# general means inequality

The power means inequality is a generalization of arithmetic-geometric means inequality.

If $0\ne r\in \mathbb{R}$, the $r$-mean (or $r$-th power mean^{}) of the nonnegative
numbers ${a}_{1},\mathrm{\dots},{a}_{n}$ is defined as

$${M}^{r}({a}_{1},{a}_{2},\mathrm{\dots},{a}_{n})={\left(\frac{1}{n}\sum _{k=1}^{n}{a}_{k}^{r}\right)}^{1/r}$$ |

Given real numbers $x,y$ such that $xy\ne 0$ and $$, we have

$${M}^{x}\le {M}^{y}$$ |

and the equality holds if and only if ${a}_{1}=\mathrm{\dots}={a}_{n}$.

Additionally, if we define ${M}^{0}$ to be the
geometric mean^{} ${({a}_{1}{a}_{2}\mathrm{\dots}{a}_{n})}^{1/n}$, we have
that the inequality^{} above holds for arbitrary real numbers $$.

The mentioned inequality is a special case of this one, since ${M}^{1}$ is the arithmetic mean^{}, ${M}^{0}$ is the geometric mean and ${M}^{-1}$ is the harmonic mean^{}.

This inequality can be further generalized using weighted power means.

Title | general means inequality |

Canonical name | GeneralMeansInequality |

Date of creation | 2013-03-22 12:39:49 |

Last modified on | 2013-03-22 12:39:49 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 6 |

Author | drini (3) |

Entry type | Theorem |

Classification | msc 26D15 |

Synonym | power means inequality |

Related topic | ArithmeticGeometricMeansInequality |

Related topic | ArithmeticMean |

Related topic | GeometricMean |

Related topic | HarmonicMean |

Related topic | PowerMean |

Related topic | ProofOfArithmeticGeometricHarmonicMeansI |

Related topic | RootMeanSquare3 |

Related topic | DerivationOfZerothWeightedPowerMean |

Related topic | ProofOfArithmeticGeometricHarmonicMeansInequality |

Related topic | ComparisonOfPythagor |