LEMMA +- LEMMA

Internat. J. Math. & Math. Sci.

VOL. 19 NO. 3 (1996) 613-614 613

A NEW CRITERION

FOR

STARLIKE FUNCTIONS

LING YI DepartmentofMathematics llarbin InstituteofTechnology

IIarbin, P ft. China SHUSEN DING DepartmentofMathematics The FloridaStateUniversity Tallahassee, Florida, 32306-3027

(Received November 22, 1994 and in revised form January 19, 1995)

ABSTRACT. Inthis paperweshall getanew criterionfor starlikeness, andthehypothesis of this criterion ismuch weaker than those in

[1]

^and

[2].

KEY

WORDS AND PHRASES. Starlikefunctions, univalentfunctions, analytic^functions.

1991 AMS SUBJECT CLASSIFICATION CODES.

30C45,

30C55.

1. INTRODUCTION AND PRELIMINARIES.

LetAbe the class of functions f(z), which ^areanalyticin theunitdisc D {z

Izl

^{< 1}, with} f(0) f’(0) ^0. Let Sbe the set of starlikefunctions,S f(z)e.A, Re(zf’(z)/f(z)) >^{O, z}E

D}.

R. Singh and S. Singh in

[1]

proved that if f(z)E A and Re[f’(z)+ zf"(z)]>O, za_.D, then f(z)

s.

Recently,R. SinghandS. Singhin

[2]

provedthat if f(z) A andRe[f’(z)

+

zf"(z)] >

-,

^{z D,}

then f(z) S.

In

this paper ^we shall show that theasscrtion ofR.. Singh and S. Singh holds under a much weaker hypothesis.

LEMMA

1.

Suppose

that thefunction

:

^{C D} C satisfiesthecondition Re(iz, ^y; z)<6 for all realz,y_<

+-

^{and all zeD. Ifp(z)}

⁺

^{pz+...isanalyticin Dand}

Re(p(z), zp’(z); z) >6, for^zeD then Re(p(z))>0 in D.

A

general formof this lemmacanbe found in

[3].

^In

[4]

^theauthors gotthefollowing^result.

LEMMA

2.

Let

a>0, #< 1. If thefunctionpis analyticin D, withp(0) and Re[p(z)

+

ozp’(z)] >fl, zE D

then _Re(p(z))> (2#- 1)+ 2(1- #)F(1,

, +

^1" -1), z D, where F(a, b, c; z) isahypergeometric function. Thisresultis sharp.

By

takinga inlemma 2, weobtain

LEMMA

3. Let_{# <} 1. Ifthe functionpisanalyticin D, with p(0) and

ne

[p(z)+zp’(z)] >#, zeD

then Re(p(z)) >(2#-1)+ 2(1-#)/n2, ^z D,and the result issharp.

614 L. YI AND S. DING

2. MAIN RESULT

THEOREM. If I(z) A and Re[/’(z)

+

zf’Cz)] >

4(1 ⁱⁿ2)

+

^{2 -0.263, z D (1)}

then f(z) e^S.

PROOF.

By

usinglemma 3, from

(1)

^wehave Re(f’(z))>I- 3(1-In2)

>0,

2(I I2)

+

^{z.D. (:)}

From (2)

^andlemma 3, wehave Ref(z)

>-2

+

^{3 0.526,} z 6D.

z 2(1 In2)

+

(3)

Now,

^welet p(z) zf’(z)/l(z) and ,X(z)= f(z)/z, thenp(z) is analyticin Dandp(0)=1, Re{,(:)} >

a

A

simple computation shows that -2

+

_2(-I.

f’(=)

+

f"()

a(z)[p(=) + =’()1

(p(=), =n’(=); z), where_(u, v; z) ,X(z)(u

+

^v).Using(l), wehave Re[(p(z), zp’(z).z)]>^1- Nowforallreal::, y<

-1/2(1 +

::u),^wehave

s foreach: D.

Re [q,(/::, y; z)] (U-

x2)RelA(z)]

<

-(1 ^{+ 3x)Re}

[,(z)] <

-Re

^{[,()] (4)}

from(4)

weget for eachz D. Note

4(1 in2)

+

2 forallz D. Thus

by

^lentna1, Re[p(z)]>0 in D, that is, f(z)^qS.

REMARK.

For < 1,let R(O) {Ie.,4.Re[f’(z)

+

zf"(z)]>

,

^zeD}. Itwasprovedin

[4]

^{that if}

-m

-0.61), then/(z)

^is univalent,and theconstant ao^cannot bereplaced

f() R(o) o

.

3

byanylessone. Ourpresent theoren yieldsR

(1 (i-.)’+)

^{CS. Thus,anaturalproblemwhich}

arisesis tofind inf{B" R(B)c 8}.

REFERENCES

[I] ^{R. SINGH}

^and

S. SINGH,

Starlikeness and convexity ofcertain integrals.

Ann.

University MariaeCurie Sklodowska, Lublin,

XXXV,

16,

Sec.A(1981),

⁴⁵ 47.

[2] R. SINGH

and

S. SINGH,

Convolutionproperties ofaclass ofstarlikefunctions.pr0. A.bt.S.

106

(1)(1989),

^{145- 152.}

[3]

^{S. S.}

MILLER

and P. T.

MOCANU,

Second order differentialequationsinthecomplex plane.

J. Math. Anal. Appi.

65(1978),

^{289 305.}

[4] LING YI,

BO

GEJUN

and

LUO

SItENGZI[ENG,

On

aninequalityforsomeanalytic functions, J. Math.

nes.

andExp.

(2)13(1993),

^{259- 262.}