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Characteristic properties of Bloch functions. (English. Russian original) Zbl 0623.30043
Sib. Math. J. 28, No. 1-2, 43-46 (1987); translation from Sib. Mat. Zh. 28, No. 1(161), 61-64 (1987).
The authors give several characterizations of Bloch functions, in classical and generated form. M is a finite-dimensional complex manifold and G is a transitive group of holomorphic self-maps of M. Let f be holomorphic on M and \[ O(G,f,z_ 0) = \{f\circ g(z)-(f\circ g)(z_ 0): g\in G\}. \] If O is precompact, then f is called a Bloch function; when \(M=\Delta =\{| z| <1\}\) and G is the full Möbius group on \(\Delta\) we recover the classical Bloch space.
They introduce a class of “admissible” functions \(\phi\) on the plane \({\mathbb{C}}:\) these are positive, monotone real-valued set functions relative to bounded subsets of \({\mathbb{C}}\), with \(\phi ({\mathbb{C}})=\infty\). The main result is that f is Bloch iff sup \(\phi\) (f\(\circ g(V))<\infty\) (sup over \(g\in G)\), where \(V\subset {\mathbb{C}}\) has compact closure.
This result applies to several situations. One is when M is a hyperbolic manifold with the Kobayashi metric; then \({\mathcal B}\) coincides with Lipschitz functions.

30D45 Normal functions of one complex variable, normal families
30G30 Other generalizations of analytic functions (including abstract-valued functions)
Full Text: DOI
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