Proving that the expectation is always negative I am interested in if the expected value $$E_{X_3}\left[\log\frac{f_1(y)}{f_2(y)}\right]<0$$ is always be negative or not. Here, I have $3$ random variables $X_1,X_2,X_3$, corresponding to the densities $f_1,f_2,f_3$ with their Cumulative distribution functions satifying $F_1(Y)>F_2(Y)>F_3(Y)$.
If I rewrite the expectation, I have $$\int_{\mathbb{R}}f_3(y) \log\frac{f_1(y)}{f_2(y)}\mathrm{d}y$$
If I take the expectation with respect to $X_1$ then I have $$E_{X_1}\left[\log\frac{f_1(y)}{f_2(y)}\right]>0$$
If I take the expectation with respect to $X_2$ then I have $$E_{X_2}\left[\log\frac{f_1(y)}{f_2(y)}\right]<0$$
due to stochastical ordering ${X_3}_{ST}>{X_2}_{ST}>{X_1}_{ST}$ since $F_1(Y)>F_2(Y)>F_3(Y)$ I suppose that I must have $$E_{X_3}\left[\log\frac{f_1(y)}{f_2(y)}\right]<0$$
do you have any ideas how I can prove this?
Thanks for your help.
Additional Information: If $F_1(Y)>F_2(Y)>F_3(Y)$ then $E[X_3]>E[X_2]>E[X_1]$
 A: (These are remnants of a wrong try... Finding a case when $I>0$ is still open.)
If $X_i=\frac{X}{a_i}$ with $X$ standard exponential and $a_3\lt a_2\lt a_1$, then $f_i(x)=a_i\mathrm e^{-a_ix}$ on $x\gt0$ hence
$$
\log\left(\frac{f_1(x)}{f_2(x)}\right)=\log\left(\frac{a_1}{a_2}\right)-\left(a_1-a_2\right)\,x,
$$
and the expectation sought for is
$$
I=E\left(\log\left(\frac{f_1(X_3)}{f_2(X_3)}\right)\right)=\log\left(\frac{a_1}{a_2}\right)-\frac{a_1-a_2}{a_3}.
$$
Thus,
$$
I\lt\log\left(\frac{a_1}{a_2}\right)-\frac{a_1-a_2}{a_2}\lt0.
$$
A: For some ${X_3}_{ST}>{X_2}_{ST}>{X_1}_{ST}>0$, I claim that $$E[X_3^N]>E[X_2^N]>E[X_1^N],\quad N\geq 1$$ For $X_3=a_3X$, $X_2=a_2X$, $X_1=a_1X$, with $a_3>a_2>a_1$ it is obvious. For the general case I have no proof about it. For the moment, I can only suppose that It also holds for the general case.
Based on this assumption I can create any positive function $g_i$ as some linear combinations of $X_i$, $i=1,2,3$.
$$\int_{\Omega}f_3(X_3)\{a_1X_3^N+a_2X_3^{N-1}...\}\mathrm{d}\mu>\int_{\Omega}f_2(X_2)\{a_1X_2^N+a_2X_2^{N-1}...\}\mathrm{d}\mu>\int_{\Omega}f_1(X_1)\{a_1X_1^N+a_2X_1^{N-1}...\}\mathrm{d}\mu$$
Applying this to the logarithm function $$\log\frac{f_1(y)}{f_2(y)}$$ we have $$\int_{\log\frac{f_1(y)}{f_2(y)}>1}f_3(y)\log\frac{f_1(y)}{f_2(y)}\mathrm{d}y>\int_{\log\frac{f_1(y)}{f_2(y)}>1}f_2(y)\log\frac{f_1(y)}{f_2(y)}\mathrm{d}y>\int_{\log\frac{f_1(y)}{f_2(y)}>1}f_1(y)\log\frac{f_1(y)}{f_2(y)}\mathrm{d} y$$
Similarly we have 
$$\int_{\log\frac{f_1(y)}{f_2(y)}<1}f_1(y)\log\frac{f_1(y)}{f_2(y)}\mathrm{d}y>\int_{\log\frac{f_1(y)}{f_2(y)}<1}f_2(y)\log\frac{f_1(y)}{f_2(y)}\mathrm{d}y>\int_{\log\frac{f_1(y)}{f_2(y)}<1}f_3(y)\log\frac{f_1(y)}{f_2(y)}\mathrm{d} y$$
For the conlusion I was thinking about combining both and perhaps if necessary to use Jensen's inequality. Am I on the right track or do i have problems even at the start where I did the assumption?
