Young's Inequality for Increasing Functions

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

Let $f: \closedint 0 {a_0} \to \closedint 0 {b_0}$ 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:
 * $\displaystyle ab \le \int_0^a \map f u \rd u + \int_0^b \map {f^{-1}} v \rd v$

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

Equality
Equality holds in the above $b = \map f a$.

Also see

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