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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 $r$ th weighted power mean of the $x_i$ as:

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

When all the $w_i=\frac{1}{n}$ we get the standard power mean. The weighted power mean is a continuous function of $r$ and taking limit when $r\to0$ gives us $$M_w^0=x_1^{w_1}x_2^{w_2}\cdots w_n^{w_n}.$$

We can weighted use power means to generalize the power means inequality: If $w$ is a set of weights, and if $r<s$ then $$M_w^r \leq M_w^s.$$




"weighted power mean" is owned by drini. [ owner history (1) ]
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See Also: arithmetic-geometric-harmonic means inequality, arithmetic mean, geometric mean, harmonic mean, power mean, proof of arithmetic-geometric-harmonic means inequality, root-mean-square, proof of general means inequality, derivation of geometric mean as the limit of the power mean


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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
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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 8188 times total.

Classification:
AMS MSC26B99 (Real functions :: Functions of several variables :: Miscellaneous)

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Additional References by smithpith on 2009-04-25 18:16:01
You may also be interested to read about averages, weighted averages, and means in the following book and article.

* Jane Grossman, Michael Grossman, Robert Katz. "Averages: A New Approach", ISBN 0977117049, 1983. (Available for reading at Google Book Search:
http://books.google.com/books?q=%22Non-Newtonian+Calculus%22&btnG=Search+Books&as_brr=0).

* Michael Grossman and Robert Katz. "A new approach to means of two positive numbers", International Journal of Mathematical Education in Science and Technology, Volume 17, Number 2, March 1986, pages 205 - 208.

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Correct? by Patak on 2005-01-30 15:38:19
This inequality becomes an equality if and only if x1 = x2 =... = xn
Hence there must be =<
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