Newton's inequalities


In mathematics, the Newton inequalities are named after Isaac Newton. Suppose a1a2, ..., an are real numbers and let denote the kth elementary symmetric polynomial in a1a2, ..., an. Then the elementary symmetric means, given by

satisfy the inequality

If all the numbers ai are non-zero, then 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


  • 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" (PDF). 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. The American Mathematical Monthly, Vol. 76, No. 8. 76 (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.