# Newton's inequalities

In mathematics, the Newton inequalities are named after Isaac Newton. Suppose a1a2, ..., an are real numbers and let ${\displaystyle \sigma _{k}}$ denote the kth elementary symmetric function in a1a2, ..., an. Then the elementary symmetric means, given by

${\displaystyle S_{k}={\frac {\sigma _{k}}{\binom {n}{k}}}}$

satisfy the inequality

${\displaystyle S_{k-1}S_{k+1}\leq S_{k}^{2}}$

If all the numbers ai are nonzero, then equality holds if and only if all the numbers ai are equal. S1 is the arithmetic mean, and Sn is the n-th power of the geometric mean.

• Niculescu, Constantin (2000). "A New Look at Newton's Inequalities". Journal of Inequalities in Pure and Applied Mathematics. 1 (2). |article= ignored (help)