# Hölder inequality

The *Hölder inequality ^{}* concerns

*vector p-norms*: given $1\le p$, $q\le \mathrm{\infty}$,

$$\text{If}\frac{1}{p}+\frac{1}{q}=1\text{then}|{x}^{T}y|\le {||x||}_{p}{||y||}_{q}$$ |

An important instance of a Hölder inequality is the *Cauchy-Schwarz inequality*.

There is a version of this result for the ${L}^{p}$ spaces (http://planetmath.org/LpSpace).
If a function $f$ is in ${L}^{p}(X)$, then the ${L}^{p}$-norm of $f$ is denoted
${||f||}_{p}$.
Given a measure space^{} $(X,\U0001d505,\mu )$, if $f$ is in ${L}^{p}(X)$ and $g$ is in ${L}^{q}(X)$ (with $1/p+1/q=1$), then
the Hölder inequality becomes

${\parallel fg\parallel}_{1}={\displaystyle {\int}_{X}}|fg|d\mu $ | $\le $ | ${\left({\displaystyle {\int}_{X}}{|f|}^{p}d\mu \right)}^{\frac{1}{p}}{\left({\displaystyle {\int}_{X}}{|g|}^{q}d\mu \right)}^{\frac{1}{q}}$ | ||

$=$ | ${\parallel f\parallel}_{p}{\parallel g\parallel}_{q}$ |

Title | Hölder inequality |

Canonical name | HolderInequality |

Date of creation | 2013-03-22 11:43:06 |

Last modified on | 2013-03-22 11:43:06 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 27 |

Author | PrimeFan (13766) |

Entry type | Theorem |

Classification | msc 15A60 |

Classification | msc 55-XX |

Classification | msc 46E30 |

Classification | msc 42B10 |

Classification | msc 42B05 |

Synonym | Holder inequality^{} |

Synonym | Hoelder inequality |

Related topic | VectorPnorm |

Related topic | CauchySchwartzInequality |

Related topic | CauchySchwarzInequality |

Related topic | ProofOfMinkowskiInequality |

Related topic | ConjugateIndex |

Related topic | BoundedLinearFunctionalsOnLpmu |

Related topic | ConvolutionsOfComplexFunctionsOnLocallyCompactGroups |

Related topic | LpNormIsDualToLq |