Comparison of hyperbolic metric and triangular ratio metric in a square

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 $s_K(x,y)$.

💡 Research Summary

The paper investigates the relationship between two intrinsic metrics defined on a planar square K: the hyperbolic distance ρ_K (the distance induced by the hyperbolic metric of the domain) and the triangular ratio metric s_K, which is defined by
( s_K(x,y)=\frac{|x-y|}{\displaystyle\sup_{z\in\partial K}(|x-z|+|z-y|)}) for (x\neq y) and (s_K(x,x)=0).
For general simply‑connected domains D of hyperbolic type, it is known that there exist constants C₁ and C₂ such that
( C₁,s_D(x,y)\le \tanh!\big(\rho_D(x,y)/2\big)\le C₂,s_D(x,y)) for all x,y∈D.
Previous work (referenced as