Is this family normal?

Question

Suppose that $R$ is a region, $a \in R$ and $\mathcal{F}=\{f \in H(R) : \mbox{$|f(a)|<1$ and $0,1 \notin f(R)$}\}$. Is this family normal? I guess it is, but I fail to find the reasoning behind.

Answer 1

Montel's normality criterion states that a family of holomorphic functions which all omit the same two values, is normal.

In this context, “normality” means that every sequence in $\cal F$ has a subsequence which either

• converges locally uniformly to a holomorphic function, or
• converges locally uniformly to $\infty$.

(See e.g. Daniel's answer at The functions $\{f_n(x) = n\}$ are analytic and each miss the points $-2, -3$. But, they are not a normal family. So what am I missing. Thanks.).

Your additional criterion $|f(a)| < 1$ for all $f \in \cal F$ ensures that the second case can not occur.