Monotonicity and Comparison Results for Conformal Invariants

Baernstein II Al, Alexander Solynin

Research output: Contribution to journalArticlepeer-review

Abstract

Abstract. Let a1, . . . , aN be points on the unit circle T with aj = 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 condition δ(Ω) ≤ 2π/n so that the Poincar´e metric λΩ(0) of Ω at the origin is as small as possible? In this paper we answer this question by showing that λΩ(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
JournalRevista Matematica Iberoamericana
StatePublished - Jan 2013

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