# MacLaurin’s inequality

Let ${a}_{1},{a}_{2},\mathrm{\dots},{a}_{n}$ be positive real numbers , and define the sums ${S}_{k}$ as follows :

$$ |

Then the following chain of
inequalities^{} is true :

$${S}_{1}\ge \sqrt{{S}_{2}}\ge \sqrt[3]{{S}_{3}}\ge \mathrm{\cdots}\ge \sqrt[n]{{S}_{n}}$$ |

Note : ${S}_{k}$ are called the averages^{} of the elementary symmetric^{} sums

This inequality is in fact important because it shows that the arithmetic-geometric mean inequality is nothing but a consequence of a chain of stronger inequalities

Title | MacLaurin’s inequality |
---|---|

Canonical name | MacLaurinsInequality |

Date of creation | 2013-03-22 13:19:28 |

Last modified on | 2013-03-22 13:19:28 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 7 |

Author | Mathprof (13753) |

Entry type | Definition |

Classification | msc 26D15 |