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
By Monotone Function is Riemann Integrable, $f$ and $f^{-1}$ are Riemann integrable.

Let $b = f \left({a}\right)$.

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

Consider any subdivision $P = \left\{{x_0, x_1, \ldots, x_n}\right\}$ of the closed real interval $\left[{0 \,.\,.\, a}\right]$.

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

is a subdivision of $\left[{0 \,.\,.\, b}\right]$.

We have that:

Similarly:
 * $L \left({P, f}\right) + U \left({ f P, f^{-1} }\right) = ab$

Let $\epsilon \in \R_{>0}$ be an arbitrary strictly positive real number.

By the definition of the Riemann integral, there exist subdivisions $P_+$ and $P_-$ of $\left[{0 \,.\,.\, a}\right]$ such that:
 * $ab = U \left({P_+, f}\right) + L \left({f P_+, f^{-1}}\right) < A + \epsilon$
 * $ab = L \left({P_-, f}\right) + U \left({f P_-, f^{-1}}\right) > A - \epsilon$

The result follows from Real Plus Epsilon.

Also see

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