On Subclass of Analytic Univalent Functions Associated with Negative Coefficients

Ma'Moun Harayzeh Al-Abbadi,Maslina Darus

doi:10.1155/2010/343580

Abstract

M. H. Al-Abbadi and M. Darus (2009) recently introduced a new generalized derivative operatorμλ1,λ2n,m, which generalized many well-known operators studied earlier by many different authors. In this present paper, we shall investigate a new subclass of analytic functions in the open unit diskU={z∈ℂ:|z|<1}which is defined by new generalized derivative operator. Some results on coefficient inequalities, growth and distortion theorems, closure theorems, and extreme points of analytic functions belonging to the subclass are obtained.

Highlights

Introduction and DefinitionsLet A x denote the class of functions of the form ∞fz z akzk, ak is complex number, kx[1] and x ∈ N {1, 2, 3, ...}, which are analytic in the open unit disc U {z ∈ C : |z| < 1} on the complex plane C; note that A 1 A and A x ⊆ A 1
Kx1 and x ∈ N {1, 2, 3, ...}, which are analytic in the open unit disc U {z ∈ C : |z| < 1} on the complex plane C; note that A 1 A and A x ⊆ A 1
In 1 Silverman investigated the subclasses of T 1 denoted by S∗T α and CT α for 0 ≤ α < 1

Summary

Introduction and Definitions

Fz z akzk, ak is complex number, kx[1] and x ∈ N {1, 2, 3, ...}, which are analytic in the open unit disc U {z ∈ C : |z| < 1} on the complex plane C; note that A 1 A and A x ⊆ A 1. Let Sα∗ x and Cα x be the classes of S x consisting of functions, respectively, starlike of order α and convex of order α in U, for 0 ≤ α < 1. Let T x denote the subclass of S x consisting of functions of 2 the form. The authors in 2 have recently introduced a new generalized derivative operator μnλ1,m,λ2 as follows. By making use of the generalized derivative operator μnλ1,m,λ2 , the authors introduce a new subclass as follows. . .} and λ2 ≥ λ1 ≥ 0, let Hnλ1,m,λ2 x, α be the subclass of S x consisting of functions f satisfying. . TH10,,0λ2 1, α ≡ TH1λ,11,0 1, α ≡ TH10,,m0 1, α ≡ TH01,,20 1, α ≡ CT α , 1.9 class of convex function of order α with negative coefficients. We introduced and studied by Al-Shaqsi and Darus 13

Coefficient Inequalities

Growth and Distortion Theorems

Extreme Points