Let $G$ be a connected open set of the complex plane, let $F \subset G$ be a relatively closed set and $D$ be a proper dense set in $F$. Let $g_n : G \rightarrow \mathbb{C}$ be a sequence of analytic functions. Assume $g_n$ converges uniformly in $G \smallsetminus F$ and that converges pointwisely in $D$. Can we deduce that $g_n$ converges uniformly in $G$? Can we estate a counterexample?
Thank you!