Bloch function

Let $ D $ be the open unit disc in $ \mathbf C $. A holomorphic function $ f $ on $ D $ is called a Bloch function if it has the property that

$$ \tag{a1 } \left | {f ^ \prime ( z ) } \right | ( 1 - \left | z \right | ^ {2} ) < C, $$

for a positive constant $ C $, independent of $ z \in D $. The Bloch norm of $ f $ is $ \| f \| _ {\mathcal B} = | {f ( 0 ) } | +C _ {f} $, where $ C _ {f} $ is the infimum of the constants $ C $ for which (a1) holds. The Bloch norm turns the set of Bloch functions into a Banach space, $ {\mathcal B} $, and $ C _ {f} $ is a Möbius-invariant semi-norm on $ {\mathcal B} $( cf. also Fractional-linear mapping).

Bloch functions appear naturally in connection with Bloch's theorem. Call a disc in $ \mathbf C $ in the image of $ f $ 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 $ B $( the Bloch constant) such that the image of every holomorphic function $ f $ with $ f ( 0 ) = 0 $, $ f ^ \prime ( 0 ) = 1 $ contains the schlicht disc $ \{ w : {| w | < B } \} $.

A disc automorphism leads to schlicht discs of radius at least $ B | {f ^ \prime ( z ) } | ( 1 - | z | ^ {2} ) $ about $ f ( z ) $. 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 $ { \mathop{\rm BMO} } $( cf. $ { \mathop{\rm BMO} } $- space), are in $ {\mathcal B} $.

ii) $ {\mathcal B} $ coincides with the class of analytic functions that are in $ { \mathop{\rm BMO} } $ of the disc.

iii) $ {\mathcal B} $ is the largest Möbius-invariant space of holomorphic functions on $ D $ 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 $ f $ is Bloch, then $ \{ {f \circ \tau } : {\tau \in { \mathop{\rm AUT} } ( D ) } \} $ 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 $ \Omega \subset \mathbf C ^ {n} $. This can be done by replacing (a1) by the estimates

$$ \left | {f ^ \prime ( P ) \zeta } \right | < C F ^ \Omega ( P, \zeta ) . $$

Here $ F ^ \Omega ( P, \zeta ) $ denotes the Kobayashi metric of $ \Omega $ at $ P $ in the direction $ \zeta $. (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