power mean

The $r$-th power mean of the numbers $x_1,x_2,\mathrm{\dots },x_n$ is defined as:

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

The arithmetic mean is a special case when $r=1$. The power mean is a continuous function of $r$, and taking limit when $r\to 0$ gives us the geometric mean:

$$M^0(x_1,x_2,\mathrm{\dots },x_n)=\sqrt[n]{x_1{x}_2\mathrm{\cdots }{x}_n}.$$

Also, when $r=-1$ we get

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

the harmonic mean.

A generalization of power means are weighted power means.

Title	power mean
Canonical name	PowerMean
Date of creation	2013-03-22 11:47:17
Last modified on	2013-03-22 11:47:17
Owner	drini (3)
Last modified by	drini (3)
Numerical id	14
Author	drini (3)
Entry type	Definition
Classification	msc 26D15
Classification	msc 16D10
Classification	msc 00-01
Classification	msc 34-00
Classification	msc 35-00
Related topic	WeightedPowerMean
Related topic	ArithmeticGeometricMeansInequality
Related topic	ArithmeticMean
Related topic	GeometricMean
Related topic	HarmonicMean
Related topic	GeneralMeansInequality
Related topic	RootMeanSquare3
Related topic	ProofOfGeneralMeansInequality
Related topic	DerivationOfZerothWeightedPowerMean
Related topic	DerivationOfHarmonicMeanAsTheLimitOfThePowerMean