Schottky-Landau growth estimates for \(s\)-normal families of holomorphic mappings
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 \(X\).
scientific article; zbMATH DE number 126339
scientific article