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# MacLaurin’s inequality

Let $a_{1},a_{2},\ldots,a_{n}$ be positive real numbers , and define the sums $S_{k}$ as follows :

$S_{k}=\frac{\displaystyle\sum_{{1\leq i_{1}<i_{2}<\cdots<i_{k}\leq n}}a_{{i_{1% }}}a_{{i_{2}}}\cdots a_{{i_{k}}}}{\displaystyle{n\choose k}}$ |

Then the following chain of inequalities is true :

$S_{1}\geq\sqrt{S_{2}}\geq\sqrt[3]{S_{3}}\geq\cdots\geq\sqrt[n]{S_{n}}$ |

Note : $S_{k}$ 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

Keywords:

Young's Inequality

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Definition

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Reference

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## Mathematics Subject Classification

26D15*no label found*

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