Assume that $f$ is an entire function with $f(0)=0$. Consider the family of functions {$f_{n}$} where $f_{n}$ is the $n^{th}$ iterate of $f$, i.e. $f_{n}=f \circ f \circ \ldots \circ f$ ($n$ times).
I'm trying to show that if |f^{'}(0)|<1, then there is an open set $U$ containing zero so that the family {$f_{n}$} is normal on $U$. I have been able to show |f_{n}^{'}(0)|<1 and not much more.
Lastly if we take |f^{'}(0)|>1 then there does not exist an open set $U$ containing zero such that {$f_{n}$} is normal, but I'm not sure how to show this either. I'm pretty sure we can do this problem without relying on Fundamental Normality Test. Thank you for the help and guidance.