holomorphic functions with special condition

Question

i want to find all holomorphic funtions $ f: \mathbb{C} \rightarrow \mathbb{C}$ with $ f(0)=0$ and $ f(f(f(f(z))))=z \forall z \in \mathbb C$

I would say rotations like $f(z)= |z| e^{i(arg z + \frac{\pi}{2})} $ can satisfy the conditions? Am I right?

Answer 1

Such a map is necessarily an automorphism of $\mathbb{C}$ and hence of the form $f(z) = az+b$. From here it is easy to see that only rotations qualify. So yes, you already found all of them.

EDIT: Seems like I slightly misread the question. Yes, rotations satisfy this as long as the angles add up to a multiple of $2\pi$.