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

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 $ϕ\in \mathcal{A}$, let $\mathcal{W}_{H}^{-}(ϕ):=\{f=h+\bar{g} \in \mathcal{H}:h-g=ϕ\}$ and $\mathcal{W}_{H}^{+}(ϕ):=\{f=h+\bar{g} \in \mathcal{H}:h+g=ϕ\}$ 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 $f_2 \in \mathcal{W}_{H}^{-}(ϕ)$. A similar analysis is carried out if $f_1 \in \mathcal{W}_{H}^{-}(z)$ and $f_2 \in \mathcal{W}_{H}^{+}(ϕ)$. Examples of univalent harmonic mappings constructed by way of convolution are also presented.