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

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... Show more