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 |
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Hauptverfasser: | , , |
Format: | Artikel |
Sprache: | eng |
Schlagworte: | |
Online-Zugang: | Volltext |
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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. |
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ISSN: | 0037-4466 1573-9260 |
DOI: | 10.1134/S0037446621040121 |