A subclass of harmonic univalent mappings with a restricted analytic part

TL;DR

This paper introduces a new subclass of univalent harmonic mappings with a restricted analytic part, analyzing its properties such as coefficient bounds, growth, distortion, and Bloch's constant.

Contribution

It defines a novel subclass of harmonic univalent functions with restricted analytic parts and investigates its fundamental geometric and analytic properties.

Findings

Coefficient estimates established

Growth and distortion properties derived

Bounds for Bloch's constant obtained

Abstract

In this article, a subclass of univalent harmonic mapping is introduced by restricting its analytic part to lie in the class $\mathcal{S}^{\delta}[\alpha]$, $0\leq \alpha < 1$, $-\infty < \delta < \infty$ which has been introduced and studied by Kumar \cite{Kumar87} (see also \cite{Mishra95}, \cite{MishraChoudhury95}, \cite{MishraDas96}, \cite{MishraGochhayat06}). Coefficient estimations, growth and distortion properties, area theorem and covering estimates of functions in the newly defined class have been established. Furthermore, we also found bounds for the Bloch's constant for all functions in that family.