Comparison of hyperbolic metric and triangular ratio metric in a square

TL;DR

This work compares the hyperbolic metric and the triangular ratio metric on the square $K=[-1,1]^2$, proving the sharp bound $\tanh(\rho_K(x,y)/2) \le C(\lambda_0)\, s_K(x,y)$ for all $x,y\in K$, where $C(\lambda_0)=\mathcal{K}(\sqrt{2}/2)\approx1.85407$. The authors combine a maximum-principle analysis of the ratio $g(x,y)=\tanh(\rho_K(x,y)/2)/s_K(x,y)$ with a conformal mapping from the unit disk to the square given by $f(z)=C^{-1}\int_0^z \frac{d\zeta}{\sqrt{1+\zeta^4}}$, together with elliptic-integral relations to control preimages and regions where $s_K$ is minimized. They show the bound is sharp with the constant $C(\lambda_0)$, where $\lambda_0=3-2\sqrt{2}$ and $C(\lambda_0)=\mathcal{K}(\sqrt{2}/2)\approx1.85407$, and identify the center of the square as the limiting attainment point. The approach extends previous results for rectangles to the square, providing exact constants and a detailed structural description of where the maximum ratio occurs.

Abstract

Let $K$ be a square in the plane and $ρ_K(x,y)$ be the hyperbolic distance between $x$, $y\in K$. Denote by $s_K(x,y)$ the triangular ratio metric in $K$; for $x\neq y$ the value of $s_K(x,y)$ equals the ratio of the Euclidean distance $|x-y|$ between $x$, $y\in K$ to the value $\sup_{z\in \partial K}(|x-z|+|z-y|)$. We obtain a sharp estimate for the ratio of $þ(ρ_K(x,y)/2)$ to $s_K(x,y)$.

Figures (1)

• Figure 1: Decomposition of a square into parts $G_j$ depending on the location of the point $z$ at which the infimum \ref{['4d']} is attained, under the fixed point $x$.

Theorems & Definitions (23)

• Theorem 1
• Lemma 1
• proof
• Lemma 2
• proof
• Corollary 1
• Lemma 3
• proof
• Lemma 4
• proof