Result: linearly invariant families of holomorphic mappings of a ball the dimension reduction method: Linearly invariant families of holomorphic mappings of a ball. The dimension reduction method

Title:
linearly invariant families of holomorphic mappings of a ball the dimension reduction method: Linearly invariant families of holomorphic mappings of a ball. The dimension reduction method
Authors:
V. V. Starkov, P. Liczberski
Source:
Siberian Mathematical Journal. 42(4):715-730
Publisher Information:
Russian Academy of Sciences - RAS (Rossiĭskaya Akademiya Nauk - RAN), Siberian Branch (Sibirskoe Otdelenie), Sobolev Insitute of Mathematics (Institut Matematiki Im. S. L. Soboleva), Novosibirsk, 2001.
Publication Year:
2001
Subject Terms:
Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.), Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables, Bloch function, distortion of a locally biholomorphic map, extension theorem, geometric properties of holomorphic mappings, Other generalizations of function theory of one complex variable, theory of univalent functions
Document Type:
Academic journal Article
File Description:
application/xml
ISSN:
0037-4466
DOI:
10.1023/a:1010497431634
Access URL:
https://zbmath.org/1882362
https://link.springer.com/article/10.1023/A%3A1010497431634
Accession Number:
edsair.dedup.wf.002..46dccb5dc5e06b99fcced043b3cd6ba7
Database:
OpenAIRE

Further Information

Summary: The notion of a linearly invariant family of mappings of a ball in \(\mathbb C^m\) was introduced in the article [\textit{J.~A.~Pfaltzgraff}, Complex Variables, Theory Appl. 33, No. 1-4, 239-253 (1997; Zbl 0912.32017)]. It generalizes the classical case \(m=1\) studied previously by Ch.~Pommerenke and other authors. In the above-mentioned article, J.~A.~Pfaltzgraff obtained and used a false equality, (5.3). Moreover, this equality was used in some other papers, for example, [\textit{J.~A.~Pfaltzgraff} and \textit{T.~J.~Suffridge}, Ann. Univ. Mariae Curie-Sklodowska, Sect. A 53, 193-207 (1999; Zbl 0996.32006)]. We propose the dimension reduction method which makes it possible to obtain correct proofs for the theorems of the above-mentioned papers in which equality (5.3) was originally used. We also obtain new results on linearly invariant families of mappings of a ball. The idea of the method proposed is simple and consists in reducing a problem posed for linearly invariant families in \(\mathbb C^m\) to a problem for the classical case of a disk \((m=1)\).