Consider the following integral,
$$K(\alpha)=\int_\mathbb{R}\log^2(g/f)(g/f)^\alpha f \, \mathrm{d}\mu$$ where $\alpha\geq 0$, $\,\mu$ is some measure and $f,g$ are some distinct continuous probability density functions on reals.
Prove either of the two:
Claim 1: $\lim_{\alpha\rightarrow\infty}K(\alpha)=\infty$
Claim 2: $K(\alpha)$ takes its minimum value on $\alpha\in[0,1]$.
I think
$\lim_{\alpha\rightarrow\infty}K(\alpha)\neq 0$ can prove both claims.