An inequality related to Möbius transformations

Abstract

The open unit ball $\mathbb{B}=\{\mathbf{v}\in {\mathbb{R}}^n:\Vert \mathbf{v}\Vert <1\}$ is endowed with Möbius addition ${\oplus }_M$ defined by $$\mathbf{u}{\oplus }_M\mathbf{v}=\frac{(1+2⟨\mathbf{u},\mathbf{v}⟩+\Vert \mathbf{v}{\Vert }^2)\mathbf{u}+(1-\Vert {\mathbf{u}}^2)\mathbf{v}}{1+⟨\mathbf{u},\mathbf{v}⟩+\Vert \mathbf{u}{\Vert }^2\Vert \mathbf{v}{\Vert }^2\Vert }$$ for all $\mathbf{u},\mathbf{v}\in \mathbf{B}$. In this article, we prove the inequality $$\frac{\Vert \mathbf{u}\Vert -\Vert \mathbf{v}\Vert }{1+\Vert \mathbf{u}\Vert \Vert \mathbf{v}\Vert }\le \Vert \mathbf{u}{\oplus }_M\mathbf{v}\Vert \le \frac{\Vert \mathbf{u}\Vert +\Vert \mathbf{v}\Vert }{1-\Vert \mathbf{u}\Vert \Vert \mathbf{v}\Vert }$$ in $\mathbb{B}$. This leads to a new metric on $\mathbb{B}$ defined by $$d_T(\mathbf{u},\mathbf{v})={\tan }^{-1}\Vert -\mathbf{u}{\oplus }_M\mathbf{v}\Vert ,$$ 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.

14 pages