# 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 http://planetmath.org/node/5575an attachment.

Title Young’s inequality YoungsInequality 2013-03-22 13:19:25 2013-03-22 13:19:25 rspuzio (6075) rspuzio (6075) 8 rspuzio (6075) Theorem msc 26D15 YoungInequality