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.