Decreasing tail integrals for nonnegative random variable $X$ Let $X$ be a nonnegative random variable with density function $f(x)$, distribution function $F(x)$, survival function $S(x)=1-F(x)$ and finite first and second moments. Let also
$$\ell(x):=\frac{1}{xS(x)}\int_{x}^{+\infty}S(u)du,\quad\quad \mu(x):=\frac{1}{S(e^x)}\int_{e^x}^{+\infty}\frac{1}{u}S(u)du$$
I want to show or find a counterexample that: $\mu(x)$ is decreasing in $x>-\infty$ if and only if $\ell(x)$ is decreasing in $x>0$.
Since $\frac1u=\ln{(u)}'$, I am trying to use a variable transformation, $t\to \ln{u}$ in $\mu(x)$. This is similar to considering the variable transformation $X\to \ln{(X)}$. However, from the result, I cannot get the equivalence. So, I am currently thinking of constructing a counterexample, possibly using the Pareto distribution with $x_m=1$ and $\alpha>1$ (using the Wikipedia notation) which has $S(x)=x^{-a}$, but again, I cannot get a counterexample.
In fact, I expect the answer to be no (so, I am more inclined to look for a counterexample). The reason is that $\ell(x)$ is constant for the Pareto distribution. However, $\mu(x)$ would be constant for the exponential distribution, where exponential = $\ln{}$(Pareto), if there was not the $1/u$ term inside the integral. On the other hand, the $1/u$ is also decreasing, so maybe it is possible to show only the one direction and find a counterexample for the other direction.
 A: This iff statement is false. Indeed, this iff statement can be restated as follows:

If
$$\ell(x):=\frac1{xS(x)}\int_x^\infty S(u)\,du\quad\text{and}\quad 
m(x):=\frac1{S(x)}\int_x^\infty\frac{S(u)}{u}\,du,$$
then $\ell$ is decreasing on $(0,\infty)$ iff $m$ is decreasing on $(0,\infty)$.

To show that the highlighted statement is false,
let
$$a:=\frac{757}{256},\ b:=-1,\ c:=\frac{729}{8192}$$
and then let
$$\tilde S(x):=x (a + b x + c x^2) e^{-x}$$
for $x\ge6$, with $\tilde S(x):=S(6)$ for $x\in[0,6)$, and then let
$$S(x):=\tilde S(x)/\tilde S(0).$$
Then $S$ is a positive nonincreasing to $0$ function on $[0,\infty)$ with $S(0)=1$, so that $S$ is a survival function. Moreover, on $(0,6]$ the function $S$ is constant and hence $\ell$ is obviously decreasing. Also, $\ell'<0$ on $[6,\infty)$. So, $\ell$ is decreasing on $(0,\infty)$. However, $m'(6)=\frac{17141}{2597784}>0$, and hence the function $m$ is not decreasing on $(0,\infty)$. $\quad\Box$

Here is an image of the Mathematica notebook with the corresponding calculations:

