Bulletin of the
Korean Mathematical Society
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Bull. Korean Math. Soc. 2015; 52(1): 105-123

Printed January 31, 2015

https://doi.org/10.4134/BKMS.2015.52.1.105

Copyright © The Korean Mathematical Society.

On harmonic convolutions involving a vertical strip mapping

Raj Kumar, Sushma Gupta, Sukhjit Singh, and Michael Dorff

Sant Longowal Institute of Engineering and Technology, Sant Longowal Institute of Engineering and Technology, Sant Longowal Institute of Engineering and Technology, Brigham Young University

Abstract

Let $ f_\beta=h_\beta+\overline{g}{_\beta}$ and $F_a=H_a+\overline{G}_a$ be harmonic mappings obtained by shearing of analytic mappings $$h_\beta+g_\beta={1}/{(2i{\sin}\beta)}\log\left({(1+ze^{i\beta})}/{(1+ze^{-i\beta})}\right),~0<\beta<\pi$$ and $H_a+G_a={z}/{(1-z)}$, respectively. Kumar \emph{et al.} \cite{ku and gu} conjectured that if $\omega(z)=e^{i\theta}z^n (\theta\in\mathbb{R},\,\, n\in \mathbb{N})$ and $ \omega_a(z)={(a-z)}/{(1-az)},\,a\in(-1,1)$ are dilatations of $f_\beta$ and $F_a$, respectively, then $F_a\widetilde\ast f_\beta \, \in S_H^0$ and is convex in the direction of the real axis, provided $a\in \left[{(n-2)}/{(n+2)},1\right)$. They claimed to have verified the result for $n=1,2,3$ and $4$ only. In the present paper, we settle the above conjecture, in the affirmative, for $\beta=\pi/2$ and for all $n\in \mathbb{N}$.

Keywords: univalent harmonic mapping, vertical strip mapping, harmonic convolution

MSC numbers: 30C45

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