Univalence of horizontal shear of Cesàro type transforms

This manuscript investigates the classical problem of determining conditions on the parameters $\alpha ,\beta \in \mathbb{C}$ for which the integral transform $$C_{\alpha \beta }[\phi ](z):={\int }_0^z(\frac{\phi (\zeta )}{\zeta (1-\zeta {)}^{\beta }}{)}^{\alpha }d\zeta$$ is also univalent in the unit disk, where $\phi$ is a normalized univalent function. Additionally, whenever $\phi$ belongs to some subclasses of the class of univalent functions, the univalence features of the harmonic mappings corresponding to $C_{\alpha \beta }[\phi ]$ 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.