On a family of pseudohyperbolic disks

TL;DR

This paper investigates the geometric properties of pseudohyperbolic disks within the unit disk, explicitly determining their envelope using function theoretic methods, which enhances understanding of hyperbolic geometry in complex analysis.

Contribution

It provides an explicit description of the envelope of pseudohyperbolic disks with fixed radius and variable centers in the unit disk, using function theoretic tools.

Findings

Explicit envelope of pseudohyperbolic disks determined

Connection between hyperbolic geometry and function theory clarified

Methodology applicable to related geometric problems

Abstract

Hyperbolic geometry plays an important role within function theory of the disk. For example, via the Schwarz-Pick Lemma, the isometries of the unit disk $\mathbb D$ with respect to this geometry are the conformal self-maps of $\mathbb D$. In this elementary classroom note, we are interested in the collection of the pseudohyperbolic disks $D_\rho(x,r)$ (with fixed radius $r$ and variable hyperbolic centers $-1<x<1$) and determine explicitely with function theoretic tools the enveloppe of these disks.