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Mathematics > Complex Variables

arXiv:2210.01262 (math)
[Submitted on 3 Oct 2022 (v1), last revised 30 Mar 2023 (this version, v2)]

Title:Geometric properties of Blaschke-like maps on domains with a conic boundary

Authors:Masayo Fujimura, Yasuhiro Gotoh
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Abstract:For a circle $ C $ contained in the unit disk, the necessary and sufficient condition for the existence of a triangle inscribed in the unit circle and circumscribed about $ C $ is known as Chapple's formula. The geometric properties of Blaschke products of degree 3 given by Daepp et al. (2002) and Frantz (2004) allow us to extend Chapple's formula to the case of ellipses in the unit disk. The main aim of this paper is to provide a further extension of Chapple's formula. Introducing a Blaschke-like map of a domain whose boundary is a conic, we extend their results to the case where the outer curve is an ellipse or a parabola. Moreover, we also give some geometrical properties for the Blaschke-like maps of degree $ d $.
Comments: 20 pages, 7 figures
Subjects: Complex Variables (math.CV)
MSC classes: 30C20, 30J10
Cite as: arXiv:2210.01262 [math.CV]
  (or arXiv:2210.01262v2 [math.CV] for this version)
  https://doi.org/10.48550/arXiv.2210.01262
Journal reference: Computational Methods and Function Theory, 24, 2, (2024) , 389--413
Related DOI: https://doi.org/10.1007/s40315-023-00499-z

Submission history

From: Masayo Fujimura [view email]
[v1] Mon, 3 Oct 2022 22:56:05 UTC (1,392 KB)
[v2] Thu, 30 Mar 2023 21:51:17 UTC (1,406 KB)
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