I am having a hard time figuring out where to start and what results to use to address the following question:
Suppose $f(z)$ is analytic in the unit disc $D=\{z:|z|<1\}$ and continuous in the corresponding closed disc $\overline{D}=\{z:|z|\leq 1\}$. Suppose also $\frac{f(z)}{z^{7}}$ can be extended to be analytic in all of $D$, including the origin. If $|f(z)|\leq 9$ in $\overline{D}$, what is the maximal value that $|f(0.2-0.5i)|$ can assume under these conditions?
For your information, I am reading E.B. Saff & A.D. Snider's Fundamentals of Complex Analysis.
Clearly, as $g(z)=z^{-7}f(z)$ is analytic, then $$ \sup_{\lvert z\rvert\le 1}\lvert g(z) \rvert=\sup_{\lvert z\rvert\le 1}\lvert\,f(z) \rvert\le 9. $$ Thus $\lvert\, f(z) \rvert\le9\lvert z \rvert^7$, and hence $$ \lvert\,f(.2+.5i)\rvert\le 9\lvert.2+.5i\rvert^7. $$ This value is achieved for $f(z)=9z^7$.