Lipschitz constants for a hyperbolic type metric under Möbius transformations. (English) Zbl 07893393
Summary: Let \(D\) be a nonempty open set in a metric space \((X,d)\) with \(\partial D\neq\emptyset\). Define \[h_{D,c}(x,y)=\log\bigg (1+c\frac{d(x,y)}{\sqrt{d_D(x)d_D(y)}}\bigg),\] where \(d_D(x)=d(x,\partial D)\) is the distance from \(x\) to the boundary of \(D\). For every \(c\geq 2\), \(h_{D,c}\) is a metric. We study the sharp Lipschitz constants for the metric \(h_{D,c}\) under Möbius transformations of the unit ball, the upper half space, and the punctured unit ball.
MSC:
| 51M10 | Hyperbolic and elliptic geometries (general) and generalizations |
| 30C65 | Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations |
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