On the one hand, Young's inequality, in the form of
$$ ab \leq \frac{a^p}{p}+\frac{b^q}{q} $$ where $p$ and $q$ are Hölder conjugates, can be seen to be easily rearranged to be a restatement of either Bernoulli's inequality, which is a statement that the power function $x^p$ is convex for $p>1$, or else of AM-GM, which may itself be seen as a statement of the convexity of the exponential function, via Jensen's inequality. Versions of those arguments can be seen at this question (see robjohn's answer for the Bernoulli argument, Ben's for the AM-GM).
On the other hand, there is a generalized form of Young's inequality, having the form
$$ ab \leq \int_0^a f(x)\,dx + \int_0^b f^{-1}(y)\, dy. $$
This is mentioned on wikipedia as well as user17762's answer to this question. It reduces the special case above when $f(x) = x^{p-1}.$ This form seems to have nothing to do with convexity of the power function or convexity of the exponential function, and is rather just a corollary of the additivity of the measure on the plane.
How is it that the more general result doesn't require any analytic hypotheses about convexity of the functions like the power function? What is the actual content of the general Young's inequality? What are the hypotheses? Can I use this discrepancy in hypotheses to turn basic properties of the integral into a proof of the convexity of power functions or exponentials?
