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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>, for which if <math>\forall x,y\in \mathbb{R}^d </math> where <math>x</math> is [[majorization|majorized]] by <math>y</math>, then <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 implication is not true, but all Schur-convex functions are symmetric (under permutations of the arguments).
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| == Schur-concave function ==
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| A function <math>f</math> is 'Schur-concave' if its negative,<math>-f</math>, is Schur-convex.
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| ==A simple criterion==
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| If <math>f</math> is Schur-convex and all first partial derivatives exist, then the following holds, where <math> f_{(i)}(x) </math> denotes the partial derivative with respect to <math> x_i </math>:
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| :<math> (x_1 - x_2)(f_{(1)}(x) - f_{(2)}(x)) \ge 0
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| </math> for all <math> x </math>. Since <math> f </math> is a symmetric function, the above condition implies all the similar conditions for the remaining indexes!
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| == Examples ==
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| * <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.
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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.
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| * The [[Rényi entropy]] function is also Schur-concave.
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| * <math> \sum_{i=1}^d{x_i^k},k \ge 1 </math> is Schur-convex.
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| * 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>.
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| * 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.
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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.
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| * Some probability examples: If <math> X_1, \dots, X_n </math> are exchangeable random variables, then the function
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| :<math> \text{E} \prod_{j=1}^n X_j^{a_j} </math>
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| is Schur-convex as a function of <math> a=(a_1, \dots, a_n) </math>, assuming that the expectations exist.
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| * The [[Gini coefficient]] is strictly Schur concave.
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| ==See also==
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| * [[majorization]]
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| * [[Quasiconvex function]]
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| * [[Convex function]]
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| [[Category:Convex analysis]]
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| [[Category:Inequalities]]
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| {{mathanalysis-stub}}
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Alyson Meagher is the title her parents gave her but she doesn't like when individuals use her full name. He is an purchase clerk and it's some thing he truly enjoy. I've always cherished residing in Mississippi. To climb is something I really enjoy doing.
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