Newton's inequalities

In mathematics, the Newton inequalities are named after Isaac Newton. Suppose a₁, a₂, ..., a_n are real numbers and let $\sigma _{k}$ denote the kth elementary symmetric function in a₁, a₂, ..., a_n. Then the elementary symmetric means, given by

$S_{k}={\frac {\sigma _{k}}{\binom {n}{k}}}$

satisfy the inequality

$S_{k-1}S_{k+1}\leq S_{k}^{2}$

with equality if and only if all the numbers a_i are equal. Note that S₁ is the arithmetic mean, and S_n is the n-th power of the geometric mean.

See also

• Maclaurin's inequality

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. 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). {{cite journal}}: |article= ignored (help)

External links

• Journal of Inequalities in Pure and Applied Mathematics