Abstract
We study a new generalized version of the point pair function defined with a constant α > 0. We prove that this function is a quasi-metric for all values of α > 0 and compare it to several hyperbolic-type metrics, such as the j∗-metric, the triangular ratio metric, and the hyperbolic metric. Most of the inequalities presented here have the best possible constants in terms of α. Furthermore, we research the distortion of the generalized point pair function under conformal and quasiregular mappings.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Chen, P. Hariri, R. Klén, and M. Vuorinen, Lipschitz conditions, triangular ratio metric, and quasiconformalmaps, Ann. Acad. Sci. Fenn., Math., 40(2):683–709, 2015.
D. Dautova, S. Nasyrov, O. Rainio, and M. Vuorinen, Metrics and quasimetrics induced by point pair function, Bull. Braz. Math. Soc. (N.S.), 53(4):1377–1401, 2022.
M. Fujimura, M. Mocanu, and M. Vuorinen, Barrlund’s distance function and quasiconformal maps, Complex Var. Elliptic Equ., 66(8):1225–1255, 2021.
F.W. Gehring and B.G. Osgood, Uniform domains and the quasi-hyperbolic metric, J. Anal. Math., 36:50–74, 1979.
A.A. Gol’dberg and T.V. Strochik, Conformal mapping of certain kinds of half-strips, Lith. Math. J., 16:177–181, 1976.
P. Hariri, R. Klén, andM. Vuorinen, Conformally InvariantMetrics and QuasiconformalMappings, Springer, Cham, 2020.
P. Hariri, M. Vuorinen, and X. Zhang, Inequalities and bilipschitz conditions for triangular ratio metric, Rocky Mt. J. Math., 47(4):1121–1148, 2017.
P. Hästö, A new weighted metric: The relative metric I, J. Math. Anal. Appl, 274:38–58, 2002.
A. Janušauskas, Analog of mappings, harmonic in the sense of M. A. Lavrent’ev, Lith. Math. J., 24:286–292, 1984.
E. Kirjackis, On Chebyshev systems of functions holomorphic in the unit disk, Lith. Math. J., 45:192–199, 2005.
O. Rainio, Intrinsic quasi-metrics, Bull. Malays. Math. Sci. Soc. (2), 44(5):2873–2891, 2021.
O. Rainio, Intrinsic metrics under conformal and quasiregular mappings, Publ. Math. Debr., 101(1–2):189–215, 2022.
O. Rainio and M. Vuorinen, Introducing a new intrinsic metric, Results Math., 77(2):71, 2022.
O. Rainio and M. Vuorinen, Triangular ratio metric under quasiconformal mappings in sector domains, Comput. Methods Funct. Theory, 23(2):269–293, 2023.
Funding
Open Access funding provided by University of Turku (UTU) including Turku University Central Hospital.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The research was funded by Finnish Culture Foundation.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Rainio, O. Inequalities for the generalized point pair function. Lith Math J 63, 396–410 (2023). https://doi.org/10.1007/s10986-023-09603-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10986-023-09603-1