# arithmetic-geometric-harmonic means inequality

Let ${x}_{1},{x}_{2},\mathrm{\dots},{x}_{n}$ be positive numbers. Then

$\mathrm{max}\{{x}_{1},{x}_{2},\mathrm{\dots},{x}_{n}\}$ | $\ge $ | $\frac{{x}_{1}+{x}_{2}+\mathrm{\cdots}+{x}_{n}}{n}$ | ||

$\ge $ | $\sqrt[n]{{x}_{1}{x}_{2}\mathrm{\cdots}{x}_{n}}$ | |||

$\ge $ | $\frac{n}{\frac{1}{{x}_{1}}+\frac{1}{{x}_{2}}+\mathrm{\cdots}+\frac{1}{{x}_{n}}}$ | |||

$\ge $ | $\mathrm{min}\{{x}_{1},{x}_{2},\mathrm{\dots},{x}_{n}\}$ |

The equality is obtained if and only if ${x}_{1}={x}_{2}=\mathrm{\cdots}={x}_{n}$.

There are several generalizations^{} to this inequality^{} using power means^{} and weighted power means.

Title | arithmetic-geometric-harmonic means inequality |

Canonical name | ArithmeticgeometricharmonicMeansInequality |

Date of creation | 2013-03-22 11:42:32 |

Last modified on | 2013-03-22 11:42:32 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 22 |

Author | drini (3) |

Entry type | Theorem^{} |

Classification | msc 00A05 |

Classification | msc 20-XX |

Classification | msc 26D15 |

Synonym | harmonic-geometric-arithmetic means inequality |

Synonym | arithmetic-geometric means inequality |

Synonym | AGM inequality |

Synonym | AGMH inequality |

Related topic | ArithmeticMean |

Related topic | GeometricMean |

Related topic | HarmonicMean |

Related topic | GeneralMeansInequality |

Related topic | WeightedPowerMean |

Related topic | PowerMean |

Related topic | RootMeanSquare3 |

Related topic | ProofOfGeneralMeansInequality |

Related topic | JensensInequality |

Related topic | DerivationOfHarmonicMeanAsTheLimitOfThePowerMean |

Related topic | MinimalAndMaximalNumber |

Related topic | ProofOfArithm |