Newton's inequalities

In mathematics, the Newton inequalities are named after Isaac Newton. Suppose a₁, a₂, ..., a_n are real numbers and let $\sigma_k$ denote the kth elementary symmetric function in a₁, a₂, ..., a_n. Then the elementary symmetric means, given by

$S_k = \frac{\sigma_k}{\binom{n}{k}}$

satisfy the inequality

$S_{k-1}S_{k+1}\le S_k^2$

with equality if and only if all the numbers a_i are equal. Note that S₁ is the arithmetic mean, and S_n is the n-th power of the geometric mean.

See also

• Maclaurin's inequality

References

• D.S. Bernstein Matrix Mathematics: Theory, Facts, and Formulas (2009 Princeton) p. 55

External links

• Journal of Inequalities in Pure and Applied Mathematics