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On the hyperbolic metric of certain domains

Aimo Hinkkanen, Matti Vuorinen

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 $n\geq 1$ such that $\lim_{n\to\infty} a_n=0$ and for all $n$ we have $ |a_{n+1}| \geq \frac{1}{2} |a_n| $, and if $G={\mathbb D} \setminus E$ is connected and $0\in \partial G$, then there is a constant $c>0$ such that for all $z\in G$ we have $ λ_{G } (z) \geq c/|z| $ where $λ_{G } (z)$ is the density of the hyperbolic metric in $G$.

On the hyperbolic metric of certain domains

Abstract

We prove that if is a compact subset of the unit disk in the complex plane, if contains a sequence of distinct points for such that and for all we have , and if is connected and , then there is a constant such that for all we have where is the density of the hyperbolic metric in .
Paper Structure (10 sections, 2 theorems, 54 equations)

This paper contains 10 sections, 2 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['th1']}
  3. Preliminary observations.
  4. Closest boundary point on the unit circle.
  5. Closest boundary point inside the unit disk.
  6. The case $|z-\zeta|\geq \delta/2$.
  7. The case $|z-\zeta|< \delta/2$ and $|z|\geq \delta/2$.
  8. The case $|z-\zeta| < \delta/2$ and $|z| < \delta/2$, subcase $|z-\zeta| \leq |z|/8$.
  9. The case $|z-\zeta| < \delta/2$ and $|z| < \delta/2$, subcase $|z-\zeta| > |z|/8$.
  10. An example

Key Result

Theorem 1.1

Let $E$ be a compact subset of ${\mathbb D}$ and suppose that $E$ contains a sequence of distinct points $a_n\not= 0$ for $n\geq 1$ such that $\lim_{n\to\infty} a_n=0$ and such that for all $n$ we have Suppose that $G={\mathbb D} \setminus E$ is connected, hence a domain, and that $0\in \partial G$. Then there is a constant $c>0$ such that for all $z\in G$ we have We may take where and

Theorems & Definitions (2)

  • Theorem 1.1
  • Theorem 1.2