Comparing the relative entropies of some stochastically ordered distributions Motivation of this question:
This question is related to the expected stopping time of a stochastic process under two hypotheses. Especially, it answers the question "how many more samples are required such that a sequential test stops when there is a model missmatch compared to the case when there is no missmatch". I found that the ratio is ${D(f_0,f_1)}/{D(g_0,g_1)}$ if the null hypothesis is correct and ${D(f_1,f_0)}/{D(g_1,g_0)}$ if the alternative hypothesis is correct. I know that both should be greater than $1$ because $g_0$ and $g_1$ are gradually getting closer to each other when we increase the degree of missmatch which indicates that their distance is also getting smaller and smaller. However, I was not able to show it analytically.
My question is as follows:

Given $4$ distinct densities $f_0,f_1$ and $g_0,g_1$ on $\mathbb{R}$ with their cumulative distribution functions $F_0,F_1$ and $G_0,G_1$ satisfying
$$F_0(y)>G_0(y)>G_1(y)>F_1(y),\quad  \forall y$$
Show that
$$\int_{-\infty}^{\infty}f_1(y)\ln\frac{f_1}{f_0}(y)\mathrm{d}y>\int_{-\infty}^{\infty}g_1(y)\ln\frac{g_1}{g_0}(y)\mathrm{d}y$$
holds.

Note that the equation above is based on the distance
$$D(h_1,h_0)=\int_{-\infty}^{\infty}h_1(y)\ln\frac{h_1}{h_0}(y)\mathrm{d}y$$
In addition to this, I wonder if it is possible to generalize the result for any distance measure $D$ between two densities? for example
$$D^2(h_0,h_1)=\frac{1}{2}\int \left(\sqrt{h_0(y)}-\sqrt{h_1(y)}\right)^2 \mathrm{d}y$$
or
$$D(h_0,h_1)=\frac{1}{2}\int |h_0(y)-h_1(y)| \mathrm{d}y$$
For the first question I tried to separate the integrals to $(-\infty,y]$ and $[y,\infty)$ but then I was stuck and couldnt continue with the solution.
EDIT: here is an example for Gaussian densities

It can be seen that $g_0,g_1$ are between $f_0,f_1$ and closer to each other and $D$ is getting smaller.
Thanks for reading this post and any possible help.
 A: Here is a hint leading to the conclusion that the inequality between relative entropies that you suggest cannot hold in general. Call $\gamma_t$ the gaussian density with mean $t$ and variance $1$ and consider the densities
$$
f^{(t)}_0=\tfrac12(\gamma_0+\gamma_{2t}),\qquad g_0^{(t)}=\tfrac12(\gamma_1+\gamma_{2t}),\qquad g_1^{(t)}=\tfrac12(\gamma_t+\gamma_{2t}),\qquad f_1^{(t)}=\gamma_{2t}.
$$
Then $F_1^{(t)}\lt G_1^{(t)}\lt G_0^{(t)}\lt F_0^{(t)}$ for every $t\gt1$, as desired, and I am fairly confident that the relative entropies are such that $D(f_1^{(t)},f_0^{(t)})\lt D(g_1^{(t)},g_0^{(t)})$, at least for $t$ large enough, for the following reason.
Renormalizing when $t\to+\infty$, the distributions of respective densities $f_0^{(t)}$, $g_0^{(t)}$, $g_1^{(t)}$ and $f_1^{(t)}$ converge to the measures
$$
\mu_0=\tfrac12(\delta_0+\delta_2),\qquad\nu_0=\tfrac12(\delta_0+\delta_2),\qquad\nu_1=\tfrac12(\delta_1+\delta_2),\qquad\mu_1=\delta_2,
$$
respectively, and $D(\mu_1,\mu_0)=\log2$ while $D(\nu_1,\nu_0)=+\infty$. Since $D(f_1^{(t)},f_0^{(t)})\to D(\mu_1,\mu_0)$ and $D(g_1^{(t)},g_0^{(t)})\to D(\nu_1,\nu_0)$ when $t\to+\infty$, the conclusion follows.
A: My work with the constraint when $f_1/f_0$ and $g_1/g_0$ are increasing:

I tried to come up with some densities. First, I thought perhaps I could consider Lagrangian type of optimization for the given inequality
$$\int_{-\infty}^{\infty}f_1(y)\ln\frac{f_1}{f_0}(y)\mathrm{d}y>\int_{-\infty}^{\infty}g_1(y)\ln\frac{g_1}{g_0}(y)\mathrm{d}y$$
I would minimize the left hand side and maximize the right hand side jointly. Since this would be the worst case, and if in this case no such density would exist then I would be done. BUT, relative entropy is convex and maximizing the right hand side therefore makes no sense. Then, I decided to make some conclusions. You can see them below as well.
$f_0$ and $f_1$ should be as close as possible to each other while $g_0$ and $g_1$ should be as far as possible.
Based on this idea, the $\arg \max g_0$ should be greater than $\arg \max f_0$. There are two ways to do this. In the first case $\max g_0>\max f_0$ or $\max g_0<\max f_0$.
Then I decided to plot them. You can see the plot above. The higher variance $g_0$ and $g_1$ need to have asymmetric tails to satisfy the constraints. Whenever I consider lower variance pairs $g_0,g_1$, namely when $\max g_0>\max f_0$ is true, then the areas $A,B,C,D$ should satisfy the given criteria. This case might have the potential to be a counterexample but I dont know such asymmetric densities on the Real numbers. For example on $[0,\infty]$ there is Rayleigh distribution but on reals?
