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

中文翻译：

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正多边形上柯西变换的星象性和凸性

为了$n\geq 3$， 让$Q_n\subset \mathbb {C}$是任意正则n边多边形。我们证明了柯西变换$F_{Q_n}$的归一化二维 Lebesgue 测度$Q_n$是单价的和星状的但不是凸的$\widehat {\mathbb {C}}\setminus Q_n$.