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 ${\displaystyle e_{k}}${\displaystyle e_{k}} denote the kth elementary symmetric polynomial in a₁, a₂, ..., a_n. Then the elementary symmetric means, given by

${\displaystyle S_{k}={\frac {e_{k}}{\binom {n}{k}}},}${\displaystyle S_{k}={\frac {e_{k}}{\binom {n}{k}}},}

satisfy the inequality

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

• Maclaurin's inequality

• Hardy, G. H.; Littlewood, J. E.; Pólya, G. (1952). Inequalities. Cambridge University Press. ISBN 978-0521358804.

• 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". Philosophical 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. 76 (8). The American Mathematical Monthly, Vol. 76, No. 8: 905–909. doi:10.2307/2317943. JSTOR 2317943.
• Niculescu, Constantin (2000). "A New Look at Newton's Inequalities". Journal of Inequalities in Pure and Applied Mathematics. 1 (2). Article 17.

• Isaac Newton
• Inequalities
• Symmetric functions