Geometric properties of Blaschke-like maps on domains with a conic boundary
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913362961498112 |
|---|---|
| author | Fujimura, Masayo Gotoh, Yasuhiro |
| author_facet | Fujimura, Masayo Gotoh, Yasuhiro |
| contents | 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 $. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_01262 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Geometric properties of Blaschke-like maps on domains with a conic boundary Fujimura, Masayo Gotoh, Yasuhiro Complex Variables 30C20, 30J10 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 $. |
| title | Geometric properties of Blaschke-like maps on domains with a conic boundary |
| topic | Complex Variables 30C20, 30J10 |
| url | https://arxiv.org/abs/2210.01262 |