Subdomain geometry of hyperbolic type metrics

TL;DR

This paper investigates the relationships between hyperbolic-type metrics in a domain and its subdomains, establishing conditions for stronger monotonicity and inverse inequalities under certain hypotheses.

Contribution

It provides new results on domain monotonicity and inverse inequalities for quasihyperbolic and distance ratio metrics in subdomains of Euclidean spaces.

Findings

Distances in subdomains are larger than in the original domain.

Under certain conditions, stronger monotonicity results are established.

Inverse inequalities are proven under special hypotheses.

Abstract

Given a domain $G \subsetneq \Rn$ we study the quasihyperbolic and the distance ratio metrics of $G$ and their connection to the corresponding metrics of a subdomain $D \subset G$. In each case, distances in the subdomain are always larger than in the original domain. Our goal is to show that, in several cases, one can prove a stronger domain monotonicity statement. We also show that under special hypotheses we have inequalities in the opposite direction.