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

A [[Meromorphic function|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|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|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 [[#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====
<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====
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.