arithmetic-geometric-harmonic means inequalityArithmeticGeometricMeansInequality2001-08-18 00:48:282004-06-05 22:23:00Theoreminequalitymeanarithmetic meangeometric meanharmonic mean\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic}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*}
The equality is obtained if and only if $x_1=x_2=\cdots = x_n$.
There are several generalizations to this inequality using power means and weighted power means.