Two-point distortion bounds for biholomorphic mappings of the ball in \(\mathbb C^n\)
In this interesting paper, the author obtains lower and upper two-point distortion theorems for families \({\mathcal F}\) of biholomorphic mappings on the Euclidean unit ball \(\mathbb{B}\) in \(\mathbb{C}^n\). To this end, the author uses the notions of (trace) order and norm order for linearly invariant families (LIF) of locally biholomorphic mappings on \(\mathbb{B}\), as well as properties of the Carathéodory distance on the unit ball \(\mathbb{B}\). Let \( {\mathcal B}(\mathbb{B})\) be the family of biholomorphic mappings on \(\mathbb{B}\). Also, let \(C_{\mathbb B}\) be the Carathéodory distance on \({\mathbb B}\). If \({\mathcal M}\) is a linearly invariant family on \(\mathbb{B}\), let \(\|\mathrm{ord}\|{\mathcal M}\) be the norm order of \({\mathcal M}\). One of the main results of this paper is given in Theorem 3.1, in which the author proves that if \({\mathcal F}\subseteq {\mathcal B}(\mathbb{B})\) is such that \(\beta=\|\mathrm{ord}\|\Lambda[{\mathcal F}]<\infty\), then for all \(f\in {\mathcal F}\) and \(a,b\in \mathbb{B}\), the following lower bound holds: \[ \|f(b)-f(a)\|\geq \psi_{n,\beta}(C_{\mathbb B}(a,b))\max\left\{\frac{1}{\|(Df(a)D\varphi_a(0))^{-1}\|}, \frac{1}{\|(Df(b)D\varphi_b(0))^{-1}\|}\right\}. \] Note that \(\psi_{n,\beta}:[0,\infty)\to [0,\infty)\) is the increasing function given by \[ \psi_{n,\beta}(x)=\int_0^x\frac{e^{-2(2n-1)\beta t}}{\cosh^{n-1}t}dt, \] and \(\varphi_a\) is the automorphism of \(\mathbb{B}\) given by \[ \varphi_a(z)=\frac{a-P_az-s_aQ_az}{1-\langle z,a\rangle},\quad z\in\mathbb{B}, \] where \(P_a\) is the orthogonal projection of \(\mathbb{C}^n\) onto span\(\{a\}\), \(Q_a=I-P_a\) and \(s_a=\sqrt{1-\|a\|^2}\). In Theorem 3.2, the author obtains an upper bound for \(\|f(b)-f(a)\|\), for all \(f\in {\mathcal F}\) and \(a,b\in \mathbb{B}\), where \({\mathcal F}\subseteq {\mathcal B}(\mathbb{B})\). In the sixth section, the author proves two-point distortion results for subfamilies of \({\mathcal B}(\mathbb{B})\), by using the notion of (trace) order.
linearly invariant families of locally biholomorphic mappings
scientific article; zbMATH DE number 6391254
scientific article