Here are a few more thoughts on this intriguing problem. First,

(reversing the order of integration in the double integral). Therefore

.

Next, let A denote the set of all infinitely-differentiable function on the positive reals whose derivatives alternate in sign:

-1)^kf^{(k)}(x)>0\text{ for all }x>0\text{ and all }k\geqslant0\}" alt="A := \{f\in C^\infty(\mathbb{R}_{>0})

-1)^kf^{(k)}(x)>0\text{ for all }x>0\text{ and all }k\geqslant0\}" />.

**Proposition.** If

then

.

*Outline proof:* The n'th derivative of f is of the form

, where

. Also, you can see by differentiating

that

. It is then straightforward to prove by induction (using the condition

) that the functions

alternate in sign, with

for all x>0 and all n.

It follows from that Proposition and the previous remarks that if the function

is in A then so is the function

.

I suspect that the function u(z) is indeed in A, but I haven't been able to prove that. However, this seems like a step in the right direction, because u(z) is a considerably simpler-looking function than f(z).

One final thought: functions whose derivatives alternate in sign seem to occur in

**Boltzmann's approach** to the theory of entropy. I wonder if that is where this problem originated?