Superadditivity Explained

f is superadditive iff(x + y) \geq f(x) + f(y) for all x and y in the domain of f. a _{1,} a _{2,} \ldots is called superadditive if it satisfies the inequality$$a\_\; \backslash geq\; a\_n\; +\; a\_m$$for all m and n.

The term "superadditive" is also applied to functions from a boolean algebra to the real numbers where

P(X \lor Y) \geq P(X) + P(Y), such as lower probabilities.

Examples of superadditive functions

• The map

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

• The determinant is superadditive for nonnegative Hermitian matrix, that is, if

A, B \in Mat _{n(\Complex)} are nonnegative Hermitian then \det(A + B) \geq \det(A) + \det(B). This follows from the Minkowski determinant theorem, which more generally states that \det( ⋅ ) ^{1/n} is superadditive (equivalently, concave)^[1] for nonnegative Hermitian matrices of size n : If A, B \in Mat _{n(\Complex)} are nonnegative Hermitian then \det(A + B) ^{1/n} \geq \det(A) ^{1/n} + \det(B) ^{1/n} . H(x) is superadditive for all real numbers x, y with x, y \geq 1 . 5031 .

• Mutual information

Properties

If

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

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.^[3]

Lemma: (Fekete) For every superadditive sequence

a _{1,} a _{2,} \ldots, the limit \lim a _{n/n} is equal to the supremum \sup a _{n/n.} (The limit may be positive infinity, as is the case with the sequence 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

m and 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]}

References

Notes

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

Notes and References

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