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Newton's inequalities

## Summary

In mathematics, the Newton inequalities are named after Isaac Newton. Suppose a1a2, ..., an are real numbers and let ${\displaystyle e_{k}}$ denote the kth elementary symmetric polynomial in a1a2, ..., an. Then the elementary symmetric means, given by

${\displaystyle S_{k}={\frac {e_{k}}{\binom {n}{k}}},}$

satisfy the inequality

${\displaystyle S_{k-1}S_{k+1}\leq S_{k}^{2}.}$

If all the numbers ai are non-zero, then equality holds if and only if all the numbers ai are equal.

It can be seen that S1 is the arithmetic mean, and Sn is the n-th power of the geometric mean.