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 $h(z)+e^{-2i\gamma }g(z)=z/(1-e^{i\gamma }z)$ having analytic dilatation $e^{i\theta }{z}^n(0\le \theta <2\pi )$, with the mapping $f_{a,\alpha }=h_{a,\alpha }+{\overline{g}}_{a,\alpha }$, where $h_{a,\alpha }(z)=(z/(1+a)-e^{i\alpha }{z}^2/2)/(1-e^{i\alpha }z{)}^2$, $g_{a,\alpha }(z)=(a{e}^{2i\alpha }z/(1+a)-e^{3i\alpha }{z}^2/2)/(1-e^{i\alpha }z{)}^2$ 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 $(a-z^2)/(1-a{z}^2)$ 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.