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Keywords:
hyperbolic geometry; hyperbolic metric; intrinsic geometry; Möbius metric; quasiregular mapping; triangular ratio metric
Summary:
The Möbius metric $\delta _G$ is studied in the cases, where its domain $G$ is an open sector of the complex plane. We introduce upper and lower bounds for this metric in terms of the hyperbolic metric and the angle of the sector, and then use these results to find bounds for the distortion of the Möbius metric under quasiregular mappings defined in sector domains. Furthermore, we numerically study the Möbius metric and its connection to the hyperbolic metric in polygon domains.
References:
[1] Chen, J., Hariri, P., Klén, R., Vuorinen, M.: Lipschitz conditions, triangular ratio metric, and quasiconformal maps. Ann. Acad. Sci. Fenn., Math. 40 (2015), 683-709. DOI 10.5186/aasfm.2015.4039 | MR 3409699 | Zbl 1374.30069
[2] Gehring, F. W., Hag, K.: The Ubiquitous Quasidisk. Mathematical Surveys and Monographs 184. AMS, Providence (2012). DOI 10.1090/surv/184 | MR 2933660 | Zbl 1267.30003
[3] Gehring, F. W., Osgood, B. G.: Uniform domains and the quasi-hyperbolic metric. J. Anal. Math. 36 (1979), 50-74. DOI 10.1007/BF02798768 | MR 0581801 | Zbl 0449.30012
[4] Gehring, F. W., Palka, B. P.: Quasiconformally homogeneous domains. J. Anal. Math. 30 (1976), 172-199. DOI 10.1007/BF02786713 | MR 0437753 | Zbl 0349.30019
[5] Hariri, P., Klén, R., Vuorinen, M.: Local convexity of metric balls. Monatsh. Math. 186 (2018), 281-298. DOI 10.1007/s00605-017-1142-y | MR 3808654 | Zbl 1395.51019
[6] Hariri, P., Klén, R., Vuorinen, M.: Conformally Invariant Metrics and Quasiconformal Mappings. Springer Monographs in Mathematics. Springer, Cham (2020). DOI 10.1007/978-3-030-32068-3 | MR 4179585 | Zbl 1450.30003
[7] Hariri, P., Vuorinen, M., Zhang, X.: Inequalities and bi-Lipschitz conditions for the triangular ratio metric. Rocky Mt. J. Math. 47 (2017), 1121-1148. DOI 10.1216/RMJ-2017-47-4-1121 | MR 3689948 | Zbl 1376.30019
[8] Hästö, P.: A new weighted metric: The relative metric. I. J. Math. Anal. Appl. 274 (2002), 38-58. DOI 10.1016/S0022-247X(02)00219-6 | MR 1936685 | Zbl 1019.54011
[9] Hästö, P., Ibragimov, Z., Minda, D., Ponnusamy, S., Swadesh, S.: Isometries of some hyperbolic-type path metrics, and the hyperbolic medial axis. In the Tradition of Ahlfors-Bers. IV Contemporary Mathematics 432. AMS, Providence (2007), 63-74. DOI 10.1090/conm/432 | MR 2342807 | Zbl 1147.30029
[10] Lindén, H.: Quasihyperbolic geodesics and uniformity in elementary domains. Ann. Acad. Sci. Fenn. Math. Diss. 146 (2005), 50 pages. MR 2183008 | Zbl 1092.51003
[11] Nasser, M. M. S., Rainio, O., Vuorinen, M.: Condenser capacity and hyperbolic perimeter. Comput. Math. Appl. 105 (2022), 54-74. DOI 10.1016/j.camwa.2021.11.016 | MR 4345260 | Zbl 07447596
[12] Rainio, O.: Intrinsic quasi-metrics. Bull. Malays. Math. Sci. Soc. (2) 44 (2021), 2873-2891. DOI 10.1007/s40840-021-01089-9 | MR 4296316 | Zbl 1476.51019
[13] Rainio, O., Vuorinen, M.: Introducing a new intrinsic metric. Result. Math. 77 (2022), Article ID 71, 18 pages. DOI 10.1007/s00025-021-01592-2 | MR 4379104 | Zbl 07490182
[14] Rainio, O., Vuorinen, M.: Triangular ratio metric under quasiconformal mappings in sector domains. To appear in Comput. Methods Funct. Theory (2022). DOI 10.1007/s40315-022-00447-3 | MR 4584499
[15] Seittenranta, P.: Möbius-invariant metrics. Math. Proc. Camb. Philos. Soc. 125 (1999), 511-533. DOI 10.1017/S0305004198002904 | MR 1656825 | Zbl 0917.30015
[16] Väisälä, J.: Lectures on $n$-Dimensional Quasiconformal Mappings. Lecture Notes in Mathematics 229. Springer, Berlin (1971). DOI 10.1007/BFb0061216 | MR 0454009 | Zbl 0221.30031
[17] Vuorinen, M.: Conformal Geometry and Quasiregular Mappings. Lecture Notes in Mathematics 1319. Springer, Berlin (1988). DOI 10.1007/BFb0077904 | MR 0950174 | Zbl 0646.30025
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