I do not know if the following is true: A bounded region $U$ is simply connected if and only if for any holomorphic function $f$ on $U$ and any closed curve $\gamma\subseteq{U}$ we have $\int_{\gamma}f(z)dz=0$. The "only if" part follows from the homotopy of curves theorem, but I don't know how to begin with the "if" part, intuitively I think it is because the integral condition implies that $U$ has no "holes", and this implies $U$ is biholomorphic to an open ball.
Or to prove that the interior of a closed curve is simply connected.
Thanks