# Monotonicity and comparison results for conformal invariants

### Albert Baernstein II

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

Texas Tech University, Lubbock, USA

## Abstract

Let $a_1,\dots,a_N$ be points on the unit circle $\mathbb{T}$ with $a_j=e^{i\theta_j}$, where $0=\theta_1\le\theta_2\le\dots\le \theta_N=2\pi$. Let $\Omega=\overline{\mathbb{C}}\setminus\{a_1,\dots,a_N\}$ and let $\Omega^*$ be $\overline{\mathbb{C}}$ with the $n$-th roots of unity removed. The maximal gap $\delta(\Omega)$ of $\Omega$ is defined by $\delta(\Omega)=\max\{\theta_{j+1}-\theta_j:\,0\le j\le N-1\}$. How should one choose $a_1,\dots,a_N$ subject to the condition $\delta(\Omega)\le 2\pi/n$ so that the Poincaré metric $\lambda_\Omega(0)$ of $\Omega$ at the origin is as small as possible? In this paper we answer this question by showing that $\lambda_\Omega(0)$ is minimal when $\Omega=\Omega^*$. 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