# Shapiro inequality

Let $n$ be a positive integer and let ${x}_{1},\mathrm{\dots},{x}_{n}$ be positive reals. If $n$ is even and $n\le 12$, or $n$ is odd and $n\le 23$, then

$$\sum _{i=1}^{n}\frac{{x}_{i}}{{x}_{i+1}+{x}_{i+2}}\ge \frac{n}{2},$$ |

where the subscripts are to be understood modulo $n$.

The particular case of $n=3$ is also known as Nesbitts inequality.

Title | Shapiro inequality |
---|---|

Canonical name | ShapiroInequality |

Date of creation | 2013-03-22 13:43:15 |

Last modified on | 2013-03-22 13:43:15 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 7 |

Author | Koro (127) |

Entry type | Theorem |

Classification | msc 26D05 |

Synonym | Shapiro’s inequality^{} |

Related topic | NesbittsInequality |