Some Properties for an Integral Operator Defined by Generalized Hypergeometric Function

Ibtisam Aldawish,Maslina Darus

doi:10.1155/2014/134910

Abstract

We define a new general integral operator for meromorphic functions involving the generalized hypergeometric function. Furthermore, we study the characterization and other properties for this operator.

Highlights

Let Ω denote the class of functions of the form f (z) = 1 z + ∞ ∑akzk, k=1 (1)which are analytic in the punctured open unit disk U∗ = {z : z ∈ C, 0 < |z| < 1} = U \ {0}.We say that a function f ∈ Ω is meromorphic starlike of order η (0 ≤ η < 1) and belongs to the class Ω∗(η), if it satisfies the following inequality:−Re ( zf󸀠 (z) ) > η. f (z) (2)
Analogous to the differential operator defined in [9] which involves the q-hypergeometric functions on the normalized analytic functions, we define the following differential operator Nnr,s[αi, βj; q] : Ω → Ω on the space of meromorphic functions in the class Ω by
With the aid of differential operator Nnr,s[αi, βj; q]f(z) given by (10), we define a new subclass of functions in Ω as follows

Summary

Introduction

Let Ω denote the class of functions of the form f (z) We say that a function f ∈ Ω is meromorphic starlike of order η (0 ≤ η < 1) and belongs to the class Ω∗(η), if it satisfies the following inequality: Let ΩN(ξ) (ξ > 1, z ∈ U) be the subclass of Ω, consisting of the functions f, which satisfy the inequality zf󸀠󸀠 (z) f󸀠 (z)

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Results

Conclusion