On the Hyperbolic Metric of Certain Domains

We prove that if E is a compact subset of the unit disk D in the complex plane, if E contains a sequence of distinct points an≠0 for n≥1 such that limn→∞an=0 and for all n we have |an+1|≥|an|/2, and if G=D\E is connected and 0∈∂G, then there is a constant c>0 such that for all z∈G we have λG(z)≥c/|z| where λG(z) is the density of the hyperbolic metric in G.