On harmonic convolutions involving a vertical strip mapping
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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 $n\in \mathbb{N}$.
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 $n\in \mathbb{N}$.
참고문헌 (12)
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