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# Young’s inequality

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

$ab\leq\int_{{0}}^{a}\phi(x)dx+\int_{{0}}^{b}\phi^{{-1}}(y)dy$ |

Equality only holds when $b=\phi(a)$. This inequality can be demonstrated by drawing the graph of $\phi(x)$ 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 an attachment.

Keywords:

Young's Inequality

Related:

YoungInequality

Type of Math Object:

Theorem

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Reference

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26D15*no label found*

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Apr 15

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

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Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier