Question: Let $f$ be an analytic function in a neighborhood of the closure of $\mathbb{D}$. Show that if $f$ is real on all the boundary of $\mathbb{D}$, then $f$ must be constant.
Thoughts: I feel like this should be done using maximum modulus principle, but I am not quite sure how to do it. Since $f$ is analytic in a neighborhood of the closure of $\mathbb{D}$, maybe we can use some partial derivative stuff since C-R equations must be satisfied and the fourt partial derivatives are continuous... Any thoughts or help is greatly appreciated!
MMP is the right approach. Let $g(z)=e^{if(z)}$. Note that $|g(z)|=1$ for all $z$ on the boundary. Use MMP for $g$ and $\frac 1 g$ to show that $|g|$ is a constant. This implies that $g$ is itself a constant. Now conclude that $f$ is a constant.