Comparison and Möbius quasi-invariance properties of Ibragimov’s metric. (English) Zbl 1502.30130
Summary: For a domain \(D \subsetneq{\mathbb{R}}^n \), Ibragimov’s metric is defined as
\[
u_D(x,y) = 2\, \log \frac{|x-y|+\max \{d(x),d(y)\}}{\sqrt{d(x)\,d(y)}}, \quad x,y \in D,
\]
where \(d(x)\) denotes the Euclidean distance from \(x\) to the boundary of \(D\). In this paper, we compare Ibragimov’s metric with the classical hyperbolic metric in the unit ball or in the upper half space, and prove sharp comparison inequalities between Ibragimov’s metric and some hyperbolic type metrics. We also obtain several sharp distortion inequalities for Ibragimov’s metric under some families of Möbius transformations.
MSC:
| 30F45 | Conformal metrics (hyperbolic, Poincaré, distance functions) |
| 51M10 | Hyperbolic and elliptic geometries (general) and generalizations |
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