Newton's inequalities

In mathematics, the Newton inequalities are named after Isaac Newton. Suppose a₁, a₂, ..., a_n are non-negative real numbers and let $e_k$ denote the kth elementary symmetric polynomial in a₁, a₂, ..., a_n. Then the elementary symmetric means, given by

$S_k=\frac{e_k}{(\genfrac{}{}{0ex}{}{n}{k})},$

satisfy the inequality

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

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

References

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