Newton's inequalities

In mathematics, the Newton inequalities are named after Isaac Newton. Suppose a1, a2, ..., an are non-negative real numbers and let $$e_k$$ denote the kth elementary symmetric polynomial in a1, a2, ..., an. Then the elementary symmetric means, given by


 * $$S_k = \frac{e_k}{\binom{n}{k}},$$

satisfy the inequality


 * $$S_{k-1}S_{k+1} \le S_k^2.$$

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.