# Does stochastical ordering imply ordering in distance?

Given $4$ distinct densities $f_0,f_1$ and $g_0,g_1$ on $\mathbb{R}$ with $f_1/f_0$ increasing and $g_1/g_0$ non-decreasing and their cumulative distribution functions $F_0,F_1$ and $G_0,G_1$ satisfying

$$F_0(y)>G_0(y)>G_1(y)>F_1(y),\quad \forall y$$

Is it true that

$$\int_{-\infty}^{\infty}f_1(y)\ln\frac{f_1}{f_0}(y)\mathrm{d}y>\int_{-\infty}^{\infty}g_1(y)\ln\frac{g_1}{g_0}(y)\mathrm{d}y$$

If yes, can this result be generalized to any arbitrary distance in the palce of relative entropy above?

From ordering I mean a general expression like $F_0>F_1>...>F_N$ and the inner pairs have lower distances.

I asked a simplified version of the same question at MSE and can be reached from here.

Thank you very much. I am looking forward to hear any possible hint.

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I proved this question. However, I still need $f_1/g_1$ and $g_0/f_0$ are increasing. I guess a counterexample for the general version above can be found. –  Seyhmus Güngören Sep 9 at 15:34