On the Hyperbolic Metric of Certain Domains

: Hinkkanen Aimo, Vuorinen Matti

Publisher: Springer Science and Business Media Deutschland GmbH

: 2024

: Computational Methods and Function Theory

: 2195-3724

DOI: https://doi.org/10.1007/s40315-023-00518-z

: https://link.springer.com/article/10.1007/s40315-023-00518-z

: https://arxiv.org/abs/2303.08238

: https://arxiv.org/abs/2303.08238v1

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 lim n→∞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.