Directional convexity of combinations of harmonic half-plane and strip mappings
Commun. Korean Math. Soc. 2022 Vol. 37, No. 1, 125-136
Published online May 13, 2021
Printed January 31, 2022
Subzar Beig, Vaithiyanathan Ravichandran
Baramulla--193 123; Tiruchirappalli -- 620 015
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 $\tilde{f}$.
Keywords : Harmonic mappings, directional convexity, harmonic shear, linear combination, strip mappings
MSC numbers : 31A05, 30C45
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