A new approach for the univalence of certain integral of harmonic mappings

TL;DR

This paper investigates conditions on complex parameters for which certain integral transforms of harmonic mappings are univalent, extending classical results to harmonic functions using shear construction techniques.

Contribution

It introduces a novel approach to determine univalence of integral transforms of harmonic mappings via shear construction, expanding classical univalence criteria.

Findings

Identifies specific complex values of  for univalence of integral transforms

Extends classical univalence results to harmonic mappings

Utilizes shear construction method for analysis

Abstract

The principal goal of this paper is to extend the classical problem of find the values of $\alpha\in \C$ for which the mappings, either $F_\alpha(z)=\int_0^z(f(\zeta)/\zeta)^\alpha d\zeta$ or $f_\alpha(z)=\int_0^z(f'(\zeta))^\alpha d\zeta$ are univalent, whenever $f$ belongs to some subclasses of univalent mappings in $\D$, but in the case of harmonic mappings, considering the \textit{shear construction} introduced by Clunie and Sheil-Small in \cite{CSS}.