Newton's inequalities explained

In mathematics, the Newton inequalities are named after Isaac Newton. Suppose a₁, a₂, ..., a_n are non-negative real numbers and let

e _{k} denote the kth elementary symmetric polynomial in a₁, a₂, ..., a_n. Then the elementary symmetric means, given byS _{k} = (e _{k})/(\binom{n) {k}},

satisfy the inequality

S _{k-1} S _{k+1} \le S ^{2 .} _{k}

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

• Book: Hardy, G. H. . Littlewood, J. E. . Pólya, G. . Inequalities . Cambridge University Press . 1952 . 978-0521358804.
• Book: Newton , Isaac . Arithmetica universalis: sive de compositione et resolutione arithmetica liber . 1707.
• D.S. Bernstein Matrix Mathematics: Theory, Facts, and Formulas (2009 Princeton) p. 55
• Maclaurin . C. . A second letter to Martin Folks, Esq.; concerning the roots of equations, with the demonstration of other rules in algebra . Philosophical Transactions . 36 . 1729 . 59–96 . 10.1098/rstl.1729.0011 . 407–416. free .
• Whiteley . J.N. . On Newton's Inequality for Real Polynomials . The American Mathematical Monthly . 76 . 1969 . 905–909 . 10.2307/2317943 . 8 . 2317943 . The American Mathematical Monthly, Vol. 76, No. 8.
• Niculescu . Constantin . A New Look at Newton's Inequalities . Journal of Inequalities in Pure and Applied Mathematics . 1 . 2 . 2000 . Article 17 .