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?