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STARLIKENESS AND CONVEXITY OF CAUCHY TRANSFORMS ON REGULAR POLYGONS

Part of: Classical measure theory Miscellaneous topics of analysis in the complex domain Geometric function theory

Published online by Cambridge University Press:  05 October 2020

PENG-FEI ZHANG  and
XIN-HAN DONG
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
*
e-mail: xhdonghnsd@163.com
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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 $\widehat {\mathbb {C}}\setminus Q_n$ .

Keywords

starlikenessconvexityunivalenceCauchy transformregular polygons

MSC classification

Primary: 30E20: Integration, integrals of Cauchy type, integral representations of analytic functions
Secondary: 28A80: Fractals 30C55: General theory of univalent and multivalent functions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 103 , Issue 2 , April 2021 , pp. 291 - 302
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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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