A New Subclass of Analytic Functions Defined by Generalized Ruscheweyh Differential Operator

Serap Bulut

doi:10.1155/2008/134932

Abstract

We investigate a new subclass of analytic functions in the open unit disk which is defined by generalized Ruscheweyh differential operator. Coefficient inequalities, extreme points, and the integral means inequalities for the fractional derivatives of order of functions belonging to this subclass are obtained.

Highlights

Throughout this paper, we use the following notations:N : {1, 2, 3, . . .}, N0 : N ∪ {0}, R−1 : {u ∈ R : u > −1}, R0−1 : R−1 \ {0}.Let A denote the class of all functions of the form ∞fz z anzn, n2 which are analytic in the open unit disk U : {z ∈ C : |z| < 1}
N2 which are analytic in the open unit disk U : {z ∈ C : |z| < 1}
For fj ∈ A given by fj z z an,j zn j 1, 2, 1.3

Summary

Introduction

Throughout this paper, we use the following notations:. Journal of Inequalities and Applications the Hadamard product or convolution f1∗f2 of f1 and f2 is defined by f1∗f2 z z an,1an,2zn. Using the convolution 1.4 , Shaqsi and Darus 1 introduced the generalization of the Ruscheweyh derivative as follows. N∈N , 1.7 and where a n is the Pochhammer symbol or shifted factorial defined in terms of the Gamma function by a n: Γa n Γa. 1.9 and for λ 0, we obtain uth Ruscheweyh derivative introduced in 2 , Rm0 Rm. Using the generalized Ruscheweyh derivative operator Ruλ, we define the following classes. Basic properties of the class Sλ u, v; α are studied, such as coefficient bounds, extreme points, and integral means inequalities for the fractional derivative

Coefficient inequalities

Extreme points

The main integral means inequalities for the fractional derivative