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Homesubadditivity

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# subadditivity

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

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

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

###### Lemma ([1]).

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

Similarly, a function $f(x)$ is *subadditive* if

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

The analogue of Fekete lemma holds for subadditive 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. Zbl 0338.00001.
- 2 Michael J. Steele. Probability theory and combinatorial optimization, volume 69 of CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, 1997. Zbl 0916.90233.

## Mathematics Subject Classification

39B62*no label found*

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