Solving a Complex Problem: Proving a Function Reduces to a Polynomial

[AlbertEinstein]

Hi all.

The problem is "Prove that a function which is analytic in the whole plane and satisfies an inequality |f(z)| < |z|^n for some n and sufficiently large |z| reduces to a polynomial." I do not understand what I need to show that the function reduces to a polynomial.

Any help will be appreciated.
Thanks.

 

[Eynstone]

A more general theorem holds,that no entire function dominates another.. Check out :
http://en.wikipedia.org/wiki/Liouville's_theorem_(complex_analysis )

 

[some_dude]

Hi all.

The problem is "Prove that a function which is analytic in the whole plane and satisfies an inequality |f(z)| < |z|^n for some n and sufficiently large |z| reduces to a polynomial." I do not understand what I need to show that the function reduces to a polynomial.

Any help will be appreciated.
Thanks.

Write $$f(z) = \sum_{m=0}^{\infty} a_m\ z^{m}$$

(which you can do since it's entire).

Show that |f(z)| < |z|^n implies for some natural number N,

$$a_m = 0$$

for any m >= N.

 

[ampat]

You can apply Cauchy Integral Formula to show that f^(n+m)(a) = 0 for any complex number a in C and m>=1.