TITLE: Maximum Modulus Theorem and Annulus
QUESTION [5 upvotes]: 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.

REPLY [2 votes]: As suggested, my comment above will be posted as an answer to seek clarification.
If $z=1$, then it the inequality holds. If $z=2$, the inequality is still true. The Maximum Modulus Theorem states a non-constant analytic function in a region $D$ does not have any interior maximum points. Using $\frac{f(z)}{z^2}$, then it does not have any interior maximum points; hence, it assumes its maximum modulus at its boundary points?