A sequence {an}n=1 is called subadditive if it satisfies the inequality

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

Similarly, a function f(x) is subadditive if

f(x+y)f(x)+f(y)  for all x and y.

The analogue of Fekete lemma holds for subadditive 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 theoremsMathworldPlanetmath in analysis, 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 subadditivity
Canonical name Subadditivity
Date of creation 2013-03-22 13:52:22
Last modified on 2013-03-22 13:52:22
Owner bbukh (348)
Last modified by bbukh (348)
Numerical id 10
Author bbukh (348)
Entry type Definition
Classification msc 39B62
Synonym subadditive
Related topic Superadditivity