# zbMATH — the first resource for mathematics

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.

##### MSC:
 30D45 Normal functions of one complex variable, normal families 30G30 Other generalizations of analytic functions (including abstract-valued functions)
Full Text:
##### References:
 [1] J. M. Anderson, J. Clunie, and C. Pommerenke, ?On Bloch functions and normal functions,? J. Reine Angew. Math.,270, No. 1, 12-37 (1974). · Zbl 0292.30030 [2] J. A. Cima, ?The basic properties of Bloch functions,? Int. J. Math. Sci.,2, No. 3, 369-413 (1979). · Zbl 0423.30022 · doi:10.1155/S0161171279000314 [3] C. Pommerenke, ?On Bloch functions,? J. London Math. Soc.,2, No. 3, 689-695 (1970). · Zbl 0199.39803 [4] S. Yamashita, ?Criteria for a function to be Bloch,? Bull. Austral. Math. Soc.,21, No. 1, 223-227 (1980). · Zbl 0424.30025 · doi:10.1017/S0004972700006043 [5] R. M. Timoney, ?A necessary and sufficient condition for Bloch functions,? Proc. Am. Math. Soc.,71, No. 2, 263-266 (1978). · Zbl 0408.30038 · doi:10.1090/S0002-9939-1978-0481012-7 [6] R. M. Timoney, ?Bloch functions in several variables,? Bull. London Math. Soc.,12, No. 4, 241-267 (1980). · Zbl 0428.32018 · doi:10.1112/blms/12.4.241 [7] B. V. Shabat, Introduction to Complex Analysis [in Russian], Part 2, Nauka, Moscow (1976).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.