# Newton's inequalities

In mathematics, the Newton inequalities are named after Isaac Newton. Suppose a1a2, ..., an are real numbers and let $\sigma_k$ denote the kth elementary symmetric function in a1a2, ..., an. 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 ai are equal. Note that S1 is the arithmetic mean, and Sn is the n-th power of the geometric mean.

## References

• Newton, Isaac (1707). Arithmetica universalis: sive de compositione et resolutione arithmetica liber.
• D.S. Bernstein Matrix Mathematics: Theory, Facts, and Formulas (2009 Princeton) p. 55
• Maclaurin, C. (1729). "A second letter to Martin Folks, Esq.; concerning the roots of equations, with the demonstration of other rules in algebra,". Phil. Transactions, 36 (407–416): 59–96. doi:10.1098/rstl.1729.0011.
• Whiteley, J.N. (1969). "On Newton's Inequality for Real Polynomials". The American Mathematical Monthly (The American Mathematical Monthly, Vol. 76, No. 8) 76 (8): 905–909. doi:10.2307/2317943. JSTOR 2317943.
• Niculescu, Constantin (2000). 17. "A New Look at Newton's Inequalities". Journal of Inequalities in Pure and Applied Mathematics 1 (2).