Young's Inequality for Increasing Functions/Equality

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 $b = \map f a$ :
 * $\ds a b = \int_0^a \map f u \rd u + \int_0^b \map {f^{-1} } v \rd v$

where $\ds \int$ denotes the Darboux integral.

Sufficient Condition
By Monotone Real Function is Darboux Integrable, $f$ and $f^{-1}$ are Darboux integrable.

Let $b = \map f a$.

Define:
 * $\ds A = \int_0^a \map f u \rd u + \int_0^b \map {f^{-1} } v \rd v$

Consider any subdivision $P = \set {x_0, x_1, \ldots, x_n}$ of the closed real interval $\closedint 0 a$.

Then:
 * $f P = \set {\map f {x_0}, \map f {x_1}, \ldots, \map f {x_n} }$

is a subdivision of $\closedint 0 b$.

We have that:

Similarly:
 * $\map L {P, f} + \map U {f P, f^{-1} } = a b$

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

By the definition of the Darboux integral, there exist subdivisions $P_+$ and $P_-$ of $\closedint 0 a$ such that:
 * $a b = \map U {P_+, f} + \map L {f P_+, f^{-1} } < A + \epsilon$
 * $a b = \map L {P_-, f} + \map U {f P_-, f^{-1} } > A - \epsilon$

The result follows from Real Plus Epsilon.

Also see

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