How many functions are known to be both Abel functions and Bernstein functions ?
An abel function is the $\alpha$ function that solves an Abel equation , see
https://en.m.wikipedia.org/wiki/Abel_equation
And a Bernstein function is this :
https://en.m.wikipedia.org/wiki/Bernstein%27s_theorem_on_monotone_functions
So a Bernstein function is the same as a complete monotone function. Or total monotone function. It is just terminology.
This question has also been posted on the tetration forum by my friend.
http://math.eretrandre.org/tetrationforum/showthread.php?tid=1163