Some Properties of Fractional Calculus and Linear Operators Associated with Certain Subclass of Multivalent Functions

Sh Khosravianarab,S R Kulkarni,O P Ahuja

doi:10.1155/2009/450632

Abstract

We investigate several distortion inequalities involving fractional calculus, Ruscheweyh derivatives, and some well-known integral operators. In special cases, the results presented in this paper provide new approaches to several previously known results.

Highlights

Let A p denote the class of functions f of the form ∞f z zp ap kzp k p ∈ N {1, 2, 3, . . .}, k1 which are analytic in the open unit disc U {z ∈ C : |z| < 1}
Let K p denote the subclass of A p consisting of all functions f of the form
The Ruscheweyh derivative Dδ p−1 has been studied by several authors; for example, see 1, 2

Summary

Introduction

F z zp ap kzp k p ∈ N {1, 2, 3, . . .} , k1 which are analytic in the open unit disc U {z ∈ C : |z| < 1}. Let K p denote the subclass of A p consisting of all functions f of the form f z zp − ap kzp k p ∈ N {1, 2, 3, . G ∈ A p , given by f z : zp ap kzp k, g z : zp bp kzp k, k1 the Hadamard product or convolution of f and g is defined by f ∗ g z : zp ap kbp kzp k. The Ruscheweyh derivative Dδ p−1 has been studied by several authors; for example, see 1, 2. For β < 1, γ ≥ 0, p ∈ N, and δ > −p, let Tδ,p,γ β consist of functions f ∈ K p so that. We investigate several distortion inequalities involving fractional calculus, Ruscheweyh derivative, and some well-known integral operators defined on the class Tδ,p,γ β. Throughout this section, we assume that δ p ≥ 1

Definitions and Lemmas

Distortion Inequalities of Convolution Operators

Distortion Inequalities of Integral Operator