Let $\Omega$ be a simply connected open set in the complex plane and $\gamma$ be a simple path inside $\Omega$. Suppose $f\_n$ is a sequence of holomorphic functions converging pointwise to 0 on $\gamma$. Does it imply that $f\_n$ converges pointwise on the region enclosed by $\gamma$?



Does pointwise convergence of holomorphic functions on the boundary imply pointwise convergence in the interior?
For a counterexample, let $\gamma$ be the unit circle. Let $$A\_n = \{z \in \gamma:\; \text{Im}(z) \in [-1,0] \cup [1/n, 1]\}$$ By Runge's theorem there is a polynomial $f\_n$ such that $|f\_n| < 1/n$ on $A\_n$ but $f\_n(0) = (-1)^n$. We then have
$f\_n \to 0$ pointwise on $\gamma$ but $f\_n(0)$ does not converge.


