Pointwise convergence of sequences of holomorphic functions to holomorphic functions

Let $(f_{n})_{n \in \mathbb{N}}$ be a sequence of holomorphic functions on the open unit disc $D$ in $\mathbb{C}$, and suppose that this sequence converges pointwise to a function $f$. By Osgood's theorem one can conclude then that there is an open and dense subset $V$ of the disc, so that the function $f$ is holomorphic there and that the convergence of the sequence is locally uniform on $V$. If we further suppose that the limit function $f$ is also holomorphic on the entire disc $D$, is it then possible to conclude that the sequence $(f_{n})_{n \in \mathbb{N}}$ converges locally uniformly to $f$ on $D$?

Solution 1:

Let $K_n$ be the union of $\{ 0 \}$, the line segment $[2/n, 1]$ and the compact set $\{ z \in \mathbb{C} : |z| \le 1 \text{ and } \operatorname{dist}(z,\mathbb{R}_+) \ge 1/n \}$. Let $L_n = K_n \cup \{ 1/n \}$. (You should probably draw a picture...) Note that the complement of $L_n$ is connected.

Let $f_n$ be a holomorphic function on a (non-connected) neighborhood of $L_n$ such that $f_n = 0$ on $K_n$ and $f_n(1/n) = 1$. By Runge's theorem, there is a polynomial $p_n$ such that $|p_n - f_n| < 1/n$ on $L_n$. It's not hard to see that $p_n \to 0$ (pointwise) on the unit disc, but the convergence is not uniform on any neighborhood of $0$ since $p_n(1/n) \approx 1$. (In fact the convergence is not uniform near any point on the positive real axis.)