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| In [[mathematics]], '''Maclaurin's inequality''', named after [[Colin Maclaurin]], is a refinement of the [[inequality of arithmetic and geometric means]].
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| Let ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub> be [[positive number|positive]] [[real number]]s, and for ''k'' = 1, 2, ..., ''n'' define the averages ''S''<sub>''k''</sub> as follows:
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| :<math> S_k = \frac{\displaystyle \sum_{ 1\leq i_1 < \cdots < i_k \leq n}a_{i_1} a_{i_2} \cdots a_{i_k}}{\displaystyle {n \choose k}}.</math>
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| The numerator of this fraction is the [[elementary symmetric polynomial]] of degree ''k'' in the ''n'' variables ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub>, that is, the sum of all products of ''k'' of the numbers ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub> with the indices in increasing order. The denominator is the number of terms in the numerator, the [[binomial coefficient]] <math>\scriptstyle {n\choose k}.</math> | |
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| Maclaurin's inequality is the following chain of [[inequalities]]:
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| :<math> S_1 \geq \sqrt{S_2} \geq \sqrt[3]{S_3} \geq \cdots \geq \sqrt[n]{S_n}</math>
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| with equality if and only if all the ''a''<sub>''i''</sub> are equal.
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| For ''n'' = 2, this gives the usual inequality of arithmetic and geometric means of two numbers. Maclaurin's inequality is well illustrated by the case ''n'' = 4:
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| : <math>
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| \begin{align}
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| & {} \quad \frac{a_1+a_2+a_3+a_4}{4} \\[8pt]
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| & {} \ge \sqrt{\frac{a_1a_2+a_1a_3+a_1a_4+a_2a_3+a_2a_4+a_3a_4}{6}} \\[8pt]
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| & {} \ge \sqrt[3]{\frac{a_1a_2a_3+a_1a_2a_4+a_1a_3a_4+a_2a_3a_4}{4}} \\[8pt]
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| & {} \ge \sqrt[4]{a_1a_2a_3a_4}.
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| \end{align}
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| </math>
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| Maclaurin's inequality can be proved using the [[Newton's inequalities]].
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| ==See also== | |
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| * [[Newton's inequalities]]
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| * [[Muirhead's inequality]]
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| * [[Generalized mean inequality]]
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| ==References==
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| *{{cite book
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| | last = Biler
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| | first = Piotr
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| | coauthors = Witkowski, Alfred
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| | title = Problems in mathematical analysis
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| | publisher = New York, N.Y.: M. Dekker
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| | date = 1990
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| | pages =
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| | isbn = 0-8247-8312-3
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| }}
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| {{PlanetMath attribution|id=3835|title=MacLaurin's Inequality}}
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| [[Category:Real analysis]]
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| [[Category:Inequalities]]
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| [[Category:Symmetric functions]]
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