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

TL;DR

This paper constructs generalized harmonic univalent mappings, provides coefficient bounds, and offers a counterexample to the Clunie and Sheil-Small coefficient conjecture, leading to improved bounds for the class of harmonic univalent functions.

Contribution

It introduces generalized harmonic univalent mappings, constructs counterexamples, and refines existing coefficient bounds for harmonic univalent functions.

Findings

Counterexample invalidates the original conjecture.

Improved bounds for coefficients of harmonic univalent functions.

Construction of generalized harmonic univalent mappings.

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 $\mathcal{S}_{\mathcal{H}}$.