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 $f=h+\overline{g}\in {\mathcal{S}}_{\mathcal{H}}$ with the help of these examples we improve the conjecture bounds of class ${\mathcal{S}}_{\mathcal{H}}$.