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 ofinequalities 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