Superadditivity
In mathematics, a function is superadditive if
for all and in the domain of
Similarly, a sequence is called superadditive if it satisfies the inequality
for all and
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
- The map is a superadditive function for nonnegative real numbers because
- 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 for nonnegative Hermitian matrices of size : If are nonnegative Hermitian then
- Horst Alzer proved that Hadamard's gamma function is superadditive for all real numbers with
- Mutual information
Properties
If is a superadditive function whose domain contains then To see this, simply set and in the defining inequality.The negative of a superadditive function is subadditive.
Fekete's lemma
The major reason for the use of superadditive sequences is the following lemma due to Michael Fekete.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.