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Intern. Math. Journal, Vol. 4, 2003, ro. 2) 119 - 125

On a Criteria for Starlikeness V Ravichandran Department of Computer Applications Sri VenkateswaraCollegeof Engineering Sriperumbudur602 105, India Maslina Darus Schoolof Mathematical Sciences Faculty of Scienceand Technology Universiti KebangsaanMalaysia Bangi 43600SelangorD.E., Malaysia Abstract A function f (z) _ z * a2z2+ "' definedon ltl (convex)if Re (zf ' (z)l f Q)) obtain somesufficientconditionsfor functions definedin terms of the ratio of 1* 'f"Q)lfl(z) and tf'Q)lf(z) to satisfytf'(r)lf @ < h(t). Mathematics

Subject Classification: Primary 30C45

Keywords: Starlike function, differentialsubordination

1

Introduction and Definitions

Let A denotethe classof all functionsof the form f (r) - z*azz2 +. . . defined onA: { r ; l t l < 1 } . A f u n c t i o n / e " 4 i s s t a r l i k (ec o n v e xo )f o r d e r a i f Re (zf'(r)lf(r)) > a (Re(1+ rf"(t)lf'Q)) > "). Denotethe classof starlike functionsof ordera by S-(o) andwe write S-(0) as,S*.Let -1 < B < A < L Then (

{ r). A , s . l A ,: B #

.#,

z €^ }

and Obradovic In particular,we have^9-[1- 2a,-l] - S-(a). Silvermanf4] and Tuneskif2]havestudiedthe propertiesof functionsdefinedin termsof the

120

V Ravichandran and Maslina Darus

quotient(I+zf"Q)lf'Q))lQf'Q)lf @).In facr,silverman[4]haveobrained the order of starlikeness for functionsin the class

)

t -z1)|11 b , z e L , 0 < b < 1. f G - vu : 4, e rt AA, ,l rl +#zzff',(, ze )) lf f! ,Q(_ ) | f'-'I ) In fact, the classGa can be definedby subordinationas follows: - '.l r } . G u : { , , A ! J : ! " Q ) l f ' Q ) <,Lr t+abz z) , z tet L : ' ,w, o(- ' a (

zJ' QVfQ)

Obradovic and Tuneski[2]have obtained subordination result for functions in Ga and also the functions for which

I + zf"Q)lf'Q) zf'(z)lfQ)

l1--

2z

(t + z1z'

In this paper, we obtain somesubordinationresultsfor functionsdefinedin

termsof theratioof 1+ ,f"Q)lf'Q) andzf'(r)lf Q) to satisfyzf'(r)lf Q) < h(r). Our results extends the results of Silverman[4] and Obradovic and Tuneski[2]. Also we obtain results for the class S.IA, Bl and certain other subclassof functions ( defined by using Ruscheweyhderivatives and Salagean derivatives). Note that the convolutionof two function f (r) - z * Df:, anzn andsQ)z + L L z b n z n i s t h e f u n c t i o nf * g d e f i n e d b y( f * g ) ( r ) _ z + L!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!f:2 a,nbnzn.The Ruscheweyh derivative of order d > - 1 is defned by

' D o f( r ) - f ( ' ) * (1 z)6+t' The Salageanderivative of a function /(z), denoted by D^f(z)

is defined by

D * f ( r )- f ( r ) * ( z * i n ^ a , r ^ ) . n=2

I t i s a l s oe a s yt o s e et h a t D ' f ( r ) - f ( r ) , D ' f ( r ) - z f ' ( t ) a n dD " f ( t ) " ( D " - ' f ( r ) ) '. To proveour results,we needthe followingresults: Theorem A Let F(r) and G(z) b, analytici,nL, and F(0) - G(0) If H(r) - zG'(z) 'isstarl'ikei,nL andzF'(z) < zG'(z), thenF(r) < G(r). Of course,the abovetheoremis a specialcaseof the followingmoregeneral: Theorem B Let q(z) be uni,ualent and let $ be analytr,cr,na domai,nD conta'in'ing q(A) If p(r) is analyt,ic andzq'(z)d@Q))r,sstarh,ke, then z p ' ( z ) d ( p Q )<) r q ' Q ) d Q @) + p ( z ) < q ( z ) and q(z) is the bestdom'inant. The abovetwo resultsare due to Miller and Mocanoand for moredetails, we referthe book bv Miller and Mocanull],

721

On a Criteria for Starlikeness

Main Results

2

We begin with the following: 'in A', Lemma 1 Let h(z) be starlike i,n L, and h(0) : 0. If p(r) is analytic P(0) : 1 and

*p (.r,) 'n ) ?q ()r ) ' _ h e ) ,

then

p(z)
Proof.

Let p1(r) - I lpQ). Thenwe have- zp'(4 lpQ)z : zPtt(z)and hence

-he) 42.'n?) q(r)'

p(z)2 becomes

- zprQ) < h(r). By using Theorem A, we have the result. Note that the result can also be proved by using Theorem B also. Also the following special caseis interesting: Corollary 1 Let-t<

B
1 . I f p ( r ) ' i s a n a l y t i , c zAn, p ( 0 ) - 1 a n d

zp'(z) , (A - B), -

W then

1 t +A r y '

I*Az , \ plz)< I+Bz.

When A - 1 and B - - 1, the aboveresult reducesto the following:

Corollary2Ifp(,)_1*cz*..|i,sanlyticinLo"d,ffi< Rep(z) > 0. Theorem 1 Let h(z) bestarliketn L, and h(0) - 0. If f (r) € A satzsfi,es

fg,) - D6+)f(z)lD,f (z) < 1+ 5 + h(z), l@+ Dn5+2 ( D o + tf ( r ) ) ,

then

' P5+' fP .,' Lft- - f h(d*arf ''.J lo q At@

I22

V Ravichandran and Maslina Darus

Proof.Definep(z)_w.Clearlyp(0)-1.Usingtheidentity , ( D 6f Q D ' - ( d + l ) r d + ' f ( r ) - o o 67 Q ) , we have zp,(z) p(z) and hence

(d + 2 )3 :{ !-g- -r (d+1)

zp'(z) p(z)2

ot+ r f (z) _WT

which impliesthat zp'(z) -

-

l

p(r)z

r (d+ \no+zyQ) P6+tf(r)lDof(') - ( d 1 ) < h ( " ) . + L ( D a +ft ( r ) ) ,

By using Lemma 1, we have , \ : plz)

p6+rf(z\

Dofd

<

h(rilorf-, q l

l'-1,"

By takin1 h(z) - bz we have the following: Corollary

3 tf f (r) e A sat'isfies

- P6+tf(4lDof(r) [(d+ 2)oo+z7Q) (Dt+t f (')), then

p6+Lf(t)

-DoW

-,

-'l.#,

r

1.t 6r'

In part'icular, we haue

o ,eD o * 'f ( ' ) > 1 K D6fe) l+b Corollary 4 Let h(z) bestarlzkei,nL, and h(0) - 0. If f e A and

I + z-f"(1/f'(z) < 1 + h(r), ,f'(r)lf Q)

then rf'(r)lfQ) < [r - fi ry or]

123

On a Criteria for Starlikeness

Corollary 5 If f (r) e A satisfies

p6f (z)lp6+2 (A - B)z f (z)- +rp6+tf (z))_ d + 1 * '6+2 '(d+2)(1 (Do+tf(t)), *Az)z' then P 6 + L (f r ) , I *

Wt

Az

r+B/

Our next resultgivesthe subordinationresultfor the classS.IA, Bl. C o r o l l a r y 6 I ff e A a n d I + zf"(z)lf'Q) t 1t @ - B)z trr

W

G+Azy'

thenf (t) € ,S*lA,Bl. By taking A - 0 and B - b in the above corollary 6, we have the following result of Obradovic and Tuneski[2]: CorollaryT If f eGo,0 < b < 1. Thenrf'(z)lf@ <|lQ+bz) result is sharp.

andthe

of this result, we have [4]: As a consequence

G uC S .( 1 l ( t + b ) ) . Also by taking A : I, B and Tuneski[2]: CorollaryS lf f e Aand I+zf"(z)lf'Q) 2 j_L

2z

wlI-rG+zY, thenf (t) € ,S*. The class^9S-(a) of stronglystarlikefunctionsof ordero consistsof functions f e A satisfying I z f ' ( z \. 1 e i T

s; lu"(=#)l J\.) | I

In fact, this condition is equivalentto the following:

tf'(r) , /r*r\"

7T;'\--,) For this class,we havethe following:

, z€a

124

V Ravichandran and Maslina Darus

Corollaryg lf f e Aand I+zft'(z)lf'Q) 2az
f (") -Pm+t n-m - , f, f h(rt) arf-' . Lt-ro q I Let p(z) - #W.

Proof.

Then a computationshowsthat

zp'(z) _ z(D^+rf (z))' z(D* f (r)), D*+, f (r) _ - --

i6

D-+ry6-

D;Kr-

Dn+r f e)

pm+rf (r)

Therefore z p ' ( z )_

9::f(z)D*f(z)

1,

;W:ffi-1
L

By Lemma 1, w€ have

D T I J Q.. f t - h ( ' i o r f - ' / \ : -n^f p(z) f ,J e) t^ Jo rt This completesthe proof. By using Theorem B, we have the following: Lemma 2 Let q(z) be uniualent and,zq'(z)q(z) be starlzke.If p(t) r,sanalytlc 'in 4,, then p ( z ) z p ' ( t ) < q ( z ) z q ' ( z )+ p ( z ) < q ( z ) . The function q(z) - (E)''' is convexand q(z)zq'(z) starlike. Thereforethe Lemma 2 gives the following: Theorem 3 If p(r) - 1* cz*...is

p(z) zp'(z)-

C+

a n a l y t , iacn d , t h e n

+ pQ). (-)'

P

,le

-

42 is

125

On a Criteria for StarliJreness As a special case,we have the following: Corollaryl0 If f eAand

' 't'Q)\'l ( zl t ,l\ 1 1 ){ l

then

s- zy'

'I',?) ., (r + zlttz

'\I-z/ f(r) or equ'iualently f (r) is stronglystarli,keof order 1/2. ACKNOWLEDGEMENTS. The work presentedherewas supportedby IRPA grant 09-02-02-0080-EA208, Malaysia.

References [1] S. S. Miller, P. T. Mocanu, Differential Subordinations:Theory and Applications. Pure and Applied Mathemat'icsNo. 225, Marcel Dekker, New York, (2000). l2l M. Obradovid, N. Tuneski, On the starlike criteria defined Silverman, Zesz. Nauk. Poli,tech.Rzesz.,Mat. 181(24) (2000), 59-64. t3] S. Ozaki, M. Nunokawa, The Schwarzianderivative and univalent functions, Proc. Am. Math. Soc. 33(1972),392-394. t4] H. Silverman,Convexand starlikecriteria, Int. J. Math. Math. Sci.22(l) ( 1 9 9 9,)7 5 - 7 9 . Received: July 10, 2003