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Comparison and Möbius Quasi-invariance Properties of Ibragimov’s Metric

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Abstract

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.

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Acknowledgements

This research was partly supported by National Natural Science Foundation of China (NNSFC) under Grant No.11771400 and No.11601485, and Science Foundation of Zhejiang Sci-Tech University (ZSTU) under Grant No.16062023 -Y.

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  1. School of Science, Zhejiang Sci-Tech University, Hangzhou, 310018, China

    Xiaoxue Xu, Gendi Wang & Xiaohui Zhang

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  1. Xiaoxue Xu
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  2. Gendi Wang
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  3. Xiaohui Zhang
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Correspondence to Gendi Wang.

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Communicated by Pekka Koskela.

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Xu, X., Wang, G. & Zhang, X. Comparison and Möbius Quasi-invariance Properties of Ibragimov’s Metric. Comput. Methods Funct. Theory 22, 609–627 (2022). https://doi.org/10.1007/s40315-021-00414-4

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