On Certain Class of Analytic Functions Related to Cho-Kwon-Srivastava Operator

Motivated by a multiplier transformation and some subclasses of meromorphic functions which were defined by means of the Hadamard product of the Cho-Kwon-Srivastava operator, we define here a similar transformation by means of the Ghanim and Darus operator. A class related to this transformation will be introduced and the properties will be discussed.


Introduction
Let Σ denote the class of meromorphic functions f z normalized by a n z n , 1.1 which are analytic in the punctured unit disk U {z : 0 < |z| < 1}.For 0 ≤ β, we denote by S * β and k β the subclasses of Σ consisting of all meromorphic functions which are, respectively, starlike of order β and convex of order β in U cf. e.g., 1-4 .
For functions f j z j 1; 2 defined by a n,j z n , 1.2 International Journal of Mathematics and Mathematical Sciences we denote the Hadamard product or convolution of f 1 z and f 2 z by a n,1 a n,2 z n .1.3 Let us define the function φ α, β; z by for β / 0, −1, −2, . .., and α ∈ C/{0}, where λ n λ λ 1 n 1 is the Pochhammer symbol.We note that φ α, β; z is the well-known Gaussian hypergeometric function.
Let us put Corresponding to the functions φ α, β; z and q λ,μ z and using the Hadamard product for f z ∈ Σ, we define a new linear operator L α, β, λ, μ f z on by

1.8
The meromorphic functions with the generalized hypergeometric functions were considered recently by Dziok and Srivastava 5, 6 , Liu 7 , Liu and Srivastava 8-10 , and Cho and Kim 11 .
For a function f ∈ L α, β, λ, μ f z , we define and, for k 1, 2, 3, . .., Furthermore, we say that a function The main object of this paper is to present several inclusion relations and other properties of functions in the classes Σ μ,k α,β,λ A, B and Σ μ,k, α,β,λ A, B which we have introduced here.

Main Results
We begin by recalling the following result popularly known as Jack's Lemma , which we will apply in proving our first inclusion theorem.

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Lemma 2.1 see Jack's Lemma 22 .Let the (nonconstant) function w z be analytic in U with w 0 0. If |w z | attains its maximum value on the circle |z| r < 1 at a point z 0 ∈ U, then where γ is a real number and γ ≥ 1.
where the function w z is either analytic or meromorphic in U, with w 0 0. By using 2.4 and 1.11 , we have Upon differentiating both sides of 2.5 with respect to z logarithmically and using the identity 1.11 , we obtain We suppose now that max and apply Jack's Lemma, we thus find that By writing w z 0 e iθ 0 ≤ θ < 2π 2.9 and setting z z 0 in 2.6 , we find after some computations that

2.10
Set Then, by hypothesis, we have 2.12 which, together, imply that View of 2.13 and 2.10 would obviously contradict our hypothesis that Hence, we must have and we conclude from 2.4 that The proof of Theorem 2.2 is thus complete.
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Properties of the Class f ∈ Σ μ,k, α,β,λ A, B
Throughout this section, we assume further that α, β > 0 and We first determine a necessary and sufficient condition for a function f ∈ Σ of the form 1.13 to be in the class f ∈ Σ μ,k, α,β,λ A, B of meromorphically univalent functions with positive coefficients.
where, for convenience, the result is sharp for the function f z given by for all z / 0.
Proof.Suppose that the function f ∈ Σ is given by 1.13 and is in the class Σ μ,k, α,β,λ A, B .Then, from 1.13 and 1.12 , we find that Since |R z | ≤ |z| for any z, therefore, we have

International Journal of Mathematics and Mathematical Sciences 7
Choosing z to be real and letting z → 1 through real values, 3.5 yields which leads us to the desired inequality 3.2 .
Conversely, by applying hypothesis 3.2 , we get where the equality holds true for the function f z given by 3.3 .
Next, we prove the following growth and distortion properties for the class

3.9
Each of these results is sharp with the extremal function f z given by 3.3 .

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Thus, for 0 < |z| r < 1 and utilizing 3.10 , we have

3.11
Also from Theorem 3.1, we get

3.13
This completes the proof of Theorem 3.3.
We conclude this section by determining the radii of meromorphically univalent starlikeness and meromorphically univalent convexity of the class Σ μ,k, α,β,λ A, B .We state our results as in the following theorems.

3.14
The equality is attained for the function f z given by 3.3 .
Proof.It suffices to prove that Hence, 3.16 holds true if with the aid of 3.18 and 3.2 , it is true to have

3.20
This completes the proof of Theorem 3.4.

3.21
The equality is attained for the function f z given by 3.

3 .
International Journal of Mathematics and Mathematical SciencesProof.By using the technique employed in the proof of Theorem 3.4, we can show that |z| < r 2 , with the aid of Theorem 3.1.Thus, we have the assertion of Theorem 3.5.