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SOME BEST POSSIBLE LIPSCHITZ CONSTANTS FOR THE DISTANCE RATIO METRIC
Last modified: 2013-12-06
Abstract
\begin{abstract}
We study expansion/contraction properties of some common classes
of mappings of the Euclidean space ${\mathbb R}^n, n\ge 2\,,$ with
respect to the distance ratio metric. The first main case is the
behavior of M\"obius transformations of the unit ball ${\mathbb
B}^n$ in ${\mathbb R}^n$ onto itself. In the second main case we
study the behavior of bounded analytic functions or polynomials of
the unit disk. In both cases sharp constants are obtained.
For $n=2$ we also propose some open problems.
\end{abstract}
We study expansion/contraction properties of some common classes
of mappings of the Euclidean space ${\mathbb R}^n, n\ge 2\,,$ with
respect to the distance ratio metric. The first main case is the
behavior of M\"obius transformations of the unit ball ${\mathbb
B}^n$ in ${\mathbb R}^n$ onto itself. In the second main case we
study the behavior of bounded analytic functions or polynomials of
the unit disk. In both cases sharp constants are obtained.
For $n=2$ we also propose some open problems.
\end{abstract}
Keywords
Moebius transformation, Lipschitz constants