Young's Inequality for Increasing Functions/Equality

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

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

Sufficient Condition

 * $P = \left\{{x_0, x_1, \ldots, x_n}\right\}$


 * $Q = \left\{{f \left({x_0}\right), f \left({x_1}\right), \ldots, f \left({x_n}\right)}\right\}$


 * $\displaystyle L \left({P, f}\right) + U \left({Q, f^{-1}}\right) = a f \left({a}\right) = U \left({P, f}\right) + L \left({Q, f^{-1}}\right)$

The result follows from Monotone Function is Riemann Integrable.

Also see

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