Carlson's theorem and tetration

[mike3]

I found the following easy uniqueness theorem that characterizes the regular tetrational of the base \( b = \eta = e^{1/e} \), and perhaps also the whole regular tetrational (with attracting fixed point) (though base-\( \eta \) is particularly interesting since it seems that both the regular and non-regular (i.e. Kneser's, etc.) method approach the same tetrational at this base.),

Hm, you mean that the super-exponential is bounded on the positive real axis by e and is imaginary periodic and hence is (exponentially) bounded on the right halfplane ...
This is realy a nice finding, Mike!

Yeah, though for base \( \eta \) there is no imaginary periodicity, but still, bounded on the right half plane (actually, the conditions you give that it should be bounded on the positive real axis and on the imaginary do not imply ("hence") it is bounded on the right half plane alone -- see, e.g. translations of the entire "upper regular iteration" along the imaginary axis).

I guess it can be generalized to arbitrary regular superfunctions as they are always of the form \( \eta(\pm e^{\kappa z}) \) for some function \( \eta \) analytic at 0.

Yes, provided the fixed point is attracting and positive real.