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Note on the Coefficient Conjecture of Clunie and Sheil-Small on the univalent harmonic mapping

Omendra Mishra, Asena Çetinkaya

Abstract

In this article, we construct generalized harmonic univalent mappings and find its coefficients bounds. We present the counterexample to validate the coefficient conjecture proposed by Clunie and Sheil-Small for the class of functions $f=h+\overline{g}\in \mathcal{S}_{\mathcal{H}}$ with the help of these examples we improve the conjecture bounds of class $\mathcal{S}_{\mathcal{H}}$.

Note on the Coefficient Conjecture of Clunie and Sheil-Small on the univalent harmonic mapping

Abstract

In this article, we construct generalized harmonic univalent mappings and find its coefficients bounds. We present the counterexample to validate the coefficient conjecture proposed by Clunie and Sheil-Small for the class of functions with the help of these examples we improve the conjecture bounds of class .
Paper Structure (2 sections, 1 theorem, 30 equations)

This paper contains 2 sections, 1 theorem, 30 equations.

Table of Contents

  1. Introduction
  2. Coefficient Conjecture of univalent harmonic mappings

Key Result

Lemma 1.1

A locally univalent harmonic function $f=h+\overline{g}$ in $\mathbb{D}$ is a univalent harmonic mapping of $\mathbb{D}$ onto a domain convex in the direction $\varphi$ if and only if $h-e^{2i\varphi }g$ is a univalent analytic mapping of $\mathbb{D}$ onto a domain convex in the direction $\varphi$.

Theorems & Definitions (6)

  • Lemma 1.1
  • Conjecture 2.1
  • proof
  • Conjecture 2.2
  • proof
  • Conjecture 2.3