In mathematics, the Newton inequalities refer to a set of mathematical inequalities related to mathematical series. These inequalities are named after Isaac Newton who proved the theorem in 1707. Suppose a<sub>1</sub>, a<sub>2</sub>, ..., a<sub>n</sub> are non-negative real numbers and let denote the kth elementary symmetric polynomial in a<sub>1</sub>, a<sub>2</sub>, ..., a<sub>n</sub>. Then the elementary symmetric means, given by
satisfy the inequality
Equality holds if and only if all the numbers a<sub>i</sub> are equal.
It can be seen that S<sub>1</sub> is the arithmetic mean, and S<sub>n</sub> is the n-th power of the geometric mean.