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

Abstract

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.