Newton's inequalities

In mathematics, the Newton inequalities are named after Isaac Newton. Suppose a₁, a₂, ..., a_n are real numbers and let ${\displaystyle e_{k}}$ denote the kth elementary symmetric polynomial in a₁, a₂, ..., a_n. 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 a_i are non-zero, then equality holds if and only if all the numbers a_i are equal.

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