Univalence of horizontal shear of Cesàro type transforms

Abstract

This manuscript investigates the classical problem of determining conditions on the parameters α,β∈C for which the integral transformCαβ[φ](z):=∫0z(φ(ζ)ζ(1−ζ)β)αdζ 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 Cαβ[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.