Newton's inequalities

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.^[1] Suppose a₁, a₂, ..., a_n are non-negative 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

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

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.

See also

• Maclaurin's inequality