Two-point distortion bounds for biholomorphic mappings of the ball in \(\mathbb C^n\)
type: Item
instance of: scholarly article
title: Two-point distortion bounds for biholomorphic mappings of the ball in \(\mathbb C^n\)
zbMATH Open document ID: 1322.32014
published in: Journal of Mathematical Analysis and Applications
publication date: 2015-01-27T00:00:00Z
review text: 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 u…
reviewed by: G. Kohr
Mathematics Subject Classification ID: 32H02
zbMATH DE Number: 6391254
zbMATH Keywords: biholomorphic mappings
zbMATH Keywords: Carathéodory distance
zbMATH Keywords: two-point distortion
zbMATH Keywords: linearly invariant families of locally biholomorphic mappings
author: Jerry R. jun. Muir
MaRDI profile type: Publication
full work available at URL: j.jmaa.2014.02.059
OpenAlex ID: W2094564300
cites work: Ein Verzerrungssatz für schlichte Funktionen
cites work: Variation of tensor powers and spectrat
cites work: Distortion theorems for biholomorphic convex mappings on bounded convex circular domains
cites work: Boundary Behavior of the Caratheodory and Kobayashi Metrics on Strongly Pseudoconvex Domains in C n with Smooth Boundary
cites work: Q4444884
cites work: Growth and two-point distortion for biholomorphic mappings of the ball
cites work: Linear invariance of locally biholomorphic mappings in the unit ball of a JB\(^\ast\)-triple
cites work: Growth and distortion theorems for linearly invariant families on homogeneous unit balls in \(\mathbb C^n\)
cites work: Extensions of \(L^d\)-Loewner chains to higher dimensions
cites work: On a theorem of Grace
cites work: Two-point distortion theorems for univalent functions
cites work: Function theory of several complex variables
cites work: Two-point distortion for univalent functions
cites work: Distortion of locally biholomorphic maps of the N-Ball
cites work: Linear Invariance, Order and Convex Maps in
cites work: Norm order and geometric properties of holomorphic mappings in \(\mathbb{C}^n\)
cites work: Linear-invariante Familien analytischer Funktionen. I
cites work: Linear-invariante Familien analytischer Funktionen. II
cites work: Convex mappings on the unit ball of \(\mathbb C^ n\)
cites work: Q3958740
DOI: 10.1016/J.JMAA.2014.02.059
description: scientific article; zbMATH DE number 6391254