# Newton's inequalities

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.