Intrinsic Geometry and Boundary Structure of Plane Domains

Given a nonempty compact set in a proper subdomain of the complex plane, we denote the diameter of and the distance from to the boundary of by and , respectively. The quantity is invariant under similarities and plays an important role in geometric function theory. In case  has the hyperbolic distan...

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Veröffentlicht in:Siberian mathematical journal 2021-07, Vol.62 (4), p.691-706
Hauptverfasser: Rainio, O., Sugawa, T., Vuorinen, M.
Format: Artikel
Sprache:eng
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Zusammenfassung:Given a nonempty compact set in a proper subdomain of the complex plane, we denote the diameter of and the distance from to the boundary of by and , respectively. The quantity is invariant under similarities and plays an important role in geometric function theory. In case  has the hyperbolic distance , we consider the infimum of the quantity over compact subsets of with at least two points, where  stands for the hyperbolic diameter of  . Let the upper half-plane be  . We show that  is positive if and only if the boundary of is uniformly perfect and for all , with equality holding precisely when is convex.
ISSN:0037-4466
1573-9260
DOI:10.1134/S0037446621040121