Young's Inequality for Increasing Functions

Lawrence C. Paulson

31 January 2022


Young's inequality states that $$ ab \leq \int_0^a f(x)dx + \int_0^b f^{-1}(y) dy $$ where $a\geq 0$, $b\geq 0$ and $f$ is strictly increasing and continuous. Its proof is formalised following the development by Cunningham and Grossman. Their idea is to make the intuitive, geometric folklore proof rigorous by reasoning about step functions. The lack of the Riemann integral makes the development longer than one would like, but their argument is reproduced faithfully.
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