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 $a_n≠0$ for $n\ge 1$ such that ${\lim }_{n\to \mathrm{\infty }}{a}_n=0$ and for all $n$ we have $\mathrm{\mid }{a}_{n+1}\mathrm{\mid }\ge \frac{1}{2}\mathrm{\mid }{a}_n\mathrm{\mid }$, and if $G=\mathbb{D}\setminus E$ is connected and $0\in \mathrm{\partial }G$, then there is a constant $c>0$ such that for all $z\in G$ we have ${\lambda }_G(z)\ge c\mathrm{/}\mathrm{\mid }z\mathrm{\mid }$ where ${\lambda }_G(z)$ is the density of the hyperbolic metric in $G$.