Correction : “On a class of starlike functions”

(1)

C ^OMPOSITIO M ^ATHEMATICA

R ^AM S ^INGH

Correction : “On a class of starlike functions”

Compositio Mathematica, tome 21, n

^o

2 (1969), p. 230-231

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(2)

230

Correction

On a class of starlike functions

by Ram

Singh

Compositio Mathematica 19 (1968), ^78-82

COMPOSITIO MATHEMATICA, ^Vol.21, ^Fasc.2, 1969, pag. 230-231 Wolters-Noordhoff Publishing

Printed in the Netherlands

Introduction

In a recent paper

(Comp. ^Math.,

¹⁹

(1968), 78-82)

^{with the}

above title we

investigated

^the

^{class S}

^of^functions

that are

analytic

and starlike in

Izi

¹^and

satisfy

the condition

|{zf’(z)/f(z)}-1| ~

¹ⁱⁿ

|z|

^1.^Theorem⁴^ofthat paper,

giving

the radius of

convexity

of the class

S,

^contains^{an error}^and^as

such the result obtained is not

sharp.

^Below^we

give

^the^correct

proof

of that theorem.

THEOREM. Each

function f(z) ^{~ S}

maps

onto a convex domain.

PROOF. If

f(z) ~ S,

^{we can}^write

when

y(z)

^is

analytic in Izl

¹ and satisfies

|03C8(z)| ~

¹^and

11’(0)

⁼^0.

Logarithmic

differentiation of

(1) yields:

Making

^useof the fact that

|03C8(z)| ~ Izi

^and

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231

we find that

Therefore, f(z)

^is^convex^if

or

To show that

(3-y5)j2

^{is the}^exact^{radius of}

convexity,

^we

consider the function

A little calculation shows that for this function

vanishes when z ⁼

(5-3)/2.

(Oblatum 30-1-69) Punjabi University, Patiala, India

Correction : “On a class of starlike functions”

C OMPOSITIO M ATHEMATICA

R AM S INGH