Young's Inequality for Increasing Functions/Equality

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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 $b = f \left({a}\right)$ if and only if:

$\displaystyle ab = \int_0^a f \left({u}\right) \rd u + \int_0^b f^{-1} \left({v}\right) \rd v$

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


Proof

Necessary Condition


Sufficient Condition

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


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

Define:

$\displaystyle A = \int_0^a f \left({u}\right) \rd u + \int_0^b f^{-1} \left({v}\right) \ rd v$


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:

\(\displaystyle U \left({P, f}\right) + L \left({f P, f^{-1} }\right)\) \(=\) \(\displaystyle \sum_{k \mathop = 1}^n \left({x_k - x_{k - 1} }\right) f \left({x_k}\right) + \sum_{k \mathop = 1}^n \left({f \left({x_k}\right) - f \left({x_{k - 1} }\right)}\right) f^{-1} \left({f \left({x_{k - 1} }\right)}\right)\) because $f$ and $f^{-1}$ are increasing
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 1}^n \left({x_k f \left({x_k}\right) - x_{k - 1} f \left({x_{k - 1} }\right)}\right)\) by Summation is Linear
\(\displaystyle \) \(=\) \(\displaystyle a b\)

Similarly:

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


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:

$a b = U \left({P_+, f}\right) + L \left({f P_+, f^{-1} }\right) < A + \epsilon$
$a b = L \left({P_-, f}\right) + U \left({f P_-, f^{-1} }\right) > A - \epsilon$

The result follows from Real Plus Epsilon.

$\Box$


Source of Name

This entry was named for William Henry Young.


Also see