Newton's inequalities

In mathematics, the Newton inequalities are named after Isaac Newton. Suppose a₁, a₂, ..., a_n are non-negative real numbers and let [math]\displaystyle{ e_k }[/math] denote the kth elementary symmetric polynomial in a₁, a₂, ..., a_n. Then the elementary symmetric means, given by

[math]\displaystyle{ S_k = \frac{e_k}{\binom{n}{k}}, }[/math]

satisfy the inequality

[math]\displaystyle{ S_{k-1}S_{k+1} \le S_k^2. }[/math]

Equality holds if and only if all the numbers a_i are equal.

It can be seen that S₁ is the arithmetic mean, and S_n is the n-th power of the geometric mean.

See also

• Maclaurin's inequality

References

• 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". Philosophical Transactions 36 (407–416): 59–96. doi:10.1098/rstl.1729.0011. https://zenodo.org/record/1432210. 
• 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. 
• Niculescu, Constantin (2000). "A New Look at Newton's Inequalities". Journal of Inequalities in Pure and Applied Mathematics 1 (2): Article 17. http://www.emis.de/journals/JIPAM/article111.html?sid=111.