1 Introduction and Definition

Let Σ denote the class of meromorphic functions of the form:

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which are analytic in the open unit disk

A function f ∈ Σ is meromorphic starlike of order β,denoted by S∗(β),if Re(0≤ β＜ 1,z∈ U∗).A function f∈ Σ is meromorphic convex of order β,denoted by

Let φ be an analytic function with positive real part in the open unit disk U,φ(0)=1,φ′(0) ＞ 0 and φ(U)be symmetric with respect to the real axis.The Taylor’s series expansion of such function is of the form

Aouf[1]introduced and studied the classwhich consists of functions f(z)∈Σ for

For the functions

let(f∗g)(z)be the Hadamard product or convolution of f(z)and g(z)defined by

The generalized hypergeometric functionlFmfor a1, ···,al,d1, ···,dmsuch that dj0,−1,···for j=1,2,···,m,and z ∈ C is defined in[2]as follows:

with l≤ m+1,l,m ∈ N,where the Pochhammer symbol(ν)n(or the shifted factorial since(1)n=n!)is given in terms of the gamma function as

For the positive real values a1, ···,al,d1, ···,dmsuch that dj0,−1,···for j=1,2,···,m,by using the Gaussian hypergeometric function given by(1.4),we thus obtain

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where

(see[3]–[5],and also the more recent works[6]–[8]dealing extensively with Dziok-Srivastava operator).

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We note that:

Lemma 1.2[15] If ω(z)∈ Ω and ω(z)=c1z+c2z2+ ···,z ∈ U,then

(ii) The differential operator2I1(n+1,1;1;z)=Dnf(z)(n ∈ N+)was studied by Ruscheweyh[10];

(iii) The operator2I1(a,1;c;z)=L(a,c)f(z)was studied by Carson and Shaffer[11];

(iv) The operator2I1(ν,1; λ +1;z)=Iλ,νf(z)(λ ＞ −1, ν ＞ 0)was studied by Choi et al.[12].

Let H denote the class of function f(z)of the form

as asserted.An examination of the proof shows that for the first case,the equality is attained when c1=0 and c2=1,where the function inl(α,φ)is given by

Liu and Cui[13]defined a class Mα(φ)consisting of functions f(z)∈ S for which

Analogous to the class Mα(φ),by using the operatorlImf(z),we define the classas follows:

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Definition 1.1 Let φ(z)=1+B1z+B2z2+B3z3+ ···A function f(z)∈ Σ is in the class

It is interesting to note that,for l=2,m=1,a1=d1,a2=1,the classlM∗m(α,φ)reduces to the following new subclasses.

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Example 1.1 Fora function f(z)∈Σ given by(1.1)is said to be in the class M(α,φ),if the following conditions are satis fied:

In order to derive our main results,we have to recall here the following lemmas.

Lemma 1.1[14] Let Ω be the class of analytic functions ω,normalized by ω(0)=0,

satisfying condition|ω(z)|＜ 1.If ω(z)∈ Ω and ω(z)=c1z+c2z2+ ···,z ∈ U,then

(i) The differential operator2I1(a,b;c;z)=()(z)(a,b∈ C,c∈ Z+)was studied by Hohlov[9];

for any complex number t.The result is sharp for the functions ω(z)=z2or ω(z)=z.

Lemma 1.3[16] If ω(z)∈ Ω and ω(z)=c1z+c2z2+ ···,z ∈ U,then

which completes the proof.

2 Main Results

Theorem 2.1Let φ(z)=1+B1z+B2z2+B3z3+···If f(α,φ)and l≤m+1,l,m∈N,then

and

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Proof.Let f∈l(α,φ).Then there is an analytic function ω(z)=c1z+c2z2+···such that

Since

and

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it follows from(2.1)–(2.3)that

Then,we see that

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and,by Lemma 1.2,we have

Also,if B1=0,then

Then,by using(1.6),(2.7)and Lemma 1.2,we have

By using(1.6)and Lemma 1.1,we get

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Moreover,by using(1.6),(2.6)and Lemma 1.1,we have

For t＜ −1 or t＞ 1,the equality holds if and only if ω(z)=z or one of its rotation.For−1 ＜ t＜ 1,the equality holds if ω(z)=z2or one of its rotation.The equality holds for t= −1 if and only ifor one of its rotation.For t=1,the equality holds if and only ifor one of its rotation.The above upper bound for−1＜t＜1 is sharp,and it can be improved as follows:

Now,we consider the Fekete-Szegö problem for the classl(α,φ).

Theorem 2.2Let φ(z)=1+B1z+B2z2+B3z3+···If f∈l(α,φ)and l≤m+1,then forµ∈C we have

For eachµ,there is a function inl(α,φ)such that the equality holds.

Taking into account(2.6),(1.6)and Lemma 1.1,we obtain

Proof.Applying(2.6),we have

where

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which is analytic in the open unit disc U,and S be the subclass of H consisting of functions which are analytic and univalent in U.

and for the second case,when c1=1 and c2=0,we have

respectively.

By using Lemma 1.3,we can obtain the following theorem.

Theorem 2.3 Let φ(z)=1+B1z+B2z2+B3z3+ ···(Bi＞ 0,i∈ N+,α ＞).Ifand l≤m+1,l,m∈N,then forµ ∈R,we have

where

For eachµ,there is a function inl(α,φ)such that the equality holds.

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Proof.By using(2.7),(1.6)and Lemma 1.3 successively,we have

where

To show that these results are sharp,we define the functions such that

It is clear that the functions LetIf u＜u1or u＞u2,then the equality occurs for the function or one of its rotations.For u1＜u＜u2,the equality is attained if and only if f isor one of its rotations.If u=u1,then the equality holds for the functionor one of its rotations.If u=u2,then the equality is obtained for the functionor one of its rotations.

Using arguments similar to those in the proof of Theorem 2.3,we obtain the following theorem.

Theorem 2.4 Let φ(z)=1+B1z+B2z2+B3z3+ ···(Bi＞ 0,i∈ N+,α ＞)and.If f(z)given by(1.1)belongs to the classthen forµ∈R,we have

whereµ1andµ2are given in Theorem 2.3.

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