Problem statement:
Let be an entire function and set . Determine all entire functions such that is a compact normal family on .
Defs: I'm including these as I'm clear on the definition of normal and compact, but unsure of the definition for "compact normal". Please comment if you think I have a definition in error.
Compact: I'm assuming the problem's author implies that we have sequential compactness, i.e., every sequence from has a limit in the set.
Normal: Every sequence from has a subsequence that converges uniformly on any compact subset of .
So compact normal means that any sequence from has a subsequence that converges uniformly on any compact subset of to a function in .
Ideas:
Any function is entire since it is the composition of two entire functions, and .
Examples: The constant functions work trivially.
In the class of non-constant entire functions, polynomials fail, fails.
I don't see how to make the general argument. Liouville's theorem identifies that any non-constant functions are unbounded. So if we can prove must be bounded we're done. Montel's theorem gives us a normal family if we have locally uniformly bounded. This doesn't imply f bounded though.
I'm sure that I'm overlooking something ... thank you for your help.