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 which ifall <math>\forall x,y\in \mathbb{R}^d </math> wheresuch that <math>x</math> is [[majorization|majorized]] by <math>y</math>, thenone 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 implication is not true, but all Schur-convex functions are symmetric (under permutations of the arguments).

A function <math>''f</math>'' is 'Schur-concave' if its negative,<math>- −''f</math>'', is Schur-convex.

== Schur-concave function ==

A function <math>f</math> is 'Schur-concave' if its negative,<math>-f</math>, is Schur-convex.

==A simpleProperties 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 <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>:

:<math> (x_1 - x_2)(f_{(1)}(x) - f_{(2)}(x)) \ge 0

</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!

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.

== 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.

*If The<math> [[Shannong entropy]]</math> is a convex function defined on a real interval, then <math> \sum_{i=1}^d{P_in \cdotg(x_i) \log_2{\frac{1}{P_i}}}</math> is Schur-concaveconvex.

=== Schur–Ostrowski criterion ===

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

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 <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 ==

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

* <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 [[Rényi entropy]] funtionfunction 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|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 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 [[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 ==

* 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

{{reflist}}

[[Elementary symmetric polynomial|Elementary symmetric function]]s are Schur-convex, when <math> x_i > 0 </math>.

== See also ==

* A natural interpretation of [[majorization]] is that if <math> x \succ y </math> then

* [[Quasiconvex function]]

[[Category:Convex analysis]]

[[Category:Inequalities (mathematics)]]