Generalized Bieberbach conjectures ?
I wanted to say that I believe tetration is related to generalizations of the Bieberbach conjecture/de Branges theorem , Koebe quarter theorem and related.

For instance Let U be the unit circle , let V be a smaller circle that touches the unit circle from the inside. The "moonshaped " area ( this has a name right ? ) Is called A. Let the open set A be mapped injectively to the complex plane by the function f(z).

Then D^n f(z) dz satisfies ....

Now by riemann mappings and inversing functions we end up with a branch for a superfunction.

And the conjectured properties must be carried along in a transformed way.

This "moonshaped conjecture " might already have a name , but if not I call it tommy's moonshine conjecture for now , although I havent figured out the details of even the precise statement.

This is something I have not seen adressed on the forum yet so I found it necc to mention it.



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