So, I want to prove that a family of functions, $f_n$, which is univalent on the unit disk for all $n$ where $f_n(0)=0$ and $f_n'(0)=1$ is a normal family.
So far, and using "Fundamental Normality Test", this is what I have: If you reduce the disk to a subdisk not containing the origin, this family omits $-1$ (by construction) and $0$. Thus the family is a normal family on this subdisk.
Now I should somehow extend this to the unit disk by the maximum principle, and is where I am having trouble.
So my real question is, "how do I extend the normality of the subdisk to the unit disk?"
Why must $f_n$ omit the value $-1$? (Edit: The OP says that's by the way he constructed the functions...)
We can do it without that. The Schwarz Lemma applied to the inverse shows there must be some value of modulus $1$ omitted by $f_n$. So there exist $\alpha_n$ with $|\alpha_n|=1$ such that if $$g_n(z)=f_n(\alpha_nz)/\alpha_n$$then $g_n$ omits the value $-1$. It's enough to show that $(g_n)$ is a normal family.
Say $0<r<R<1$. The functions $g_n$ omit the values $-1$ and $0$ on the annulus $r-\epsilon<|z|< R+\epsilon$. So the restrictions to that annulus form a normal family (note that the annulus is the union of finitely many simply connected open sets). So the $g_n$ are bounded by $M_{r,R}$ on the closed annulus $r\le |z|\le R$; hence they're also bounded by the same constant on the closed disk $|z|\le R$.