Some results on rigidity of holomorphic mappings
The rigidity property of holomorphic mappings in complex normed spaces is investigated in this paper. For two such spaces \(X\) and \(Y\), let \(D_ 1\) be a balanced domain in \(X\), \(D_ 2\) be a bounded convex domain in \(Y\), and \(f : D_ 1 \to D_ 2\) a holomorphic mapping. Three theorems are proved: The first is a generalization of the Schwarz lemma, that gives an upper bound for \(\mu_{D_ 2} (f(x))\), \(x \in D_ 1\), where \(\mu_{D_ 2}\) is the Minkowski functional of \(D_ 2\). As its corollary, a result on the extremal mappings is deduced, i.e., if \(f\) satisfies \(\| f(x) \| = \| x \|\) for all \(x \in X\), then \(f\) is linear. The second theorem concerns the lower bound for \(\mu_{D_ 2} (f(x))\). At last, the pointwise limit of a sequence of automorphisms of a bounded domain is considered; with the above two theorems it is proved that the limit mapping is one-to-one.
scientific article; zbMATH DE number 701567