The distortion of six different intrinsic metrics and quasi-metrics under
conformal and quasiregular mappings is studied in a few simple domains
GβRn. The already known inequalities between the
hyperbolic metric and these intrinsic metrics for points x,y in the unit ball
Bn are improved by limiting the absolute values of the points x,y
and the new results are then used to study the conformal distortion of the
intrinsic metrics. For the triangular ratio metric between two points
x,yβBn, the conformal distortion is bounded in terms of the
hyperbolic midpoint and the hyperbolic distance of x,y. Furthermore,
quasiregular and quasiconformal mappings are studied, and new sharp versions of
the Schwarz lemma are introduced.Comment: 25 pages, 1 figur