PlanetMath: weighted power mean

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weighted power mean

(Definition)

If $w_1,w_2,\ldots,w_n$ are positive real numbers such that $w_1+w_2+\cdots+w_n=1$ , we define the -th weighted power mean of the as:

$\displaystyle M_w^r(x_1,x_2,\ldots,x_n)=\left({w_1x_1^r+w_2x_2^r+\cdots+w_nx_n^r}\right)^{1/r}.$

When all the $w_i=\frac{1}{n}$ we get the standard power mean. The weighted power mean is a continuous function of , and taking limit when $r\to0$ gives us

$\displaystyle M_w^0=x_1^{w_1}x_2^{w_2}\cdots w_n^{w_n}.$

We can weighted use power means to generalize the power means inequality: If is a set of weights, and if then

$\displaystyle M_w^r \leq M_w^s.$

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

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Attachments:: derivation of zeroth weighted power mean (Derivation) by pbruin

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Cross-references: weights, power means inequality, limit, continuous function, power mean, real numbers, positive
There are 6 references to this entry.

This is version 7 of weighted power mean, born on 2001-10-17, modified 2005-01-30.
Object id is 267, canonical name is WeightedPowerMean.
Accessed 7411 times total.

Classification:

AMS MSC:

26B99 (Real functions :: Functions of several variables :: Miscellaneous)

Pending Errata and Addenda

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