Analytic functions on |z|<=1

May 2010
38
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?