For $f_0$ and $f_1$ two continuos probability density functions on $\mathbb{R}$, by Hölder, I know that $f_1^x f_0^{1-x}$ is integrable on $\mathbb{R}$, where $0 \leq x \leq 1$. Let $l=f_1/f_0$, then $g:=l^xf_0=f_1^x f_0^{1-x}$. Is it true that derivative of $g$, of any order, w.r.t $x$ is also integrable on $\mathbb{R}$? For example $g^{''}=\ln(l)^2 l^xf_0$?

I have a solution to this problem at mathstackexchange.com for the case when $l$ is increasing, and for $g^{''}$, here. I think it should still be true when $l$ is not necessarily increasing, and even for the higher order derivatives, but I am not able to show it.

Thanks for your help.