Differential Subordination Defined by New Generalised Derivative Operator for Analytic Functions

Ma'Moun Harayzeh Al-Abbadi,Maslina Darus

doi:10.1155/2010/369078

Abstract

A new generalised derivative operatorμλ1,λ2n,mis introduced. This operator generalised many well-known operators studied earlier by many authors. Using the technique of differential subordination, we will study some of the properties of differential subordination. In addition we investigate several interesting properties of the new generalised derivative operator.

Highlights

Introduction and PreliminariesLet A denote the class of functions of the form ∞fz z akzk, where ak is complex number, k2 which are analytic in the open unit disc U {z ∈ C : |z| < 1} on the complex plane C
K2 which are analytic in the open unit disc U {z ∈ C : |z| < 1} on the complex plane C
Let H U be the class of holomorphic function in unit disc U {z ∈ C : |z| < 1}

Summary

Introduction

Introduction and PreliminariesLet A denote the class of functions of the form ∞fz z akzk, where ak is complex number, k2 which are analytic in the open unit disc U {z ∈ C : |z| < 1} on the complex plane C. A new generalised derivative operator μnλ1,m,λ2 is introduced. The univalent function q is called a dominant of the solutions of the differential subordination, or more a dominant, if p ≺ q for all p satisfying 1.6 .

Results

Conclusion