Comparing the ratio of the expected value of a random variable to the ratio of the expected value of its stochastic process

Comparing the ratio of the expected value of a random variable to the
ratio of the expected value of its stochastic process

I wonder if $$\frac{\mathbb{E}_{X_0}[z]}{\mathbb{E}_{Y_0}[z^*]}\geq
\frac{\mathbb{E}_{X_1}[z_n]}{\mathbb{E}_{Y_1}[z_n^*]}$$ holds for $n\geq
1$ or for $n>1$ or does it hold only when $n$ large enough? if not could
you please provide me some counter examples?
Source of the question: This question is related to the Wald's sequential
test.
Explanation: I have four stochastically ordered random variables $X_0$,
$Y_0$, $Y_1$, $X_1$, with the ordering $$X_1>_{ST}Y_1>_{ST}Y_0>_{ST}X_0$$
and the corresponding density functions $f_0$, $g_0$, $g_1$, $f_1$,
respectively (e.g., $X_0\rightarrow f_0$). Furthermore, $f_1/f_0$,
$g_1/g_0$, $f_1/g_1$ and $g_0/f_0$ are all known to be some non-decreasing
functions. We can assume that all these random variables are defined on
the same probability space. The random variable $z$, $z^*$, $z_i$ and
$z_i^*$ are defined as $$z=\log\frac{f_1(x)}{f_0(x)}\quad
z^*=\log\frac{g_1(x)}{g_0(x)}\quad z_i=\log\frac{f_1(x_i)}{f_0(x_i)}\quad
z_i^*=\log\frac{g_1(x_i)}{g_0(x_i)}$$ whereas the random processes $z_n$
and $z_n^*$ with $n<\infty$ are defined as $$z_n=z_1+z_2,...,z_n,\quad
z_n^*=z_1^*+z_2^*,...,z_n^*$$ What I know: I know that
$\mathbb{E}_{X_0}[z]>\mathbb{E}_{Y_0}[z^*]$ and
$\mathbb{E}_{X_1}[z_n]>\mathbb{E}_{Y_1}[z_n^*]$ are true due to
stochastical ordering. When $n=1$ the Problem is reduced to
$$\frac{D(f_0,f_1)}{D(g_0,g_1)}\geq \frac{D(f_1,f_0)}{D(g_1,g_0)}$$ where
$D(\cdot,\cdot)$ is the relative entropy (KL-divergence or KL-Distance).
Since $D$ is not a symmetric distance measure, in general, I can not claim
that the inequality holds with the equality case. However, the
stochastical ordering as well as non-decreasing likelihood ratios (e.g.,
$f_1/f_0$), can provide this. I also think that the inequality can not
hold with the ineautliy case because of the symmetry, either it should
hold with equality or it shouldnt hold. When $n>1$, I dont have any idea.
I dont have any experince, to be honest, with stochastic processes.
If you have questions, especially if there is a missing information or
something wrong, I will be happy to answer them.
If you have any opinion about the problem, I will be even more happy than
the previous case.
If you make some efforts to make me happy, then I will also do the same!
Thanks!