|
|
|
|
MacLaurin's inequality
|
(Definition)
|
|
|
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
|
"MacLaurin's inequality" is owned by Mathprof. [ full author list (3) | owner history (3) ]
|
|
(view preamble | get metadata)
| Keywords: |
Young's Inequality |
|
|
Cross-references: stronger, consequence, arithmetic-geometric mean, symmetric, averages, inequalities, chain, sums, real numbers, positive
There is 1 reference to this entry.
This is version 4 of MacLaurin's inequality, born on 2002-12-26, modified 2007-05-26.
Object id is 3835, canonical name is MacLaurinsInequality.
Accessed 4703 times total.
Classification:
| AMS MSC: | 26D15 (Real functions :: Inequalities :: Inequalities for sums, series and integrals) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|