Newton's inequalities

Newton's inequalities

In mathematics, the Newton inequalities are named after Isaac Newton. Suppose a1a2, ..., an are real numbers and let σk 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 simply the arithmetic mean, and Sn is of course the geometric mean.

See also

References

  • 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). http://jipam.vu.edu.au/. 

External links


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