Commun. Korean Math. Soc. 2022; 37(1): 125-136
Online first article May 13, 2021 Printed January 31, 2022
https://doi.org/10.4134/CKMS.c200470
Copyright © The Korean Mathematical Society.
Subzar Beig, Vaithiyanathan Ravichandran
Baramulla--193 123; Tiruchirappalli -- 620 015
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
2019; 34(4): 1223-1228
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