ON MEROMORPHIC FUNCTIONS DEFINED BY A NEW CLASS OF LIU-SRIVASTAVA INTEGRAL OPERATOR SYED

In this work, we introduce and explore certain new subclasses of meromorphic functions. We aim to study some important properties such as coefficient estimates, growth rate and partial sums for these newly defined subclasses. It is important to mentioned that our results are generalization of number of existing results.


Introduction
Let p denote the class of p-valent meromorphic function of the form: Here we are listing some important subclasses of meromorphic functions which will be used in our subsequal work. In 1936, Roberston [23] introduced the classes of meromorphic starlike and meromorphic convex functions of order α. By M S (α) we mean the subclass of 1 consisting of all meromorphic starlike functions of order α. Analytically A closely related class of meromorphic convex functions of order α is denoted by M C (α) and defined as: In 1952, Kaplan [16] introduced and studied an important class of analytic functions in the open unit disc U known as close-to-convex functions. A function λ belongs to 1 is in class M K (α, β), of meromorphic close-to-convex functions of order α and type β, if there exist δ (ω) ∈ M S (β) and Many differential and integral operators can be written in terms of convolution of certain holomorphic functions. Let δ (ω) ∈ p and having series representation of the form then convolution (Hadamard product) is denoted by λ * δ and defined as: where λ (ω) as given by (1.1).
Following the current work of Liu and Srivastava [18] (see also [1]- [6]), now we defined the integral operator given below The above integral operator converts into the following operator when p = 1 It can be easily verified from (1.8) For more details see [7-9, 12, 15, 20, 21, 24].

Main Results
In this section, in present the work to acquire sufficient conditions in which (1.13) gives the function λ (ω) within the class N * m p (a, b; d, S, T ), as well as demonstrates that this condition is required for function which belong to this class. In our first theorem, we begin with the necessary and sufficient condition for function λ in N * m p (a, b; d, S, T ). We also prove some other related theorems.
Proof. Assuming that (2.1) holds true, we obtain since the above inequality is genuine for all ω ∈ U , let the value of ω on the real axis.
Which complete the proof.
The result is sharp for the function Growth and distortion bounds for functions belonging to the class N * m p (a, b; d, S, T ) will be given in the following result: Theorem 2.2. If a function λ (ω) given by (1.1) is in the class N * m p (a, b; d, S, T ) then for |ω| = r, we have: Proof. In view of Theorem 2.2, we have Therefore, . Now, by differentiating(1.13), we have We have thus completed the proof. Theorem 2.3. Let the function λ (ω) given by (1.13) is in the class N * m p (a, b; d, S, T ). Then we have (i) λ is meromorphically starlike of order q in the disc |ω| < r 3 , that is (ii) λ is meromorphically convex of order q in the disc |ω| < r 4 , that is Proof. (i) In order to the inequality (2.9), we set ωλ (ω) Then we have Thus, by Theorem 2.1, the inequality (2.11) will be true if The last inequality leads us immediately to the disc |ω| < r 3 , where r 3 is given by (2.9).
(ii) in order to prove the second affirmation of Theorem 2.3, we find from (1.1) that: Thus we have desired inequality: Thus, by Theorem 2.1, the inequality (2.12) will be true if The last inequality readily yields the disc |ω| < r 4 , where r 4 is given by (2.10), which complete the proof. Proof. Let the function are in N * m p (a, b; d, S, T ), it suffices to show that the function h defined by is in the class N * m p (a, b; d, S, T ). Since

In view of Theorem 2.1, we have
which show that h (ω) ∈ N * m p (a, b; d, S, T ), which is required.
Proof. Let the function λ (ω) be expressed in the form given by (2.13), then and for this function, we have the condition (2.1) is satisfied. Thus, λ ∈ N * m p (a, b; d, S, T ). Conversely, we suppose that λ ∈ N * m p (a, b; d, S, T ). Since This completes the assertion of Theorem 2.5.

Conclusion
In our current investigation, we have presented and studied thoroughly some new subclasses of p−valent functions related with meromorphic convex and meromorphic starlike functions, in connection with the integral operator given by (1.7). We have obtained sufficient and necessary conditions in relation to these classes, including growth and distortion theorem along with a radius problem. The technique and ideas of this paper may stimulate further research in the theory of multivalent meromorphic functions.

Conflicts of Interest:
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.