Some Properties of Certain Integral Operators on New Subclasses of Analytic Functions with Complex Order

We define new subclasses of -valent meromorphic functions with complex order. We prove some properties for certain integral operators on these subclasses.


Introduction
Let U {z ∈ C : |z| < 1} be the open unit disc in the complex plane C, U * U \ {0}, the punctured open unit disk.Let Σ p denote the class of meromorphic functions of the form which are analytic and p-valent in U * .
For p 1, we obtain the class of meromorphic functions Σ.
We say that a function f ∈ Σ p is the meromorphic p-valent starlike of order α 0 ≤ α < p and belongs to the class f ∈ Σ p α , if it satisfies the inequality A function f ∈ Σ p is the meromorphic p-valent convex function of order α 0 ≤ α < p , if f satisfies the following inequality: Most recently, Mohammed and Darus 19 introduced the following two general integral operators of p-valent meromorphic functions Σ p :

Main Results
In this section, considering the above new subclasses we obtain for the integral operators F 1,γ 1 ,...,γ n z and J 1,γ 1 ,...,γ n z some sufficient conditions for a family of functions f i to be in the the above new subclasses. where γ j β j − p .

2.2
Proof.A differentiation of F p,γ 1 ,...,γ n z which is defined in 1.16 , we get

2.3
Then from 2.3 , we obtain

2.4
By multiplying 2.4 with z yield, γ j zf j z f j z p .

2.5
That is equivalent to γ j .

2.7
Taking the real part of both terms of 2.7 , we have

2.8
Sine That is,

2.10
Then This completes the proof. where γ j β j − p .

2.14
Then from 2.14 , we obtain 2.17 Taking the real part of both terms of 2.17 , we have

2.18
Sine f j ∈ Σ pN β j , b , we get