Univalence of horizontal shear of Ces\`aro type transforms

TL;DR

This paper explores conditions on complex parameters for which a specific integral transform preserves univalence in the unit disk, and examines the univalence of related harmonic mappings, using Becker's criteria and shear construction.

Contribution

It provides new conditions on parameters for the univalence of Cesàro type transforms and analyzes their harmonic mappings within subclasses of univalent functions.

Findings

Derived parameter conditions for univalence of the transform.

Established univalence of harmonic mappings related to the transform.

Presented examples of non-trivial univalent harmonic mappings.

Abstract

This manuscript investigates the classical problem of determining conditions on the parameters $\alpha,\beta \in \mathbb{C}$ for which the integral transform $$C_{\alpha\beta}[\varphi](z):=\int_{0}^{z} \bigg(\frac{\varphi(\zeta)}{\zeta (1-\zeta)^{\beta}}\bigg)^\alpha\,d\zeta $$ is also univalent in the unit disk, where $\varphi$ is a normalized univalent function. Additionally, whenever $\varphi$ belongs to some subclasses of the class of univalent functions, the univalence features of the harmonic mappings corresponding to $C_{\alpha\beta}[\varphi]$ and its rotations are derived. As applications to our primary findings, a few non-trivial univalent harmonic mappings are also provided. The primary tools employed in this manuscript are Becker's univalence criteria and the shear construction developed by Clunie and Sheil-Small.