I have the following question:
Suppose $f$ is entire, $f (0) = 3 + 4i$, and $|f(z)| \leq 5$ in $\mathbb{D}$. Find $f'(0)$.
Does the following work? If we consider $f$ on $D_r(0)$ (disk of radius $r$ centered at $0$) where $0<r<1$, then since $|f(0)| = 5$ and $|f(z)| \leq 5$ on $\mathbb{D}$, and hence $D_r(0)$ we see that $|f|$ attains a maximum in $D_r(0)$. Therefore, by the maximum principle $f$ is constant on $D_r(0)$, namely $f \equiv 3 + 4i$ on $D_r(0)$. Since $f$ is entire, by the identity principle this means that $f \equiv 3 + 4i$ on $\mathbb{C}$ and therefore $f'(0) = 0$