Subadditive function

From Wikipedia, the free encyclopedia

In mathematics, a subadditive function is a function f : A → B, having a domain A and a codomain B that are both closed under addition, with the following property:

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

for all x and y in A.

An example is the square root function, having the non-negative real numbers as domain and codomain, since for all x and y that are at least 0 we have:

$\sqrt{x+y}\leq \sqrt{x}+\sqrt{y}.$

A sequence { a_n }, n ≥ 1, is called subadditive if it satisfies the inequality

$(1) \qquad a_{n+m}\leq a_n+a_m$

for all m and n. The major reason for use of subadditive sequences is the following lemma due to M. Fekete.

Lemma: For every subadditive sequence { a_n }, n ≥ 1, the limit lim a_n/n exists and is equal to inf a_n/n. (The limit may be negative infinity.)

The analogue of Fekete's lemma holds for subadditive functions as well. (The limit then may be positive infinity: consider the sequence a_n = $log n!$ .)

There are extensions of Fekete's lemma that do not require equation (1) 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.¹

[edit] See also

Triangle inequality

[edit] References

György Pólya and Gábor Szegö. (1976). Problems and theorems in analysis, volume 1. Springer-Verlag, New York. ISBN 0-387-05672-6.
Michael J. Steele (1997). Probability theory and combinatorial optimization. SIAM, Philadelphia. ISBN 0-89871-380-3.