Monotonicity and comparison results for conformal invariants

Albert Baernstein, Alexander Yu Solynin

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6 Scopus citations


Let a1, . . . , aN be points on the unit circle T with a j = eiθj , where 0 = θ1 ≤ θ2 ≤ · · · ≤ θN = 2π. Let Ω = C \ {a1, . . . , aN} and let Ω* be C with the n-th roots of unity removed. The maximal gap δ(Ω) of Ω is defined by δ(Ω) = max{θj+1 - θj: 0 ≤ j ≤ N - 1}. How should one choose a1, . . . , aN subject to the condition δ(Ω) ≤ 2π/n so that the Poincaré metric λΩ(0) of Ω at the origin is as small as possible? In this paper we answer this question by showing that λ Ω(0) is minimal when Ω = Ω*. Several similar problems on the extremal values of the harmonic measures and capacities are also discussed.

Original languageEnglish
Pages (from-to)91-113
Number of pages23
JournalRevista Matematica Iberoamericana
Issue number1
StatePublished - 2013


  • Capacity
  • Comparison theorem
  • Harmonic measure
  • Hyperbolic metric


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