# 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].)

How to Cite This Entry:
Bloch function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bloch_function&oldid=46084
This article was adapted from an original article by J. Wiegerinck (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article