View source for Newton's inequalities

In [[Mathematics|mathematics]], the '''Newton inequalities''' are named after [[Biography:Isaac Newton|Isaac Newton]].  Suppose ''a''<sub>1</sub>,&nbsp;''a''<sub>2</sub>,&nbsp;...,&nbsp;''a''<sub>''n''</sub> are non-negative [[Real number|real number]]s and let <math>e_k</math> denote the ''k''th [[Elementary symmetric polynomial|elementary symmetric polynomial]] in ''a''<sub>1</sub>,&nbsp;''a''<sub>2</sub>,&nbsp;...,&nbsp;''a''<sub>''n''</sub>.  Then the '''elementary symmetric means''', given by
:<math>S_k = \frac{e_k}{\binom{n}{k}},</math>

satisfy the [[Inequality (mathematics)|inequality]]

:<math>S_{k-1}S_{k+1} \le S_k^2.</math>

Equality holds [[If and only if|if and only if]] all the numbers ''a''<sub>''i''</sub> are equal. 

It can be seen that ''S''<sub>1</sub> is the [[Arithmetic mean|arithmetic mean]], and ''S''<sub>''n''</sub> is the ''n''-th power of the [[Geometric mean|geometric mean]].

==See also==

* [[Maclaurin's inequality]]

==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.&nbsp;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}}

[[Category:Inequalities]]
[[Category:Symmetric functions]]

{{Sourceattribution|Newton's inequalities}}