Intrinsic Geometry and Boundary Structure of Plane Domains

O. Rainio, T. Sugawa, M. Vuorinen

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)691-706
Number of pages16
JournalSiberian Mathematical Journal
Issue number4
Publication statusPublished - 2021 Jul


  • 517.54
  • condenser capacity
  • hyperbolic metric
  • uniformly perfect set

ASJC Scopus subject areas

  • Mathematics(all)


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