Schur-convex function: Difference between revisions

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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.

=== 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 {{nowrap|<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>

<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 {{nowrap|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|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>.

* <math> \sum_{i=1}^d{x_i^k},0 < k < 1 </math> is Schur-concave.

* The function <math> f(x) = \prod_{i=1}^d x_i </math> is Schur-concave, when we assume all <math> x_i > 0 </math>. In the same way, all the [[Elementary symmetric polynomial|elementary 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 moreless 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 [[median absolute deviation]] is not.

* 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)]]