superadditivity

A sequence ${\{a_n\}}_{n=1}^{\mathrm{\infty }}$ is called superadditive ifit satisfies the inequality

$$a_{n+m}\ge a_n+a_m\mathit{\hspace{1em}\hspace{1em}}\text{for all }n\text{ and }m.$$(1)

The major reason for use of superadditive sequences is the followinglemma due to Fekete.

Lemma ([1]).

For every superadditive sequence ${\mathrm{\{}{a}_n\mathrm{\}}}_{n\mathrm{=}\mathrm{1}}^{\mathrm{\infty }}$ the limit$\lim \mathit{}{a}_n\mathrm{/}n$ exists and is equal to $\sup \mathit{}{a}_n\mathrm{/}n$.

Similarly, a function $f(x)$ is superadditive if

$$f(x+y)\ge f(x)+f(y)\mathit{\hspace{1em}\hspace{1em}}\text{for all }x\text{ and }y.$$

The analogue of Fekete lemma holds for superadditive functions aswell.

There are extensions of Fekete lemma that do not require (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 lemma if some kind of both super- and subadditivity is present. A good exposition of this topic may be found in [2].