Young's inequality for integrals states that if $f:[0,\infty) \to [0,\infty)$ is a continuous, strictly increasing function with $f(0)=0$, then for any $a\ge 0$ and for any $b$ in $f$'s image we have that $$ab \le \int_0^a f(x) dx+\int_0^b f^{-1}(x) dx.$$
However, we also have the relation in this post Show rigorously that the sum of integrals of $f$ and of its inverse is $bf(b)-af(a)$ . This made me wonder if there is a more general version of Young's inequality (i.e. instead of having an integral of $f$ from $0$ to some $a$ maybe we could have an integral of $f$ from some $a$ to some $b$ $+$ some integral involving $f^{-1}$?).
EDIT : I believe that the most general inequality we can get is the following :
Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous, strictly increasing function. Then for any $a, b \in\mathbb{R} $ and $c, d$ in $f$'s image we have that $$\int_a^b f(x) dx + \int_c^d f^{-1}(x) dx\ge bd-ac$$ and the equality occurs when $c=f(a)$ and $d=f(b) $
Am I right?