Schur-convex function

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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. Every function that is convex and symmetric is also Schur-convex. The opposite implication is not true, but all Schur-convex functions are symmetric (under permutations of the arguments).[1]

Schur-concave function[edit]

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

Schur-Ostrowski criterion[edit]

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

for all

holds for all 1≤ijd.[2]

Examples[edit]

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

References[edit]

  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. Convex Functions, Partial Orderings, and Statistical Applications. Academic Press. p. 333. ISBN 9780080925226. 

See also[edit]