Prove that there exists an analytic function $f : D → D$ such that $f(1/2) = f(−1/2)$

Question

This is an old qualifying exam problem that I am working on. I would appreciate some help. Thank you.

Prove that there exists an analytic function $f : D → D$ such that $f(1/2) = f(−1/2)$ and $f′(z) \neq 0$ for all $z ∈ D.$

Answer 1

Expanding the hint by Daniel Fischer: $f(z) = c \exp(2\pi iz)$ works when $|c|$ is small enough so that $f(D)\subset D$.