Newton's inequalities

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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.

See also[edit]


  • 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). "A New Look at Newton's Inequalities". Journal of Inequalities in Pure and Applied Mathematics 1 (2).  |article= ignored (help)

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