An inequality related to M\"{o}bius transformations

Abstract

The open unit ball $\mathbb{B} = \{\mathbf{v}\in\mathbb{R}^n\colon\|\mathbf{v}\|<1\}$ is endowed with M\"{o}bius addition $\oplus_M$ defined by $$\mathbf{u}\oplus_M\mathbf{v} = \dfrac{(1 + 2\langle\mathbf{u},\mathbf{v}\rangle + \|\mathbf{v}\|^2)\mathbf{u} + (1 - \|\mathbf{u}^2)\mathbf{v}}{1 + \langle\mathbf{u},\mathbf{v}\rangle + \|\mathbf{u}\|^2\|\mathbf{v}\|^2\|}$$ for all $\mathbf{u},\mathbf{v}\in \mathbf{B}$. In this article, we prove the inequality $$ \dfrac{\|\mathbf{u}\|-\|\mathbf{v}\|}{1+\|\mathbf{u}\|\|\mathbf{v}\|}\leq \|\mathbf{u}\oplus_M \mathbf{v}\| \leq \dfrac{\|\mathbf{u}\|+\|\mathbf{v}\|}{1-\|\mathbf{u}\|\|\mathbf{v}\|} $$ in $\mathbb{B}$. This leads to a new metric on $\mathbb{B}$ defined by $$d_T(\mathbf{u},\mathbf{v}) = \tan^{-1}{\|-\mathbf{u}\oplus_M\mathbf{v}\|},$$ which turns out to be an invariant of M\"{o}bius transformations on $\mathbb{R}^n$ carrying $\mathbb{B}$ onto itself. We also compute the isometry group of $(\mathbb{B}, d_T)$ and give a parametrization of the isometry group by vectors and rotations.