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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b1106201.png" /> be the open unit disc in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b1106202.png" />. A [[Holomorphic function|holomorphic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b1106203.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b1106204.png" /> is called a Bloch function if it has the property that
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b1106205.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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{{TEX|done}}
  
for a positive constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b1106206.png" />, independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b1106207.png" />. The Bloch norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b1106208.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b1106209.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b11062010.png" /> is the infimum of the constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b11062011.png" /> for which (a1) holds. The Bloch norm turns the set of Bloch functions into a [[Banach space|Banach space]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b11062012.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b11062013.png" /> is a Möbius-invariant [[Semi-norm|semi-norm]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b11062014.png" /> (cf. also [[Fractional-linear mapping|Fractional-linear mapping]]).
+
Let  $  D $
 +
be the open unit disc in  $  \mathbf C $.  
 +
A [[Holomorphic function|holomorphic function]] $  f $
 +
on  $  D $
 +
is called a Bloch function if it has the property that
  
Bloch functions appear naturally in connection with Bloch's theorem. Call a disc in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b11062015.png" /> in the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b11062016.png" /> schlicht if it is the univalent image of some open set (cf. [[Univalent function|Univalent function]]). Bloch's theorem can be stated as follows. There is a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b11062017.png" /> (the ''[[Bloch constant]]'') such that the image of every holomorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b11062018.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b11062019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b11062020.png" /> contains the schlicht disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b11062021.png" />.
+
$$ \tag{a1 }
 +
\left | {f  ^  \prime  ( z ) } \right | ( 1 - \left | z \right |  ^ {2} ) < C,
 +
$$
  
A disc automorphism leads to schlicht discs of radius at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b11062022.png" /> about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b11062023.png" />. The radii of the schlicht discs of Bloch functions are therefore bounded.
+
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|Banach space]],  $  {\mathcal B} $,
 +
and  $  C _ {f} $
 +
is a Möbius-invariant [[Semi-norm|semi-norm]] on  $  {\mathcal B} $(
 +
cf. also [[Fractional-linear mapping|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|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.
 
The following properties of Bloch functions are well-known.
  
i) Bounded holomorphic functions, and moreover analytic functions with boundary values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b11062024.png" /> (cf. [[BMO-space|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b11062025.png" />-space]]), are in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b11062026.png" />.
+
i) Bounded holomorphic functions, and moreover analytic functions with boundary values in $  { \mathop{\rm BMO} } $(
 +
cf. [[BMO-space| $  { \mathop{\rm BMO} } $-
 +
space]]), are in $  {\mathcal B} $.
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b11062027.png" /> coincides with the class of analytic functions that are in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b11062028.png" /> of the disc.
+
ii) $  {\mathcal B} $
 +
coincides with the class of analytic functions that are in $  { \mathop{\rm BMO} } $
 +
of the disc.
  
iii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b11062029.png" /> is the largest Möbius-invariant space of holomorphic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b11062030.png" /> that possesses non-zero continuous functionals that are also continuous with respect to some Möbius-invariant semi-norm, cf. [[#References|[a3]]].
+
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. [[#References|[a3]]].
  
iv) Bloch functions are normal, i.e., if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b11062031.png" /> is Bloch, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b11062032.png" /> is a [[Normal family|normal family]].
+
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|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. [[#References|[a1]]], [[#References|[a2]]].)
 
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. [[#References|[a1]]], [[#References|[a2]]].)
  
The concept of a Bloch function has been extended to analytic functions of several complex variables on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b11062033.png" />. This can be done by replacing (a1) by the estimates
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b11062034.png" /></td> </tr></table>
+
$$
 +
\left | {f  ^  \prime  ( P ) \zeta } \right | < C F  ^  \Omega  ( P, \zeta ) .
 +
$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b11062035.png" /> denotes the Kobayashi metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b11062036.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b11062037.png" /> in the direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110620/b11062038.png" />. (Cf. [[#References|[a2]]], [[#References|[a4]]], [[#References|[a5]]].)
+
Here $  F  ^  \Omega  ( P, \zeta ) $
 +
denotes the Kobayashi metric of $  \Omega $
 +
at $  P $
 +
in the direction $  \zeta $.  
 +
(Cf. [[#References|[a2]]], [[#References|[a4]]], [[#References|[a5]]].)
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.M. Anderson,  J. Clunie,  Ch. Pommerenke,  "On Bloch functions and normal functions"  ''J. Reine Angew. Math.'' , '''270'''  (1974)  pp. 12–37</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S.G. Krantz,  "Geometric analysis and function spaces" , ''CBMS'' , '''81''' , Amer. Math. Soc.  (1993)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L. Rubel,  R. Timoney,  "An extremal property of the Bloch space"  ''Proc. Amer. Math. Soc.'' , '''43'''  (1974)  pp. 306–310</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Timoney,  "Bloch functions in several complex variables, I"  ''Bull. London Math. Soc.'' , '''12'''  (1980)  pp. 241–267</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Timoney,  "Bloch functions in several complex variables, II"  ''J. Reine Angew. Math.'' , '''319'''  (1980)  pp. 1–22</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.M. Anderson,  J. Clunie,  Ch. Pommerenke,  "On Bloch functions and normal functions"  ''J. Reine Angew. Math.'' , '''270'''  (1974)  pp. 12–37</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S.G. Krantz,  "Geometric analysis and function spaces" , ''CBMS'' , '''81''' , Amer. Math. Soc.  (1993)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L. Rubel,  R. Timoney,  "An extremal property of the Bloch space"  ''Proc. Amer. Math. Soc.'' , '''43'''  (1974)  pp. 306–310</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Timoney,  "Bloch functions in several complex variables, I"  ''Bull. London Math. Soc.'' , '''12'''  (1980)  pp. 241–267</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Timoney,  "Bloch functions in several complex variables, II"  ''J. Reine Angew. Math.'' , '''319'''  (1980)  pp. 1–22</TD></TR></table>

Latest revision as of 10:59, 29 May 2020


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
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