Convexity in one direction of convolutions and linear combination of harmonic functions

We show that the convolution of the harmonic function $f=h+\bar{g}$, where $h(z)+e^{-2i\gamma }g(z)=z\mathrm{/}(1-e^{i\gamma }z)$ having analytic dilatation $e^{i\theta }{z}^n(0\le \theta <2\pi )$, with the mapping $f_{a,\alpha }=h_{a,\alpha }+{\bar{g}}_{a,\alpha }$, where $h_{a,\alpha }(z)=(z\mathrm{/}(1+a)-e^{i\alpha }{z}^2\mathrm{/}2)\mathrm{/}(1-e^{i\alpha }z{)}^2$, $g_{a,\alpha}(z)=(a {e}^{2{i}\alpha}z/(1+a)-{e}^{3{i}\alpha}z^2/2)/(1-{e}^{{i}\alpha}z)^2$ 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 $(a-z^2)\mathrm{/}(1-a{z}^2)$ 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.