s

= 1.

The condition of

Theorem 5.8 is fulfilled and the statement is contained already in (2.119). A more general case has been studied in [59J, [68J: equation

(P

n

= Legendre

polynomials)

the solutions of the

154

are given by (6 = Laplace operator) w

where

f

= l n (f)

is harmonic in

U.

As in section 1 we consider just those solutions

which are generated by analytic functions

f.

This equation fulfills the as-

sumptions of Theorem 5.8 and - with a considerable amount of calculation - one can show that the corresponding functions n

=1

ft' 0

<

t < 1.

we have again our standard example with n-IEOREH 5.9:

w = In(f)

s

Lb wuvale.VL:t in.

are convex univalent.

For

= 1. U i6 a.n.d on.ly

in

f E K.

It is then a consequence of Theorem 5.4 and the Riemann mapping theorem that a domain w

= l n (f)

5.3.

G is convex if and only if there exists a univalent solution

of (5.28) with

w(U)

= G.

Extension of Schwarz' Lemma We discuss the following problem:

(5.29)

w

=l

a

(£)

assume

=

fulfills (5.30)

Iw(z)1 ~ 1. z E U .

Is it true that (5.31)

[w(z)1 ~ d m(lzl)

155

holds for

0

<::

Iz)

<::

I?

THEOREM 5.10: (5.29), (5.30)

~6

We give a necessary and sufficient condition for (5.31), (5.31) holdo 60n 6~xed

and only

Izl

= p and any

w 4~6y~ng

~6

00

(5.32)

d (p) + 2 l d k(P) cos k~ ~ 0, ~ E m k=l m+

PROOF:

In view of Theorem 5.2, (5.30) holds if and only if

thus (5.31) - for

Izl

=P

R. If(z)1 ~ Izml

and But

- holds if and only if

An app lication of Wilf's theorem (see the remark following Corollary 4.1) now shows that (5.31) holds if and only if d m+k (p)

00

L

(5.32)

k=O

dm(p)

k

z

ER

which is equivalent to (5.32). In case of our standard example (for a proof see [59, p.216J).

For

p

a (zz), (5.32) is valid for s fixed and large

0 <:: p <::

1

m we can show using

(5.5) that d m+k _ l (p) d (p) m

and thus (5.33) by Theorem 2.10.

<

d m+k +1 (p) -d'":"'m-(~p""")-

Since (5.33) holds for

m = 0, 0

<::

p

<::

1

(see

(5.15)), it is not unlikely that the extension (5.31) of the classical Schwarz lemma holds in general.

This is an open problem.

156

Additional remarks

5.4.

Theorem 5.2 has many other applications, we mention two of them.

A ~ot~o~ w = TaCf), weD) = 1,

THEOREM 5.11: -t~

u -t6

bildy

a~d o~ty

me.a~uJr..e.

]1

h~;th~ pnopeJtty.

-til

f

o~

au

(5.34 )

~uc.h

w(z)

h~ po~~ve n~

I~ .th~ c.~e.

.the.ne.

e.x.~;to

pant

a pnoba-

.that

=

(

1r;1=1 The proof is immediate from Theorem 5.2 and the classical Herglotz representation.

(5.34) generalizes this representation formula.

Another classical result which extends to let

p

be a polynomial of degree

is due to Lax [32J:

a

n nonvanishing in

U.

Then

IIp'lI ~ ~ IIpll .

(5.35) THEOREM 5.12: no nvavU.o fUng -tn

U)

Let p be. a

poty~om-i.at

06 de.gne.e.

~

n.

I6

w

= Ta(p)

~

we. have

a

II~ were

(5.36) PROOF:

~

T (f)

By Theorem 5.2 we get:

i~

p'" 0

) II s ~ IIwll .

in

U, IIwll = lip II·

Furthermore,

= re i~ and again from Theorem 5.2,

d

IId~

were

i~

)11

= Ilzp'll·

Using (5.35) we obtain (5.36).

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SUBJECT INDEX

Bernstein inequality. 125. 127. 130 Cesaro means. 142 Closed convex hull. 16 Coefficient body. 77. 98 Complete. 15 Conjecture Bieberbach. 66. 81. 130. 134 Mandelbrojt-Schiffer, 13, 98, 132 P6lya-Schoenberg. 13, SO, 84, 152 Robertson, 136 Robinson. 98 Rogosinski. 130. 134 Sheil-Small. 130, 134, 136 Wilf. 84 Continued fractions. 56 Convolution, 11 de la Vallee-Poussin. 84 Dual hull. lOS Dual set. 13 Duality principle. 13. 15 Extreme points. 17 Functions Bazilevic, 46. 64. 78 Boundary rotation at most kIT, 47, 64 Close-to-convex, 46, 63, 68, 78, 80, 83. 116. 131. 143 Close-to-convex of order S. 47 Convex. 49. 153 Convex of order a. 61 Convexity generating, 85, 91 Hynergeometric. 52. 60 Kernel. 106 Linear accessible. 70, 80 Prestarlike, 47. 48. 67, 95, 100, 116 Rational, 103

Spiral-like. 79. 94. 99, 131. 135. 143 Starlike. 94. 97, 143 Starlike of order a. 28. 4S Typically real. 12, 97 Hadamard product. 11 Herglotz formula. 18. 23, 37 Jacobi polynomials. 59. 139 Kaplan classes, 33 Libera transform. 68 Linear invariant, 81 Neighborhood. 9S Operators Bound preserving. 121 Continuous linear, 106 Hull preserving. 123 Invertible. 107 Origin preserving. 106 Range preserving, 121. 142 Self-inverse polynomials. 129. 133 Shadow of identity. 143 Subordination, 84, 129 Subordination chain. 85. 88. 149 Theorem of Caratheodory, 91, 151 Enestrom-Kakeya. 136 Hopf. 89. 148. 151 Hurwitz. 26. 108 Jack, 28 Julia-Wolff. 28 Riemann. IS4 Szego. 12. 26. 36. 104. 121. 140 Test set, 19

LIST OF SYMBOLS AND ABBREVIATIONS

A , 12

H , 22

Sa ,46

AO ' 13

Ha , 23

Sea) , 148

Al ' 45

K , 84

T , 82

As '

K ,61 a

T

A(a) , 146

KCa) , 151

TCa, S) , 28

B(a,S) , 46

KCa,S) ,

T

Bn ,

A

143

121

Co. ' 46

,71

a

32

15

,146

a

U , 12

-

M , 48

W

co , 16

147

,

Z

c.g. , 91

r2, r20 '

c.s.c. , 88

P

n

106

£

,26

75

*

g , 11

V*, V** , 13

~

Pn

26

--;- , 15

V(a,S) , 30

Pn ,

129

~>

, 20

Vy

p~a.s) , S9 Q (operator) , 111

*z

,21

f(o) , 17

q (polynomial)

~

£(-1) , 41, 55

R

II '

RCa,S) , 118

Vy ,

30

48

'"

it '

52 147

a

,48

S , 13

J

125

, 35

-< ,

62

v , 111

LES PRESSES DE L'UNIVERSITE DE MONTREAL C.P. 6128, Montreal, succ. «An, Que., Canada H3C 3J7

EXTRAIT DU CATALOGUE

MatMmatlques

COLLECTION « SEMlNAlRE DE MATHEMA TlQUES SUPERlEURES) 1. 2. 3. 4. 5. 6.

1. 8. 9. 10. 11.

12. 13. 14.

15. 16. 11. 18.

19. 20. 21. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 31. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55.

Problemes aux Iimites dans les equations aux derivees partielles. Jacques L. LIONS Theorie des algebres de Banach et des algebres localement con vexes. Lucien W AELBROECK Introduction a I'algebre homologique. Jean-Marie MARANDA Series de Fourier aleatoires. Jean-Pierre KAHANE Quelques aspects de la tbeorie des entiers algebriques. Charles PISOT Theorie des modeIes en logique mathematique. Aubert DAIGNEAULT Promenades aleatoires et mouvement brownien. Anatole JOFFE Fondements de la geometrie algebrique moderne. Jean DIEUDONNE Theorie des valuations. Paulo RIBENBOIM Categories non abeliennes. Peter HILTON, Tudor GANEA, Heinrich KLEISLI, Jean-Marie MARANDA, Howard OSBORN Homotopie et cobomologie. Beno ECKMANN Integration dans les groupes topologiques. Geoffrey FOX Unicite et convexite dans les probJemes differentiels. Shmuel AGMON Axiomatique des fonctions harmoniques. Marcel BRELOT Probh~mes non lineaires. Felix E. BROWDER Equations elliptiques du second ordre a coefficients discontinus. Guido STAMP ACCHIA Problemes aux Iimites non homogenes. Jose BARROS-NETO Equations differentieiles abstraites. Samuel ZAlDMAN Equations aux derivees partielles. Robert CARROL, George F.D. DUFF, Jbran FRIBERG, Jules GOBERT, Pierre GRISVARD, Jindrich NECAS et Robert SEELEY L' Aigebre logique et ses rapports avec la thoorie des relations. Roland FRA1SSE Logical Systems Containing Only a Finite Number of Symbols. Leon HENKIN Representabilite et definissabilite dans les algebres transformationneUes et dans les algebres polyadiques. Leon LEBLANC Modeles transitjfs de la tbiorie des ensembles de Zermelo-Fraenkel. Andrzej MOSTOWSKI Theorie de l'approximation des fonctions d'une variable complexe. Wolfgang H.I. FUCHS Les Fonctions multivalentes. Walter K. HAYMAN Fonctionnelles analytiques et fonctions entieres (n variables), Pierre LELONG Applications of Functional Analysis to Extremal Problems for Polynomials. Qazi Ibadur RAHMAN Topics In Complex Manifolds. Hugo ROSSI Theorie de I'inference statistique robuste. Peter J. HUBER Aspects probabilistes de la theorie du potentiel. Mark KAC Theorie asymptotique de la decision statistique. Lucien M. LECAM Processus aleatoires gaussiens. Jacques NEVEU Nonparametric Estimation. Constance van EEDEN K-Theorie. Max KAROUBI Differential Complexes. Joseph J. KOHN Varietes hilbertiennes : aspects geometriques. Nicolaas H. KUIPER Deformations of Compact Complex Manifold. Masatake KURANISHI Grauert's Theorem on Direct Images of Coherent Sbeaves. Raghavan NARASIMHAN Systems of Linear Partial Differential Equations and Deformation of Pseudo group Structures. A. KUMPERA et D.C. SPENCER Analyse globale. P. LIBERMANN, K.D. ELWORTHY, N. MOULlS, K.K. MUKHERJEA, N. PRAKASH, G. LUSZTIC et W. SHIH Algebraic Space Curves. Shreeram S. ABHYANKAR Tbooremes de representabilite pour les espaces algebriques. Michael ARTIN Groupes de Barsotti-Tate et cristaux de Dieudonne. Alexandre GROTHENDlECK On Flat Extensions of a Ring. Masayoshi NAGATA Introduction ala theorie des sites et son application a la construction des preschemas quotients. Masayoshi MIYANISHI Methodes logiques en geometrie diophantienne. Shuichi TAKAHASHI Index Theorems of Atiyah - Bott - Patodi and Curvature Invariants. Ravindra S. KULKARNI Numerical Methods in Statistical Hydrodynamics. Alexandre CHORIN Introduction a la theorie des hypergraphes. Claude BERGE Automatb, a Language for Mathematics. Nicolaas G. DE BRUUN Logique des topos (Introduction a la theorie des topos elementaires). Dana SCHLOMIUK La Serie genera trice exponentielle dans les problemes d'enumeration. Dominique FOATA Feuilletages: resultats anciens et nouveaux (Painleve, Hector et Martinet). Georges H. REEB

56. Finite Embedding Theorems for Partial Designs and Algebras. Charles C. LINDNER et Trevor EVANS 57. Minimal Varieties in Real and Complex Geometry. H. Blaine LAWSON, Jr 58. La Theorie des points fixes et ses applications Ii I'analyse. Kazimierz GEBA, Karol BORSUK, Andrzej JANKOWSKI et Edward ZHENDER 59. Numerical Analysis of the Finite Element Method. Philippe G. CIARLET 60. Methodes numeriques en mathematiques appliquees. J.F. Giles AUCHMUTY, Michel CROUZEIX, Pierre JAMET, Colette LEBAUD. Pierre LESAINT et Bertrand MERCIER 6l. Analyse numerique matricielle. Paul ARMINJON 62. Problemes d'optimisation en calcul des probabilites. Serge DUBUC 63. Chaines de Markov sur les permutations. Gerard LETAC 64. Geometrie differentieJie stochastique. Paul MALLIA VIN 65. Numerical Methods for Solving Time-Dependent Problems for Partial Differential Equations. Hemz-Otto KREISS 66. Difference Sets in Elementary Abelian Groups. Paul CAMION 67. Groups in Physics: Collective Model of the Nucleus; Canonical Transformation in Quantum Mechanics. Marcos MOSHINSKY 68. Points fixes pour les applications compactes : espaces de Lefschetz et la theorie de I'indice. Andrzej GRANAS 69. Set Theoretic Methods in Homological Algebra and Abelian Groups. Paul EKLOF 70. Abelian p-Groups and Mixed Groups. Laszlo FUCHS 71. Integral Representations and Structure of Finite Group Rings. Klaus W. ROGGENKAMP 72. Homological Invariants of Modules over Commutative Rings. Paul ROBERTS 73. Representations of Valued Graphs. VlastimiJ DLAB 74. Groupes abeliens sans torsion. Khalid BEN ABDALLAH 75. Lie Groups, Lie Algebras and Representation Theory. Hans ZASSENHAUS 76. Birational Geometry for Open Varieties. Shigeru lIT AKA 77. Lectures on Hilbert Modular Surfaces. Friedrich HIRZEBRUCH, Gerard van der GEER 78. Complex Geometry in Mathematical Physics. R.O. WELLS, Jr. 79. Lectures on Approximation and Value Distribution. Tord GANELIUS, Walter K. HAYMAN, Donald J. NEWMAN 80. Sur la topologie des surfaces complexes compactes. Srinivasacharyulu KILAMBI, Gottfried BARTHEL, Ludger KAUP 8l. Topics in Polynomial and Rational Interpolation and Approximation. Richard S. VARGA 82. Approximation uniforme qualitative sur des ensembles non bornes. Paul M. GAUTHIER, Walter HENGARTNER 83. Convolutions in Geometric Function Theory. Stephan Ruscheweyh. 84. Characteristic Properties of Quasidisks. Frederick W. Gehring.

COLLECTION «CHAIRE A ISENSTADT» Physical Aspects of Lie Group Theory. Robert HERMANN Quelques problemes mathematiques en physique statistique. Mark KAC La Transformation de Weyl et la fonction de Wigner : une forme alternative de la mecanique quantique. Sybren DE GROOT Sur quelques questions d'analyse, de mecanique et de controle optimal. Jacques Louis LIONS Mariages stables et leurs relations avec d'autres problemes combinatoires. Donald E. KNUTH Symetries, jauges et variHes de groupe. Yuval NE'EMAN La Theorie des sous-gradients et ses applications Ii I'optimisation. Fonctions con vexes et non convexes. R. Tyrrel ROCKAFELLAR

Acheve d'imprimer en aout 1982 sur les presses des Ateliers GraphlQues Marc Veilleux Inc. Cap St-Ignace, Que.