Linear combinations of a class of harmonic univalent mappings

Abstract

A planar harmonic mapping is a complex-valued function $f: \mathbb{U} \rightarrow \mathbb{C}$ of the form $f(x+iy) = u(x,y) + iv(x,y)$, where $u$ and $v$ are both real harmonic.  Such a function can be written as $f = h + \overline{g}$, where $h$ and $g$ are both analytic; the function $\omega = g'/h'$ is called the dilatation of $f$.  We consider linear combinations of planar harmonic mappings that are the vertical shears of the asymmetrical vertical strip mappings $\varphi(z) = \tfrac{1}{2i\sin\alpha}\log\left(\frac{1+ze^{i\alpha}}{1+ze^{-i\alpha}}\right)$ with various dilatations where $\alpha_{j}\in[\frac{\pi}{2}, \pi)$.  We prove sufficient conditions for the linear combination of this class of harmonic univalent mappings to be univalent and convex in the direction of the imaginary axis.