Real-analytic tetration uniqueness criterion?

[mike3]

I assume mike3 meant :

\( \Gamma(\theta(x)) \)

As for the numerical testing , did mike test for x > 0 or x >-2 or a Taylor series at some x ? Or x > some C ?

If his conjecture means for x > -2 then it is quite a strong idea ,
but Im not sure about existance since going from 0 to 1 is not convex.

SO I assume for x > 0.

Right ?

regards

tommy1729

Um, it should be \( \Gamma(x) \theta(x) \). \( \Gamma(\theta(x)) \) is a 1-periodic function itself and not at all a solution of the Gamma function equations!

Convexity is over odd derivatives (for tetration), and even derivatives (for gamma function).