Let $D$ denote the open unit disk around the origin in the complex plane. Let $f:D\rightarrow D$ holomorphic and $f$, $f^\prime$ extend continuously to $\overline{D}$. Let $u$ be the real part of $f$.
If $f$ attains a maximum at $z$ (which must be on the unit circle by the maximum modulus principle), is it true that $u=\mathrm{Re}(f)$ also has a maximum at $z$?
I think yes, but I don't know how to prove it. I started saying that the tangential derivative of $u$ at $z$ must be $0$, but that's not enough..