  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.
