Bounded type (mathematics)
Open-knowledge reference entry
In mathematics, a function defined on a region of the complex plane is said to be of bounded type if it is equal to the ratio of two analytic functions bounded in that region. But more generally, a function is of bounded type in a region
Ω
{\displaystyle \Omega }
if and only if
f
{\displaystyle f}
is analytic on
Ω
{\displaystyle \Omega }
and
log
+
|
f
(
z
)
|
{\displaystyle \log ^{+}|f(z)|}
has a harmonic majorant on
Ω
,
{\displaystyle \Omega ,}
where
log
+
(
x
)
=
max
[
0
,
log
(
x
)
]
{\displaystyle \log ^{+}(x)=\max[0,\log(x)]}
. Being the ratio of two bounded analytic functions is a sufficient condition for a function to be of bounded type (defined in terms of a harmonic majorant), and if
Ω
{\displaystyle \Omega }
is simply connected the condition is also necessary.
The class of all such
f
{\displaystyle f}
on
Ω
{\displaystyle \Omega }
is commonly denoted
N
(
Ω
)
{\displaystyle N(\Omega )}
and is sometimes called the Nevanlinna class for
Ω
{\displaystyle \Omega }
. The Nevanlinna class includes all the Hardy classes.
Functions of bounded type are not necessarily bounded, nor do they have a property called "type" which is bounded. The reason for the name is probably that when defined on a disc, the Nevanlinna characteristic (a function of distance from the centre of the disc) is bounded.
Clearly, if a function is the ratio of two bounded functions, then it can be expressed as the ratio of two functions which are bounded by 1:
f
(
z
)
=
P
(
z
)
/
Q
(
z
)
{\displaystyle f(z)=P(z)/Q(z)}
The logarithms of
|
1
/
P
(
z
)
|
{\displaystyle |1/P(z)|}
and of
|
1
/
Q
(
z
)
|
{\displaystyle |1/Q(z)|}
are non-negative in the region, so
log
|
f
(
z
)
|
=
log
|
1
/
Q
(
z
)
|
−
log
|
1
/
P
(
z
)
|
≤
log
|
1
/
Q
(
z
)
|
{\displaystyle {\begin{aligned}\log |f(z)|&=\log |1/Q(z)|-\log |1/P(z)|\\&\leq \log |1/Q(z)|\end{aligned}}}
log
+
|
f
(
z
)
|
=
max
[
0
,
log
|
f
(
z
)
|
]
≤
max
(
0
,
log
|
1
/
Q
(
z
)
|
)
≤
log
|
1
/
Q
(
z
)
|
≤
−
ℜ
(
log
Q
(
z
)
)
.
{\displaystyle {\begin{aligned}\log ^{+}|f(z)|&=\max[0,\log |f(z)|]\\&\leq \max(0,\log |1/Q(z)|)\\&\leq \log |1/Q(z)|\\&\leq -\Re \left(\log Q(z)\right).\end{aligned}}}
The latter is the real part of an analytic function and is therefore harmonic, showing that
log
+
|
f
(
z
)
|
{\displaystyle \log ^{+}|f(z)|}
has a harmonic majorant on Ω.
For a given region, sums, differences, and products of functions of bounded type are of bounded type, as is the quotient of two such functions as long as the denominator is not identically zero.
Examples
Polynomials are of bounded type in any bounded region.
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