On convex combinations of harmonic mappings

[摘要] Let $\psi_{\mu,\nu}(z)=(1-2\cos\nu e^{i\mu}z+e^{2i\mu}z^{2})^{-1}$, $\mu,\nu\in[0,2\pi)$ and p be an analytic mapping with $\operatorname{Re} p>0$ on the open unit disk. We consider the sense-preserving planar harmonic mappings $f=h+\overline{g}$, which are shears of the mapping $\int_{0}^{z} \psi_{\mu,\nu}(\xi) p(\xi)\,{d}\xi$ in the direction μ. These mappings include the harmonic right half-plan mappings, vertical strip mappings, and their rotations. For various choices of dilatations $g'/h'$ of f, sufficient conditions are found for the convex combinations of these mappings to be univalent and convex in the direction μ.

[学科分类] 电力

[关键词] Harmonic mappings;Convex combination;Directional convexity