Real-analytic tetration uniqueness criterion?

[tommy1729]

Im very convinced its not unique.

Many reasons.

Asymptotics , interpolation , " fake function theory " ,...

In fact I posted similar conjectures years ago and came to reconsider them.

Hence I posted " TPID 16 " and a related uniqueness claim for exp^[1/2].

See :

http://math.eretrandre.org/tetrationforu...hp?tid=881

http://math.eretrandre.org/tetrationforu...hp?tid=879

( were the suggestion in post 2 is considered solid now )

Somewhat different :

http://math.eretrandre.org/tetrationforu...hp?tid=842

---
Here I was a bit upset because I posted these things first

http://math.eretrandre.org/tetrationforu...hp?tid=503
post 13.
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http://math.eretrandre.org/tetrationforu...hp?tid=474

http://math.eretrandre.org/tetrationforu...hp?tid=484

and probably more posts.
( I currently posted more than 13% of this forum so I dont remember them all so well )

So the idea is far from completely new.
But its intresting.
I hope you get more responses than I did.

As a remark : Notice this type of uniqueness was not sufficient for a unique real-analytic gamma function in the past.
Gamma also grows fast and satisfies a functional equation !
See the Bohr-Möllerup and Wielandt theorems.

If you want to replace convex with log convex , probably better to replace with arc2sinh or so ... which then again resembles my conjectures.
Imho the lenght idea is also underrated although I owe some credit to gottfried.

A few members here have already proved analogues of the Wielandt for tetration.

regards

tommy1729