Schur-convex function: Difference between revisions

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In mathematics, a '''Schur-convex function''', also known as '''S-convex''', '''isotonic function''' and '''order-preserving function''' is a [[function (mathematics)|function]] <math>f: \mathbb{R}^d\rightarrow \mathbb{R}</math> that for all <math>x,y\in \mathbb{R}^d </math> such that <math>x</math> is [[majorization|majorized]] by <math>y</math>, one has that <math>f(x)\le f(y)</math>. Named after [[Issai Schur]], Schur-convex functions are used in the study of [[majorization]]. Every function that is [[Convex function|convex]] and [[Symmetric function|symmetric]] is also Schur-convex. The opposite [[Strict conditional|implication]] is not true, but all Schur-convex functions are symmetric (under permutations of the arguments).<ref>{{cite book|last1=Roberts|first1=A. Wayne|last2=Varberg|first2=Dale E.|title=Convex functions|date=1973|publisher=Academic Press|___location=New York|isbn=9780080873725|page=258}}</ref>

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

== Schur-concave function ==

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

== Properties ==

== Schur-Ostrowski criterion==

Every function that is [[Convex function|convex]] and [[Symmetric function|symmetric]] (under permutations of the arguments) is also Schur-convex.

Every Schur-convex function is symmetric, but not necessarily convex.<ref>{{cite book |last1=Roberts |first1=A. Wayne |url=https://archive.org/details/convexfunctions0000robe |title=Convex functions |last2=Varberg |first2=Dale E. |date=1973 |publisher=Academic Press |isbn=9780080873725 |___location=New York |page=[https://archive.org/details/convexfunctions0000robe/page/258 258] |url-access=registration}}</ref>

If ''f'' is symmetric and all first partial derivatives exist, then

''f'' is Schur-convex if and only if

If <math>f</math> is (strictly) Schur-convex and <math>g</math> is (strictly) monotonically increasing, then <math>g\circ f</math> is (strictly) Schur-convex.

<math>(x_i - x_j)\left(\frac{\partial f}{\partial x_i} - \frac{\partial f}{\partial x_j}\right) \ge 0 </math> for all <math>x \in \mathbb{R}^d</math>

* If <math> g </math> is a convex function defined on a real interval, then <math> \sum_{i=1}^n g(x_i) </math> is Schur-convex.

holds for all 1≤''i''≠''j''≤''d''.<ref>{{cite book|last1=E. Peajcariaac|first1=Josip|last2=L. Tong|first2=Y.|title=Convex Functions, Partial Orderings, and Statistical Applications|publisher=Academic Press|isbn=9780080925226|page=333}}</ref>

=== Schur-OstrowskiSchur–Ostrowski criterion ===

If ''f'' is symmetric and all first partial derivatives exist, then

''f'' is Schur-convex if and only if

: <math>(x_i - x_j)\left(\frac{\partial f}{\partial x_i} - \frac{\partial f}{\partial x_j}\right) \ge 0 </math> for all <math>x \in \mathbb{R}^d</math>

holds for all 1≤''<math>1\le i''≠'',j''≤''\le d''</math>.<ref>{{cite book|last1=E. Peajcariaac|first1=Josip|last2=L. Tong|first2=Y.|title=Convex Functions, Partial Orderings, and Statistical Applications|date=3 June 1992 |publisher=Academic Press|isbn=9780080925226|page=333}}</ref>

== Examples ==

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* The [[Shannon entropy]] function <math>\sum_{i=1}^d{P_i \cdot \log_2{\frac{1}{P_i}}}</math> is Schur-concave.

* The [[Rényi entropy]] function is also Schur-concave.

* <math>x \mapsto \sum_{i=1}^d{x_i^k},k \ge 1 </math> is Schur-convex if <math>k \geq 1</math>, and Schur-concave if <math>k \in (0, 1)</math>.

* The function <math> f(x) = \prod_{i=1}^nd x_i </math> is Schur-concave, when we assume all <math> x_i > 0 </math>. In the same way, all the [[Elementary symmetric polynomial|Elementaryelementary symmetric function]]s are Schur-concave, when <math> x_i > 0 </math>.

* A natural interpretation of [[majorization]] is that if <math> x \succ y </math> then <math> x </math> is less spread out than <math> y </math>. 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 [[Medianmedian absolute deviation]] is not.

* If <math> g </math> is a convex function defined on a real interval, then <math> \sum_{i=1}^n g(x_i) </math> is Schur-convex.

* A probability example: If <math> X_1, \dots, X_n </math> are [[exchangeable random variables]], then the function <math> \text{E} \prod_{j=1}^n X_j^{a_j} </math> is Schur-convex as a function of <math> a=(a_1, \dots, a_n) </math>, assuming that the expectations exist.

* The [[Gini coefficient]] is strictly Schur convex.

== References ==

{{Reflistreflist}}

== See also ==

* [[Quasiconvex function]]

[[Category:Convex analysis]]

[[Category:Inequalities (mathematics)]]