(zf)' FOR / UNIVALENT - American Mathematical Society

PROCEEDINGS OF AMERICAN MATHEMATICAL SOCIETY Volume 53, Number 2, December

1975

ON THE RADIUS OF STARLIKENESS

OF (zf)' FOR / UNIVALENT ROGER W. BARNARD ABSTRACT. functions.

Let

For

S be the

a given

standard

function

|z| < 1, let r(f) be the radius

class

/, f'z)

of normalized

univalent

2 +... , regular for of /. In 1947, R. M.

— z + a2z

of starlikeness

Robinson considered the combination g ,(z) = (z/)'/2 for /f S, He found a lower bound of .38 for r(g,) for all / £ S. He noted that the standard Koebe function k, k(z) = z(l - z) , has its Kg.) equal to Vi. A question that has been asked since Robinson's paper is whether '/2 is the minimum r(g.) for all / in S. It is shown here that this is not the case by giving examples of functions / whose r(g ,) is less

than V2.

Introduction. regular

in

Let

U, let

of starlikeness

normalized

U = [z: \z\ < 1}.

rAf)

be the radius

of f (see

functions

Hayman

subclass

[10] considered

of S of starlike

the combination

bound of .38 for r.(gi) noted

hence,

not starlike

that

Let

showed

that

spectively. These

case

have

results.

ness

and,

for all

f £ S.

Presented

of

g,(|.z|

function

Let

S

R. M. Robinson

He found

a lower

author

We give

Vi-

subclass

from the Koebe

hence,

f £ S.

function

i.e.,

[3] for definition).

been

the Koebe

/i obtained

(zf) /2,

Goluzin

The Koebe

the sharpness the

< r where

it f £ S*, K, then,

results

In each

in

In 1947,

for / £ S.

for z, \z\ < Vi, while |z|

standard

If (see

+ •••,

the radius

S be the class

and univalent

Koebe

rAgk) = r^g,),

univalent

K be the

in

Let

functions.

= (zf) /2

the standard

in any disk

convex

g,(z)

r.(f)

for all f £ S. He noted that g,(z) 4 0 for all z, \z\

has gk(-Vi) = 0 and that tion and,

/, f(z) = z + a2z of / and

[4] for definitions).

/, f(z) = z +• • • , regular

be the standard

< Vi. He also

For a given

of univalence

been

[6], [9].

has

asked

radius

given

is whether

of starlike-

for all functions

g .=

is not the smallest / whose

received

r.(g.)

by the editors

r.(g.)

is less

October

7,

1974. AMS (MOS) subject Key words

classifications

and phrases.

Univalent

(1970). functions,

likeness.

Primary 30A32. starlike

function,

radius

of star-

Copyright © 1975. American Mathematical Society

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

385

386

R. W. BARNARD

than

A-

less

than

The

function

example We also

h for which

tion does even

first

0.445.

not give

circularly

= r0(gi,)

is a nonsymmetric give

rjg.)

is less

the minimum

symmetric,

is the minimum

function

as a second

than

rjg,)

functions radius

example

0.493-

tor all in S.

/.

having

a circularly

Therefore

the Koebe

/ in the class The

of univalence

question

of g,

rAg/A symmetric

func-

of symmetric,

or

as to whether

for all

A

/ in S is still

open. Example S such

1.

that

point

A