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

Comparison and Möbius Quasi-invariance Properties of Ibragimov’s Metric For a domain D⊊Rn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$D \subsetneq {\mathbb {R}}^{n}$$\end{document}, Ibragimov’s metric is defined as uD(x,y)=2log|x-y|+max{d(x),d(y)}d(x)d(y),x,y∈D,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}\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}\end{document}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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

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

, Volume 22 (3): 19 – Sep 1, 2022
19 pages      /lp/springer-journals/comparison-and-m-bius-quasi-invariance-properties-of-ibragimov-s-y8iyfnSdK1
Publisher
Springer Journals
Copyright © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-021-00414-4
Publisher site
See Article on Publisher Site

### Abstract

For a domain D⊊Rn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$D \subsetneq {\mathbb {R}}^{n}$$\end{document}, Ibragimov’s metric is defined as uD(x,y)=2log|x-y|+max{d(x),d(y)}d(x)d(y),x,y∈D,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}\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}\end{document}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.

### Journal

Computational Methods and Function TheorySpringer Journals

Published: Sep 1, 2022

Keywords: Ibragimov’s metric; Hyperbolic metric; Hyperbolic type metrics; Möbius transformations; 30F45 (51M10)

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