On the Hyperbolic Metric of Certain Domains

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