Assume that $f: \mathbb{C} \rightarrow \mathbb{C}$ is holomorphic and fulfils $f(\partial B_r(0)) \subseteq \mathbb{R}$ for some $r>0$. Is it then true that $f$ is constant?
I really think it is. To prove this, I desperately tried to apply the identity theorem to $\mathrm{Re}f$. The latter did not work because I can not assume the real part to be holomorphic. The Cauchy-Integral-Formula did not help me either. I suppose that the proof is a lot more tedious. Cauchy-Riemann-DEs did not get me anywhere, too, because $\partial B_r(0)$ is not open.
Can someone help me out or is there some non-constant $f$ that I did not find?