Convolution properties of univalent harmonic mappings convex in one direction

Abstract

Let $\ast$ and $\tilde{\ast }$ 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 $e^{i{\hat{I}}^3}$ and $\ddot{I}•$ is an analytic map in the class DCP, then $f\tilde{\ast }\ddot{I}•=h\tilde{\ast }\ddot{I}•+\overline{g\tilde{\ast }\ddot{I}•}$ is also convex in the direction of $e^{i{\hat{I}}^3}$, provided $f\tilde{\ast }\ddot{I}•$ is locally univalent and sense-preserving. In the present paper we obtain a general condition under which $f\tilde{\ast }\ddot{I}•$ is locally univalent and sense-preserving. Some interesting applications of the general result are also presented.

7 Pages