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Applied Mathematics Letters

Applied Mathematics Letters 15 (2002) 63-69

PERGAMON

Close-To-Convexity, of Certain

www.elsevier.com/locate/aml

Starlikeness, and Convexity Analytic Functions s. OWA

Department of Mathematics, Kinki University Higashi-Osaka, Osaka 577-8502, Japan owaQmath.kindai.ac.jp M. NUNOKAWA Department of Mathematics, Gunma University Aramaki (Maebashi), Gunma 371-8510, Japan nunokawaQedu. gunma-u. ac jp H. SAITOH Department of Mathematics, Gunma National College of Technology Toriba (Maebashi), Gunma 371-8530, Japan saitohQnat.gunma-ct.ac.jp H. M. SRIVASTAVA Department of Mathematics and Statistics, University of Victoria Victoria, British Columbia V8W 3P4, Canada [email protected] (Received

December

2000; accepted

January

2001)

Abstract-The main object of the present paper is to derive several sufficient conditions for close-to-convexity, starlikeness, and convexity of certain (normalized) analytic functions. Relevant connections of some of the results obtained in this paper with those in earlier works are also provided. @ 2001 Elsevier Science Ltd. All rights reserved. Keywords-Analytic Subordination principle,

functions, Starlike functions, Univalent functions.

1. INTRODUCTION Let

A

denot,e

the

class

of functions

f

functions,

Convex functions,

AND DEFINITIONS

normalized

f(z)

Close-to-convex

= z+

hy

2 a,, ZR,

(1.1)

7t=2

which

are

analytic

in the

open

unit

disk

U := ‘The present

investigation

tinder Grant-in-Aid neering

Research

at the spring

was supported,

for General Council

meeting

Scientific

of Canada

{z

in part,

IzI
a, (Z E 24; 0 2 cr < 1)

,

(1.3)

and

where.

> cy, (Z E u; 0 2 a: < 1; g E Kc) )

(1.4)

for convenience,

s* := s*(o),

K := K(O),

Next, with a view to recalling the principle the functions f and g be analytic in U. Then there exists a function

h, analytic

and

c := C(0).

(1.5)

of subordination between analytic functions, let we say that the function f is subordinate to g if

in U, with

h(0) = 0

Ih(

and

< 1,

(z E U) 1

(1.6)

such that f(z) We denote

this subordination

= 9 (h(z)) >

(1.7)

by f(z)

In particular,

(z E U) .

if the function

g is univalent

+ 9(z).

(1.8)

in U, the subordination

(1.8) is equivalent

to (cf. [l,

11. 1901) f(O) = 9(O)

and

f(U)

c s(U).

(1.9)

results involving univalence and starRecently, Singh and Singh [4] p roved several interesting likeness of functions f E A. In our attempt here to generalize these results of Singh and Singh [4], we are led naturally to several sufficient conditions for close-to-convexity, starlikeness, and convexity of functions f E A. The following lemma (popularly known as Jack’s lemma) will be required in our present investigation. LEMMA j~(z)I

1. (See [5,6].) Let the (nonconstant)

where c is a real number

2.

function W(Z) be analytic in U with w(0) = 0. If value on the circle Iz/ = T < 1 at a point zo E U, then

attains its maximum

and c 2 1.

SUFFICIENT

Our first result functions f E A. THEOREM

CONDITIONS (Theorem

1. Let the function

1 below)

FOR provides

CLOSE-TO-CONVEXITY

a sufficient

condition

for close-to-convexity

of

f E A satisfy the inequality

(2.1)

Certain Analytic

65

Functions

Then (z E u: 0 2 cy < 1) )

(2.2)

or equivalently,

(F >,

fEC PROOF.

We begin

by defining

f’(z) = Then,

clearly,

a function

1+

now that there exists a point

by applying

0 5 Q < 1).

(2.4)

Lemma

(i E U).

- 1 + w(z)’

(2.5)

zo E U such that

and

when

iw(z)i < 1,

IzI < lzoi.

(2.6)

1, we have (c 2 1; ‘w (~0) = e”; Q E IR) .

20 w’ (20) = cw (x0), Thus,

z E u;

zw’( 2)

QZW’(Z)

1 +ckw(z)

lw (zo)l = 1 Then,

(w(z) # -1;



in U with w(0) = 0. We also find from (2.4) that

w is analytic

1 + -= z.f”b) f’(z)

Suppose

(2.3)

w by

1+ &W(Z) 1+ w(z)

(0 5 cy < 1).

(2.7)

we find from (2.5) and (2.7) that ITi

( 1

Zof"

+

f'

(zo) (zo)

)=lf%(g+?o) ca(a+cosQ)

=1+

-- c 2

1+a2+2ckcos6J 1+3a (20 E u; 0 5 f2 < 1) , = 2(1 + ck) ’


x+1 -

1, we shall obtain

2

- 1) (2 - X)

+ 2 (1 + x* - 2x cos f3) ’

(zo E u; 1 < x < 3) !

which yields the inequality ZOfll(ZO)

>

5x - 1 2(x+1)’

fl(iO)

=

X+1

CJ% 1 + (

Since (3.6) obviously

)

contradicts

(

(20EU;1