Analytic functions on a disk attaining the maximum of absolute value at $0$. Find all functions $f$ which are analytic in the region $|z|\le 1$ and are such that $f(0)=3$ and $|f(z)|\le 3$ for all z such that $|z|<1$.
How to do this? I know the maximum principle, which says that the maximum value of $|f(z)|$ is attained on $|z|=1 $. 

I believe due to the maximum modulus Theorem the answer is just f(z)=3 ....since the maximum is not attained on the boundary it must be a constant........
