Newton's inequalities

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In mathematics, the Newton inequalities are named after Isaac Newton. Suppose a1, a2, ..., an are real numbers and let

σk

denote the kth elementary symmetric function in a1, a2, ..., an. Then the elementary symmetric mean given by

S_k = \frac{\sigma_k}{\binom{n}{k}}

satisfies the inequality

S_{k-1}S_{k+1}\le S_k^2

with equality if and only if all the numbers ai are equal.

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[edit] References

  • Newton, Isaac (1707). Arithmetica universalis: sive de compositione et resolutione arithmetica liber. 
  • 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: 59–96. 
  • Whiteley, J.N. (1969). "On Newton's Inequality for Real Polynomials". The American Mathematical Monthly 76: 905–909. doi:10.2307/2317943. 
  • Niculescu, Constantin (2000). "A New Look at Newton's Inequalities". Journal of Inequalities in Pure and Applied Mathematics 1 (2). 

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