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The matrix and chordal norms of Möbius transformations. (English) Zbl 0666.30016

Complex analysis, Artic. dedicated to Albert Pfluger, 51-59 (1988).
[For the entire collection see Zbl 0649.00011.]
For a Möbius transformation f in \(\bar R^ n=R^ n\cup \{\infty \}\) let \(q(f)=\sup \{q(f(x),x):\) \(x\in \bar R^ n\}\) where q stands for the spherical chordal metric of \(\bar R^ n\). Next let \(\tilde f\) be the Poincaré extension of f to \(\bar R^{n+1}\) with \(\tilde fH^{n+1}=H^{n+1}\) and let \(\rho\) be the hyperbolic metric of the half-space \(H^{n+1}\). The authors prove the following sharp inequality \[ (1)\quad \rho(\tilde f(e_{n+1}),e_{n+1})\leq \log \frac{2+d(f)}{2- d(f)};\quad e_{n+1}=(0,...,0,1). \] In the twodimensional case \((n=2)\) the authors deduce from (1) an upper bound for the matrix norm \[ \| A\|^ 2=| a|^ 2+| b|^ 2+| c|^ 2+| d|^ 2 \] of a matrix \(A=\begin{pmatrix} a&b\\ c&d \end{pmatrix}\) representing a Möbius transformation f of \(\bar R^ 2\). Finally, exploiting some results from a forthcoming paper, the authors obtain an upper bound for \(\| A-I\|\), with I the identity matrix, in terms of d(f). The inequality (1) is closely related to another one due to A. F. Beardon [The geometry of discrete groups (1983; Zbl 0528.30001), Theorem 3.6.1, p. 42].
Reviewer: M.Vuorinen

MSC:

30C99 Geometric function theory