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