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ON HARMONIC CONVOLUTIONS INVOLVING A VERTICAL STRIP MAPPING
Kumar, Raj,Gupta, Sushma,Singh, Sukhjit,Dorff, Michael Korean Mathematical Society 2015 대한수학회보 Vol.52 No.1
Let $f_{\beta}=h_{\beta}+\bar{g}_{\beta}$ and $F_a=H_a+\bar{G}_a$ be harmonic mappings obtained by shearing of analytic mappings $h_{\beta}+g_{\beta}=1/(2isin{\beta})log\((1+ze^{i{\beta}})/(1+ze^{-i{\beta}})\)$, 0 < ${\beta}$ < ${\pi}$ and $H_a+G_a=z/(1-z)$, respectively. Kumar et al. [7] 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 $F_a\tilde{\ast}f_{\beta}{\in}S^0_H$ and is convex in the direction of the real axis, provided $a{\in}[(n-2)/(n+2),1)$. 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 $n{\in}\mathbb{N}$.
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On harmonic convolutions involving a vertical strip mapping
Raj Kumar,Sushma Gupta,Sukhjit Singh,Michael Dorff 대한수학회 2015 대한수학회보 Vol.52 No.1
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 $n\in \mathbb{N}$.
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