elementary superfunctions

[bo198214]

.. \( \eta(e^{b^x}) \) ..is a regular superfunction at some fixed point with \( f'(z_0)=b \).
The only exception would be that we have a complex fixed point (of a real polynomial) with a real derivative \( b \), is that possible?

I do not understand the question. Is \( \eta \) allowed to be signular at the fixed point?

\( \eta=P \).

We could find numerically the superfunction for basefunction H(z)=1+z^2, describe the properties, and nominate it to the Mathematical Community as new element of the set of Special Functions.