Integral transform for Logharmonic mappings

TL;DR

This paper extends classical integral transform techniques to logharmonic mappings, analyzing conditions for univalence and generalizing the shear construction method within geometric function theory.

Contribution

It introduces an extension of the shear construction to logharmonic mappings and investigates the univalence of integral transforms in this context.

Findings

Extended shear construction for logharmonic mappings.

Identified conditions for univalence of integral transforms.

Generalized classical methods to a broader class of functions.

Abstract

Bieberbach's conjecture was very important in the development of Geometric Function Theory, not only because of the result itself, but also due to the large amount of methods that have been developed in search of its proof, it is in this context that the integral transformations of the type $f_\alpha(z)=\int_0^z(f(\zeta)/\zeta)^\alpha d\zeta$ or $F_\alpha(z)=\int_0^z(f'(\zeta))^\alpha d\zeta$ appear. In this notes we extend the classical problem of finding the values of $\alpha\in\mathbb{C}$ for which either $f_\alpha$ or $F_\alpha$ are univalent, whenever $f$ belongs to some subclasses of univalent mappings in $\mathbb D$, to the case of logharmonic mappings, by considering the extension of the \textit{shear construction} introduced by Clunie and Sheil-Small in \cite{CSS} to this new scenario.