TITLE: Normal Families
QUESTION [7 upvotes]: Suppose $\mathcal{F}$ is a famliy of analytic functions of the unit disc.   Suppose also that
$( Re(f(z)) )^2 \ne ( Im(f(z)) ) $for all $|z|<1$ and all $f \in \mathcal{F}$.   
It follows from the Fundamental Normality test   that $\mathcal{F}$ is a normal family. 
Is there a for elementary way of showing $\mathcal{F}$ is a normal family without invoking the Fundamental Normality Test?

REPLY [5 votes]: Actually, you can avoid uniformisation, and just use the following simple normality criterion:
Theorem (Montel). If there is a nonempty open set $U$ such that all functions in $\mathcal{F}$ omit all points of $U$, then $\mathcal{F}$ is normal.
(By postcomposing with a M\"obius transformation, we can assume that the family $\mathcal{F}$ is uniformly bounded, and the claim follows e.g. from Marty's theorem.)
Since each of your functions omits one of the two complementary domains of the mentioned parabola, your family is the union of two normal families, and hence itself normal.