Liouville's Theorem Derivative solving? I believe this is a Liouville's Theorem problem, but I am unsure as to how to solve it.  

Assume that $|f(z)|< |z^2 + 3z +1|$ for all $z$, and that $f(1) = 2$. Evaluate $f '(2)$, and explain your answer. 


You are quite correct that this is a Liouville's theorem problem. The function
$$g(z) := \frac{f(z)}{z^2 + 3z + 1}$$
(where $z^2 + 3z + 1 \ne 0$) is bounded and has an analytic extension to all of $\mathbb{C}$; hence, it is constant. This implies that $f(z)$ is a scalar multiple of $z^2 + 3z + 1$, and the fact that $f(1) = 2$ tells you what that multiple is. Now computing the derivative is an elementary calculation.

As a remark, it's a nice result that whenever $f$ and $g$ are analytic functions so that there is a constant $c$ with $|f(z)| \le c |g(z)|$ for all $z$, $f$ and $g$ are multiples of each other.
