Convolution properties of univalent harmonic mappings convex in one direction

Abstract

Let $\ast$ and $\widetilde {\ast}$ denote the convolution of two analytic maps and that of an analytic map and a harmonic map respectively. Pokhrel [1] proved that if $f = h+\overline{g}$ is a harmonic map convex in the direction of $e^{i\gamma}$ and $\phi$ is an analytic map in the class DCP, then $f\widetilde{\ast} \phi= h\widetilde{\ast}\phi + \overline{g\widetilde{\ast}\phi}$ is also convex in the direction of $e^{i\gamma}$, provided $f\widetilde{\ast}\phi$ is locally univalent and sense-preserving. In the present paper we obtain a general condition under which $f\widetilde{\ast} \phi$ is locally univalent and sense-preserving. Some interesting applications of the general result are also presented.