# 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 $a_{1},…,a_{N}$ be points on the unit circle $T$ with $a_{j}=e_{iθ_{j}}$, where $0=θ_{1}≤θ_{2}≤⋯≤θ_{N}=2π$. Let $Ω=C∖{a_{1},…,a_{N}}$ and let $Ω_{∗}$ be $C$ with the $n$-th roots of unity removed. The maximal gap $δ(Ω)$ of $Ω$ is defined by $δ(Ω)=max{θ_{j+1}−θ_{j}:0≤j≤N−1}$. How should one choose $a_{1},…,a_{N}$ subject to the condition $δ(Ω)≤2π/n$ so that the Poincaré metric $λ_{Ω}(0)$ 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.

## 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