On the hyperbolic metric of certain domains

Abstract

We prove that if $E$ is a compact subset of the unit disk ${\mathbb D}$ in the complex plane, if $E$ contains a sequence of distinct points $a_n\not= 0$ for $n\geq 1$ such that $\lim_{n\to\infty} a_n=0$ and for all $n$ we have $ |a_{n+1}| \geq \frac{1}{2} |a_n| $, and if $G={\mathbb D} \setminus E$ is connected and $0\in \partial G$, then there is a constant $c>0$ such that for all $z\in G$ we have $ \lambda_{G } (z) \geq c/|z| $ where $\lambda_{G } (z)$ is the density of the hyperbolic metric in $G$.