Directional Convexity of Combinations of Harmonic Half-Plane and Strip Mappings

Abstract

For $k=1,2$, let $f_k=h_k+\overline{g_k}$ be normalized harmonic right half-plane or vertical strip mappings. We consider the convex combination $\hat{f}=\eta f_1+(1-\eta )f_2=\eta h_1+(1-\eta )h_2+\overline{\overline{\eta }{g}_1+(1-\overline{\eta })g_2}$ and the combination $\tilde{f}=\eta h_1+(1-\eta )h_2+\overline{\eta g_1+(1-\eta )g_2}$. For real $\eta$, the two mappings $\hat{f}$ and $\tilde{f}$ are the same. We investigate the univalence and directional convexity of $\hat{f}$ and $\tilde{f}$ for $\eta \in \mathbb{C}$. Some sufficient conditions are found for convexity of the combination $\tilde{f}$.