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The Apollonian metric aD of a domain \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$D\subset {\overline R}^{n}$\end{document} is rarely conformal. In fact, if it is conformal at one point then D is, up to a Möbius transformation, a complement of a convex body of constant width and if it is conformal at two points then D is a ball. We consider a quantity that measures the deviation of aD from being conformal. This quantity is essential in comparing the Apollonian metric to hyperbolic and quasihyperbolic metrics. We show that this quantity is invariant under Möbius transformations and compute it for some standard domains. We then use it to obtain sharp estimates between any two of the Apollonian, hyperbolic and quasihyperbolic metrics on such domains.
Computational Methods and Function Theory – Springer Journals
Published: Mar 1, 2004
Keywords: Apollonian metric; hyperbolic metric; quasihyperbolic metric; Möbius maps; 30F45; 30C65
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