Young's Inequality for Increasing Functions

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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:

$\displaystyle ab \le \int_0^a \map f u \rd u + \int_0^b \map {f^{-1}} v \rd v$

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


Equality

Equality holds in the above if and only if $b = \map f a$.


Proof

The blue colored region corresponds to $\displaystyle \int_0^a \map f u \rd u$ and the red colored region to $\displaystyle \int_0^b\map {f^{-1}} v \rd v$.



Source of Name

This entry was named for William Henry Young.


Also see