Starlikeness and convexity of Cauchy transforms on regular polygons

Authors

  • P.-F. Zhang Hunan Normal University
  • X.-H. Dong Hunan Normal University

Keywords:

starlikeness, convexity, univalence, Cauchy transform, regular polygons

Abstract

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

Author Biographies

P.-F. Zhang, Hunan Normal University

Department of Mathematics

X.-H. Dong, Hunan Normal University

Department of Mathematics

Published

2021-03-10

Issue

Vol. 103 No. 2 (2021)

Section

Articles
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