A sequence $\{a_{n}\}_{n=1}^{\infty}$ is called superadditive if it satisfies the inequality

 $a_{n+m}\geq a_{n}+a_{m}\qquad\text{for all n and m}.$ (1)

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

Lemma ([1]).

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

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

 $f(x+y)\geq f(x)+f(y)\qquad\text{for all x and y}.$

The analogue of Fekete lemma holds for superadditive functions as well.

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].

References

• 1 György Polya and Gábor Szegö. Problems and theorems in analysis, volume 1. 1976. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0338.00001Zbl 0338.00001.
• 2 Michael J. Steele. Probability theory and combinatorial optimization, volume 69 of CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, 1997. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0916.90233Zbl 0916.90233.