Functional Inequalities for Metric-Preserving Functions with Respect to Intrinsic Metrics of Hyperbolic Type

We obtain functional inequalities for functions which are metric-preserving with respect to one of the following intrinsic metrics in a canonical plane domain: hyperbolic metric or some restrictions of the triangular ratio metric, respectively, of a Barrlund metric. The subadditivity turns out to be an essential property, being possessed by every function that is metric-preserving with respect to the hyperbolic metric and also by the composition with some specific function of every function that is metric-preserving with respect to some restriction of the triangular ratio metric or of a Barrlund metric. We partially answer an open question, proving that the hyperbolic arctangent is metric-preserving with respect to the restrictions of the triangular ratio metric on the unit disk to radial segments and to circles centered at origin.

Intrinsic Quasi-Metrics

The point pair function $$p_G$$ p G defined in a domain $$G\subsetneq {\mathbb {R}}^n$$ G ⊊ R n is shown to be a quasi-metric, and its other properties are studied. For a convex domain $$G\subsetneq {\mathbb {R}}^n$$ G ⊊ R n , a new intrinsic quasi-metric called the function $$w_G$$ w G is introduced. Several sharp results are established for these two quasi-metrics, and their connection to the triangular ratio metric is studied.