Definition 1   Let $G \subset \mathbb{C}$ be a region and let $\gamma$ be a connected subset of $\partial G$ (boundary of $G$ , then $\gamma$ is a free analytic boundary arc of $G$ if for every point $\zeta \in \gamma$ there is a neighbourhood $U$ of $\zeta$ and a conformal equivalence $h \colon {\mathbb{D}} \to U$ (where ${\mathbb{D}}$ is the unit disc) such that $h(0) = \zeta$ $h(-1,1) = \gamma \cap U$ and $h({\mathbb{D}}_+) = G \cap U$ (where ${\mathbb{D}}_+$ is all the points in the unit disc with non-negative imaginary part).