Using Liouville's Theorem to Show that Bounded Entire Functions are Polynomials

[shebbbbo]

Let f: ℂ→ ℂ be an entire function. If there is some nonnegative integer m and positive constants M,R such that

|f(z)| ≤ M|z|^m, for all z such that |z|≥ R,

show that f is a polynomial of degree less that or equal to m.


im really lost on this question. i feel like because there is an inequality sign that i may have to use the ML inequality but I've tried that and i didnt get very far? am i going in the right direction?

any help or hints would be appreciated :-)

thanks

 

[Bacle]

It seems like it may be an application of Liouville. Every time you have entire and bounded together, consider Liouville's theorem.

Maybe you can argue that for balls of fixed radius 1,2,3,.. |f| is bounded if the degree of the poly. is ≥ m, by , e.g., consider the Taylor expansion for f , which is global in ℂ , but let me think about it some more.