| Version 2 |
Version 1 |
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Let $a_1,a_2,\ldots,a_n$ be positive real numbers , and define the sums
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Let $a_1,a_2,\ldots,a_n \in \mathbb{R}_+$ be real numbers , and define the sums
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| $S_k$ as follows : |
$S_k$ as follows : |
| $$ S_k = \frac{\displaystyle \sum_{ 1\leq i_1 < i_2 < \cdots < i_k \leq n}a_{i_1} a_{i_2} |
$$ 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}}$$ |
\cdots a_{i_k}}{\displaystyle {n \choose k}}$$ |
| Then the following chain of |
Then the following chain of |
| inequalities is true : |
inequalities is true : |
| $$ S_1 \geq \sqrt{S_2} \geq \sqrt[3]{S_3} \geq \cdots \geq \sqrt[n]{S_n}$$ |
$$ S_1 \geq \sqrt{S_2} \geq \sqrt[3]{S_3} \geq \cdots \geq \sqrt[n]{S_n}$$ |
| \textbf{Note} : $S_k$ are called the averages of the elementary symmetric sums |
\textbf{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 |
\\ This inequality is in fact important because it shows that the $AM-GM$ |
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inequality is nothing but a consequence of a chain of stronger inequalities |
