Communications of the
Korean Mathematical Society CKMS

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 $ ilde{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 $ ilde{f}$ are the same. We investigate the univalence and directional convexity of $hat{f}$ and $ ilde{f}$ for $etainmathbb{C}$. Some sufficient conditions are found for convexity of the combination $ ilde{f}$.

Keywords: Harmonic mappings, directional convexity, harmonic shear, linear combination, strip mappings

MSC numbers: 31A05, 30C45