Carlson's theorem and tetration

[bo198214]

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).

Ya, right of course it needs not only be bounded on the real axis but on the strip around the real axis of period width. But then how you prove the boundedness of the superexponential?