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 all <math>x,y\in \mathbb{R}^d </math> such that <math>x</math> is [[majorization|majorized]] by <math>y</math>, one has that <math>f(x)\le f(y)</math>. d is the dimensionality of the real space \mathbb{R} (please correct and clarify all terms!) 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 ==