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

TL;DR

This paper extends Chapple's formula to Blaschke-like maps on domains with conic boundaries, such as ellipses and parabolas, and explores their geometric properties and higher-degree cases.

Contribution

It generalizes Chapple's formula to conic-boundary domains and analyzes geometric properties of Blaschke-like maps of various degrees.

Findings

Extended Chapple's formula to elliptical and parabolic boundaries.

Derived geometric properties of Blaschke-like maps.

Analyzed properties of higher-degree Blaschke-like maps.

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 $.