Locally uniformly convergence

1. Apr 24, 2013

hedipaldi

1. The problem statement, all variables and given/known data
Let f(z) be holomorphic in the unit disc B(0,1),such thaf f(0)=0.Prove that the series Ʃf(z^n) is locally uniformly convergence in B(0,1).

2. Relevant equations
locally uniformly convergence:if it is uniformly convergence in a neibourhood of each point of B(0,1).

3. The attempt at a solution
At each point z in the disc there is an r such that the closure of B(z,r) is contained in B(0,1).I tried to use the cauchy formula for finding bounds on the derivatives so that the required series will be uniformly convergence in B(z,r),from Weierstrass M-test.I proved that it uniformly converges in B(0,r) for certain r.

Last edited: Apr 24, 2013
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