Schur-convex function: Difference between revisions

Revision as of 03:35, 20 September 2012

Revision as of 09:08, 5 February 2013

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

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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.

* Some probability examples: If <math> X_1, \dots, X_n </math> are exchangableexchangeable 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.