How can one determine $$\max_{f_0,f_1}\frac{\int_{\mathbb{R}}f_1(y)^xf_0(y)^{1-x}\log\left(\frac{f_1(y)}{f_0(y)}\right)\mbox{d}y}{\int_{\mathbb{R}}f_1(y)^xf_0(y)^{1-x}\mbox{d}y}$$
given
$$\int_{\mathbb{R}}f_1(y)^xf_0(y)^{1-x}\mbox{d}y=a$$ where $x$ is a given number in $(0,1)$ and $f_0$ and $f_1$ are some positive density functions on $\mathbb{R}$?
Note that $0<a<1$ holds due to the integral of a positive function and the Cauchy-Schwarz inequality.
It would be also nice to know the minimum as well based on the same condition. Please let me know if you have any idea related to any possible solution of this problem.
Thank you very much!!