Inclusion Relationships for Certain Subclasses of Meromorphic Functions Defined by Using the Extended Multiplier Transformations

which are analytic in the punctured open unit diskU∗ {z : z ∈ C and 0 < |z| < 1} U \ {0}. We denote by ∑ S η , ∑ K η , and ∑ C η, β 0 ≤ η, β < 1 the subclasses of consisting of all meromorphic functions which are, respectively, starlike of order η inU, convex of order η inU, and close-to-convex of order η and type β inU see 1–3 . Let M be the class of all function φ which are analytic and univalent in U and for which φ U is convex with


Introduction
Let denote the class of functions of the form: We denote by S η , K η , and C η, β 0 ≤ η, β < 1 the subclasses of consisting of all meromorphic functions which are, respectively, starlike of order η in U, convex of order η in U, and close-to-convex of order η and type β in U see 1-3 . Let M be the class of all function ϕ which are analytic and univalent in U and for which ϕ U is convex with ϕ 0 1, Re ϕ z > 0 z ∈ U . 1.2

ISRN Mathematical Analysis
For two functions f and g analytic in U, we say that f is subordinate to g and write f ≺ g in U or f z ≺ g z , if there exists a Schwarz function w z , which is analytic in U with w 0 0 and |w z | < 1 z ∈ U , such that f z g w z . It is known that Furthermore, if the function g is univalent in U see, 4, page 4 , Making use of the principle of subordination between analytic functions, we define the subclasses S η; ϕ , K η, ϕ , and C η, β; ϕ, ψ of the class for 0 ≤ η, β < 1 and ϕ, ψ ∈ M, which are defined by respectively. For special choices for the parameters η and β as well as for special choices for the function ϕ and ψ, we will obtain various subclasses of meromorphic function of the above classes see 5-7 . For m ∈ N 0 N ∪ {0} N {1, 2, . . .} , we define the multiplier transformation J m λ, for functions f ∈ see 8, 9 with p 1 by we define a new function ϕ m,μ λ, z in terms of the Hadamard product or convolution by Essentially Choi et al. 14 motivated the Choi-Saigo-Srivastava operator for analytic functions, which includes an integral operator considered earlier by Noor 15 and others 16-18 ; we now introduce the operator I m μ λ, : → , which is defined here by; We note that It is easily verified from the definition of the operator I m μ λ , that Next, by using the operator I m μ λ, defined by 1.10 , we introduce the following subclasses of meromorphic functions: 1.13

ISRN Mathematical Analysis
We also note that

1.15
The main object of this paper is to investigate several inclusion properties of the classes mentioned above. Some applications involving integral operator are also considered.

Inclusion Properties Involving the Operator I m μ λ,
The following lemmas will be required in our investigation.
where the function p z is analytic in U with p 0 1. Then, by applying 1.11 in 2.8 , we obtain Differentiating 2.9 logarithmically with respect to z and multiplying the resulting equation by z, we have we see that Proof. Applying 1.14 and Theorem 2.3, we observe that

2.19
Then, for the function classes defined by