Certain Families of Multivalent Analytic Functions Associated with Iterations of the Owa-Srivastava Fractional Differintegral Operator

A K Mishra,S N Kund

doi:10.1155/2014/915385

Abstract

By making use of a multivalent analogue of the Owa-Srivastava fractional differintegral operator and its iterations, certain new families of analytic functions are introduced. Several interesting properties of these function classes, such as convolution theorems, inclusion theorems, and class-preserving transforms, are studied.

Highlights

Let A denote the class of analytic functions in the open unit diskU = {z : z ∈ C, |z| < 1} (1)and let Ap be the subclass of A consisting of functions represented by the following Taylor-Maclaurin’s series: ∞f (z) = zp + ∑akzk+p (p ∈ N := {1, 2, 3, . . .}) . (2) k=1In a recent paper Patel and Mishra [1] studied several interesting mapping properties of the fractional differintegral operator: Ω(zλ,p) : Ap 󳨀→ Ap (p ∈ N, −∞ < λ < p + 1, z ∈ U), (3) defined byΩ(zλ,p)f (z) := zp +
Investigations have been initiated only recently regarding iterations of certain transforms defined on classes of analytic and meromorphic functions
Al-Oboudi and Al-Amoudi [9, 10] investigated properties of certain classes of analytic functions associated with conical domains, by making use of the operator Dnt,λ

Summary

Introduction

Let Ap be the subclass of A consisting of functions represented by the following Taylor-Maclaurin’s series:. The transformation D(tλ,p)(n, m) includes, among many, the following two previously studied interesting operators as particular cases. Investigations have been initiated only recently regarding iterations of certain transforms defined on classes of analytic and meromorphic functions. Al-Oboudi and Al-Amoudi [9, 10] investigated properties of certain classes of analytic functions associated with conical domains, by making use of the operator Dnt ,λ. In the sequel to these current investigations, in the present paper, we define the following subclass of Ap associated with the iterated operator D(tλ,p)(n, m) and investigate its several interesting properties. The function f ∈ Ap is said to be in the class Hnp,m(λ, t; h) if the following subordination condition is satisfied:. The function class Hnp,m(λ, t; h) includes several previously studied subclasses of A as particular cases. We find inclusion theorems and study behavior of the LiberaLivingston integral operator

Some More Definitions and Preliminary Lemmas

Convolution Results

Properties of the Libera-Livingston Transform

Inclusion Theorems