I'm looking at old complex analysis exams and am stuck on the following. Suppose f(z) is holomorphic on $D(0,2)$ and continuous on its closure.
Suppose the $|f(z)|\le 16$ on the closure of $D(0,2)$ and is non-constant and |f(0)|=1. Show $f$ cannot have more than 4 zeros in $D(0,1)$ .
I found this technique Upper bound for zeros of holomorphic function , but it doesn't seem to apply to this problem.