A1 Refereed original research article in a scientific journal
THE VISUAL ANGLE METRIC AND QUASIREGULAR MAPS
Authors: Wang GD, Vuorinen M
Publisher: AMER MATHEMATICAL SOC
Publication year: 2016
Journal: Proceedings of the American Mathematical Society
Journal name in source: PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
Journal acronym: P AM MATH SOC
Volume: 144
Issue: 11
First page : 4899
Last page: 4912
Number of pages: 14
ISSN: 0002-9939
DOI: https://doi.org/10.1090/proc/13188
Abstract
The distortion of distances between points under maps is studied. We first prove a Schwarz-type lemma for quasiregular maps of the unit disk involving the visual angle metric. Then we investigate conversely the quasi-conformality of a bilipschitz map with respect to the visual angle metric on convex domains. For the unit ball or half space, we prove that a bilipschitz map with respect to the visual angle metric is also bilipschitz with respect to the hyperbolic metric. We also obtain various inequalities relating the visual angle metric to other metrics such as the distance ratio metric and the quasihyperbolic metric.
The distortion of distances between points under maps is studied. We first prove a Schwarz-type lemma for quasiregular maps of the unit disk involving the visual angle metric. Then we investigate conversely the quasi-conformality of a bilipschitz map with respect to the visual angle metric on convex domains. For the unit ball or half space, we prove that a bilipschitz map with respect to the visual angle metric is also bilipschitz with respect to the hyperbolic metric. We also obtain various inequalities relating the visual angle metric to other metrics such as the distance ratio metric and the quasihyperbolic metric.
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