I am studying for a qualifying exam on Gamelin's Complex Analysis Chapters 1-11 and am stuck on the following past exam question:
Let $\phi(n): \mathbb{N} \to \mathbb{R}$ such that $\lim_{n\to \infty} \frac{\phi(n)}{n}=0$. Let $F$ be the family of analytic functions $f$ defined on neighborhoods of $0$ such that the coefficients of the Taylor series $f(z) = \sum_{n=0}^{\infty} a_nz^n$ satisfy $|a_n| \le e^{\phi (n)}$. Show that the Taylor series of $f \in F$ converges on the unit disk $D$ and the family $F$ is normal on $D$.
My goal was to compute the radius of convergence to be 1. I attempted to look at $f'(z)$ since that has the same radius of convergence as $f$ and was hoping to get a bound on $n|a_n|$ using $\phi$.
The family being normal should follow after showing $f$ converges since $f$ is the uniform limit of continuous functions and thus continuous and bounded on compact sets. Montel's theorem shows every sequence has a normally convergent subsequence on $D$.
Could someone give me a hint on how to show $f$ converges on $D$?