Consider a probability measure $\mu$ on a set $X$. Let $p,q \in (1, \infty)$, $f \in L^{pq} \cap L^1$ (so also $f\in L^p \cap L^q$) by non-negative. Can we say anything about the relationship between
$\int f^{qp} d\mu + \left(\int f d\mu\right)^{qp}$
and
$\left(\int f^p d\mu \right)^q + \left(\int f^q d\mu \right)^p ?$
In other words, is there an inequality saying that one of these two quantities is greater than or equal to the other under certain circumstances? It seems as if there should be a way to deduce something like this from Hölders/Jensens inequalities, but I have been unable to do so.