Newton's inequalities
In mathematics, the Newton inequalities are named after Isaac Newton. Suppose a1, a2, ..., an are non-negative real numbers and let denote the kth elementary symmetric polynomial in a1, a2, ..., an. Then the elementary symmetric means, given by
:
satisfy the inequality
:
Equality holds if and only if all the numbers ai are equal.
It can be seen that S1 is the arithmetic mean, and Sn is the n-th power of the geometric mean.
See also
References
- {{cite book
| author = Hardy, G. H.
|author2=Littlewood, J. E.
|author3=Pólya, G.
| title = Inequalities
| publisher = Cambridge University Press
| date = 1952
| isbn = 978-0521358804
}}
- {{cite book
| last = Newton
| first = Isaac
| title = Arithmetica universalis: sive de compositione et resolutione arithmetica liber
| year = 1707
}}
- D.S. Bernstein Matrix Mathematics: Theory, Facts, and Formulas (2009 Princeton) p. 55
- {{cite journal
| last = Maclaurin
| first = C.
| title = A second letter to Martin Folks, Esq.; concerning the roots of equations, with the demonstration of other rules in algebra
| journal = Philosophical Transactions
| volume = 36
| year = 1729
| pages = 59–96
| doi = 10.1098/rstl.1729.0011
| issue = 407–416
| url = https://zenodo.org/record/1432210
| doi-access = free
}}
- {{cite journal
| last = Whiteley
| first = J.N.
| title = On Newton's Inequality for Real Polynomials
| journal = The American Mathematical Monthly
| volume = 76
| year = 1969
| pages = 905–909
| doi = 10.2307/2317943
| issue = 8
| jstor = 2317943
| publisher = The American Mathematical Monthly, Vol. 76, No. 8
}}
- {{ cite journal
|last = Niculescu
|first = Constantin
|title = A New Look at Newton's Inequalities
|journal = Journal of Inequalities in Pure and Applied Mathematics
|volume = 1
|issue = 2
|year = 2000
|at = Article 17
|url = http://www.emis.de/journals/JIPAM/article111.html?sid=111
}}
{{Isaac Newton}}