On the Hyperbolic Metric of Certain Domains

Aimo Hinkkanen; Matti Vuorinen

doi:10.1007/s40315-023-00518-z

On the Hyperbolic Metric of Certain Domains

Aimo Hinkkanen, Matti Vuorinen

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

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 a_n≠ 0 for n≥ 1 such that lim _n_→_∞a_n= 0 and for all n we have | a_n₊₁| ≥ | a_n| / 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 λ_G(z) ≥ c/ | z| where λ_G(z) is the density of the hyperbolic metric in G.

Original language	English (US)
Pages (from-to)	129-138
Number of pages	10
Journal	Computational Methods and Function Theory
Volume	25
Issue number	1
Early online date	Jan 17 2024
DOIs	https://doi.org/10.1007/s40315-023-00518-z
State	Published - Mar 2025

Keywords

Analytic function
Conformal invariant
Hyperbolic metric

ASJC Scopus subject areas

Analysis
Computational Theory and Mathematics
Applied Mathematics

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10.1007/s40315-023-00518-z

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AU - Vuorinen, Matti

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N2 - 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.

AB - 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.

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KW - Conformal invariant

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On the Hyperbolic Metric of Certain Domains

Abstract

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