SOME FAMILIES OF MEROMORPHIC p − VALENT FUNCTIONS INVOLVINGA NEW OPERATOR DEFINED BY GENERALIZED MITTAG-LEFFLER FUNCTION

SOME FAMILIES OF MEROMORPHIC p− VALENT FUNCTIONS INVOLVINGA NEW OPERATOR DEFINED BY GENERALIZED MITTAG-LEFFLER FUNCTION M. K. Aouf* and T. M. Seoudy * Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt mkaouf127@yahoo.com Department of Mathematics, Faculty of Science, Fayoum University, Fayoum 63514, Egypt tms00@fayoum.edu.eg Receive 10/2/2018, Revised , Accepted 21/5/2018


Introduction
Let p be the class of functions of the form: which are analytic and p−valent in the punctured unit disk U * = U\{0}, where U = {z : z ∈ C, |z| < 1}.If f and g are analytic functions in U, we say that f is subordinate to g, written f ≺ g if there exists a Schwarz function w, which (by definition) is analytic in U with w(0) = 0 and |w(z)| < 1 for all z ∈ U, such that f (z) = g(w(z)), z ∈ U. Furthermore, if the function g is univalent in U, then we have the following equivalence (see [3] and [4]): f (z) ≺ g(z) ⇔ f (0) = g(0) and f (U ) ⊂ g(U ).
We note that and For −1 ≤ A < B ≤ 1 and z ∈ U * , Mogra [8] defined the following subclass of p as follows and Srivastava et al. [9] defined the following subclass of p as follows and Srivastava [12]).Using the operator T γ,k p,α,β , we define the classes S γ,k p,α,β (A, B) and K γ,k p,α,β (A, B) as follows: and We notice that To prove our results we need the following lemmas. where (1.17) In this paper, we obtain some interesting results for the families S γ,k p,α,β (A, B) and K γ,k p,α,β (A, B) of meromorphic p−valent functions defined by the operator T γ,k p,α,β .
Theorem 2.1.The function f (z) defined by (1.1) is in the class S γ,k p,α,β (A, B) if and only if Proof.From Lemma 1.1, we find that f ∈ S γ,k p,α,β (A, B) if and only if Proof.From Lemma 1.2, we find that f ∈ K γ,k p,α,β (A, B) if and only if Now it can be easily shown that ) ) Proof.Since The result follows from Theorem 2.1.
Using the same technique, we can also prove the following theorem.
Proof.Let f ∈ S γ+1,k p,α,β (A, B) and define the function we see that g(z) is analytic in U with g(0) = 1.Using the identity (1.7) in (2.10) we have Differentiating (2.11) logarithmically and using (2.10), we have (2.12) Simple computations show that the inequality {−h(z) + γ k + p} > 0 can be written in the form which is equivalent to (2.9).Since the function h(z) is a convex function, then applying Lemma 1.3, we see that the subordination (2.12) implies g(z) ≺ h(z).This completes the proof of Theorem 2.5.
Theorem 2.5 yields the following theorem.

Lemma 1 . 1 .
[1,2] The function f (z) defined by (1.1) is in the class S p [A, B] if and only if