Holomorphic in an open set containing the closed unit disc $\mathbb D$ and real on the boundary of $\mathbb D$

Question

If f(z) is holomorphic in an open set containing the closed unit disc, and if f($e^{i\theta})$ is real for all $\theta$ in $\mathbb R$, then prove that f(z) is constant.

I came across this problem in the exercises of a book on Complex Analysis. I need two clarifications about this problem. I think the word "connected" is missing in the problem in the domain of f(z). Is the problem true for all open sets or only for domains? For the domain of f for which the problem is true, how do I prove the result? What is given is that the function is real on the boundary of the unit disc $\mathbb D$. This means that the imaginary part of f say v is zero on $\partial\mathbb D$. From here how do I prove that f is constant? Please help.

Answer 1

It is obviously false if the domain is not connected: Take $f(z)=1$ for $|z|<2$ and $f(z)=0$ for $|z-10|<1$.

For the proof when the domain is connected consider $g(z)=e^{if(z)}$ and note that $|g(z)|=1$ for all $z$ with $|z|=1$. Do you know how to show that $g$ is a constant? [This has been proved many times on this site].