PlanetMath: MacLaurin's inequality

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MacLaurin's inequality

(Definition)

Let $a_1,a_2,\ldots,a_n$ be positive real numbers , and define the sums as follows :

$\displaystyle S_k = \frac{\displaystyle \sum_{ 1\leq i_1 < i_2 < \cdots < i_k \leq n}a_{i_1} a_{i_2} \cdots a_{i_k}}{\displaystyle {n \choose k}}$

Then the following chain of inequalities is true :

$\displaystyle S_1 \geq \sqrt{S_2} \geq \sqrt[3]{S_3} \geq \cdots \geq \sqrt[n]{S_n}$

Note :

are called the averages of the elementary symmetric sums
This inequality is in fact important because it shows that the arithmetic-geometric mean inequality is nothing but a consequence of a chain of stronger inequalities

"MacLaurin's inequality" is owned by Mathprof. [ full author list (3) | owner history (3) ]

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Keywords:

Young's Inequality

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Cross-references: consequence, arithmetic-geometric mean, symmetric, averages, inequalities, chain, sums, real numbers, positive
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This is version 4 of MacLaurin's inequality, born on 2002-12-26, modified 2007-05-26.
Object id is 3835, canonical name is MacLaurinsInequality.
Accessed 4081 times total.

Classification:

AMS MSC:

26D15 (Real functions :: Inequalities :: Inequalities for sums, series and integrals)

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