A NOTE ON CERTAIN SUBCLASS OF CLOSE-TO-CONVEX FUNCTIONS

Internat. J. Math. & Math. Sci.

VOL. ii NO. 3

(1988)

493-496

493

A NOTE ON CERTAIN SUBCLASS OF CLOSE-TO-CONVEX FUNCTIONS

SHIGEYOSHI OWA

Department of Mathematics

Kinki University Higashi-Osaka, Osaka 577, Japan

LIU

^LIQUAN

Department

of Mathematics Heilongjiang University

Harbin, China

WANCANG MA Department

of Mathematics

Northwest University Xian, China

(Received June 25, 1987 and in revised form July 31,

1987)

ABSTRACT. The object of the present paper is to show a result for functions belonging to class

P’(I-e,0)

which is a subclass of close-to-convex functions in the unit disk U.

KEY WORDS AND PHRASES. Close-to-Convex of order

,

Class

P’(e),

Class

P’(I-,0),

subordination.

1980 AMS SUBJECT CLASSIFICATION CODE. 30C45.

I. INTRODUCTION

Let A denote the class of functions of the form

f(z)

z

+ anZ

^(I.i)

nffi2

which are analytic in the unit disk U {z:

Izl

^{< i}. A function f(z) belonging to A}

is said to be in the class

P’(=) (according

to Goodman

[4])

if and only if it satisfies the condition

Re{f’(z)}

^>

= ^(1.2)

for some (0 ^< i) and for all z e U. Note that

P’(a)

the subclass of close-to- convex functions of order e in the unit disk U. Further, let

P’(I-e,0)

(according to Goodman

[4])

be the subclass of A consisting of all functions which satisfy the condition

If’(z) 11

^{< 1 e}

^(1.3)

for some (0

<

^< i) and for all z U.

494 S. OWA, L. LIQUAN, and W.

MA

It is clear that

P’(l-a,0)

is the subclass of

P’(a)

for 0 & I. Nunokawa, Fukui, Owe, Saitoh and Sekine [I] showed that functions in

P’(l-e,0)

are starlike in

Izl

^{< r}I, where r is the root of the equation

log

1

(21(3-=))

² (r-

(I-=)r212)

²

2

+ Sin-l((1-a)r) .

Also, Fukui, Owe, Ogawa and Nunokawa [2] proved that functions in

P’(a)

are starlike in

Izl

^{< r}2, where r

2 ^is the smallest root in

[0,i)

of the equation

-1

2(1-a)r

1

Sin

+

log

.

(2a-1)r

² r2

For the functions f(z) and

g(z)

belonging ^to A, we say that f(z) is subordinate to

g(z)

in D if there exists an analytic function

w(z)

in U such that

lw(z)l

^{< for}

z’

U and

f(z) g(w(z)).

We denote by

f(z)- g(z)

this subordination. In particular, if

g(z)

is univalent in U the subordination

f(z)- g(z)

is equivalent to

f(0) g(0)

and

f(U) g(U)

(cf.

[3]).

2. MAIN RESULT

In order to prove our main result, we have to recall here the following lemma due to Miller and Mocanu [5].

LEMMA. Let

q(z)

be an injective mapping of U onto Q, with

q(0)

i, such that

q(z)

is regular on U except for at most one pole on U. Let

p(z) + _pl

z

+ p2

^z2

^+...

be analytic in U with

p(z)

I. If there exists a point z

0 ^e U such that

p(z O)

^{e 8U}

and

p(Izl

^<

Iz01)

^{c Q, then}

z0p’(z 0) mw0q’(w0),

^{where m and w}₀ ^ei8

=-q-l(p(z0)).

Applying the above lemma, we derive

THEOREM. Let the function f(z) defined by

(I)

be in the class

P’(I-a,0).

Then

f(z) (I a)z

--

^I

^{+ (1.4)}

z 2

PROOF. Let

q(z)

i

+

(i

a)z/2

and

p(z) f(z)/z.

It is clear that the result holds true if

p(z)

^E for z e U.

Assume that

p(z)

for z e U and the subordination

p(z)-q(z)

does not hold in U. Then there exists a point z

0 e U such that

p(z 0)

^e

^8q(U)

^and

^p(Izl

^<

Iz01)

^c

^q(U).

Therefore, applying the lemma, we get

f’(z0) z0p’(z 0) ^{+ p(z} 0) nw0q’(w0) ^{+ q(w} 0)

m(l-)w

0

+

(l-)w 0 +I

2 2

(m

+ 1)

(l-a)w

+

0 (1.5)

CERTAIN CLASS SUBCLASS OF CLOSE-TO-CONVEX FUNCTIONS 95

where R

>-

and

[Wo[

^{I. Thus}

(R+ i) (i

-a)

[f’(z 0) 11 ^a

^{a, (1.6)}

2

which contradicts the hypothesis that f(z) e

P’(1-a,0).

So we Rust have

p(z)-q(z)

in U. This coRpletes the proof of Theorem.

Finally, we have

CORALLARY 1. Let the function

f(z)

defined by

(1.1)

be in the class

P’(1-a,0),

Then

Re

ei

^f(z) > 0,

Z

where

181

^/2

Sin-l(l )/2.

CORALLARY 2. Let the function

f(z)

defined by

(I.I)

be in the class

P’(l-a,0).

Then

f(z)

Re >0.

Z

ACKNOWLEDGEMENTS. The authors would like to thank the referee of the paper for his thoughtful encouragement and numerous helpful advlces.

REFERENCES

i. NUNOKAWA, M., FUKUI, S., OWA, S., SAITOH, H. and SEKINE, T., On the Starlike Boundary of Univalent Functions, to appear.

2. FUKUI, S., OWA, S., OGAWA, S. and

NUNOKAWA,

M.,

A Note

on a Class of Analytic Functions Satisfying

Re{f’(z)}

^>

,

Bull._

Fac.

^Edu. Wakayama Univ. Nat. SGi.

36

(1987)

13-17.

3. DUREN, P.L., Univalent Functions, Grudleheren der Mathematischen Wissenschaften 9, Springer-Verlag, New York, Berlin, Heidelberg and Tokyo, 1983.

4. GOODMAN, A.W.,

U.n.ival.ent

^Functions. Vol.

!L

Mariner Publ. CoRp. Inc., 1983.

5. MILLER, S.S. and MOCANU, P.T., Second Order Differential Inequalitiesln the Complex Plane, J.

Math.

LAnal,_AppI.. 65

(1978)

289-305.