Title: Directional convexity of harmonic mappings,
Abstract: 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 $\phi(z):=\int_0^z
(1/(1-2\xi\textit{e}^{\textit{i}\mu}\cos\nu+\xi^2\textit{e}^{2\textit{i}\mu}))\textit{d}\xi$,
where $\mu$ and $\nu$ are real numbers. For any real number $\alpha$ and
harmonic function $f=h+\overline{g}$, define an analytic function
$f_{\alpha}:=h+\textit{e}^{-2\textit{i}\alpha}g$. Let $\mu_1$ and $\mu_2$
$(\mu_1+\mu_2=\mu)$ be real numbers, and $f=h+\overline{g}$ and
$F=H+\overline{G}$ be locally-univalent and sense-preserving harmonic functions
such that $f_{\mu_1}*F_{\mu_2}=\phi$. It is shown that the convolution $f*F$ is
univalent and convex in the direction of $-\mu$, provided it is locally
univalent and sense-preserving. Also, local-univalence of the above convolution
$f*F$ is shown for some specific analytic dilatations of $f$ and $F$.
Furthermore, if $g\equiv0$ and both the analytic functions $f_{\mu_1}$ and
$F_{\mu_2}$ are convex, then the convolution $f*F$ is shown to be convex. These
results extends the work done by Dorff \textit{et al.} to a larger class of
functions.