An Exercise about Luarent Series

May 2010
3
0
It's an exercise in Luarent Series.How to prove it?

f(Z)=1/z+\(\displaystyle sum_{n=1}^{infinity}a_{n}z^n\) is univalent and holomorphic in domain B(0,1)\{1}. Prove that
\(\displaystyle sum_{n=1}^{infinity}n\left|a_{n}\right|^2\) <=1

Is it called Area Theorem?
 

Bruno J.

MHF Hall of Honor
Jun 2009
1,266
498
Canada
You probably mean \(\displaystyle B(0,1)-\{0\}\) (the punctured unit disc)?

This is a fairly standard proof which can be found easily. Check back if you don't understand the proof!
 
May 2010
3
0
I didn't find the proof actually. But the professor has gave the proof after I handed in my homework(Crying)