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The visual angle metric and quasiregular maps


Authors: Gendi Wang and Matti Vuorinen
Journal: Proc. Amer. Math. Soc. 144 (2016), 4899-4912
MSC (2010): Primary 30C65; Secondary 30F45
DOI: https://doi.org/10.1090/proc/13188
Published electronically: June 17, 2016
MathSciNet review: 3544538
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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 quasiconformality 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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Additional Information

Gendi Wang
Affiliation: School of Sciences, Zhejiang Sci-Tech University, Hangzhou 310018, People’s Republic of China
Email: gendi.wang@zstu.edu.cn

Matti Vuorinen
Affiliation: Department of Mathematics and Statistics, University of Turku, Turku 20014, Finland
Email: vuorinen@utu.fi

DOI: https://doi.org/10.1090/proc/13188
Received by editor(s): May 4, 2015
Received by editor(s) in revised form: January 19, 2016
Published electronically: June 17, 2016
Additional Notes: The research of both authors was supported by the Academy of Finland, Project 2600066611
The first author was also supported by the Turku University Foundation, the Academy of Finland, Project 268009, and the Science Foundation of Zhejiang Sci-Tech University(ZSTU)
The authors thank Dr. Xiaohui Zhang for useful discussions and helpful comments and the referee for valuable corrections.
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2016 American Mathematical Society