Monotonicity and comparison results for conformal invariants

Abstract

Let $a_1,\dots ,a_N$ be points on the unit circle $\mathbb{T}$ with $a_j=e^{i{\theta }_j}$, where $0={\theta }_1\le {\theta }_2\le \cdots \le {\theta }_N=2\pi$. Let $\Omega =\overline{\mathbb{C}}\setminus \{a_1,\dots ,a_N\}$ and let ${\Omega }^{\ast }$ 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 $\Omega ={\Omega }^{\ast }$. Several similar problems on the extremal values of the harmonic measures and capacities are also discussed.