Superadditivity

In mathematics, a function $f$ is superadditive if

$f(x+y) \geq f(x) + f(y)$

for all $x$ and $y$ in the domain of $f.$

Similarly, a sequence $a_1, a_2, \ldots$ is called superadditive if it satisfies the inequality

$a_{n+m} \geq a_n + a_m$

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 $f(x + y) = (x + y)^2 = x^2 + y^2 + 2 x y = f(x) + f(y) + 2 x y \ge f(x) + f(y).$

The determinant is superadditive for nonnegative Hermitian matrix, that is, if $A, B \in \text{Mat}_n(\Complex)$ are nonnegative Hermitian then $\det(A + B) \geq \det(A) + \det(B).$ This follows from the Minkowski determinant theorem, which more generally states that $\det(\cdot)^{1/n}$ is superadditive (equivalently, concave)M. Marcus, H. Minc (1992). [https://books.google.com/books?id=hLHKwSNqLOcC A survey in matrix theory and matrix inequalities]. Dover. Theorem 4.1.8, page 115. for nonnegative Hermitian matrices of size $n$ : If $A, B \in \text{Mat}_n(\Complex)$ are nonnegative Hermitian then $\det(A + B)^{1/n} \geq \det(A)^{1/n} + \det(B)^{1/n}.$
Horst Alzer proved{{cite journal|author=Horst Alzer|title=A superadditive property of Hadamard's gamma function|journal=Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg |publisher=Springer|year=2009|volume=79 |pages=11–23 |doi=10.1007/s12188-008-0009-5|s2cid=123691692 }} that Hadamard's gamma function $H(x)$ is superadditive for all real numbers $x, y$ with $x, y \geq 1.5031.$ {{Cite OEIS|A381340|2=Decimal value of c > 1.5 for which H(2*c) = 2*H(c) for H = Hadamard's gamma function}}
Mutual information

Properties

If $f$ is a superadditive function whose domain contains $0,$ then $f(0) \leq 0.$ To see this, simply set $x=0$ and $y=0$ 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.{{cite journal|last=Fekete|first=M.|title=Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten|journal=Mathematische Zeitschrift|volume=17|issue=1|year=1923|pages=228–249|doi=10.1007/BF01504345|s2cid=186223729 }}

: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).{{cite book|author=Michael J. Steele|title=Probability theory and combinatorial optimization|publisher=SIAM, Philadelphia|year=1997|isbn=0-89871-380-3|url-access=registration|url=https://archive.org/details/probabilitytheor0000stee}}{{cite video|author=Michael J. Steele|title=CBMS Lectures on Probability Theory and Combinatorial Optimization|publisher=University of Cambridge|year=2011|url=http://sms.cam.ac.uk/collection/1189351}}

References

Notes

{{cite book|author=György Polya and Gábor Szegö.|title=Problems and theorems in analysis, volume 1|publisher=Springer-Verlag, New York|year=1976|isbn=0-387-05672-6}}

Category:Mathematical analysis

Category:Sequences and series

Category:Types of functions

Superadditivity

Examples of superadditive functions

Properties

Fekete's lemma

See also

References