PlanetMath: arithmetic-geometric-harmonic means inequality

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arithmetic-geometric-harmonic means inequality

(Theorem)

Let $x_1,x_2,\ldots,x_n$ be positive numbers. Then

$\displaystyle \max\{x_1,x_2,\ldots,x_n\}$	$\displaystyle \ge$	$\displaystyle \frac{x_1+x_2+\cdots+x_n}{n}$
	$\displaystyle \ge$	$\displaystyle \sqrt[n]{x_1 x_2\cdots x_n}$
	$\displaystyle \ge$	$\displaystyle \frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}}$
	$\displaystyle \ge$	$\displaystyle \min\{x_1,x_2,\ldots,x_n\}$

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.

"arithmetic-geometric-harmonic means inequality" is owned by drini. [ owner history (1) ]

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Other names:

harmonic-geometric-arithmetic means inequality, arithmetic-geometric means inequality, AGM inequality, AGMH inequality

Keywords:

inequality, mean, arithmetic mean, geometric mean, harmonic mean

Attachments:: proof of arithmetic-geometric-harmonic means inequality (Proof) by drini
proof of arithmetic-geometric means inequality (Proof) by mathcam
proof of arithmetic-geometric-harmonic means inequality (Proof) by Mathprof

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Cross-references: weighted power means, power means, inequality, equality, numbers, positive
There are 11 references to this entry.

This is version 5 of arithmetic-geometric-harmonic means inequality, born on 2001-08-18, modified 2004-06-05.
Object id is 25, canonical name is ArithmeticGeometricMeansInequality.
Accessed 28408 times total.

Classification:

26D15 (Real functions :: Inequalities :: Inequalities for sums, series and integrals)

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