TY - JOUR

T1 - Intrinsic Geometry and Boundary Structure of Plane Domains

AU - Rainio, O.

AU - Sugawa, T.

AU - Vuorinen, M.

N1 - Funding Information:
The authors were supported in part by the JSPS KAKENHI (Grant JP17H02847).
Publisher Copyright:
© 2021, Pleiades Publishing, Ltd.

PY - 2021/7

Y1 - 2021/7

N2 - Given a nonempty compact set E in a proper subdomain Ω of the complex plane,we denote the diameter of E andthe distance from E to the boundary of Ω by d(E) and d(E,∂Ω), respectively.The quantity d(E)/d(E,∂Ω) is invariant under similarities andplays an important role in geometric function theory.In case Ω has the hyperbolic distance hΩ(z,w), we consider the infimum k(Ω) ofthe quantity hΩ(E)/log(1+d(E)/d(E,∂Ω)) over compact subsetsEof Ω with at least two points, where hΩ(E) stands forthe hyperbolic diameter of E.Let the upper half-plane be H. We show that k(Ω) is positive if and only if theboundaryof Ω is uniformly perfect and k(Ω) ≤ k(H) for all Ω, with equality holding precisely when Ω is convex.

AB - Given a nonempty compact set E in a proper subdomain Ω of the complex plane,we denote the diameter of E andthe distance from E to the boundary of Ω by d(E) and d(E,∂Ω), respectively.The quantity d(E)/d(E,∂Ω) is invariant under similarities andplays an important role in geometric function theory.In case Ω has the hyperbolic distance hΩ(z,w), we consider the infimum k(Ω) ofthe quantity hΩ(E)/log(1+d(E)/d(E,∂Ω)) over compact subsetsEof Ω with at least two points, where hΩ(E) stands forthe hyperbolic diameter of E.Let the upper half-plane be H. We show that k(Ω) is positive if and only if theboundaryof Ω is uniformly perfect and k(Ω) ≤ k(H) for all Ω, with equality holding precisely when Ω is convex.

KW - 517.54

KW - condenser capacity

KW - hyperbolic metric

KW - uniformly perfect set

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U2 - 10.1134/S0037446621040121

DO - 10.1134/S0037446621040121

M3 - Article

AN - SCOPUS:85112654791

VL - 62

SP - 691

EP - 706

JO - Siberian Mathematical Journal

JF - Siberian Mathematical Journal

SN - 0037-4466

IS - 4

ER -