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).<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>
== Schur-concave function ==
A function <math>''f</math>'' is 'Schur-concave' if its negative,<math> -''f</math>'', is Schur-convex.
== Schur-Ostrowski criterion==
If <math>''f</math>'' is symmetric and all first partial derivatives exist, then
<math>''f</math>'' 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>
