MacLaurin's inequality

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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

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