Schur-convex function

Summary

In mathematics, a Schur-convex function, also known as S-convex, isotonic function and order-preserving function is a function that for all such that is majorized by , one has that . Named after Issai Schur, Schur-convex functions are used in the study of majorization.

A function f is 'Schur-concave' if its negative, −f, is Schur-convex.

Properties

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Every function that is convex and symmetric (under permutations of the arguments) is also Schur-convex.

Every Schur-convex function is symmetric, but not necessarily convex.[1]

If   is (strictly) Schur-convex and   is (strictly) monotonically increasing, then   is (strictly) Schur-convex.

If   is a convex function defined on a real interval, then   is Schur-convex.

Schur–Ostrowski criterion

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If f is symmetric and all first partial derivatives exist, then f is Schur-convex if and only if

  for all  

holds for all  .[2]

Examples

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  •   is Schur-concave while   is Schur-convex. This can be seen directly from the definition.
  • The Shannon entropy function   is Schur-concave.
  • The Rényi entropy function is also Schur-concave.
  •   is Schur-convex if  , and Schur-concave if  .
  • The function   is Schur-concave, when we assume all  . In the same way, all the elementary symmetric functions are Schur-concave, when  .
  • A natural interpretation of majorization is that if   then   is less spread out than  . So it is natural to ask if statistical measures of variability are Schur-convex. The variance and standard deviation are Schur-convex functions, while the median absolute deviation is not.
  • A probability example: If   are exchangeable random variables, then the function   is Schur-convex as a function of  , assuming that the expectations exist.
  • The Gini coefficient is strictly Schur convex.

References

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  1. ^ Roberts, A. Wayne; Varberg, Dale E. (1973). Convex functions. New York: Academic Press. p. 258. ISBN 9780080873725.
  2. ^ E. Peajcariaac, Josip; L. Tong, Y. (3 June 1992). Convex Functions, Partial Orderings, and Statistical Applications. Academic Press. p. 333. ISBN 9780080925226.

See also

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