An Inequality Related to Möbius Transformations

Abstract

The open unit ball $\mathbb {B} = \{\mathbf {v}\in \mathbb {R}^n\colon \|\mathbf {v}\|<1\}$ is endowed with Möbius addition ⊕_M defined by

$$\displaystyle \mathbf {u}\oplus _M\mathbf {v} = \dfrac {(1 + 2\langle \mathbf {u},\mathbf {v}\rangle + \|\mathbf {v}\|{ }^2)\mathbf {u} + (1 - \|\mathbf {u}\|{ }^2)\mathbf {v}}{1 + 2\langle \mathbf {u},\mathbf {v}\rangle + \|\mathbf {u}\|{ }^2\|\mathbf {v}\|{ }^2}, $$

for all $\mathbf {u},\mathbf {v}\in \mathbb {B}$. In this article, we prove the inequality

$$\displaystyle \dfrac {\|\mathbf {u}\|-\|\mathbf {v}\|}{1+\|\mathbf {u}\|\|\mathbf {v}\|}\leq \|\mathbf {u}\oplus _M \mathbf {v}\| \leq \dfrac {\|\mathbf {u}\|+\|\mathbf {v}\|}{1-\|\mathbf {u}\|\|\mathbf {v}\|} $$

in $\mathbb {B}$. This leads to a new metric on $\mathbb {B}$ defined by

$$\displaystyle d_T(\mathbf {u},\mathbf {v}) = \tan ^{-1}{\|-\mathbf {u}\oplus _M\mathbf {v}\|}, $$

which turns out to be an invariant of Möbius 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.