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'arithmetic-geometric-harmonic means inequality'
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| Title of object: |
arithmetic-geometric-harmonic means inequality |
| Canonical Name: |
ArithmeticGeometricMeansInequality |
| Type: |
Theorem |
| Created on: |
2001-08-18 00:48:28 |
| Modified on: |
2002-06-07 00:24:39 |
| Classification: |
msc:26D15 |
| Keywords: |
inequality, mean, arithmetic mean, geometric mean, harmonic mean |
| Synonyms: |
arithmetic-geometric-harmonic means inequality=harmonic-geometric-arithmetic means inequality
arithmetic-geometric-harmonic means inequality=arithmetic-geometric means inequality
arithmetic-geometric-harmonic means inequality=AGM inequality
arithmetic-geometric-harmonic means inequality=AGMH inequality |
Revision comment (for changes between this and next version):
| Changes for correction #4482 ('equality'). |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic} |
Content:
Let $x_1,x_2,\ldots,x_n$ be positive numbers.
Then
\begin{eqnarray*}
\max\{x_1,x_2,\ldots,x_n\} &\ge& \frac{x_1+x_2+\cdots+x_n}{n}\\
&\ge& \sqrt[n]{x_1 x_2\cdots x_n} \\
&\ge& \frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}}\\
&\ge& \min\{x_1,x_2,\ldots,x_n\}
\end{eqnarray*}
There are several generalizations to this inequality using power means and weighted power means. |
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