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'MacLaurin's inequality'
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| Title of object: |
MacLaurin's inequality |
| Canonical Name: |
MacLaurinsInequality |
| Type: |
Definition |
| Created on: |
2002-12-26 15:13:10 |
| Modified on: |
2007-05-26 17:38:05 |
| Classification: |
msc:26D15 |
| Keywords: |
Young's Inequality |
Revision comment (for changes between this and next version):
| Changes for correction #11669 ('capitalization of title'). |
Preamble:
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\usepackage{amssymb}
\usepackage{amsmath}
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%\usepackage{psfrag}
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%\usepackage{graphicx}
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%\usepackage{amsthm}
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Content:
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 :
$$ S_1 \geq \sqrt{S_2} \geq \sqrt[3]{S_3} \geq \cdots \geq \sqrt[n]{S_n}$$
\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 |
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