Entire functions and polynomials

1. Mar 18, 2009

WWGD

Hi: I am trying to show:

If f is analytic in C (i.e., f is entire) and : |z|>1 implies |f(z)|>1.
Prove that f(z) is a polynomial.

I have tried using the fact that f(z)=Suma_nz^n (Taylor series) valid in the whole of C,

and derive a contradiction assuming |f(z)|>1 for |z|>1 . I checked that the "standard"

non-poly. functions, e.g., e^z, sinz , were not counterexamples (they're not).

Obviously, if this result were false for f(z) non-polynomial, then f(z) would need to

have all its zeros( if any) inside of S^1, which is not impossible), and any poly.

satisfying above condition should also have its zeros inside of S^1.

I have considered using the fact that any f analytic has a singularity at oo (follows

from Liouville's), but this only works if the singularity is an essential singularity; is

this true, that any non-poly. analytic function has an essential singularity at oo ?.

Any other ideasa?

2. Mar 19, 2009

yyat

The important fact in this case, which follows from Casorati-Weierstrass, is that the singularity at $$\infty$$ is not an essential one (so it is a pole or removable). A pole at $$\infty$$ can be "removed" by adding a suitable polynomial, just like a pole in $$\mathbb{C}$$ can be removed by substracting the principal part. The result then follows from Liouville.

btw, the only meromorphic functions $$\bar{\mathbb{C}} \to \bar{\mathbb{C}}$$ are rational functions.

3. Mar 19, 2009

WWGD

Yes, thanks, I realized it right after posting it.

Are you refering to the Mobius maps (az+b)/(cz+d) ?. I don't see how that
relates here.

4. Mar 19, 2009

yyat

By "rational map" I mean any quotient of polynomials (the Möbius transformations are the bijective ones). My statement implies that the answer to

is yes.

5. Mar 19, 2009

matt grime

Please don't assume this will lead to the right answer, but the first thing that occurs to me is that it might be easier to think of a different function. Define

g(z) = 1/f(1/z)

this is a function that maps the unit disk into itself.

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