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Elementary symmetric mean

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Title: Elementary symmetric mean  
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Subject: Mean, Contraharmonic mean
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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{\sigma_k}{\binom{n}{k}}

satisfy the inequality

S_{k-1}S_{k+1}\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.

See also


  • D.S. Bernstein Matrix Mathematics: Theory, Facts, and Formulas (2009 Princeton) p. 55

External links

  • Journal of Inequalities in Pure and Applied Mathematics
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