Subadditivity

In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. There are numerous examples of subadditive functions in various areas of mathematics, particularly norms and square roots. Additive maps are special cases of subadditive functions. A subadditive function is a function , having a domain A and an ordered codomain B that are both closed under addition, with the following property: An example is the square root function, having the non-negative real numbers as domain and codomain, since we have: A sequence , is called subadditive if it satisfies the inequality for all m and n. This is a special case of subadditive function, if a sequence is interpreted as a function on the set of natural numbers. Note that while a concave sequence is subadditive, the converse is false. For example, randomly assign with values in , then the sequence is subadditive but not concave. A useful result pertaining to subadditive sequences is the following lemma due to Michael Fekete. The analogue of Fekete's lemma holds for superadditive sequences as well, that is: (The limit then may be positive infinity: consider the sequence .) There are extensions of Fekete's lemma that do not require the inequality to hold for all m and n, but only for m and n such that Moreover, the condition may be weakened as follows: provided that is an increasing function such that the integral converges (near the infinity). 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. Besides, analogues of Fekete's lemma have been proved for subadditive real maps (with additional assumptions) from finite subsets of an amenable group and further, of a cancellative left-amenable semigroup. If f is a subadditive function, and if 0 is in its domain, then f(0) ≥ 0.