Coefficient Problems in the Subclasses of Close-to-Star Functions

Adam Lecko,Young Jae Sim

doi:10.1007/s00025-019-1030-y

Adam Lecko, Young Jae Sim

Open Access

https://doi.org/10.1007/s00025-019-1030-y

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Abstract

For two subclasses of close-to-star functions we estimate early logarithmic coefficients, coefficients of inverse functions, Hankel determinant H_{2,2} and Zalcman functional J_{2,3}. All results are sharp.

Highlights

Given r > 0, let Dr := {z ∈ C: |z| < r}, and let D := D1
Let S be the subclass of A of all univalent functions and S∗ be the subclass of S of all starlike functions, namely, f ∈ S∗ if f ∈ A and zf (z)
Denote by CST the class of all close-to-star functions introduced by Reade [30]

Summary

Introduction

Given r > 0, let Dr := {z ∈ C: |z| < r}, and let D := D1. Let D := {z ∈ C: |z| ≤ 1} and T := ∂D. A function f ∈ A is called close-to-star if there exist g ∈ S∗ and β ∈ R such that eiβf (z) g(z)

Results

Logarithmic Coefficients

Zalcman Functional and Hankel Determinant

Inverse Coefficients