Subadditive function
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A sequence { an }, n ≥ 1, is called subadditive if it satisfies the inequality
- <math>(1) \qquad a_{n+m}\leq a_n+a_m<math>
for all m and n. The major reason for use of subadditive sequences is the following lemma due to Fekete.
- Lemma: For every subadditive sequence { an }, n ≥ 1, the limit lim an/n exists and equal to inf an/n.
Similarly, a function f(x) is subadditive if
- <math>f(x+y)\leq f(x)+f(y)<math>
for all x and y in the domain of f.
The analogue of Fekete lemma holds for subadditive functions as well.
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. A good exposition of this topic may be found in [2].