PlanetMath: power mean

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power mean

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The -th power mean of the numbers $x_1,x_2,\ldots,x_n$ is defined as:

$\displaystyle M^r(x_1,x_2,\ldots,x_n)=\left(\frac{x_1^r+x_2^r+\cdots+x_n^r}{n}\right)^{1/r}.$

The arithmetic mean is a special case when . The power mean is a continuous function of , and taking limit when $r\to0$ gives us the geometric mean:

$\displaystyle M^0(x_1,x_2,\ldots,x_n)=\sqrt[n]{x_1 {x_{2}} \cdots x_n}.$

Also, when we get

$\displaystyle M^{-1}(x_1,x_2,\ldots,x_n)=\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}}$

A generalization of power means are weighted power means.

"power mean" is owned by drini. [ owner history (1) ]

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Attachments:: derivation of geometric mean as the limit of the power mean (Derivation) by Mathprof

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Cross-references: weighted power means, harmonic mean, geometric mean, limit, continuous function, arithmetic mean, numbers
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This is version 9 of power mean, born on 2001-10-17, modified 2004-02-01.
Object id is 266, canonical name is PowerMean.
Accessed 6717 times total.

Classification:

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

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