Schur-convex function: Difference between revisions

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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.

=== Schur-OstrowskiSchur–Ostrowski criterion ===

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>

<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>

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== References ==

{{Reflistreflist}}

== See also ==

* [[Quasiconvex function]]

[[Category:Convex analysis]]

[[Category:Inequalities (mathematics)]]