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Erschienen in:
2021 | OriginalPaper | Buchkapitel
An Inequality Related to Möbius Transformations
verfasst von : Themistocles M. Rassias, Teerapong Suksumran
Erschienen in: Approximation Theory and Analytic Inequalities
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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 for all \(\mathbf {u},\mathbf {v}\in \mathbb {B}\). In this article, we prove the inequality in \(\mathbb {B}\). This leads to a new metric on \(\mathbb {B}\) defined by 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.
$$\displaystyle \mathbf {u}\oplus _M\mathbf {v} = \dfrac {(1 + 2\langle \mathbf {u},\mathbf {v}\rangle + \|\mathbf {v}\|{ }^2)\mathbf {u} + (1 - \|\mathbf {u}\|{ }^2)\mathbf {v}}{1 + 2\langle \mathbf {u},\mathbf {v}\rangle + \|\mathbf {u}\|{ }^2\|\mathbf {v}\|{ }^2}, $$
$$\displaystyle \dfrac {\|\mathbf {u}\|-\|\mathbf {v}\|}{1+\|\mathbf {u}\|\|\mathbf {v}\|}\leq \|\mathbf {u}\oplus _M \mathbf {v}\| \leq \dfrac {\|\mathbf {u}\|+\|\mathbf {v}\|}{1-\|\mathbf {u}\|\|\mathbf {v}\|} $$
$$\displaystyle d_T(\mathbf {u},\mathbf {v}) = \tan ^{-1}{\|-\mathbf {u}\oplus _M\mathbf {v}\|}, $$