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STARLIKENESS AND CONVEXITY OF CAUCHY TRANSFORMS ON REGULAR POLYGONS

Published online by Cambridge University Press:  05 October 2020

PENG-FEI ZHANG
Affiliation:
Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China), College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan, China e-mail: pfzhang@link.cuhk.edu.hk
XIN-HAN DONG*
Affiliation:
Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China), College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan, China

Abstract

For $n\geq 3$ , let $Q_n\subset \mathbb {C}$ be an arbitrary regular n-sided polygon. We prove that the Cauchy transform $F_{Q_n}$ of the normalised two-dimensional Lebesgue measure on $Q_n$ is univalent and starlike but not convex in $\widehat {\mathbb {C}}\setminus Q_n$ .

Type
Research Article
Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

This research is supported in part by the NNSF of China (No. 11831007).

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