Let $f_n:\mathbb R \rightarrow \mathbb R$ be a sequence of functions such thateach $f_n$ can be extended to an entire function $g_n: \mathbb C \rightarrow \mathbb C$.
Let$$\Omega = \{ x + y i \in \mathbb C : x, y \in \mathbb R, |y| \le x^2, |x|\le 1\}.$$
We assume that $g_n$ converges uniformly on $\Omega$.Then it is easy to see that $f_n$ converges to a continuous function $f:[-1, 1] \rightarrow \mathbb R$ on $[-1, 1]$and $f$ is real-analytic on $(-1, 0) \cup (0, 1)$.
Suppose that $f(x)$ is also real-analytic at $x=0$. Then there is a neighborhood $D$ of $0$ in $\mathbb C $ so that $f$ can be extended to an analytic function $F$ on $D$.
My question is:
Does $g_n$ converge to $F$ on $D$?