Suppose we have a function, f, analytic on the punctured disc,
ie: the set of all z such that 0 < |z| < 1.

What form must f have if we know that f(z^3)=f(z)^3?

I have tried many times to write a Laurent series expansion of f and find some recurrence relation.
This is a total mess. I suspect that f should have to be zero or +/- some power of z. But, I do not know.

If anyone could help I would appreciate it greatly.