Uniformization by rectangular domains: A path from slits to squares

Alexander Yu Solynin, Nadeesha C. Vidanage

Research output: Contribution to journalArticlepeer-review

Abstract

Let Σ(Ω) be the class of functions [Formula presented] univalent on a finitely connected domain Ω, ∞∈Ω⊂C‾. By a classical result due to H. Grötzsch, the function f0 maximizing ℜa1 over the class Σ(Ω) maps Ω onto C‾ slit along horizontal segments. Recently, M. Bonk found a similar extremal problem, which maximizer f1∈Σ(Ω) maps Ω onto a domain on C‾, whose complementary components are squares. In this note, we discuss a parametric family of extremal problems on the class Σ(Ω) with maximizers fm, 0<m<1, mapping Ω onto domains on C‾, whose complementary components are rectangles with horizontal and vertical sides and with module m.

Original languageEnglish
Article number123927
JournalJournal of Mathematical Analysis and Applications
Volume486
Issue number2
DOIs
StatePublished - Jun 15 2020

Keywords

  • Canonical domain
  • Conformal mapping
  • Extremal problem
  • Rectangle
  • Uniformization

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