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.
Categories: math.CV