On a family of pseudohyperbolic disks

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.

arXiv comment8 pages, elementary classroom paper in hyperbolic geometry resp. function theory