JournalsrmiVol. 29, No. 1pp. 91–113

Monotonicity and comparison results for conformal invariants

  • Albert Baernstein II

    Washington University, St. Louis, USA
  • Alexander Yu. Solynin

    Texas Tech University, Lubbock, USA
Monotonicity and comparison results for conformal invariants cover
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Abstract

Let a1,,aNa_1,\dots,a_N be points on the unit circle T\mathbb{T} with aj=eiθja_j=e^{i\theta_j}, where 0=θ1θ2θN=2π0=\theta_1\le\theta_2\le\dots\le \theta_N=2\pi. Let Ω=C{a1,,aN}\Omega=\overline{\mathbb{C}}\setminus\{a_1,\dots,a_N\} and let Ω\Omega^* be C\overline{\mathbb{C}} with the nn-th roots of unity removed. The maximal gap δ(Ω)\delta(\Omega) of Ω\Omega is defined by δ(Ω)=max{θj+1θj:0jN1}\delta(\Omega)=\max\{\theta_{j+1}-\theta_j:\,0\le j\le N-1\}. How should one choose a1,,aNa_1,\dots,a_N subject to the condition δ(Ω)2π/n\delta(\Omega)\le 2\pi/n so that the Poincaré metric λΩ(0)\lambda_\Omega(0) of Ω\Omega at the origin is as small as possible? In this paper we answer this question by showing that λΩ(0)\lambda_\Omega(0) is minimal when Ω=Ω\Omega=\Omega^*. Several similar problems on the extremal values of the harmonic measures and capacities are also discussed.

Cite this article

Albert Baernstein II, Alexander Yu. Solynin, Monotonicity and comparison results for conformal invariants. Rev. Mat. Iberoam. 29 (2013), no. 1, pp. 91–113

DOI 10.4171/RMI/714