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 $a_n\ne 0$ for $n\ge 1$ such that ${\lim }_{n\to \infty }{a}_n=0$ and for all $n$ we have $|a_{n+1}|\ge \frac{1}{2}|a_n|$, 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 ${\lambda }_G(z)\ge c/|z|$ where ${\lambda }_G(z)$ is the density of the hyperbolic metric in $G$.