Abstract
A new intrinsic metric called the t-metric is introduced. Several sharp inequalities between this metric and the most common hyperbolic type metrics are proven for various domains Gsubsetneq mathbb {R}^n. The behaviour of the new metric is also studied under a few examples of conformal and quasiconformal mappings, and the differences between the balls drawn with all the metrics considered are compared by both computational and analytical means.
Highlights
In geometric function theory, one of the topics studied deals with the variation of geometric quantities such as distances, ratios of distances, local geometry and measures of sets under different mappings
Math the hyperbolic metric and, in particular, they take into account the location of the points inside the domain
We call such a generalization of the hyperbolic metric that fulfills all the properties listed in [7, p. 79] a hyperbolic type metric and any metric whose values are affected by the boundary of the domain an intrinsic metric, regardless whether it has the other properties of a hyperbolic type metric or not
Summary
One of the topics studied deals with the variation of geometric quantities such as distances, ratios of distances, local geometry and measures of sets under different mappings. The best-known example of a metric used to study intrinsic distances is the hyperbolic metric, which is the foundation of the classical hyperbolic geometry It has several desirable analytical properties but is often very difficult to define in subdomains G of an arbitrary metric space X. Because of this issue, several other newer, more general versions of the hyperbolic metric have been introduced. These generalizations share some but not all intricate features of
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