Certain Conditions for Starlikeness of Analytic Functions of Koebe Type

Saibah Siregar,Maslina Darus

doi:10.1155/2011/679704

Abstract

For α ≥ 0, λ > 0, we consider the M(α, λ) b of normalized analytic α − λ convex functions defined in the open unit disc 𝕌. In this paper, we investigate the class M(α, λ) b, that is, , with fb is Koebe type, that is, fb(z): = z/(1 − zn) b. The subordination result for the aforementioned class will be given. Further, by making use of Jack′s Lemma as well as several differential and other inequalities, the authors derived sufficient conditions for starlikeness of the class M(α, λ) b of n‐fold symmetric analytic functions of Koebe type. Relevant connections of the results presented here with those given in the earlier works are also indicated.

Highlights

Let A denote the class of normalized analytic functions of the form ∞fz z ak zk, k2 which are analytic in the open unit disk U {z : |z| < 1}
We investigate the class M α, λ b, that is, Re{ zfb z /fb z 1 − α α 1 − λ zfb z /fb z αλ 1 zfb z /fb z } > 0, with fb is Koebe type, that is, fb z : z/ 1−zn b
If the functions f and g are analytic in U, we say that the function f is subordinate to g, or g is superordinate to f written as f ≺ g if there exist a function w z analytic U, such that |w z | < 1 and z ∈ U, and w 0 0 with fzgwz in U

Summary

Introduction

Fz z ak zk , k2 which are analytic in the open unit disk U {z : |z| < 1}. as usual, let. > 0, z ∈ U , be the familiar classes of starlike functions in U and convex functions in U, respectively. The class M α was first introduced by Mocanu 1 , which was known as the class of α convex or α-starlike functions. Miller et al 2 studied this class and showed that M α is a subclass of S∗ for any real number α and that M α is a subclass of K for α ≥ 1. Motivated essentially by the aforementioned earlier works, we aim here at deriving sufficient conditions for starlikeness of n-fold symmetric function fb of the Koebe type, defined by fb z :. A function f z given by 1.1 is said to be in the class M α, λ b, for α ≥ 0, λ > 0, if the following conditions are satisfied: Re zfb z fb z. The following result popularly known as Jack’s Lemma will be required in the derivation of our result Theorem 4.1 below

Preliminaries

The Subordination Result

Applications of Differential Inequalities