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''in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f0419501.png" /> of the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f0419502.png" />''
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A [[Meromorphic function|meromorphic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f0419503.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f0419504.png" /> that can be represented in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f0419505.png" /> as the quotient of two bounded analytic functions,
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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/f/f041/f041950/f0419506.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
''in a domain  $  D $
 +
of the complex plane  $  \mathbf C $''
  
is called a function of bounded type. The class most studied is the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f0419507.png" /> of functions of bounded type in the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f0419508.png" />: A meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f0419509.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195010.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195011.png" /> if and only if its characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195012.png" /> is bounded (Nevanlinna's theorem):
+
A [[Meromorphic function|meromorphic function]]  $  f ( z) $
 +
in  $  D $
 +
that can be represented in $  D $
 +
as the quotient of two bounded analytic functions,
  
<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/f/f041/f041950/f04195013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{1 }
 +
f ( z) = \
  
<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/f/f041/f041950/f04195014.png" /></td> </tr></table>
+
\frac{g _ {1} ( z) }{g _ {2} ( z) }
 +
,\ \
 +
| g _ {1} |, | g _ {2} |  \leq  1,\ \
 +
z \in D ,
 +
$$
  
Here the sum on the right-hand side is taken over all poles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195015.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195016.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195017.png" />, and each pole is taken as many times as its multiplicity; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195018.png" /> is the multiplicity of the pole at the origin. Hence functions in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195019.png" /> are also called functions of bounded characteristic. The following sufficient condition is also of interest: If a meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195020.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195021.png" /> does not take a set of values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195022.png" /> of positive [[Capacity|capacity]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195023.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195024.png" />.
+
is called a function of bounded type. The class most studied is the class  $  N ( \Delta ) $
 +
of functions of bounded type in the unit disc  $  \Delta = \{ {z \in \mathbf C } : {| z | < 1 } \} $:  
 +
A meromorphic function f ( z) $
 +
in $  \Delta $
 +
belongs to  $  N ( \Delta ) $
 +
if and only if its characteristic  $  T ( r;  f ) $
 +
is bounded (Nevanlinna's theorem):
  
The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195025.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195026.png" /> have the following properties: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195027.png" /> has angular boundary values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195028.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195029.png" />, almost-everywhere on the unit circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195030.png" />; 2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195031.png" /> on a set of points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195032.png" /> of positive measure, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195033.png" />; 3) a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195034.png" /> is characterized by an integral representation of the form
+
$$ \tag{2 }
 +
T ( r;  f = \
 +
{
 +
\frac{1}{2 \pi }
 +
}
 +
\int\limits _ { 0 } ^ { {2 }  \pi }
 +
\mathop{\rm ln}  ^ {+} \
 +
| f ( re ^ {i \theta } ) | \
 +
d \theta +
 +
$$
  
<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/f/f041/f041950/f04195035.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$
 +
+
 +
\sum  \mathop{\rm ln}  {
 +
\frac{r}{| b _  \nu  | }
 +
} + \lambda  \mathop{\rm ln}
 +
r  \leq  C ( f ) < \infty ,\  0 < r < 1.
 +
$$
  
<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/f/f041/f041950/f04195036.png" /></td> </tr></table>
+
Here the sum on the right-hand side is taken over all poles  $  b _  \nu  $
 +
of  $  f ( z) $
 +
with  $  0 < | b _  \nu  | < r $,
 +
and each pole is taken as many times as its multiplicity; $  \lambda \geq  0 $
 +
is the multiplicity of the pole at the origin. Hence functions in the class  $  N ( \Delta ) $
 +
are also called functions of bounded characteristic. The following sufficient condition is also of interest: If a meromorphic function  $  f ( z) $
 +
in  $  \Delta $
 +
does not take a set of values  $  E $
 +
of positive [[Capacity|capacity]],  $  \mathop{\rm cap}  E > 0 $,
 +
then  $  f ( z) \in N ( \Delta ) $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195037.png" /> is the integer such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195039.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195040.png" /> is real; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195042.png" /> are the Blaschke products taken over all zeros <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195043.png" /> and poles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195044.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195045.png" /> inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195046.png" />, counted with multiplicity (cf. [[Blaschke product|Blaschke product]]); and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195047.png" /> is a singular function of bounded variation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195048.png" /> with derivative equal to zero almost-everywhere.
+
The functions  $  f ( z) $
 +
in  $  N ( \Delta ) $
 +
have the following properties: 1)  $  f ( z) $
 +
has angular boundary values  $  f ( e ^ {i \theta } ) $,  
 +
with  $  \mathop{\rm ln}  | f ( e ^ {i \theta } ) | \in L _ {1} ( \Gamma ) $,
 +
almost-everywhere on the unit circle  $  \Gamma = \{ {z \in \mathbf C } : {| z | = 1 } \} $;  
 +
2) if  $  f ( e ^ {i \theta } ) = 0 $
 +
on a set of points of  $  \Gamma $
 +
of positive measure, then  $  f ( z) \equiv 0 $;
 +
3) a function  $  f ( z) \in N ( \Delta ) $
 +
is characterized by an integral representation of the form
  
The subclass <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195049.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195050.png" /> consisting of all holomorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195051.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195052.png" /> is also of interest. A necessary and sufficient condition for a holomorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195053.png" /> to be in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195054.png" /> is that it satisfies the following condition, deduced from (2),
+
$$ \tag{3 }
 +
f ( z) = \
 +
z  ^ {m} e ^ {i \lambda }
  
<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/f/f041/f041950/f04195055.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
\frac{B _ {1} ( z; a _  \mu  ) }{B _ {2} ( z; b _  \nu  ) }
 +
\times
 +
$$
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195056.png" /> one must have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195058.png" /> in (3).
+
$$
 +
\times
 +
\mathop{\rm exp} \left \{ {
 +
\frac{1}{2 \pi }
 +
} \int\limits _ { 0 } ^ { {2 }  \pi }  \mathop{\rm ln}  | f ( e ^ {i \theta } ) |
 +
\frac{e ^
 +
{i \theta } + z }{e ^ {i \theta } - z }
 +
  d
 +
\theta + {
 +
\frac{1}{2 \pi }
 +
} \int\limits _ { 0 } ^ { {2 }  \pi }
 +
\frac{e ^ {i \theta } + z }{e ^ {i \theta } - z }
 +
  d \Phi ( \theta ) \right \} ,
 +
$$
  
Condition (4) is equivalent to the requirement that the subharmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195059.png" /> has a harmonic majorant in the whole disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195060.png" />. The condition in this form is usually taken to define the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195061.png" /> of holomorphic functions on arbitrary domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195062.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195063.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195064.png" /> has a harmonic majorant in the whole domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195065.png" />.
+
where  $  m $
 +
is the integer such that $  f ( z) = z  ^ {m} \phi ( z) $,
 +
$  \phi ( 0) \neq 0, \infty $;
 +
$  \lambda $
 +
is real;  $  B _ {1} ( z;  a _  \mu  ) $
 +
and  $  B _ {2} ( z;  b _  \nu  ) $
 +
are the Blaschke products taken over all zeros  $  a _  \mu  \neq 0 $
 +
and poles  $  b _  \nu  \neq 0 $
 +
of f ( z) $
 +
inside  $  \Delta $,
 +
counted with multiplicity (cf. [[Blaschke product|Blaschke product]]); and $  \Phi ( \theta ) $
 +
is a singular function of bounded variation on  $  [ 0, 2 \pi ] $
 +
with derivative equal to zero almost-everywhere.
  
Suppose that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195066.png" /> realizes a conformal universal covering mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195067.png" /> (i.e. a single-valued analytic function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195068.png" /> that is automorphic with respect to the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195069.png" /> of fractional-linear transformations of the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195070.png" /> onto itself corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195071.png" />). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195072.png" /> if and only if the composite function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195073.png" /> is automorphic relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195075.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195076.png" /> is a finitely-connected domain and if its boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195077.png" /> is rectifiable, then the angular boundary values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195079.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195080.png" /> exist almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195081.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195082.png" /> is summable with respect to harmonic measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195083.png" /> (for more details see the survey [[#References|[4]]]).
+
The subclass  $  N  ^ {*} ( \Delta ) $
 +
of $  N ( \Delta ) $
 +
consisting of all holomorphic functions  $  f ( z) $
 +
in  $  N ( \Delta ) $
 +
is also of interest. A necessary and sufficient condition for a holomorphic function f ( z) $
 +
to be in  $  N  ^ {*} ( \Delta ) $
 +
is that it satisfies the following condition, deduced from (2),
  
Now let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195084.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195086.png" />, be a holomorphic function of several variables on the unit polydisc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195087.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195088.png" /> be the skeleton of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195089.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195090.png" />. The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195091.png" /> of functions of bounded characteristic is defined by a condition generalizing (4):
+
$$ \tag{4 }
 +
{
 +
\frac{1}{2 \pi }
 +
}
 +
\int\limits _ { 0 } ^ { {2 }  \pi }
 +
\mathop{\rm ln}  ^ {+} \
 +
| f ( re ^ {i \theta } ) | \
 +
d \theta  \leq  \
 +
C ( f < \infty ,\ \
 +
0 < r < 1.
 +
$$
  
<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/f/f041/f041950/f04195092.png" /></td> </tr></table>
+
For  $  f ( z) \in N  ^ {*} ( \Delta ) $
 +
one must have  $  B _ {2} ( z; b _  \nu  ) \equiv 1 $,
 +
$  m \geq  0 $
 +
in (3).
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195093.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195094.png" /> is the normalized Haar measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195095.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195096.png" />. A holomorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195097.png" /> in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195098.png" /> has radial boundary values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f04195099.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f041950100.png" />, almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f041950101.png" /> with respect to Haar measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f041950102.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f041950103.png" /> is summable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f041950104.png" />. If the original definition (1) of a function of bounded type on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f041950105.png" /> is retained, then a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f041950106.png" /> of bounded type is a function of bounded characteristic, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f041950107.png" />. However, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f041950108.png" /> there are functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f041950109.png" /> that are not representable as the quotient of two bounded holomorphic functions (see [[#References|[5]]]).
+
Condition (4) is equivalent to the requirement that the subharmonic function  $  \mathop{\rm ln}  ^ {+}  | f ( z) | $
 +
has a harmonic majorant in the whole disc  $  \Delta $.
 +
The condition in this form is usually taken to define the class  $  N  ^ {*} ( D) $
 +
of holomorphic functions on arbitrary domains  $  D \subset  \mathbf C $:  
 +
$  f ( z) \in N  ^ {*} ( D) $
 +
if and only if  $  \mathop{\rm ln}  ^ {+}  | f ( z) | $
 +
has a harmonic majorant in the whole domain  $  D $.
 +
 
 +
Suppose that the function  $  w = w ( z) $
 +
realizes a conformal universal covering mapping  $  \Delta \rightarrow D $(
 +
i.e. a single-valued analytic function on  $  \Delta $
 +
that is automorphic with respect to the group  $  G $
 +
of fractional-linear transformations of the disc  $  \Delta $
 +
onto itself corresponding to  $  D $).  
 +
Then  $  f ( w) \in N  ^ {*} ( D) $
 +
if and only if the composite function  $  f ( w ( z)) $
 +
is automorphic relative to  $  G $
 +
and  $  f ( w ( z)) \in N  ^ {*} ( \Delta ) $.  
 +
If  $  D $
 +
is a finitely-connected domain and if its boundary  $  \partial  D $
 +
is rectifiable, then the angular boundary values  $  f ( \zeta ) $,
 +
$  \zeta \in \partial  D $,
 +
of  $  f ( z) \in N  ^ {*} ( D) $
 +
exist almost-everywhere on  $  \partial  D $,
 +
and  $  \mathop{\rm ln}  | f ( \zeta ) | $
 +
is summable with respect to harmonic measure on $  \partial  D $(
 +
for more details see the survey [[#References|[4]]]).
 +
 
 +
Now let  $  f ( z) $,
 +
$  z = ( z _ {1} \dots z _ {n} ) $,
 +
$  n > 1 $,
 +
be a holomorphic function of several variables on the unit polydisc  $  \Delta  ^ {n} = \{ {z \in \mathbf C  ^ {n} } : {| z _ {j} | < 1,  j = 1 \dots n } \} $,
 +
and let  $  T  ^ {n} $
 +
be the skeleton of  $  \Delta  ^ {n} $,
 +
$  T  ^ {n} = \{ {z \in \mathbf C  ^ {n} } : {| z _ {j} | = 1,  j = 1 \dots n } \} $.
 +
The class  $  N  ^ {*} ( \Delta  ^ {n} ) $
 +
of functions of bounded characteristic is defined by a condition generalizing (4):
 +
 
 +
$$
 +
\int\limits _ {T  ^ {n} }
 +
\mathop{\rm ln}  ^ {+} \
 +
| f ( r \zeta ) | \
 +
dm _ {n} ( \zeta )  \leq  \
 +
C ( f )  <  \infty ,\ \
 +
0 < r < 1,
 +
$$
 +
 
 +
where  $  \zeta = ( \zeta _ {1} \dots \zeta _ {n} ) \in T  ^ {n} $
 +
and  $  m _ {n} ( \zeta ) $
 +
is the normalized Haar measure on  $  T  ^ {n} $,
 +
$  m _ {n} ( T  ^ {n} ) = 1 $.  
 +
A holomorphic function f ( z) $
 +
in the class $  N  ^ {*} ( \Delta  ^ {n} ) $
 +
has radial boundary values $  \lim\limits _ {r \rightarrow 1 }  f ( r \zeta ) = f ( \zeta ) $,  
 +
$  \zeta \in T  ^ {n} $,  
 +
almost-everywhere on $  T  ^ {n} $
 +
with respect to Haar measure $  m _ {n} $,  
 +
and $  \mathop{\rm ln}  | f ( \zeta ) | $
 +
is summable on $  T  ^ {n} $.  
 +
If the original definition (1) of a function of bounded type on $  D = \Delta  ^ {n} $
 +
is retained, then a function f ( z) $
 +
of bounded type is a function of bounded characteristic, $  N ( \Delta  ^ {n} ) \subset  N  ^ {*} ( \Delta  ^ {n} ) $.  
 +
However, for $  n > 1 $
 +
there are functions $  g ( z) \in N  ^ {*} ( \Delta  ^ {n} ) $
 +
that are not representable as the quotient of two bounded holomorphic functions (see [[#References|[5]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Nevanilinna,  "Analytic functions" , Springer  (1970)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''2''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.I. [I.I. Privalov] Priwalow,  "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  ''Itogi Nauk. Mat. Anal. 1963''  (1965)  pp. 5–80</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  W. Rudin,  "Function theory in polydiscs" , Benjamin  (1969)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Nevanilinna,  "Analytic functions" , Springer  (1970)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''2''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.I. [I.I. Privalov] Priwalow,  "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  ''Itogi Nauk. Mat. Anal. 1963''  (1965)  pp. 5–80</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  W. Rudin,  "Function theory in polydiscs" , Benjamin  (1969)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
One should not confuse the notion of  "function of bounded type"  as defined above with that of an [[Entire function|entire function]] of bounded type. For this reason, functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f041950110.png" /> are sometimes called functions of bounded form or have no special name at all, the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041950/f041950111.png" /> being more important.
+
One should not confuse the notion of  "function of bounded type"  as defined above with that of an [[Entire function|entire function]] of bounded type. For this reason, functions f \in N ( \Delta ) $
 +
are sometimes called functions of bounded form or have no special name at all, the class $  N  ^ {*} ( \Delta ) $
 +
being more important.

Latest revision as of 19:40, 5 June 2020


in a domain $ D $ of the complex plane $ \mathbf C $

A meromorphic function $ f ( z) $ in $ D $ that can be represented in $ D $ as the quotient of two bounded analytic functions,

$$ \tag{1 } f ( z) = \ \frac{g _ {1} ( z) }{g _ {2} ( z) } ,\ \ | g _ {1} |, | g _ {2} | \leq 1,\ \ z \in D , $$

is called a function of bounded type. The class most studied is the class $ N ( \Delta ) $ of functions of bounded type in the unit disc $ \Delta = \{ {z \in \mathbf C } : {| z | < 1 } \} $: A meromorphic function $ f ( z) $ in $ \Delta $ belongs to $ N ( \Delta ) $ if and only if its characteristic $ T ( r; f ) $ is bounded (Nevanlinna's theorem):

$$ \tag{2 } T ( r; f ) = \ { \frac{1}{2 \pi } } \int\limits _ { 0 } ^ { {2 } \pi } \mathop{\rm ln} ^ {+} \ | f ( re ^ {i \theta } ) | \ d \theta + $$

$$ + \sum \mathop{\rm ln} { \frac{r}{| b _ \nu | } } + \lambda \mathop{\rm ln} r \leq C ( f ) < \infty ,\ 0 < r < 1. $$

Here the sum on the right-hand side is taken over all poles $ b _ \nu $ of $ f ( z) $ with $ 0 < | b _ \nu | < r $, and each pole is taken as many times as its multiplicity; $ \lambda \geq 0 $ is the multiplicity of the pole at the origin. Hence functions in the class $ N ( \Delta ) $ are also called functions of bounded characteristic. The following sufficient condition is also of interest: If a meromorphic function $ f ( z) $ in $ \Delta $ does not take a set of values $ E $ of positive capacity, $ \mathop{\rm cap} E > 0 $, then $ f ( z) \in N ( \Delta ) $.

The functions $ f ( z) $ in $ N ( \Delta ) $ have the following properties: 1) $ f ( z) $ has angular boundary values $ f ( e ^ {i \theta } ) $, with $ \mathop{\rm ln} | f ( e ^ {i \theta } ) | \in L _ {1} ( \Gamma ) $, almost-everywhere on the unit circle $ \Gamma = \{ {z \in \mathbf C } : {| z | = 1 } \} $; 2) if $ f ( e ^ {i \theta } ) = 0 $ on a set of points of $ \Gamma $ of positive measure, then $ f ( z) \equiv 0 $; 3) a function $ f ( z) \in N ( \Delta ) $ is characterized by an integral representation of the form

$$ \tag{3 } f ( z) = \ z ^ {m} e ^ {i \lambda } \frac{B _ {1} ( z; a _ \mu ) }{B _ {2} ( z; b _ \nu ) } \times $$

$$ \times \mathop{\rm exp} \left \{ { \frac{1}{2 \pi } } \int\limits _ { 0 } ^ { {2 } \pi } \mathop{\rm ln} | f ( e ^ {i \theta } ) | \frac{e ^ {i \theta } + z }{e ^ {i \theta } - z } d \theta + { \frac{1}{2 \pi } } \int\limits _ { 0 } ^ { {2 } \pi } \frac{e ^ {i \theta } + z }{e ^ {i \theta } - z } d \Phi ( \theta ) \right \} , $$

where $ m $ is the integer such that $ f ( z) = z ^ {m} \phi ( z) $, $ \phi ( 0) \neq 0, \infty $; $ \lambda $ is real; $ B _ {1} ( z; a _ \mu ) $ and $ B _ {2} ( z; b _ \nu ) $ are the Blaschke products taken over all zeros $ a _ \mu \neq 0 $ and poles $ b _ \nu \neq 0 $ of $ f ( z) $ inside $ \Delta $, counted with multiplicity (cf. Blaschke product); and $ \Phi ( \theta ) $ is a singular function of bounded variation on $ [ 0, 2 \pi ] $ with derivative equal to zero almost-everywhere.

The subclass $ N ^ {*} ( \Delta ) $ of $ N ( \Delta ) $ consisting of all holomorphic functions $ f ( z) $ in $ N ( \Delta ) $ is also of interest. A necessary and sufficient condition for a holomorphic function $ f ( z) $ to be in $ N ^ {*} ( \Delta ) $ is that it satisfies the following condition, deduced from (2),

$$ \tag{4 } { \frac{1}{2 \pi } } \int\limits _ { 0 } ^ { {2 } \pi } \mathop{\rm ln} ^ {+} \ | f ( re ^ {i \theta } ) | \ d \theta \leq \ C ( f ) < \infty ,\ \ 0 < r < 1. $$

For $ f ( z) \in N ^ {*} ( \Delta ) $ one must have $ B _ {2} ( z; b _ \nu ) \equiv 1 $, $ m \geq 0 $ in (3).

Condition (4) is equivalent to the requirement that the subharmonic function $ \mathop{\rm ln} ^ {+} | f ( z) | $ has a harmonic majorant in the whole disc $ \Delta $. The condition in this form is usually taken to define the class $ N ^ {*} ( D) $ of holomorphic functions on arbitrary domains $ D \subset \mathbf C $: $ f ( z) \in N ^ {*} ( D) $ if and only if $ \mathop{\rm ln} ^ {+} | f ( z) | $ has a harmonic majorant in the whole domain $ D $.

Suppose that the function $ w = w ( z) $ realizes a conformal universal covering mapping $ \Delta \rightarrow D $( i.e. a single-valued analytic function on $ \Delta $ that is automorphic with respect to the group $ G $ of fractional-linear transformations of the disc $ \Delta $ onto itself corresponding to $ D $). Then $ f ( w) \in N ^ {*} ( D) $ if and only if the composite function $ f ( w ( z)) $ is automorphic relative to $ G $ and $ f ( w ( z)) \in N ^ {*} ( \Delta ) $. If $ D $ is a finitely-connected domain and if its boundary $ \partial D $ is rectifiable, then the angular boundary values $ f ( \zeta ) $, $ \zeta \in \partial D $, of $ f ( z) \in N ^ {*} ( D) $ exist almost-everywhere on $ \partial D $, and $ \mathop{\rm ln} | f ( \zeta ) | $ is summable with respect to harmonic measure on $ \partial D $( for more details see the survey [4]).

Now let $ f ( z) $, $ z = ( z _ {1} \dots z _ {n} ) $, $ n > 1 $, be a holomorphic function of several variables on the unit polydisc $ \Delta ^ {n} = \{ {z \in \mathbf C ^ {n} } : {| z _ {j} | < 1, j = 1 \dots n } \} $, and let $ T ^ {n} $ be the skeleton of $ \Delta ^ {n} $, $ T ^ {n} = \{ {z \in \mathbf C ^ {n} } : {| z _ {j} | = 1, j = 1 \dots n } \} $. The class $ N ^ {*} ( \Delta ^ {n} ) $ of functions of bounded characteristic is defined by a condition generalizing (4):

$$ \int\limits _ {T ^ {n} } \mathop{\rm ln} ^ {+} \ | f ( r \zeta ) | \ dm _ {n} ( \zeta ) \leq \ C ( f ) < \infty ,\ \ 0 < r < 1, $$

where $ \zeta = ( \zeta _ {1} \dots \zeta _ {n} ) \in T ^ {n} $ and $ m _ {n} ( \zeta ) $ is the normalized Haar measure on $ T ^ {n} $, $ m _ {n} ( T ^ {n} ) = 1 $. A holomorphic function $ f ( z) $ in the class $ N ^ {*} ( \Delta ^ {n} ) $ has radial boundary values $ \lim\limits _ {r \rightarrow 1 } f ( r \zeta ) = f ( \zeta ) $, $ \zeta \in T ^ {n} $, almost-everywhere on $ T ^ {n} $ with respect to Haar measure $ m _ {n} $, and $ \mathop{\rm ln} | f ( \zeta ) | $ is summable on $ T ^ {n} $. If the original definition (1) of a function of bounded type on $ D = \Delta ^ {n} $ is retained, then a function $ f ( z) $ of bounded type is a function of bounded characteristic, $ N ( \Delta ^ {n} ) \subset N ^ {*} ( \Delta ^ {n} ) $. However, for $ n > 1 $ there are functions $ g ( z) \in N ^ {*} ( \Delta ^ {n} ) $ that are not representable as the quotient of two bounded holomorphic functions (see [5]).

References

[1] R. Nevanilinna, "Analytic functions" , Springer (1970) (Translated from German)
[2] A.I. Markushevich, "Theory of functions of a complex variable" , 2 , Chelsea (1977) (Translated from Russian)
[3] I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[4] Itogi Nauk. Mat. Anal. 1963 (1965) pp. 5–80
[5] W. Rudin, "Function theory in polydiscs" , Benjamin (1969)

Comments

One should not confuse the notion of "function of bounded type" as defined above with that of an entire function of bounded type. For this reason, functions $ f \in N ( \Delta ) $ are sometimes called functions of bounded form or have no special name at all, the class $ N ^ {*} ( \Delta ) $ being more important.

How to Cite This Entry:
Function of bounded characteristic. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Function_of_bounded_characteristic&oldid=47008
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article