Let $(X, \mathcal{A},\mu)$ a measure space such that $\mu(X) > 0$ and let $f, g : X \rightarrow (0,\infty)$ be such that $f, g, f\log(f), f\log(g) \in L^1(\mu)$. Show that $$ \|f\|_1\log \|f\|_1 + \int f \log(g) \; d\mu ~\leq~ \|f\|_1 \log\|g\|_1 + \int f \log(f) \; d\mu.$$
If $(X, \mathcal{A},\mu)$ was a probability measure, Jensen's inequality (applied to the convex function $x \log(x)$) would imply that $$ \|f\|_1\log \|f\|_1 ~\leq~ \int f \log(f) \; d\mu.$$ This suggests to me that some sort of generalized Jensen's inequality could be the key here. But I don't get my hopes up because there is no reason to believe that $\mu(X)$ is finite...
Any hint while I'm racking my brain to work out a solution ?