Directional convexity of harmonic mappings

The convolution properties are discussed for the complex-valued harmonic functions in the unit disk $\mathbb{D}$ constructed from the harmonic shearing of the analytic function $\phi(z):=\int_0^z (1/(1-2\xi$\mathit{\text{e}}$^{$\mathit{\text{i}}$\mu}\cos\nu+\xi^2$\mathit{\text{e}}$^{2$\mathit{\text{i}}$\mu}))$\mathit{\text{d}}$\xi$, where $\mu$ and $\nu$ are real numbers. For any real number $\alpha$ and harmonic function $f=h+\bar{g}$, define an analytic function $f_{\alpha }:=h+{\mathit{\text{e}}}^{-2\mathit{\text{i}}\alpha }g$. Let ${\mu }_1$ and ${\mu }_2$ $({\mu }_1+{\mu }_2=\mu )$ be real numbers, and $f=h+\bar{g}$ and $F=H+\bar{G}$ be locally-univalent and sense-preserving harmonic functions such that $f_{{\mu }_1}\ast F_{{\mu }_2}=\varphi$. It is shown that the convolution $f\ast F$ is univalent and convex in the direction of $-\mu$, provided it is locally univalent and sense-preserving. Also, local-univalence of the above convolution $f\ast F$ is shown for some specific analytic dilatations of $f$ and $F$. Furthermore, if $g\equiv 0$ and both the analytic functions $f_{{\mu }_1}$ and $F_{{\mu }_2}$ are convex, then the convolution $f\ast F$ is shown to be convex. These results extends the work done by Dorff $\mathit{\text{et}}\mathit{\text{al.}}$ to a larger class of functions.