Univalence and convexity in one direction of the convolution of harmonic mappings

Abstract

Let $\mathcal{H}$ denote the class of all complex-valued harmonic functions $f$ in the open unit disk normalized by $f(0)=0=f_{z}(0)-1=f_{\bar{z}}(0)$, and let $\mathcal{A}$ be the subclass of $\mathcal{H}$ consisting of normalized analytic functions. For $\phi \in \mathcal{A}$, let $\mathcal{W}_{H}^{-}(\phi):=\{f=h+\bar{g} \in \mathcal{H}:h-g=\phi\}$ and $\mathcal{W}_{H}^{+}(\phi):=\{f=h+\bar{g} \in \mathcal{H}:h+g=\phi\}$ be subfamilies of $\mathcal{H}$. In this paper, we shall determine the conditions under which the harmonic convolution $f_1*f_2$ is univalent and convex in one direction if $f_1 \in \mathcal{W}_{H}^{-}(z)$ and $f_2 \in \mathcal{W}_{H}^{-}(\phi)$. A similar analysis is carried out if $f_1 \in \mathcal{W}_{H}^{-}(z)$ and $f_2 \in \mathcal{W}_{H}^{+}(\phi)$. Examples of univalent harmonic mappings constructed by way of convolution are also presented.