Young's Inequality for Increasing Functions

Theorem
Let $a_0$ and $b_0$ be strictly positive real numbers.

Let $f: \left[{0 \,.\,.\, a_0}\right] \to \left[{0 \,.\,.\, b_0}\right]$ be a strictly increasing bijection.

Let $a$ and $b$ be real numbers such that $0 \le a \le a_0$ and $0 \le b \le b_0$.

Then:
 * $\displaystyle ab \le \int_0^a f \left({u}\right) \ \mathrm d u + \int_0^b f^{-1} \left({v}\right) \ \mathrm d v$

where $\displaystyle \int$ denotes the definite integral.

Equality
Equality holds in the above iff $b = f \left({a}\right)$.

Also see

 * Young's Inequality for Products
 * Young's Inequality for Convolutions