A conjectured uniqueness criteria for analytic tetration

[sheldonison]

Why do people think it is true ??
What arguments are used ?

Only boundedness seems to give uniqueness so far ?
Like bohr-mollerup.

I do not see the properties giving Uniqueness unless if those boundedness are a consequence ...

Regards

Tommy1729

From my mathstack answer, the exact solution for S(z) for bases 1<b<exp(1/e),
has the following form. Consider the increasingly good approximation z goes to infinity
\( S(z)= L+ \sum -a_n \lambda ^{nz} \;\;\lambda<1 \)
\( S(z)\approx L - \lambda ^{z} \)

For simplicity, lets look at the closely related function
\( f(z) = -\exp(-z) \)
and compare the perfectly behaved derivatives of f(z) as compared with the alternative solution
\( g(z) = -\exp(-z-\theta(z))\;\; \) where theta is 1-cyclic

The conjecture is g(z) can be shown to be not fully monotonic unless theta(z) is a constant. And then with a little bit of work, this can be used to show that S(z) is the unique completely monotonic solution to the Op's problem for bases b<exp(1/e)