# Young's Inequality for Increasing Functions

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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:

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

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

### Equality

Equality holds in the above if and only if $b = f \left({a}\right)$.

## Proof

## Source of Name

This entry was named for William Henry Young.