Comparison of hyperbolic metric and triangular ratio metric in a square

Abstract

Let $K$ be a square in the plane and $\rho_K(x,y)$ 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 $\inf_{z\in \partial K}(|x-z|+|z-y|)$. We obtain a sharp estimate for the ratio of $\th (\rho_K(x,y)/2)$ to $s_K(x,y)$.