A1 Refereed original research article in a scientific journal
Intrinsic Geometry and Boundary Structure of Plane Domains
Authors: Rainio Oona, Sugawa Toshiyuki, Vuorinen Matti
Publisher: MAIK NAUKA/INTERPERIODICA/SPRINGER
Publication year: 2021
Journal: Siberian Mathematical Journal
Journal name in source: SIBERIAN MATHEMATICAL JOURNAL
Journal acronym: SIBERIAN MATH J+
Volume: 62
Issue: 4
First page : 691
Last page: 706
Number of pages: 16
ISSN: 0037-4466
eISSN: 1573-9260
DOI: https://doi.org/10.1134/S0037446621040121(external)
Self-archived copy’s web address: https://research.utu.fi/converis/portal/detail/Publication/66954629(external)
Given a nonempty compact set E in a proper subdomain Omega of the complex plane, we denote the diameter of E and the distance from E to the boundary of Omega by d(E) and d(E, partial derivative Omega), respectively. The quantity d(E)/d(E, partial derivative Omega) is invariant under similarities and plays an important role in geometric function theory. In case Omega has the hyperbolic distance h(Omega)(z, w), we consider the infimum kappa(Omega) of the quantity h(Omega)(E)/ log(1 + d(E)/d(E, partial derivative)) over compact subsets E of Omega with at least two points, where h(Omega)(E) stands for the hyperbolic diameter of E. Let the upper half-plane be H. We show that partial derivative(Omega) is positive if and only if the boundary of Omega is uniformly perfect and partial derivative(Omega) <= kappa(H) for all Omega, with equality holding precisely when Omega is convex.
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