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''f'' is Schur-convex if and only if
<math>(x_i - x_j)(\frac{\partial f}{\partial x_i} - \frac{\partial f}{\partial x_j}) \ge 0 </math> for all <math>x \in \mathbb{R}^d</math>
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>
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== Examples ==
* <math> f(x)=\min(x) </math> is Schur-concave while <math> f(x)=\max(x) </math> is Schur-convex. This can be seen directly from the definition.
* The [[Shannon entropy]] function <math>\sum_{i=1}^d{P_i \cdot \log_2{\frac{1}{P_i}}}</math> is Schur-concave.
▲* 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> \sum_{i=1}^d{x_i^k},k \ge 1 </math> is Schur-convex.
* The function <math> f(x) = \prod_{i=1}^n 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 more 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.
▲* The function <math> f(x) = \prod_{i=1}^n 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>.
* 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.
▲* A natural interpretation of [[majorization]] is that if <math> x \succ y </math> then <math> x </math> is more 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.
▲* 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 concave.
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