Newton's inequalities

In mathematics, the Newton inequalities are named after Isaac Newton.

Suppose a1, a2, ..., an are non-negative real numbers and let

denote the kth elementary symmetric polynomial in a1, a2, ..., an.

Then the elementary symmetric means, given by satisfy the inequality 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.