# Young’s inequality

Let $\varphi :\mathbb{R}\to \mathbb{R}$ be a continuous^{} , strictly
increasing function such that $\varphi (0)=0$ . Then the following inequality^{} holds:

$$ab\le {\int}_{0}^{a}\varphi (x)\mathit{d}x+{\int}_{0}^{b}{\varphi}^{-1}(y)\mathit{d}y$$ |

Equality only holds when $b=\varphi (a)$.
This inequality can be demonstrated by drawing the graph of $\varphi (x)$
and by observing that the sum of the two areas represented by the integrals^{}
above is greater than the area of a rectangle of sides $a$ and $b$, as
is illustrated in http://planetmath.org/node/5575an attachment.

Title | Young’s inequality |
---|---|

Canonical name | YoungsInequality |

Date of creation | 2013-03-22 13:19:25 |

Last modified on | 2013-03-22 13:19:25 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 8 |

Author | rspuzio (6075) |

Entry type | Theorem |

Classification | msc 26D15 |

Related topic | YoungInequality |