Growth of some hyperbolic type distances and starlikeness of metric balls. (English) Zbl 1515.30103
Summary: The purpose of this paper is to study geometric properties of some hyperbolic type metrics, including the growth of distances along Euclidean line segments, bounds of metrics, and the starlikeness and convexity of metric balls.
MSC:
| 30F45 | Conformal metrics (hyperbolic, Poincaré, distance functions) |
| 51M10 | Hyperbolic and elliptic geometries (general) and generalizations |
References:
| [1] | Beardon, AF; Minda, D.; Ponnusamy, S.; Sugawa, T.; Vuorinen, M., The hyperbolic metric and geometric function theory, Quasiconformal Mappings And Their Applications, 9-56 (2007), New Delhi: Narosa Publishing House, New Delhi · Zbl 1208.30001 |
| [2] | Cao J., Zhang X.: Constructions of strongly hyperbolic spaces, Manuscript. (2022), 9 pp |
| [3] | Chen, J.; Hariri, P.; Klén, R.; Vuorinen, M., Lipschitz conditions, triangular ratio metric, and quasiconformal maps, Ann. Acad. Sci. Fenn. Math., 40, 683-709 (2015) · Zbl 1374.30069 · doi:10.5186/aasfm.2015.4039 |
| [4] | Hariri, P.; Klén, R.; Vuorinen, M., Local convexity of metric balls, Monatsh. Math., 186, 281-298 (2018) · Zbl 1395.51019 · doi:10.1007/s00605-017-1142-y |
| [5] | Hariri, P.; Klén, R.; Vuorinen, M.; Zhang, X., Some remarks on the Cassinian metric, Publ. Math. Debrecen., 90, 269-285 (2017) · Zbl 1399.51009 · doi:10.5486/PMD.2017.7386 |
| [6] | Hästö, P., A new weighted metric: the relative metric I, J. Math. Anal. Appl., 274, 38-58 (2002) · Zbl 1019.54011 · doi:10.1016/S0022-247X(02)00219-6 |
| [7] | Hokuni, S.; Klén, R.; Li, Y.; Vuorinen, M., Balls in the triangular ratio metric, Amer. Math. Soc., 667, 105-123 (2016) · Zbl 1362.30031 |
| [8] | Ibragimov, Z., The Cassinian metric of a domain in \(\overline{{\mathbb{R} }^n} \), Uzbek. Mat. Zh., 1, 53-67 (2009) |
| [9] | Ibragimov, Z., Hyperbolizing metric spaces, Proc. Amer. Math. Soc., 139, 4401-4407 (2011) · Zbl 1244.30066 · doi:10.1090/S0002-9939-2011-10857-8 |
| [10] | Ibragimov, Z.; Mohapatra, MR; Sahoo, SK; Zhang, X., Geometry of the Cassinian metric and its inner metric, Bull. Malays. Math. Sci. Soc., 40, 361-372 (2017) · Zbl 1366.30007 · doi:10.1007/s40840-015-0246-6 |
| [11] | Jia, G.; Wang, G.; Zhang, X., Geometric properties of the triangular ratio metric and related metrics, Bull. Malays. Math. Sci. Soc., 44, 4223-4237 (2021) · Zbl 1476.30149 · doi:10.1007/s40840-021-01163-2 |
| [12] | Katz, NN, Hyperbolic metrics on open subsets of Ptolemaic spaces with sharp parameter bounds, Proc. Amer. Math. Soc., 149, 2213-2220 (2021) · Zbl 1473.51017 · doi:10.1090/proc/15288 |
| [13] | Klén, R., Local convexity properties of j-metric balls, Ann. Acad. Sci. Fenn. Math., 33, 281-293 (2008) · Zbl 1149.30035 |
| [14] | Klén, R., Local convexity properties of quasihyperbolic balls in punctured space, J. Math. Anal. Appl., 342, 192-201 (2008) · Zbl 1166.30023 · doi:10.1016/j.jmaa.2007.12.008 |
| [15] | Klén, R., Local convexity properties of balls in Apollonian and Seittenrantas metrics, Conform. Geom. Dyn., 17, 133-144 (2013) · Zbl 1317.30028 · doi:10.1090/S1088-4173-2013-00257-9 |
| [16] | Klén, R., Hyperbolic type distances in Starlike domains, Results Math., 72, 47-69 (2017) · Zbl 1376.30031 · doi:10.1007/s00025-016-0642-8 |
| [17] | Klén, R.; Lindén, M.; Vuorinen, M.; Wang, G., The visual angle metric and Möbius transformations, Comput. Methods Funct. Theory., 14, 577-608 (2014) · Zbl 1307.30082 · doi:10.1007/s40315-014-0075-x |
| [18] | Klén, R.; Mohapatra, MR; Sahoo, SK, Geometric properties of the Cassinian metric, Math. Nachr., 290, 1531-1543 (2017) · Zbl 1392.30005 · doi:10.1002/mana.201600117 |
| [19] | Klén, R.; Vuorinen, M.; Zhang, X., Quasihyperbolic metric and Möbius transformations, Proc. Amer. Math. Soc., 142, 311-322 (2014) · Zbl 1408.30027 · doi:10.1090/S0002-9939-2013-11765-X |
| [20] | Mohapatra, MR; Sahoo, SK, Mapping properties of a scale invariant Cassinian metric and a Gromov hyperbolic metric, Bull. Aust. Math. Soc., 97, 141-152 (2018) · Zbl 1381.51015 · doi:10.1017/S0004972717000570 |
| [21] | Mohapatra, MR; Sahoo, SK, A Gromov hyperbolic metric vs the hyperbolic and other related metrics, Comput. Methods Funct. Theory, 18, 473-493 (2018) · Zbl 1402.30039 · doi:10.1007/s40315-018-0233-7 |
| [22] | Nica, B.; Špakula, J., Strong hyperbolicity, Groups Geo. Dyn., 10, 951-964 (2016) · Zbl 1368.20057 · doi:10.4171/GGD/372 |
| [23] | Rainio, O.; Vuorinen, M., Triangular ratio metric in the unit disk, Complex Var. Elliptic Equ., 67, 1299-1325 (2022) · Zbl 1494.30046 · doi:10.1080/17476933.2020.1870452 |
| [24] | Väisälä, J., Gromov hyperbolic spaces, Expo. Math., 23, 187-231 (2005) · Zbl 1087.53039 · doi:10.1016/j.exmath.2005.01.010 |
| [25] | Vuorinen, M.; Zhang, X., Distortion of quasiconformal mappings with identity boundary values, J. London Math. Soc., 90, 637-653 (2014) · Zbl 1307.30056 · doi:10.1112/jlms/jdu043 |
| [26] | Wang, G.; Vuorinen, M.; Zhang, X., On cyclic quadrilaterals in Euclidean and hyperbolic geometries, Publ. Math. Debrecen, 99, 123-140 (2021) · Zbl 1499.51017 · doi:10.5486/PMD.2021.8894 |
| [27] | Wang, G.; Xu, X.; Vuorinen, M., Remarks on the scale-invariant Cassinian metric, Bull. Malays. Math. Sci. Soc., 44, 1559-1577 (2021) · Zbl 1462.30085 · doi:10.1007/s40840-020-01011-9 |
| [28] | Xu, X.; Wang, G.; Zhang, X., Comparison and Möbius quasi-invariance properties of Ibragimov’s metric, Comput. Methods Funct. Theor., 22, 609-627 (2022) · Zbl 1502.30130 · doi:10.1007/s40315-021-00414-4 |
| [29] | Zhang, X., Comparison between a Gromov hyperbolic metric and the hyperbolic metric, Comput. Methods Funct. Theor., 18, 717-722 (2018) · Zbl 1404.30048 · doi:10.1007/s40315-018-0247-1 |
| [30] | Zhang, Z.; Xiao, Y., Strongly hyperbolic metrics on Ptolemy spaces, J. Math. Anal Appl., 478, 445-457 (2019) · Zbl 1425.30045 · doi:10.1016/j.jmaa.2019.05.036 |
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