On the hyperbolic metric of certain domains

2023/03/14 by Hinkkanen, Aimo, Vuorinen, Matti
#30C80 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2303.08238

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\not= 0 for n≥ 1 such that lim_n→∞ a_n=0 and for all n we have |a_n+1| ≥ (1)/(2) |a_n| , and if G=\mathbb D ∖ E is connected and 0∈ ∂ G, then there is a constant c>0 such that for all z∈ G we have λ_G (z) ≥ c/|z| where λ_G (z) is the density of the hyperbolic metric in G.

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