(1.School of Mathematics and Statistics,Chifeng University,Inner Mongolia 024000,China;2.School of Computer and Information Engineering,Chifeng University,Inner Mongolia 024000,China)

§1.Introduction

LetApdenote the class of functions of the form

which are analytic in the unit diskD={z:|z|＜1}.For simplicity,we writeA1=:A.

For two analytic functionsfandg,the functionfis subordinate toginD(see[1]),written as follows

if there exists an analytic functionω,withω(0)=0 and|ω(z)|＜1 such that

In particular,if the functiongis univalent in D,thenf(z)?g(z)is equivalent tof(0)=g(0)andf(D)?g(D).

Ma and Minda[2]introduced and studied the classesS?(φ)andC(φ)as below

and

whereφ(z)is an analytic function with positive real part inD,φ(D)is symmetric with respect to the real axis and starlike with respect toφ(0)=1 andφ′(0)＞0.The classS?(φ)andC(φ)include several well-known subclasses of starlike and convex functions as special case.

In the year 1970,Robertson[3]introduced the concept of quasi-subordination.For two analytic functionsfandg,the functionfis quasi-subordinate toginD,written as follows

if there exist analytic functions?andω,with|?(z)|≤1,ω(0)=0 and|ω(z)|＜1 such that

Observe that when?(z)=1,thenf(z)=g(ω(z)),so thatf(z)?g(z)inD.Also notice that ifω(z)=z,thenf(z)=?(z)g(z)and it is said thatfis majorized bygand writtenf(z)?g(z)inD.Hence it is obvious that quasi-subordination is a generalization of subordination as well as majorization.See[4-6]for works related to quasi-subordination.

Mohd and Darus[7]introduced the classes(φ)andCq(φ)as below

and

The two classes are analogous to the Ma-Minda starlike and convex classes defined in the form of quasi-subordination.

Letf(m)be then-th order ordinary differential operator,for a functionf∈Ap,that is,

wherep＞m,p∈N;n∈N0=N∪{0},z∈D.

Throughout this paper it is assumed that functionφ(z)is analytic inDwithφ(0)=1.Using the operatorf(m),we now de fine the following class ofp-valent analytic functions.

De finition 1.1Let the class(λ,b;φ)consists of functionsf(z)∈Apsatisfying the quasi-subordination

Clearly,we have the following relationship:

It is well known that then-th coefficient of a univalent functionf(z)∈Ais bounded byn(see[8]).The bounds for coefficient give information about various geometric properties of the function.Many authors have also investigated the bounds for the Fekete-Szeg? coefficient for various classes[7,9-23].In particular,some authors start to study the Fekete-Szeg? problem for various classes using quasi-subordination[7,22,23].In this paper,we obtain coefficient estimates for the functions in the above defined class.

Let ? be the class of analytic functionsω(z),normalized byω(0)=0,and satisfying the condition|ω(z)|＜1.We need the following lemmas to prove our main results.

Lemma 1.2[24]Ifω∈?,then for any complex numbert

The result is sharp for the functionsω(z)=z2orω(z)=z.

Lemma 1.3[2]Ifω∈?,then

Whent＜?1 ort＞1,equality holds if and only ifω(z)=zor one of its rotations.If?1＜t＜1,then equality holds if and only ifω(z)=z2or one of its rotations.Equality holds fort=?1 if and only ifω(z)=or one of its rotations while fort=1,equality holds if and only ifω(z)=or one of its rotations.

Also the sharp upper bound above can be improved as follows then?1＜t＜1:

and

§2.Main Results

Throughout,letf(z)=z+ap+1zp+1+ap+2zp+2+···,φ(z)=1+B1z+B2z2+···,?(z)=c0+c1z+c2z2+···,ω(z)=ω1z+ω2z2+···,B1∈RandB1＞0.

Theorem 2.1Iff(z)∈Apbelongs to(λ,b;φ),then

and,for any complex numberμ,

where

ProofIff(z)(λ,b;φ),then there exist analytic functions?(z)andω(z),with|?(z)|≤1,ω(0)=0 and|ω(z)|＜1 such that

Since

it follows from(2.3)that

Further,

where

Since?(z)is analytic and bounded inD,we have[25,page 172]

By using this fact and the well-known inequality|ω1|≤1 in(2.6)and(2.7),we get

and

Applying Lemma 1.2 and the triangle inequality to(2.8),we obtain(2.2).The result is sharp for the function

or

Forμ=0 in(2.2),we have(2.1).The proof of theorem 2.1 is complete.

Corollary 2.2[7]Iff(z)∈Abelongs to(φ),then

and,for any complex numberμ,

Corollary 2.3[7]Iff(z)∈Abelongs toCq(φ),then

and,for any complex numberμ,

Theorem 2.4Iff(z)∈Apsatis fies

then the following inequalities hold

and,for any complex numberμ,

where

ProofThe result follows by takingω(z)=zin the proof of Theorem 2.1.

Theorem 2.5Iff(z)∈Apbelongs toRpm,q(λ,b;φ),then for any real numberμandb＞0

Further,ifσ1≤μ≤σ3,then

Ifσ3≤μ≤σ2,then

For any real numberμandb＜0,

Further,ifσ2≤μ≤σ3,then

Ifσ3≤μ≤σ1,then

where

ProofWe assume thatb＞0.From(2.2),we have

Ifμ≤σ1,thent≤?1.Thus,by applying Lemma 1.3,we get the first inequality in(2.10).

Ifμ≥σ2,thent≥1.Applying Lemma 1.3,we have the last inequality in(2.10).

Whenσ1≤μ≤σ2,then|t|≤1.Thus applying Lemma 1.3,we obtain the middle inequality in(2.10).

Moreover,(2.11)and(2.12)are established by an application of Lemma 1.3.

Applying Lemma 1.3,we can prove(2.13)?(2.15)forb＜0.The proof of theorem 2.5 is complete.

Corollary 2.6Iff(z)∈Abelongs to(φ),then for any real numberμ

Further,ifσ1≤μ≤σ3,then

Ifσ3≤μ≤σ2,then

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