Young’s inequality

Let $\varphi :ℝ\to ℝ$ be a continuous , strictlyincreasing function such that $\varphi (0)=0$ . Then the following inequality holds:

$$ab\le {\int }_0^a\varphi (x)𝑑x+{\int }_0^b{\varphi }^{-1}(y)𝑑y$$

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 integralsabove is greater than the area of a rectangle of sides $a$ and $b$, asis illustrated in http://planetmath.org/node/5575an attachment.