# Showing a sequence/series of holomorphic functions is holomorphic

1. Nov 16, 2012

### TopCat

1. The problem statement, all variables and given/known data
1) Suppose a sequence of functions $f_n \in H(\Omega)\cap C(\overline{\Omega})$ converges uniformly on $\partial \Omega$. Prove that the sequence $f_n$ converges uniformly on $\overline{\Omega}$ to a function $f \in H(\Omega)\cap C(\overline{\Omega})$.

2)Suppose $f \in H(\mathbb{D})$ and $f(0)=0$. Show that $\sum_{n=1}^{\infty}f(z^n)$ defines a holomorphic function on $\mathbb{D}$.

2. Relevant equations

3. The attempt at a solution

1) I think I need to use the uniform convergence on the boundary to find a common M>0 such that each $|f_n| < M$ by Maximum Modulus, from which it follows that I have uniform convergence on the disk.

2) So f is analytic and has no constant term. I'm not quite sure how that helps or what I should even think about applying to this problem.