On Certain Subclass of Analytic Functions with Fixed Point

T Al-Hawary,B A Frasin,M Darus

doi:10.1155/2013/312387

Abstract

A new certain differential operatorTα,β,δ,λmf(z)and a subclassSm,w*(α,β,γ,δ,λ)are introduced for functions of the formf(z)=(z−w)−∑n=2∞an(z−w)nwhich are univalent in the unit discU={z∈ℂ:|z|<1}. In this paper, we obtain coefficient inequalities, distortion theorem, closure theorems, and class preserving integral operators of functions belonging to the classSm,w*(α,β,γ,δ,λ).

Highlights

Let H(U) be the set of functions which are regular in the unit disc U = {z ∈ C : |z| < 1}, A = {f ∈ H(U) : f(0) = f󸀠(0) − 1 = 0} and S = {f ∈ A : f is univalent in U}
We denote by STw(γ) the class of all starlike functions of order γ
Suppose that the functions fj(z) (j = 1, 2) defined by (35) be in the class Sm∗,w(α, β, γ, δ, λ), it is sufficient to prove that the function

Summary

Introduction

In [1], Kanas and Rønning introduced the following classes: Sw = {f ∈ A (w) : f is univalent in U} , STw = Sw∗. These classes are extensively studied by Acu and Owa [2]. We denote by STw(γ) the class of all starlike functions of order γ. For the function f(z) in the class Sw, we define the following new differential operator: T0f (z) = f (z) , Tα1,β,δ,λf (z) = (1 − β (λ − α)) f (z). With the help of the differential operator Tαm,β,δ,λ, we define the class Sm,w(α, β, γ, δ, λ) as follows. For the class Sm∗ ,w(α, β, γ, δ, λ) is considered

Coefficient Inequalities

Distortion Theorem

Radii of Starlikeness and Convexity

Closure Theorems

Integral Operators