Schur-convex function: Difference between revisions

Revision as of 21:54, 4 March 2014

Revision as of 14:57, 29 June 2015

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

==A simpleSchur-Ostrowski criterion==

If <math>f</math> is Schur-convexsymmetric 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>f</math> is Schur-convex if and only if

:<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!

<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>

holds for all 1≤''i''≠''j''≤''d''.<ref>{{cite book|last1=E. Peajcariaac|first1=Josip|last2=L. Tong|first2=Y.|title=Convex Functions, Partial Orderings, and Statistical Applications|publisher=Academic Press|isbn=9780080925226|page=333}}</ref>

== Examples ==

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* The [[Gini coefficient]] is strictly Schur concave.

== References ==

{{Reflist}}

==See also==