Expectation of log-density with respect to a different random variable

Question

Let $X$ and $Y$ be two continuous random variables with marginal density functions $f_{X}(x) $ and $f_{Y}(y) $.

Is it true that $$\ E[\log f_{X}(X)] \geq E[\log f_{Y}(X)] ?$$

Perhaps the concavity of the log function will come in handy, along with Jensen's inequality, but I am having particular trouble manipulating the $\ f_{Y}(X) $ term.

Answer 1

By Jensen's inequality, $$ \mathsf{E}\log \frac{f_Y(X)}{f_X(X)}\le \log \mathsf{E}\frac{f_Y(X)}{f_X(X)}=\log \int \frac{f_Y(x)}{f_X(x)}f_X(x)dx=0 $$