PlanetMath: Muirhead's theorem

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Muirhead's theorem

(Theorem)

Let $0\leq s_1\leq\cdots\leq s_n$ and $0\leq t_1\leq\dots\leq t_n$ be real numbers such that

$\displaystyle \sum_{i=1}^n s_i = \sum_{i=1}^n t_i\;$ and $\displaystyle \;\; \sum_{i=1}^k s_i \leq \sum_{i=1}^k t_i\;\; (k=1,\dots,n-1)$

Then for any nonnegative numbers $x_1,\dots,x_n$ ,

$\displaystyle \sum_\sigma x_1^{s_{\sigma(1)}}\dots x_n^{s_{\sigma(n)}} \geq \sum_\sigma x_1^{t_{\sigma(1)}}\dots x_n^{t_{\sigma(n)}}$

where the sums run over all permutations $\sigma$ of $\{1,2,\dots,n\}$ .

"Muirhead's theorem" is owned by Koro.

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Other names:

Muirhead's inequality

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Cross-references: permutations, sums, numbers, real numbers
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This is version 4 of Muirhead's theorem, born on 2002-12-11, modified 2005-02-25.
Object id is 3736, canonical name is MuirheadsTheorem.
Accessed 9423 times total.

Classification:

26D15 (Real functions :: Inequalities :: Inequalities for sums, series and integrals)

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