Superadditivity

• The map ${\displaystyle f(x)=x^{2}}$ is a superadditive function for nonnegative real numbers because the square of ${\displaystyle x+y}$ is always greater than or equal to the square of ${\displaystyle x}$ plus the square of ${\displaystyle y,}$ for nonnegative real numbers ${\displaystyle x}$ and ${\displaystyle y}$: ${\displaystyle f(x+y)=(x+y)^{2}=x^{2}+y^{2}+2xy=f(x)+f(y)+2xy.}$

• The determinant is superadditive for nonnegative Hermitian matrix, that is, if ${\displaystyle A,B\in {\text{Mat}}_{n}(\mathbb {C} )}$ are nonnegative Hermitian then ${\displaystyle \det(A+B)\geq \det(A)+\det(B).}$ This follows from the Minkowski determinant theorem, which more generally states that ${\displaystyle \det(\cdot )^{1/n}}$ is superadditive (equivalently, concave)[1] for nonnegative Hermitian matrices of size ${\displaystyle n}$: If ${\displaystyle A,B\in {\text{Mat}}_{n}(\mathbb {C} )}$ are nonnegative Hermitian then ${\displaystyle \det(A+B)^{1/n}\geq \det(A)^{1/n}+\det(B)^{1/n}.}$
• Horst Alzer proved[2] that Hadamard's gamma function ${\displaystyle H(x)}$ is superadditive for all real numbers ${\displaystyle x,y}$ with ${\displaystyle x,y\geq 1.5031.}$
• Mutual information

If ${\displaystyle f}$ is a superadditive function whose domain contains ${\displaystyle 0,}$ then ${\displaystyle f(0)\leq 0.}$ To see this, take the inequality at the top: ${\displaystyle f(x)\leq f(x+y)-f(y).}$ Hence ${\displaystyle f(0)\leq f(0+y)-f(y)=0.}$

The negative of a superadditive function is subadditive.

The major reason for the use of superadditive sequences is the following lemma due to Michael Fekete.[3]

Lemma: (Fekete) For every superadditive sequence ${\displaystyle a_{1},a_{2},\ldots ,}$ the limit ${\displaystyle \lim a_{n}/n}$ is equal to the supremum ${\displaystyle \sup a_{n}/n.}$ (The limit may be positive infinity, as is the case with the sequence ${\displaystyle a_{n}=\log n!}$ for example.)

The analogue of Fekete's lemma holds for subadditive functions as well. There are extensions of Fekete's lemma that do not require the definition of superadditivity above to hold for all ${\displaystyle m}$ and ${\displaystyle n.}$ There are also results that allow one to deduce the rate of convergence to the limit whose existence is stated in Fekete's lemma if some kind of both superadditivity and subadditivity is present. A good exposition of this topic may be found in Steele (1997).[4][5]

• Choquet integral
• Inner measure
• Subadditivity – property of a function where the sum of two elements in domain of function is less than the sum of function valuesPages displaying wikidata descriptions as a fallback
• Sublinear function

Notes

• György Polya and Gábor Szegö. (1976). Problems and theorems in analysis, volume 1. Springer-Verlag, New York. ISBN 0-387-05672-6.

This article incorporates material from Superadditivity on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.