Universal uniqueness criterion II

[bo198214]

...Let \( D=\mathbb{C}\setminus [x_0,\infty) \) for some \( x_0<x_1 \)...

Henryk: For tetration, I would define
\( D=\mathbb{C}\setminus (-\infty, x_0] \) for some \( x_0<x_1 \)...
then I like your proof.

Yes that was a mistype, it should read \( \mathbb{C}\setminus(-\infty,x_0] \). I change that in the original post.

I include below the small part of
http://math.eretrandre.org/tetrationforu...77#pid2477
which is picture of slog(S),

thanks, for the illustration.

1. Do you plan to polish this proof or I may include this into the paper?

Its not yet finished, we still have no universal domain \( G \) for the slog, which we need to have uniqueness independent of the specific domain \( \text{sexp}(S) \).

2. Can we claim, that some of singularities of a modified tetration are at
\( \Re(z)>-2 \) ?

Dmitrii, modified tetration is not all, if you want to consider \( g(z)=f(J(z)) \) then you need to have a \( J \) first, a \( J=f^{-1}\circ g \) that is holomorphic on \( S \) which in turn imposes conditions on the cut, set in the domain of definition of \( f^{-1} \).