A CRITERION FOR P-VALENTLY STARLIKE FUNCTIONS

Internat. J. Math. & Math. Sci.

VOL. 17 NO. (1994) 205-207

205

A CRITERION FOR P-VALENTLY STARLIKE FUNCTIONS

SHIGEYOSHIOWA MAMORU NUNOKAWA SEIICHIFUKUI

DepartmentofMathematics KinkiUniversity Higashi-Osaka, Osaka 577,Japan

Departmentof Mathematics GunmaUniversity

Aramaki, Maebashi,Gunma371,Japan

Departmentof Mathematics WakayamaUniversity Wakayama 640,Japan

(Received

December 22, 1992 andin revisedformApril19,

1993)

ABSTRACT.

The object of the present paper is to prove a criterion for p-valently starlike functions intheopen unit disk.

KEY WORDS AND PHRASES.

Analytic,open unitdisk, p-valentlystarlike.

1991 AMS

SUBJECT CLASSIFICATION CODE.

Primary,30C45.

1. INTRODUCTION.

Let

A(p) be the class of functions of theform

f(z) zp

+ E ^{an zn}

^{(pE N} {1,2,3,--. }),

(1.1)

n=p+l

whichareanalyticinthe openunitdiskU {z:

Izl

^<1}.

A

functionf(z)belongingto A(p) is said tobep-valentlystarlike in U if it satisfies

R

f zf’(z)

e

^{f(z) >0 (z}

.

^U).

^(1.2)

We

denotebyS(l) thesubclassofA(I) consistingof functions f(z) which^are p-valentlystarlike in t

(d. [ll).

Recently,Nunokawa

[4]

has shown that

THEOREM A.

If f(z) A(p) satisfiesf(z) 0(0 < z <1) and

Re

"zf’(z) I

^<

f(z)

;

then f(z)

_.

(z.U),

(1.3)

1+

In

the present paper, ^wederive anewcriterionfor the class S(p) involving the above result by Nunokawa

[4].

2.

A NEW CRITERION.

Toderiveourmain result,wehavetorecall here thefollowinglemma due to Jack

[2] (also,

due to Millerand

Mocanu [3]).

LEMMA. Let

w(z) be analytic inU with w(0)=^0. If w(z)l attains its maximumvalueon thecircle

zl

^{r< at apointz}0, thenwe canwrite

zow’(zo) W(Zo), (2.1)

wherekis areal number andk>^1.

206 S. OWA, M. NUNOKAWA AND S. FUKUI

then_f(z) S(p)and

Now,

weprove

THEOREM. If_f(z) A(p)satisfies f(z)#0(0 < < 1) and

’" t

^{l+ s’<=)}

)-(l

^>o

-7-

^{p <p (ev).}

PROOF.

Definethefunctionw(z)by

f(z) ^p(1+w(z)).

(z U),

(2.2)

(2.3)

(2.4)

Then w(z)is analyticin V and_w(0)=0. Itfollows from

(2.4)

^that

+-if(z)’

^p(1

⁺

^w(z))

⁺ + w(z’

sothat,

zS’(z)f(z) (1 ⁺ zff’(z)if(z) _]’= ⁺

V(1

^zw’(z) +

w(,)

)"

Suppose

that thereexistsapoint ₀ Vsuch that

(2.5)

I(z)

I,(z0) Then,

applying

Lemma,

^{we canwrite}

((z

o)

^{# ).}

ZoW’(Zo) kw(zo)

^(k

>=

¹⁾

W(Zo) eiO(o

#r). Thus^wehave

zoS"(zo)’

f(z) + +

eio)---

zoft(zo)

^ft(z

O) ]

p(1

+

=1+2p(1

+

cosO)

Note

that thecondition

(2.2)

^implies

s.)(

zf’(z) ¹⁺ St(z) #a (zcU),

(2.S)

where a>

+

1/4v. Therefore,

(2.7)

contradicts our condition

(2.2).

Consequently, weconclude that

IzS’(z)

that is, that f(z)S(v).

Lettingv ⁱⁿTheorem,wehave

COROLLARY.

If f(z)

_

^{A(1)satisfiesf(z)}#0(0 < [zl < l)and

(2.9)

thenf(z) S(1) and

S ^f(z),

t:S’O)(’ S’I:)7

^>0

:f’(*)

s-7-(Ty- ^{< (ev).}

(z

(2.10)

(2.11)

CRITERION FOR P-VALENTLY STARLIKE FUNCTIONS 207

ACKNOWLEDGEMENT.

The research of the first author was supported in part by

Japanese

Ministry of Education, Science and Culture under Grant-in-Aid for General Scientific Research

(No. 04640204).

REFERENCES

1.

GOODMAN, A.W.,

On the Schwarz-Christoffel transformation and p-valent ^functions, Trans. Amer. Math. Soc.68,

(1950),

204-223.

JACK,

^I.S., Functions starlike and convexoforder

,

^J. London Math. Soc. 3 (1971),469- 474.

MILLER, S.S.

z MOCANU, P.T.,

Second order differential inequalities in the complex plane, J. Math. Anal. Appl. 65

(1978),

289-305.

NUNOKAWA, M.,

Certain classofstarlikefunctions,to appear.