Muirhead's theoremMuirheadsTheorem2002-12-11 11:00:272005-02-25 18:16:46Theorem% this is the default PlanetMath preamble. as your knowledge
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% define commands hereLet $0\leq s_1\leq\cdots\leq s_n$ and $0\leq t_1\leq\dots\leq t_n$
be real numbers such that
\[\sum_{i=1}^n s_i = \sum_{i=1}^n t_i\; \mbox{and}
\;\; \sum_{i=1}^k s_i \leq \sum_{i=1}^k t_i\;\; (k=1,\dots,n-1)\]
Then for any nonnegative numbers $x_1,\dots,x_n$,
\[\sum_\sigma x_1^{s_{\sigma(1)}}\dots x_n^{s_{\sigma(n)}} \geq
\sum_\sigma x_1^{t_{\sigma(1)}}\dots x_n^{t_{\sigma(n)}}\]
where the sums run over all permutations $\sigma$ of $\{1,2,\dots,n\}$.