The function $f(z)$ is analytic in the unit disk $U = {z:|z|<1}$ and continuous in the closed unit disk. Suppose that $\frac{f(z)}{z^2}$ can be extended to be analytic in the (open) unit disk U (also at the origin). If $|f(z)| \leq 6$ in the closed unit disk , what is the maximal possible value for $f(0.4 + 0.5i)$ ?
I guess I should somehow be using the maximum modulus principle or the Cauchy estimates, but I'm not sure how. Grateful for any help with this!
Yes, you should apply the maximum principle to the function $g(z) = f(z)/z^2$. Since $|g|\le 6$ on the boundary, $|g(0.4+0.5i)|\le 6$. Put this back in terms of $f$ to get the desired upper bound.
To demonstrate its sharpness, arrange $f$ so that $g(z)\equiv 6$.