Internal point transformed in an external one?

Question

Let $f \colon \Omega \to \mathbb{C} $ be an analytic function over a connected open subset $\Omega$ of $\mathbb{C}$ and let $\gamma$ a rectifiable closed curve in $\Omega$. If $a$ is a point which is in the interior of $\gamma$, can $f(a)$ be outside the region enclosed by the transformed curve $f \circ \gamma$ ?