  Let ${\mathcal S^0}(H_{\gamma})$ denote the class of all univalent, harmonic,
sense-preserving and normalized mappings $f$ of the unit disk $\ID$ onto the
slanted half-plane $H_\gamma :=\{w:\,{\rm Re\,}(e^{i\gamma}w) >-1/2\}$ with an
additional condition $f_{\bar{z}}(0)=0$. Functions in this class can be
constructed by the shear construction due to Clunie and Sheil-Small which
allows by examining their conformal counterpart. Unlike the conformal case,
convolution of two univalent harmonic convex mappings in $\ID$ is not
necessarily even univalent in $\ID$. In this paper, we fix $f_0\in{\mathcal
S^0}(H_{0})$ and show that the convolutions of $f_0$ and some slanted
half-plane harmonic mapping are still convex in a particular direction. The
results of the paper enhance the interest among harmonic mappings and, in
particular, solves an open problem of Dorff, et. al. \cite{DN} in a more
general setting. Finally, we present some basic examples of functions and their
corresponding convolution functions with specified dilatations, and illustrate
them graphically with the help of MATHEMATICA software. These examples explain
the behaviour of the image domains.
