# weighted power mean

If ${w}_{1},{w}_{2},\mathrm{\dots},{w}_{n}$ are positive real numbers such that ${w}_{1}+{w}_{2}+\mathrm{\cdots}+{w}_{n}=1$, we define the *$r$-th weighted power mean* of the ${x}_{i}$ as:

$${M}_{w}^{r}({x}_{1},{x}_{2},\mathrm{\dots},{x}_{n})={\left({w}_{1}{x}_{1}^{r}+{w}_{2}{x}_{2}^{r}+\mathrm{\cdots}+{w}_{n}{x}_{n}^{r}\right)}^{1/r}.$$ |

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 $r\to 0$ gives us

$${M}_{w}^{0}={x}_{1}^{{w}_{1}}{x}_{2}^{{w}_{2}}\mathrm{\cdots}{w}_{n}^{{w}_{n}}.$$ |

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

$${M}_{w}^{r}\le {M}_{w}^{s}.$$ |

Title | weighted power mean |

Canonical name | WeightedPowerMean |

Date of creation | 2013-03-22 11:47:20 |

Last modified on | 2013-03-22 11:47:20 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 12 |

Author | drini (3) |

Entry type | Definition |

Classification | msc 26B99 |

Classification | msc 00-01 |

Classification | msc 26-00 |

Related topic | ArithmeticGeometricMeansInequality |

Related topic | ArithmeticMean |

Related topic | GeometricMean |

Related topic | HarmonicMean |

Related topic | PowerMean |

Related topic | ProofOfArithmeticGeometricHarmonicMeansInequality |

Related topic | RootMeanSquare3 |

Related topic | ProofOfGeneralMeansInequality |

Related topic | DerivationOfHarmonicMeanAsTheLimitOfThePowerMean |