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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

## Mathematics Subject Classification

26D15*no label found*

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