# Superadditivity

In mathematics, a function is **superadditive** if

Similarly, a sequence is called **superadditive** if it satisfies the inequality

The term "superadditive" is also applied to functions from a boolean algebra to the real numbers where such as lower probabilities.

## Examples of superadditive functions[edit]

- The map is a superadditive function for nonnegative real numbers because the square of is always greater than or equal to the square of plus the square of for nonnegative real numbers and :

- The determinant is superadditive for nonnegative Hermitian matrix, that is, if are nonnegative Hermitian then This follows from the Minkowski determinant theorem, which more generally states that is superadditive (equivalently, concave)
^{[1]}for nonnegative Hermitian matrices of size : If are nonnegative Hermitian then - Horst Alzer proved
^{[2]}that Hadamard's gamma function is superadditive for all real numbers with - Mutual information

## Properties[edit]

If is a superadditive function whose domain contains then To see this, take the inequality at the top: Hence

The negative of a superadditive function is subadditive.

## Fekete's lemma[edit]

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

**Lemma:**(Fekete) For every superadditive sequence the limit is equal to the supremum (The limit may be positive infinity, as is the case with the sequence 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 and
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]}

## See also[edit]

- 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 values
- Sublinear function – Type of function in linear algebra

## References[edit]

**^**M. Marcus, H. Minc (1992).*A survey in matrix theory and matrix inequalities*. Dover. Theorem 4.1.8, page 115.**^**Horst Alzer (2009). "A superadditive property of Hadamard's gamma function".*Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg*. Springer.**79**: 11–23. doi:10.1007/s12188-008-0009-5. S2CID 123691692.**^**Fekete, M. (1923). "Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten".*Mathematische Zeitschrift*.**17**(1): 228–249. doi:10.1007/BF01504345. S2CID 186223729.**^**Michael J. Steele (1997).*Probability theory and combinatorial optimization*. SIAM, Philadelphia. ISBN 0-89871-380-3.**^**Michael J. Steele (2011).*CBMS Lectures on Probability Theory and Combinatorial Optimization*. University of Cambridge.

**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.*