Applications of differential subordination for certain subclasses of meromorphically univalent functions defined by rapid operator


 In this work, we investigate some applications of differential subordination for the class of meromorphic univalent functions defined by rapid operator and obtained coefficient bounds, integral representations, weighted and arithmetic mean for the class Σ(A, B, µ, θ).

Given two functions f and g, which are analytic in U, the function f is said to be subordinate to g, written as if there exists a Schwarz function w analytic in U, with In [3], Athsan and Kulkarni introduced rapid operator for analytic functions and Rosy and Sunil Varma [8] modified their operator to meromorphic functions as follows.
L(n, µ, θ)a n z n , where and Γ is the familiar Gamma function.
Definition 1.2. Let A and B, (−1 ≤ B < A ≤ 1) be defined parameters. We say that a function f ∈ Σ is in the class Σ(A, B, µ, θ) if it satisfies the following subordination condition by (6) By the definition of the differential subordination (8) is equivalent to the following condition In particular, we can write Σ(1 − 2β, −1, ) = Σ(β), where Σ(β) denotes class of the functions in Σ satisfying the following condition: The aim of this study is to determine some usual properties of the geometric function theory such as coefficient bounds, integral representation, weighted mean and arithmetic mean for the class Σ(A, B, µ, θ).

Coefficient bounds
The result is sharp for the function f is given by Proof. Assume that the condition (11) is true.
The result is sharp for the function f is given by (12). A, B, µ, θ). Then , n ≥ 1.

Integral representation
In the next theorem, we obtain an integral representation for S θ µ f (z).
so that consequently, we have We can write dt.
Hence, the proof of theorem is completed.

Linear combination
In this section, we prove a linear combination for the class Σ (A, B, µ, θ).

Weighted mean
Definition 5.1. Let f and g belongs to Σ. Then the weighted mean h j (z) of f and g is given as In the following theorem we will show the weighted mean for the class Σ (A, B, µ, θ). Proof. By definition of h j (z), we get Since f, g ∈ Σ(A, B, µ, θ), by Theorem 2.1, we must prove that (1 − j)a n + 1 2 nL(n, µ, θ)a n + 1 2 Hence the proof of theorem is completed.

Proof. We have for h(z) by definition
a n,i z n .