Convolution and Convex Combination of Harmonic Mappings

Abstract

For each $a\in [0,1)$, the convolution of the right half-plane harmonic mapping having dilatation $(a-z)/(1-az)$ with the right half-plane harmonic mapping having dilatation $\left( a-z^2\right) /\left( 1-az^2\right)$ or $-(a-z)^2/(1-az)^2$ is shown to be univalent and convex in the real direction. In addition, sufficient conditions are found for the convex combination of the harmonic mappings $f_i=h_i+\overline{g_i}$, $i=1,2$ satisfying

$$\begin{aligned} h_i(z)+g_i(z)=\int _0^z\frac{(1-w^{2^{n}})(1+w^{2^n}+\alpha _i w^{2^{n-1}})}{\left( 1-w^2\right) (1+w^{2^{n+1}})}\mathrm{{d}}w,\quad -1\le \alpha _i\le 1, \quad n\ge 1, \end{aligned}$$

to be univalent and convex in the real direction.