Abstract
The Gehring-Osgood characterization for uniform Euclidean domains asserts that these are precisely the domains in which quasihyperbolic distance is bounded above by a constant multiple of the so-called relative distance. We prove that a hyperbolic plane domain is uniform if and only if its hyperbolic distance is bounded above by a constant multiple of its relative distance. Similar results hold for uniform domains in the Riemann sphere, and also for Euclidean inner uniform domains.
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Dedicated to the memory of Fred Gehring, for his passion for difficult problems
The author was supported in part by the Charles Phelps Taft Memorial Fund and by the US NSF grant DMS-15004554.
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Herron, D.A. Uniform domains and hyperbolic distance. JAMA 143, 349–400 (2021). https://doi.org/10.1007/s11854-021-0160-9
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DOI: https://doi.org/10.1007/s11854-021-0160-9