Superadditivity
property of a function
In mathematics, a function
f
{\displaystyle f}
is superadditive if
f
(
x
+
y
)
≥
f
(
x
)
+
f
(
y
)
{\displaystyle f(x+y)\geq f(x)+f(y)}
for all
x
{\displaystyle x}
and
y
{\displaystyle y}
in the domain of
f
.
{\displaystyle f.}
Similarly, a sequence
a
1
,
a
2
,
…
{\displaystyle a_{1},a_{2},\ldots }
is called superadditive if it satisfies the inequality
a
n
+
m
≥
a
n
+
a
m
{\displaystyle a_{n+m}\geq a_{n}+a_{m}}
for all
m
{\displaystyle m}
and
n
.
{\displaystyle n.}
The term "superadditive" is also applied to functions from a boolean algebra to the real numbers where
P
(
X
∨
Y
)
≥
P
(
X
)
+
P
(
Y
)
,
{\displaystyle P(X\lor Y)\geq P(X)+P(Y),}
such as lower probabilities.
Examples of superadditive functions
The map
f
(
x
)
=
x
2
{\displaystyle f(x)=x^{2}}
is a superadditive function for nonnegative real numbers because
f
(
x
+
y
)
=
(
x
+
y
)
2
=
x
2
+
y
2
+
2
x
y
=
f
(
x
)
+
f
(
y
)
+
2
x
y
≥
f
(
x
)
+
f
(
y
)
.
{\displaystyle f(x+y)=(x+y)^{2}=x^{2}+y^{2}+2xy=f(x)+f(y)+2xy\geq f(x)+f(y).}
The determinant is superadditive for nonnegative Hermitian matrix, that is, if
A
,
B
∈
Mat
n
(
C
)
{\displaystyle A,B\in {\text{Mat}}_{n}(\mathbb {C} )}
are nonnegative Hermitian then
det
(
A
+
B
)
≥
det
(
A
)
+
det
(
B
)
.
{\displaystyle \det(A+B)\geq \det(A)+\det(B).}
This follows from the Minkowski determinant theorem, which more generally states that
det
(
⋅
)
1
/
n
{\displaystyle \det(\cdot )^{1/n}}
is superadditive (equivalently, concave) for nonnegative Hermitian matrices of size
n
{\displaystyle n}
: If
A
,
B
∈
Mat
n
(
C
)
{\displaystyle A,B\in {\text{Mat}}_{n}(\mathbb {C} )}
are nonnegative Hermitian then
det
(
A
+
B
)
1
/
n
≥
det
(
A
)
1
/
n
+
det
(
B
)
1
/
n
.
{\displaystyle \det(A+B)^{1/n}\geq \det(A)^{1/n}+\det(B)^{1/n}.}
Horst Alzer proved that Hadamard's gamma function
H
(
x
)
{\displaystyle H(x)}
is superadditive for all real numbers
x
,
y
{\displaystyle x,y}
with
x
,
y
≥
1.5031.
{\displaystyle x,y\geq 1.5031.}
Mutual information
Properties
If
f
{\displaystyle f}
is a superadditive function whose domain contains
0
,
{\displaystyle 0,}
then
f
(
0
)
≤
0.
{\displaystyle f(0)\leq 0.}
To see this, simply set
x
=
0
{\displaystyle x=0}
and
y
=
0
{\displaystyle y=0}
in the defining inequality.
The negative of a superadditive function is subadditive.
Begin with the source’s own compact description: “Superadditivity” is property of a function. The dossier treats that line as a proposition to test through Superadditivity, property and function, not as a finished interpretation.
Why this record matters
The phrase “property of a function” supplies a clear boundary for inquiry. It also exposes the unanswered questions: who defined that boundary, when it became stable and which sources sit outside it.
Named sources, stable identifiers and responsible institutions provide the strongest route from overview to verifiable evidence. The source revision retrieved here is dated Feb 24, 2025. The linked authority identifier is Q922087. None of the 0 selected statements returned an explicit reference.
Overview language is designed for orientation and should not be treated as a substitute for the evidence cited beneath it. The lead is largely declarative, so disagreement and counter-evidence require a deliberate search beyond the opening account. Authority statements aid reconciliation but still require their own references, qualifiers and ranks to be checked.
How to read it
Use the entry as an orientation point, then follow its citations and revision history. Names, dates and institutional relationships should be checked against the original record.
- Subject orientation
- Search vocabulary
- Locating named sources
The closest primary source, responsible institution and strongest cited specialist reference.
Three-step research path
- Establish the record: confirm the title “Superadditivity”, its source revision and the description used here.
- Expand the search: follow Superadditivity primary sources, Superadditivity archive and Superadditivity research across catalogues and specialist indexes.
- Test the account: compare the strongest cited source with the responsible institution’s current record and note any disagreement.
Questions for further research
- Which source most directly establishes the central claim about “Superadditivity”?
- What terminology or title could unlock a more precise catalogue search?
- Which cited source is closest to the event, object or claim?
Search terms from this dossier
This entry incorporates text from “Superadditivity” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.