A sequence {an}n=1 is called superadditive if it satisfies the inequalityMathworldPlanetmath

an+man+am  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 {an}n=1 the limit liman/n exists and is equal to supan/n.

Similarly, a function f(x) is superadditive if

f(x+y)f(x)+f(y)  for all x and y.

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

There are extensionsPlanetmathPlanetmath 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].


  • 1 György Polya and Gábor Szegö. Problems and theorems in analysisMathworldPlanetmath, volume 1. 1976. 0338.00001.
  • 2 Michael J. Steele. Probability theory and combinatorial optimization, volume 69 of CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, 1997. 0916.90233.
Title superadditivity
Canonical name Superadditivity
Date of creation 2013-03-22 13:52:25
Last modified on 2013-03-22 13:52:25
Owner bbukh (348)
Last modified by bbukh (348)
Numerical id 10
Author bbukh (348)
Entry type Definition
Classification msc 39B62
Synonym superadditive
Related topic Subadditivity