The rate of convergence of the hyperbolic density on sequences of domains

TL;DR

This paper investigates how quickly hyperbolic densities converge on sequences of domains under different convergence modes, also exploring related densities like Teichmueller and three-point densities.

Contribution

It provides new results on the rates of convergence of hyperbolic density and related densities under various domain convergence scenarios.

Findings

Established convergence rates for hyperbolic density

Analyzed convergence behavior of Teichmueller and three-point densities

Extended understanding of density convergence in complex analysis

Abstract

It is known that if a sequence of domains $U_n$ converges to a domain $U$ in the Caratheodory sense then the hyperbolic densities on $U_n$ converge to the hyperbolic density on $U$. In this paper, we study the rate of convergence of the hyperbolic density under a slightly different mode of convergence. In doing so, we are led to consider two other densities on domains, the Teichmueller density and the three-point density. We obtain several results which give rates of convergence in various scenarios.