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. Suppose a1, a2, ..., an are non-negative real numbers and let e k {\displaystyle e_{k}} denote the kth elementary symmetric polynomial in a1, a2, ..., an. Then the elementary symmetric means, given by
S k = e k ( n k ) , {\displaystyle S_{k}={\frac {e_{k}}{\binom {n}{k}}},}
satisfy the inequality
S k − 1 S k + 1 ≤ S k 2 . {\displaystyle S_{k-1}S_{k+1}\leq S_{k}^{2}.}
Equality holds if and only if all the numbers ai are equal. It can be seen that S1 is the arithmetic mean, and Sn is the n-th power of the geometric mean.
See also Maclaurin's inequality
References
Other Hardy, G. H.; Littlewood, J. E.; Pólya, G. (1952). Inequalities. Cambridge University Press. ISBN 978-0521358804. {{cite book}}: ISBN / Date incompatibility (help) 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.