On the Hyperbolic Metric of Certain Domains

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 \(n\ge 1\) such that \(\lim _{n\rightarrow \infty } a_n=0\) and for all n we have \( |a_{n+1}| \ge |a_n|/2 \), 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.