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# Elementary symmetric mean

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 Title: Elementary symmetric mean Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Elementary symmetric mean

In mathematics, the Newton inequalities are named after Isaac Newton. Suppose a1a2, ..., an are real numbers and let $\sigma_k$ denote the kth elementary symmetric function in a1a2, ..., an. Then the elementary symmetric means, given by

$S_k = \frac\left\{\sigma_k\right\}\left\{\binom\left\{n\right\}\left\{k\right\}\right\}$

satisfy the inequality

$S_\left\{k-1\right\}S_\left\{k+1\right\}\le S_k^2$

with equality if and only if all the numbers ai are equal. Note that S1 is the arithmetic mean, and Sn is the n-th power of the geometric mean.