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

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}={\displaystyle \frac{(1+2⟨\mathbf{u},\mathbf{v}⟩+\mathrm{\parallel }\mathbf{v}{\mathrm{\parallel }}^2)\mathbf{u}+(1-\mathrm{\parallel }{\mathbf{u}}^2)\mathbf{v}}{1+⟨\mathbf{u},\mathbf{v}⟩+\mathrm{\parallel }\mathbf{u}{\mathrm{\parallel }}^2\mathrm{\parallel }\mathbf{v}{\mathrm{\parallel }}^2\mathrm{\parallel }}}$$ for all $$\mathbf{u}$,$\mathbf{v}$\in $\mathbf{B}$$\mathrm{.}Inthisarticle,weprovetheinequality$$ \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 byd_T($\mathbf{u}$,$\mathbf{v}$) = \tan^{-1}{|-$\mathbf{u}$\oplus_M$\mathbf{v}$|},$$whichturnsouttobeaninvariantofM\ddot{o}biustransformationson$$\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.