|
|
|
|
|
The $r$ th power mean of the numbers $x_1,x_2,\ldots,x_n$ is defined as:
$$M^r(x_1,x_2,\ldots,x_n)=\left(\frac{x_1^r+x_2^r+\cdots+x_n^r}{n}\right)^{1/r}.$$
The arithmetic mean is a special case when $r=1$ The power mean is a continuous function of $r$ and taking limit when $r\to0$ gives us the geometric mean: $$M^0(x_1,x_2,\ldots,x_n)=\sqrt[n]{x_1 {x_{2}} \cdots x_n}.$$
Also, when $r=-1$ we get $$M^{-1}(x_1,x_2,\ldots,x_n)=\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}}$$ the harmonic mean.
A generalization of power means are weighted power means.
|
"power mean" is owned by drini. [ owner history (1) ]
|
|
(view preamble | get metadata)
See Also: weighted power mean, arithmetic-geometric-harmonic means inequality, arithmetic mean, geometric mean, harmonic mean, general means inequality, root-mean-square, proof of general means inequality, derivation of zeroth weighted power mean, derivation of geometric mean as the limit of the power mean, proof of arithmetic-geometric-harmonic means inequality, 
|
|
Cross-references: weighted power means, harmonic mean, geometric mean, limit, continuous function, arithmetic mean, numbers
There are 7 references to this entry.
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 7759 times total.
Classification:
| AMS MSC: | 26D15 (Real functions :: Inequalities :: Inequalities for sums, series and integrals) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
