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2016 An application of Cohn's rule to convolutions of univalent harmonic mappings
Raj Kumar, Sushma Gupta, Sukhjit Singh, Michael Dorff
Rocky Mountain J. Math. 46(2): 559-570 (2016). DOI: 10.1216/RMJ-2016-46-2-559

Abstract

Dorff et al.~\cite {do and no} proved that the harmonic convolutions of the standard right half-plane mapping $F_0=H_0+\overline {G}_0$ (where $H_0+G_0=z/(1-z)$ and $G_0'=-zH_0'$) and mappings $f_\beta =h_\beta +\overline {g}_{\beta }$ (where $f_\beta $ are obtained by shearing of analytic vertical strip mappings with dilatation $e^{i\theta }z^n$, $n=1,2$, $\theta \in \mathbb {R}$) are in $S_H^0$ and are convex in the direction of the real axis. In this paper, by using Cohn's rule, we generalize this result by replacing the standard right half-plane mapping $F_0$ with a family of right half-plane mappings $F_a=H_a+\overline {G}_a$ (with $H_a+G_a=z/(1-z)$ and $G'_a/H'_a= {(a-z)}/{(1-az)}$, $a\in (-1,1)$) and including the cases $n=3$ and $n=4$ (in addition to $n=1$ and $n=2$) for dilatations of $f_\beta $.

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Raj Kumar. Sushma Gupta. Sukhjit Singh. Michael Dorff. "An application of Cohn's rule to convolutions of univalent harmonic mappings." Rocky Mountain J. Math. 46 (2) 559 - 570, 2016. https://doi.org/10.1216/RMJ-2016-46-2-559

Information

Published: 2016
First available in Project Euclid: 26 July 2016

zbMATH: 1359.31002
MathSciNet: MR3529083
Digital Object Identifier: 10.1216/RMJ-2016-46-2-559

Subjects:
Primary: 30C45

Rights: Copyright © 2016 Rocky Mountain Mathematics Consortium

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Vol.46 • No. 2 • 2016
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