STARLIKENESS AND CONVEXITY OF CAUCHY TRANSFORMS ON REGULAR POLYGONS

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 $\widehat {\mathbb {C}}\setminus Q_n$ .

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Bibliographic Details
Published in:Bulletin of the Australian Mathematical Society 2021-04, Vol.103 (2), p.291-302
Main Authors: ZHANG, PENG-FEI, DONG, XIN-HAN
Format: Article
Language:English
Subjects:
Convexity
Polygons
Symmetry
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Description
Summary: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 $\widehat {\mathbb {C}}\setminus Q_n$ .
ISSN:0004-9727
1755-1633
DOI:10.1017/S0004972720000696