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.