Universal uniqueness criterion II

[Kouznetsov]

I try again: If \( h \) is entire and 1-periodic and \( J(z)=z+h(z) \), then can we claim that \( \exists z \in \mathbb{C} ~:~ \Re(z)>-2 ~,~ J(z)\!=\!-2~.~ \) ?

Hm, sorry, I dont know..

ok, I do know. Conjectiure:
Let
\( h \) be entire 1-periodic non constant function,
\( h(0)=0 \),
\( |h'(z)| <1 \) for all \( z \) in some vicinity of the real axis
\( J(z)=z+h(z) \) for all complex \( z \)
Then there exist \( z \in \mathbb{C} \) such that \( \Re(z)>0 \) and \( J(z)=-2 \)

Wherefore do you need such a statement?

Yes, I do. Then we have beautiful and general proof of uniqueness of tetration as it is defined at http://en.citizendium.org/wiki/Tetration

P.S. For other participants, I repeat here the essense from one of our previous discussions.
Many times I tried to build-up an example to negate the conjecture above.
Therefore I claim this conjecture.
I agree with you, that it is not sufficient reason for such a claim.
(Although I never met a simple condition for an existing function, such that I could not provide an example.)
It would be interesting to construct the sufficient reason. (then the conjecture becomes Theorem)