Some applications of differential subordination for certain starlike functions

Rahim Kargar,Lucyna Trojnar-Spelina

doi:10.3906/mat-1903-28

Abstract

Let $\mathcal{S}^*(q_c)$ denote the class of functions $f$ analytic in the open unit disc $\Delta$, normalized by the condition $f(0)=0=f'(0)-1$ and satisfying the following inequality $\left \left(\frac{zf'(z)}{f(z)}\right)^2-1\right < c \quad (z\in\Delta, 0

Highlights

Let ∆ be the open unit disc in the complex plane C, i.e. ∆ = {z ∈ C : |z| < 1} and H(∆) be the class of functions that are analytic in ∆
The set of all univalent functions f in ∆ is denoted by U
We say that the function f ∈ A belongs to the class S∗(qc), if it satisfies the following condition ( zf ′(z) )2 − 1 < c (z ∈ ∆)

Summary

Introduction

∆ = {z ∈ C : |z| < 1} and H(∆) be the class of functions that are analytic in ∆. Let A ⊂ H(∆) be the class of functions that have the following Taylor–Maclaurin series expansion. Let f and g belong to class H(∆). The familiar class of starlike functions in ∆ is denoted by S∗. We denote by K the class of convex functions in ∆. The class of close-to-convex functions is denoted by C. We say that the function f ∈ A belongs to the class S∗(qc) , if it satisfies the following condition ( zf ′(z) )2 − 1 < c (z ∈ ∆). Theorem 2.1 Let p be an analytic function in ∆ with p(0) = 1 , |A| ≤ 1, |B| < 1, 0 < c ≤ 1.

Now consider the function ckγ

Proof Consider

Suppose that