Superadditivity

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In mathematics, a sequence { an }, n ≥ 1, is called superadditive if it satisfies the inequality

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 Michael Fekete.

Lemma: (Fekete) For every superadditive sequence { an }, n ≥ 1, the limit lim an/n exists and equal to sup an/n. (The limit may be positive infinity, for instance, for the sequence an = 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's lemma holds for subadditive functions as well. There are extensions of Fekete's lemma that do not require the definition of superadditivity above 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 Steele (1997).[1][2]

If f is a superadditive function, and if 0 is in its domain, then f(0) ≤ 0. To see this, take the inequality at the top. f(x)\leq f(x+y)-f(y). Hence f(0)\leq f(0+y)-f(y)=0

The negative of a superadditive function is subadditive.

Examples of superadditive functions

  • Mutual information
  • Horst Alzer proved [3] that Hadamard’s gamma function H(x) is superadditive for all real numbers x,y with x, y ≥ 1.5031.

See also

References

  1. ↑ Michael J. Steele (1997). Probability theory and combinatorial optimization. SIAM, Philadelphia. ISBN 0-89871-380-3. 
  2. ↑ Michael J. Steele (2011). CBMS Lectures on Probability Theory and Combinatorial Optimization. University of Cambridge. 
  3. ↑ Horst Alzer (2009). A superadditive property of Hadamard’s gamma function. Springer. doi:10.1007/s12188-008-0009-5. 
Notes
  • György Polya and Gábor Szegö. (1976). Problems and theorems in analysis, volume 1. Springer-Verlag, New York. ISBN 0-387-05672-6. 

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