Schottky-Landau growth estimates for \(s\)-normal families of holomorphic mappings
type: Item
instance of: scholarly article
title: Schottky-Landau growth estimates for \(s\)-normal families of holomorphic mappings
zbMATH Open document ID: 0766.32005
DOI: 10.1007/BF01444708
published in: Mathematische Annalen
publication date: 1993-03-03T00:00:00Z
full work available at URL: https://eudml.org/doc/164950
review text: Let \(X\), \(\overline Y\) be complex spaces and let \(Y\) be a relatively compact subspace of \(\overline Y\). A family \({\mathcal F}\subset{\mathcal O}(X,Y)\) is called \(s\)-normal if the set \(\{f\circ\varphi:\varphi\in{\mathcal O}(\Delta,X)\}\) is relatively compact in \({\mathcal O}(\Delta,\overline Y)\) in the compact-open topology \((\Delta\) denotes the unit disc in \(\mathbb{C})\). A mapping \(f\in{\mathcal O}(X,Y)\) is called normal if the family \(\{f\}\) is \(s\)-normal. The author presents various properties of \(s\)-normal families and normal mappings and their relations to other holomorphic objects. In particular, he proves the following generalization of the Schottky-Landau-Hayman growth estimates: Let \(\Psi\) be a family of holomorphic functions on a complex manifold \(X\). Then \(\Psi\) is \(s\)-normal if and only if there exists \(c>0\) such that: \[ \log(c|\psi(x)|)\leq(\log(c\|\psi\|_ Q))\exp(2k_ X(x,Q)),\quad x\in X,\;Q\subset X,\;\psi\in\Psi, \] where \(k_ X\) denotes the Kobayashi pseudodistance on \(X\).
reviewed by: Marek Jarnicki
Mathematics Subject Classification ID: 32A17
Mathematics Subject Classification ID: 32A22
Mathematics Subject Classification ID: 32F45
Mathematics Subject Classification ID: 32Q45
Mathematics Subject Classification ID: 32H25
Mathematics Subject Classification ID: 32H30
zbMATH DE Number: 126339
zbMATH Keywords: normal families
zbMATH Keywords: normal mappings
zbMATH Keywords: Schottky-Landau-Hayman growth estimates
zbMATH Keywords: holomorphic functions
author: Mikhail Zaidenberg
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 126339