## Analytic functions on |z|<=1

Hello,
I need to find all the analytic functions $\displaystyle f(z)$ defined on $\displaystyle |z|\leq1$ that displace $\displaystyle e^{i\theta} \rightarrow e^{2i\theta}$ (for every real $\displaystyle \theta$, and z is complex, of course).
How do I go about doing so?