Inequalities for the Carathéodory and Poincaré metrics in open unit balls
Let \(B\) be the unit open ball in a Banach space \(X\). The main results of the paper are the following theorems. (1) Let \(d\) be a distance on \(B\) such that \(d(a,b)\geq\varrho\big(\ell(a),\ell(b)\big)\) for all \(a, b\in B\) and all \(\ell\in X'\) with \(\|\ell\|=1\), where \(\varrho\) stands for the Poincaré distance in the unit disc \(\varDelta\). Then \[ \|a-b\|\leq2\tanh\frac{d(a,b)}2 \] for all \(a,b\in B\). Moreover, if \(a,b\in B\), \(a\neq b\), are fixed, then \[ \|a-b\|=2\tanh\frac{d(a,b)}2 \tag\dag \] iff there exists an \(\ell\in X'\) such that \(\|\ell\|=1\), \(\ell(b)=-\ell(a)\), \(\varrho\big(\ell(a),\ell(b)\big)=d(a,b)\). (2) Let \(d\) stand for the Carathéodory distance on \(B\) and let \(\alpha\) be its infinitesimal metric. Suppose that \(f:\varDelta\longrightarrow B\) is a complex geodesic, \(a:=f(-\tau)\), \(b=f(\tau)\) for some \(\tau\in(0,1)\), \(c:=f(0)\), and \(v:=f'(0)\). Then (\dag) is satisfied iff \(\alpha(c,v)=\|v\|\). (3) Assume that \(B\) is a symmetric domain and let \(d\) be the Carathéodory distance on \(B\). Then \(X\) is a Hilbert space iff \(a=\pm b\) whenever (\dag) is satisfied. (4) Assume that \(B\) is a symmetric domain. Then \(X\) is a Hilbert space iff the complex geodesics in \(B\) are uniquely determined (up to parametrization). (5) Assume that \(X\) is a Hilbert space, \(d\) is the Carathéodory distance on \(B\), \(a, b, c\) are as in (2), and \[ r:=\tanh\frac{d(a,b)}2. \] Then \[ \frac{1-\|c\|^2}{1+r^2\|c\|^2}\leq\frac{\|a-b\|}{2r}\leq\varPsi_{\|c\|^2}(r^2), \] where the right hand side is given by an effective formula. Moreover, the bounds are optimal.
scientific article; zbMATH DE number 5995321