# Liouville's theorem (extended) proof

1. Mar 18, 2012

### d2j2003

1. The problem statement, all variables and given/known data

f is an entire function with pos. constants A and m such that |f(z)| ≤ A|z|$^{m}$ for all z: |z|≥R$_{0}$

Show that f is polynomial of degree m or less

2. Relevant equations

Cauchy estimates need to be used here

|f$^{n}$(z$_{0}$)|≤$\frac{n!}{r^{n}}$max$_{z-z_{0}=r}$|f(z)| , n=0,1,2,3,....

3. The attempt at a solution
I was thinking that the right side of the inequality would just be a constant and could be treated as such.. meaning that taking the derivative of the left side would eventually result in a constant and then 0... but i'm not sure how to show that you have to take the derivative m+1 times...

2. Mar 18, 2012

### Dick

If f(z) is entire then it has a power series expansion at z=0. f(z)=a0+a1*z+a2*z^2+... $f^n(0)$ is related to an. Use your Cauchy estimate on that.

3. Mar 19, 2012

### d2j2003

ok,thats what I did and I got the answer, Thank you