On harmonic convolutions involving a vertical strip mapping
Bull. Korean Math. Soc. 2015 Vol. 52, No. 1, 105-123
https://doi.org/10.4134/BKMS.2015.52.1.105
Printed January 31, 2015
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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