# Homework Help: Complex Analysis - Normal Families

1. Jun 25, 2012

### EC92

1. The problem statement, all variables and given/known data
Let $U$ be a domain in $\mathbb{C}$ with $z_0 \in U$. Let $\mathcal{F}$ be the family of analytic functions $f$ in $U$ such that $f(z_0) = -1$ and $f(U) \cap \mathbb{Q}_{\geq 0} = \emptyset$, where $\mathbb{Q}_{\geq0}$ denotes the set of non-negative rational numbers. Is $\mathcal{F}$ a normal family?

2. Relevant equations

Montel's Theorem: A family of analytic functions on a domain is normal if it is uniformly bounded on compact subsets of the domain.

3. The attempt at a solution
I'm not sure that this is a normal family; if it's not, how would I prove it? Would I have to produce an explicit sequence of functions from the family which has no subsequence which converges uniformly (on compact subsets)?
If it is normal, I presumably need to show that it is uniformly bounded on compact subsets, which I don't know how to do either.

Thanks.