Distortion of the triangular ratio metric under Moebius transformations

Abstract

Let $\mathbb{U}$ be the unit disk in the complex plane. Denote by $s_{\mathbb{U}}(x,y)$ the triangular ratio metric in $\mathbb{U}$; the value of $s_{\mathbb{U}}(x,y)$ equals the ratio of the Euclidean distance $|x-y|$ to the value ${\inf }_{z\in \partial \mathbb{U}}(|x-z|+|z-y|)$. In the monograph by P.~Hariri, R.~Klén, and M.~Vuorinen "Conformally invariant metrics and quasiconformal mappings" (2020) the following problem was stated: for every Moebius automorphism of the unit disk, $w=f(z)=\frac{z+a}{1+za}$, $0\le a<1$, and every points $z_1$, $z_2\in \mathbb{U}$ the sharp inequality $s_{\mathbb{U}}(f(z_1),f(z_2))\le (1+a)s_{\mathbb{U}}(z_1,z_2)$ holds. We prove that the conjecture is valid.

7 pages, 1 figure