The visual angle metric and quasiregular maps (Q2821749)

This paper deals with the distortion of distances between points under maps. The authors prove a Schwarz-type lemma for quasiregular maps of the unit disk involving the visual angle metric and they study the quasiconformality of a bilipschitz map with respect to the visual angle metric on convex domains. For the unit ball or the half space, they prove that a bilipschitz map with respect to the visual angle metric is also bilipschitz with respect to the hyperbolic metric. Moreover, they also obtain various inequalities relating the visual angle metric to other metrics such as the distance ratio metric and the quasihyperbolic metric. Monotonicity properties of some quotients of special functions (like complete elliptic integrals and inverse trigonometric functions) via the monotone form of l'Hospital's rule play an important role in this study.