Homework 9: Complex Analysis

Problem 1

Let $latex V\subset \mathbb C$ be open and connected and $latex \mathcal F\subset H(V)$. If $latex \mathcal F$ is a normal family, then $latex \mathcal F' = \{f': f\in\mathcal F\}$ is also a normal family.

Problem 2

1. The converse to the result in Problem 1 is false.
2. Find a "small" hypothesis to make the converse true.

Problem 3

Let $latex V\subset \mathbb C$ be open and connected and $latex \mathcal F\subset H(V)$ a normal family. Let $latex f_n\in \mathcal F$ a sequence such that $latex f_n(z)\to f(z)$ for each $latex z\in S\subset V$, where $latex S$ has a limit point in $latex V$ and $latex f\in H(V)$. Then $latex f_n\to f$ in $latex H(V)$.

Problem 4

Let $latex V\subset \mathbb C$ be open, $latex M>0$, and 

$latex \displaystyle\mathcal F = \Big\{ f\in H(V): \iint_V |f(z)|^2 dxdy \le M \Big\}$.

Then $latex \mathcal F$ is a normal family.

Problem 5

Let $latex V\subset \mathbb C$ be open and connected and $latex \{f_n\}\subset H(V)$ a normal family such that each $latex f_n$ is never zero in $latex V$. If there exists $latex z\in V$ such that $latex f_n(z)\to 0$, then $latex f_n\to 0$ in $latex H(V)$.