A conjectured uniqueness criteria for analytic tetration

[sheldonison]

After some study of different approaches to an extension of tetration to fractional or complex heights, and many numeric experiments, I came to the following conjecture, that I am currently trying to prove:

Let \( a \) be a fixed real number in the interval \( 1 < a < e^{1/e} \). There is a unique function \( f(z) \) of a complex variable \( z \), defined on the complex half-plane \( \Re(z) > -2 \), and satisfying all of the following conditions:

* \( f(0) = 1 \).
* The identity \( f(z+1) = a^{f(z)} \) holds for all complex \( z \) in its domain (together with the first condition, it implies that \( f(n) = {^n a} \) for all \( n \in \mathbb N \)).
* For real \( x > -2, \, f(x) \) is a continuous real-valued function, and its derivative \( f'(x) \) is a completely monotone function (this condition alone implies that the function \( f(x) \) is real-analytic for \( x > -2 \)).
* The function \( f(z) \) is holomorphic on its domain.

Please kindly let me know if this conjecture has been already proved, or if you know any counter-examples to it, or if you have any ideas about how to approach to proving it.

There is a proof framework for how to show that the standard solution from the Schröder equation is completely monotonic. The framework only applies to tetration bases 1<b<exp(1/e) and does not apply to Kneser's solution for bases>exp(1/e), which is a different analytic function.
http://math.stackexchange.com/questions/...-tetration

The conjecture would be that the completely monotonic criteria is sufficient for uniqueness as well; that there are no other completely monotonic solutions. It looks like there is a lot of theorems about completely monotone functions but I am not familiar with this area of study. Can you suggest a reference? I found https://en.wikipedia.org/wiki/Bernstein%..._functions

Also, is the inverse of a completely monotone function also completely monotone? no, that doesn't work. So what are the requirements for the slog for bases<exp(1/e) given that sexp is completely monotonic?