Bloch function

Let be the open unit disc in . A holomorphic function on is called a Bloch function if it has the property that

(a1)

for a positive constant , independent of . The Bloch norm of is , where is the infimum of the constants for which (a1) holds. The Bloch norm turns the set of Bloch functions into a Banach space, , and is a Möbius-invariant semi-norm on (cf. also Fractional-linear mapping).

Bloch functions appear naturally in connection with Bloch's theorem. Call a disc in in the image of schlicht if it is the univalent image of some open set (cf. Univalent function). Bloch's theorem can be stated as follows. There is a constant (the Bloch constant) such that the image of every holomorphic function with , contains the schlicht disc .

A disc automorphism leads to schlicht discs of radius at least about . The radii of the schlicht discs of Bloch functions are therefore bounded.

The following properties of Bloch functions are well-known.

i) Bounded holomorphic functions, and moreover analytic functions with boundary values in (cf. -space), are in .

ii) coincides with the class of analytic functions that are in of the disc.

iii) is the largest Möbius-invariant space of holomorphic functions on that possesses non-zero continuous functionals that are also continuous with respect to some Möbius-invariant semi-norm, cf. [a3].

iv) Bloch functions are normal, i.e., if is Bloch, then is a normal family.

v) Boundary values of Bloch functions need not exist; also, the radial limit function can be bounded almost-everywhere, while the Bloch function is unbounded. (Cf. [a1], [a2].)

The concept of a Bloch function has been extended to analytic functions of several complex variables on a domain . This can be done by replacing (a1) by the estimates

Here denotes the Kobayashi metric of at in the direction . (Cf. [a2], [a4], [a5].)

References

[a1]	J.M. Anderson, J. Clunie, Ch. Pommerenke, "On Bloch functions and normal functions" J. Reine Angew. Math. , 270 (1974) pp. 12–37
[a2]	S.G. Krantz, "Geometric analysis and function spaces" , CBMS , 81 , Amer. Math. Soc. (1993)
[a3]	L. Rubel, R. Timoney, "An extremal property of the Bloch space" Proc. Amer. Math. Soc. , 43 (1974) pp. 306–310
[a4]	R. Timoney, "Bloch functions in several complex variables, I" Bull. London Math. Soc. , 12 (1980) pp. 241–267
[a5]	R. Timoney, "Bloch functions in several complex variables, II" J. Reine Angew. Math. , 319 (1980) pp. 1–22