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Barrlund's distance function and quasiconformal maps

Masayo Fujimura, Marcelina Mocanu, Matti Vuorinen

Abstract

Answering a question about triangle inequality suggested by R. Li, A. Barrlund introduced a distance function which is a metric on a subdomain of ${\mathbb R}^n\,.$ We study this Barrlund metric and give sharp bounds for it in terms of other metrics of current interest. We also prove sharp distortion results for the Barrlund metric under quasiconformal maps.

Barrlund's distance function and quasiconformal maps

Abstract

Answering a question about triangle inequality suggested by R. Li, A. Barrlund introduced a distance function which is a metric on a subdomain of We study this Barrlund metric and give sharp bounds for it in terms of other metrics of current interest. We also prove sharp distortion results for the Barrlund metric under quasiconformal maps.

Paper Structure

This paper contains 16 sections, 30 theorems, 165 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. On Barrlund's metric
  4. Basic properties of the Barrlund metric.
  5. The proof of Theorem \ref{['sbThm']}.
  6. Barrlund's metric for $p=1 \,.$
  7. The domain $G=\mathbb{H}$
  8. The domain $G=\mathbb{D}$
  9. Barrlund's metric for $p=2 \,.$
  10. The domain $G=\mathbb{H}$
  11. The domain $G=\mathbb{D}$
  12. Inequalities of Barrlund's metric for $p \in (1, \infty)\,.$
  13. Barrlund's metric for $p=\infty\,.$
  14. The domain $G=\mathbb{H}$.
  15. The domain $G=\mathbb{D}\,$
  16. ...and 1 more sections

Key Result

Theorem 1.3

Let $G$ be a domain in ${\mathbb{R}}^n\,$ and let $p\ge 1\,.$ Then for all points $z_1,z_2 \in G\,$

Figures (4)

  • Figure 1: Level sets $\{x+ i y: b_{{\mathbb{D}},2}(0.3, x+i y) =c \}$ for $c=0.4, 0.6, 0.8, 1.0\,$ and the unit circle. Note that for $c=1.0$ the level set meets the points $(\pm 1,0)\,$ in accordance with Theorem \ref{['cor:barrdist']}.
  • Figure 2: The oval in the figure is the boundary of $B_{\mathbb{D},2}(a;0.5)$ with $a=0.5$. The disk with center the origin indicates the upper bound in Theorem \ref{['thm322']}. The shaded region corresponds to Theorem \ref{['thm323']}.
  • Figure 3: The left and right figures indicate the case (\ref{['proof:434:1']}) and (\ref{['proof:434:2']}) respectively.
  • Figure 4: The power $\infty$ ellipse and the set $\{|z-e^{i\theta}|\leq 1-r\}\cap \mathbb{D}$.

Theorems & Definitions (71)

  • Theorem 1.3
  • Theorem 1.5: fhmv
  • Theorem 1.7
  • Remark 1.8
  • Conjecture 2.3
  • Theorem 2.5
  • proof
  • Remark 2.8
  • Definition 3.1
  • Remark 3.2
  • ...and 61 more