For a domain \( D \subsetneq {\mathbb {R}}^{n} \), Ibragimov’s metric is defined as

$$\begin{aligned} u_{D}(x,y) = 2\, \log \frac{|x-y|+\max \{d(x),d(y)\}}{\sqrt{d(x)\,d(y)}}, \quad \quad x,y \in D, \end{aligned}$$

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.

中文翻译：

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Ibragimov 度量的比较和莫比乌斯拟不变性性质

对于域\( D \subsetneq {\mathbb {R}}^{n} \)，Ibragimov 的度量定义为

$$\begin{aligned} u_{D}(x,y) = 2\, \log \frac{|xy|+\max \{d(x),d(y)\}}{\sqrt{d (x)\,d(y)}}, \quad \quad x,y \in D, \end{aligned}$$

其中d ( x ) 表示从x到D边界的欧几里得距离。在本文中，我们将 Ibragimov 度量与单位球或上半空间中的经典双曲度量进行了比较，并证明了 Ibragimov 度量与一些双曲类型度量之间的尖锐比较不等式。我们还获得了一些莫比乌斯变换家族下 Ibragimov 度量的几个尖锐的失真不等式。