Univalence of horizontal shear of Cesàro type transforms
This manuscript investigates the classical problem of determining conditions on the parameters $α,β\in \mathbb{C}$ for which the integral transform $$C_{αβ}[φ](z):=\int_{0}^{z} \bigg(\frac{φ(ζ)}{ζ(1-ζ)^β}\bigg)^α\,dζ $$ is also univalent in the unit disk, where $φ$ is a normalized univalent function. Additionally, whenever $φ$ belongs to some subclasses of the class of univalent functions, the univalence features of the harmonic mappings corresponding to $C_{αβ}[φ]$ 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.