Sufficient Conditions for Janowski Starlikeness

Rosihan M Ali,V Ravichandran,N Seenivasagan

doi:10.1155/2007/62925

Abstract

LetA,B,D,E∈[−1,1]and letp(z)be an analytic function defined on the open unit disk,p(0)=1. Conditions onA,B,D, andEare determined so that1+βzp'(z)being subordinated to(1+Dz)/(1+Ez)implies thatp(z)is subordinated to(1+Az)/(1+Bz). Similar results are obtained by considering the expressions1+β(zp'(z)/p(z))and1+β(zp'(z)/p2(z)). These results are then applied to obtain sufficient conditions for analytic functions to be Janowski starlike.

Highlights

Let A,B,D,E ∈ [−1,1] and let p(z) be an analytic function defined on the open unit disk, p(0) = 1
Let Ꮽ be the class of all analytic functions f (z) defined in the open unit disk U := {z ∈ C : |z| < 1} and normalized by the conditions f (0) = 0 = f (0) − 1
When 0 ≤ α < 1, S∗[1 − 2α, −1] =: S∗α is the familiar class of starlike functions of order α and S∗[1 − α, 0] = { f ∈ Ꮽ : |z f (z)/ f (z) − 1| < 1 − α (z ∈ U)} =: S∗(α)

Summary

Introduction

Let A,B,D,E ∈ [−1,1] and let p(z) be an analytic function defined on the open unit disk, p(0) = 1. Let Ꮽ be the class of all analytic functions f (z) defined in the open unit disk U := {z ∈ C : |z| < 1} and normalized by the conditions f (0) = 0 = f (0) − 1. Let S∗[A, B] denote the class of functions f ∈ Ꮽ satisfying the subordination z f f (z) (z)

Results

Conclusion