This is exercise 3.26 in Rudin's Real & Complex Analysis:
If $f$ is a positive measurable function on $[0,1]$, which is larger, $$\int_0^1 f(x) \log f(x) \, dx$$ or $$\int_0^1 f(s) \, ds \int_0^1 \log f(t) \, dt$$
I tried a bunch of functions and always got the first to be larger, which suggests that Hölder's inequality won't help here (at least not a direct application). I couldn't find an example that made the second larger. I'm stuck otherwise.
(This is self-study, not homework)
Clarification: The integral here is the Lebesgue integral. The only answer so far is only applicable to Riemann integrable functions.