  We show that the convolution of the harmonic function $f=h+\bar{g}$, where
$h(z)+{e}^{-2{i}\gamma}g(z)=z/(1-{e}^{{i}\gamma}z)$ having analytic dilatation
${e}^{{i}\theta} z^n (0\leq\theta<2\pi)$, with the mapping
$f_{a,\alpha}=h_{a,\alpha}+\overline{g}_{a,\alpha}$, where
$h_{a,\alpha}(z)=(z/(1+a)-{e}^{{i}\alpha}z^2/2)/(1-{e}^{{i}\alpha}z)^2$,
$g_{a,\alpha}(z)=(a
{e}^{2{i}\alpha}z/(1+a)-{e}^{3{i}\alpha}z^2/2)/(1-{e}^{{i}\alpha}z)^2$ is
convex in the direction $-(\alpha+\gamma)$. We also show that the convolution
of $f_{a,\alpha}$ with the right half-plane mapping having dilatation
$(a-z^2)/(1-az^2)$ is convex in the direction $-\alpha$. Finally, we introduce
a family of univalent harmonic mappings and find out sufficient conditions for
convexity along imaginary-axis of the linear combinations of harmonic functions
of this family.
