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 functions are special cases of subadditive functions.

## Definitions

A subadditive function is a function $f\colon A\to B$ , having a domain A and an ordered codomain B that are both closed under addition, with the following property:

$\forall x,y\in A,f(x+y)\leq f(x)+f(y).$ An example is the square root function, having the non-negative real numbers as domain and codomain, since $\forall x,y\geq 0$ we have:

${\sqrt {x+y}}\leq {\sqrt {x}}+{\sqrt {y}}.$ $(1)\qquad a_{n+m}\leq a_{n}+a_{m}$ for all m and n.

## Properties

A useful result pertaining to subadditive sequences is the following lemma due to Michael Fekete.

Fekete's Subadditive Lemma: For every subadditive sequence ${\left\{a_{n}\right\}}_{n=1}^{\infty }$ , the limit $\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{n}}$ exists and is equal to $\inf {\frac {a_{n}}{n}}$ . (The limit may be $-\infty$ .)

The analogue of Fekete's lemma holds for superadditive functions as well, that is: $a_{n+m}\geq a_{n}+a_{m}.$ (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 the inequality (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.

If f is a subadditive function, and if 0 is in its domain, then f(0) ≥ 0. To see this, take the inequality at the top. $f(x)\geq f(x+y)-f(y)$ . Hence $f(0)\geq f(0+y)-f(y)=0$ 