A metric that better approximates the hyperbolic metric

Ma, William

doi:10.1090/ecgd/368

A metric that better approximates the hyperbolic metric
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Conform. Geom. Dyn. 26 (2022), 1-9 Request permission

Abstract:

We introduce a metric, called the star metric, that gives us better upper bound of the hyperbolic metric than the Möbius invariant metric. We present various examples, and comparisons among the hyperbolic metric, the Möbius invariant metric and the star metric.

References

A. F. Beardon and D. Minda, The hyperbolic metric and geometric function theory, Quasiconformal mappings and their applications, Narosa, New Delhi, 2007, pp. 9–56. MR 2492498
Peter L. Duren, Univalent functions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR 708494
Reuven Harmelin and David Minda, Quasi-invariant domain constants, Israel J. Math. 77 (1992), no. 1-2, 115–127. MR 1194789, DOI 10.1007/BF02808014
Barbara B. Flinn and David A. Herron, Uniform estimates for the hyperbolic metric and Euclidean distance to the boundary, Michigan Math. J. 46 (1999), no. 1, 13–27. MR 1682884, DOI 10.1307/mmj/1030132355
Dennis A. Hejhal, Universal covering maps for variable regions, Math. Z. 137 (1974), 7–20. MR 349989, DOI 10.1007/BF01213931
David A. Herron, Zair Ibragimov, and David Minda, Geodesics and curvature of Möbius invariant metrics, Rocky Mountain J. Math. 38 (2008), no. 3, 891–921. MR 2426525, DOI 10.1216/RMJ-2008-38-3-891
David A. Herron, William Ma, and David Minda, A Möbius invariant metric for regions on the Riemann sphere, Future trends in geometric function theory, Rep. Univ. Jyväskylä Dep. Math. Stat., vol. 92, Univ. Jyväskylä, Jyväskylä, 2003, pp. 101–118. MR 2058115
David A. Herron, William Ma, and David Minda, Estimates for conformal metric ratios, Comput. Methods Funct. Theory 5 (2005), no. 2, 323–345. MR 2205417, DOI 10.1007/BF03321101
David A. Herron, William Ma, and David Minda, Möbius invariant metrics bilipschitz equivalent to the hyperbolic metric, Conform. Geom. Dyn. 12 (2008), 67–96. MR 2410919, DOI 10.1090/S1088-4173-08-00178-1
Ravi S. Kulkarni and Ulrich Pinkall, A canonical metric for Möbius structures and its applications, Math. Z. 216 (1994), no. 1, 89–129. MR 1273468, DOI 10.1007/BF02572311
Wan Cang Ma and David Minda, Euclidean linear invariance and uniform local convexity, J. Austral. Math. Soc. Ser. A 52 (1992), no. 3, 401–418. MR 1151295, DOI 10.1017/S1446788700035114
C. David Minda, The Hahn metric on Riemann surfaces, Kodai Math. J. 6 (1983), no. 1, 57–69. MR 698327, DOI 10.2996/kmj/1138036663
Christian Pommerenke, Linear-invariante Familien analytischer Funktionen I, Math. Ann. 155 (1964), no. 2, 108–154 (German). For a review of this paper see [MR0168751]. MR 1513275, DOI 10.1007/BF01344077