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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.

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    I corrected the spelling of "Hölder". If you can't type the "ö" character, then "oe" is an acceptable substitute. Thus "Hölder" is the same as "Hoelder", but "Holder" is a different spelling.2012-06-08
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    If the space has finite measure, and in particular if it's a probability space, and $pq>1$, then $f\in L^{pq}$ implies $f\in L^1$, i.e. $L^{pq}$ is in that case a subset of $L^1$. So there's no need to write "$L^{pq}\cap L^1$".2012-06-08
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    Since $pq$ is the largest exponent here, you can make $\int f^{qp}\,d\mu$ huge while keeping the other three as small as you wish. On the other hand, direct application of Holder's inequality gives a bound for each of the terms in the second sum in terms of $\int f^{qp}\,d\mu$.2012-06-09

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