How many methods have this property ?

[tommy1729]

How many C^oo ( or analytic ) solutions to tetration satisfy :

For all bases larger than eta for which the method works :

Property : sexp_a(x) > sexp_b(x)
For all a>b and x>1.

I used convention sexp_a(1) = a , sexp_b(1) = b here.

I could not find it here ?

I assume this property fails for base change and kneser.
I think it works for Cauchy contour method (kouznetsov?) and 2sinh method (tommy1729).

Maybe this has been said long ago.

regards

tommy1729