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| In [[mathematics]], the '''Newton inequalities''' are named after [[Isaac Newton]]. Suppose ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub> are [[real numbers]] and let <math>\sigma_k</math> denote the ''k''th [[elementary symmetric function]] in ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub>. Then the '''elementary symmetric means''', given by
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| :<math>S_k = \frac{\sigma_k}{\binom{n}{k}}</math> | |
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| satisfy the [[inequality (mathematics)|inequality]]
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| :<math>S_{k-1}S_{k+1}\le S_k^2</math> | |
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| with equality if and only if all the numbers ''a''<sub>''i''</sub> are equal. Note that ''S''<sub>1</sub> is the [[arithmetic mean]], and ''S''<sub>n</sub> is the ''n''-th power of the [[geometric mean]].
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| ==See also==
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| * [[Maclaurin's inequality]]
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| ==References==
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| *{{cite book
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| | last = Newton
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| | first = Isaac
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| | title = Arithmetica universalis: sive de compositione et resolutione arithmetica liber
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| | year = 1707
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| }}
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| *D.S. Bernstein ''Matrix Mathematics: Theory, Facts, and Formulas'' (2009 Princeton) p. 55
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| *{{cite journal
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| | last = Maclaurin
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| | first = C.
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| | title = A second letter to Martin Folks, Esq.; concerning the roots of equations, with the demonstration of other rules in algebra,
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| | journal = Phil. Transactions,
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| | volume = 36
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| | year = 1729
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| | pages = 59–96
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| | doi = 10.1098/rstl.1729.0011
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| | issue = 407–416
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| }}
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| *{{cite journal
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| | last = Whiteley
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| | first = J.N.
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| | title = On Newton's Inequality for Real Polynomials
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| | journal = The American Mathematical Monthly
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| | volume = 76
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| | year = 1969
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| | pages = 905–909
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| | doi = 10.2307/2317943
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| | issue = 8
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| | jstor = 2317943
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| | publisher = The American Mathematical Monthly, Vol. 76, No. 8
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| }}
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| *{{ cite journal
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| |last = Niculescu
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| |first = Constantin
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| |title = A New Look at Newton's Inequalities
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| |journal = Journal of Inequalities in Pure and Applied Mathematics
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| |volume = 1
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| |issue = 2
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| |year = 2000
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| |article = 17
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| |url = http://www.emis.de/journals/JIPAM/article111.html?sid=111
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| }}
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| ==External links==
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| * [http://www.emis.de/journals/JIPAM/ Journal of Inequalities in Pure and Applied Mathematics]
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| [[Category:Isaac Newton]]
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| [[Category:Inequalities]]
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| [[Category:Symmetric functions]]
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