A new approach for the univalence of certain integral of harmonic mappings

Abstract

The principal goal of this paper is to extend the classical problem of find the values of $α\in \C$ for which the mappings, either $F_{\alpha }(z)={\int }_0^z(f(\zeta )/\zeta {)}^{\alpha }d\zeta$ or $f_{\alpha }(z)={\int }_0^z(f^{\prime }(\zeta ){)}^{\alpha }d\zeta$ are univalent, whenever $f$ belongs to some subclasses of univalent mappings in $\D$, but in the case of harmonic mappings, considering the \textit{shear construction} introduced by Clunie and Sheil-Small in \cite{CSS}.