Suppose that $f$ is analytic in the annulus: $1 \leq \vert z \vert \leq 2 $, that $\vert f \vert \leq 1$ for $\vert z \vert = 1$ and that $\vert f \vert \leq 4$ for $\vert z \vert = 2$. Prove $\vert f(z) \vert \leq \vert z \vert ^2$ throughout the annulus.
I know that I would have to apply the Maximum Modulus Theorem here, but I am having trouble figuring out how to do so. Would I have to use the analyticity of $f$ in order to reach such a conclusion?
I am using the textbook Complex Analysis, Third Edition by Joseph Bak and Donald J. Newman.
Any suggestions and tips are greatly welcomed.