Which integral is greater?

Question

I'd like to find out which of these two is greater:

For a Lebesgue measurable function $f:[0,1] \rightarrow [1, \infty)$

(1): $\int_0^1 f(x)log(f(x))dx$

(2): $\int_0^1f(y)dy\int_0^1log(f(w))dw$

I've tried two things which didn't work: the first being an approximation with a step function and the second with writing $f(x) = \frac{g(x)}{1-x}$ for some $g(x):[0,\infty) \rightarrow [0,\infty),\ g(1)=0$

I'd really appreciate the help, thanks in advance!

Answer 1

Hint: The function $$t \mapsto t \log t$$ is convex. Since the measure of $(0, 1)$ is $1$, Jensen's inequality can be applied.