Superadditive

A sequence { "a_n" }, "n" ≥ 1, is called superadditive if it satisfies the inequality::$(1)\; qquad\; a\_\{n+m\}\; geq\; a\_n+a\_m$for all "m" and "n". The major reason for the use of superadditive sequences is the following lemma due to Fekete.

:Lemma: For every superadditive sequence { "a_n" }, "n" ≥ 1, the limit lim "a_n"/"n" exists and equal to sup "a_n"/"n". (The limit may be positive infinity, for instance "a_n" = $log\; n!$.)

Similarly, a function "f"("x") is "superadditive" if::$f(x+y)\; geq\; f(x)+f(y)$for all "x" and "y" in the domain of "f".

For example, $f(x)=x^2$ is a superadditive function for nonnegative real numbers because the square of $(x+y)$ is always greater than or equal to the square of $x$ plus the square of $y$, for nonnegative real numbers $x$ and $y$.

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

There are extensions of Fekete's lemma that do not require equation (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. A good exposition of this topic may be found in [2] .

References

#cite book|author=György Polya and Gábor Szegö.|title=Problems and theorems in analysis, volume 1|publisher=Springer-Verlag, New York|year=1976|id=ISBN 0-387-05672-6
#cite book|author=Michael J. Steele|title=Probability theory and combinatorial optimization|publisher=SIAM, Philadelphia|year=1997|id=ISBN 0-89871-380-3

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