  We study expansion/contraction properties of some common classes of mappings
of the Euclidean space ${\mathbb R}^n, n\ge 2\,,$ with respect to the distance
ratio metric. The first main case is the behavior of M\"obius transformations
of the unit ball in ${\mathbb R}^n$ onto itself. In the second main case we
study the polynomials of the unit disk onto a subdomain of the complex plane.
In both cases sharp Lipschitz constants are obtained.
