I've been looking in the literature for a reference on the following, but without success :
Let $\Omega_1$ be a bounded finitely connected domain in the complex plane. Suppose that the boundary of $\Omega_1$ consists of pairwise disjoint piecewise analytic Jordan curves.
A repeated application of the Riemann mapping theorem gives a biholomorphic mapping $F: \Omega_1 \rightarrow \Omega_2$, where $\Omega_2$ is a bounded finitely connected domain whose boundary consists of pairwise disjoint analytic Jordan curves. See for example Ahlfors, 3rd edition, p.252.
It is well known that the Riemann map of a Jordan domain extends to a homeomorphism on the closure. Furthermore, it also extends analytically across any analytic arc on the boundary (Ahlfors p.235).
Since $F$ is a composition of Riemann maps, it extends to a continuous function in $\overline{\Omega_1}$, the closure of $\Omega_1$, and also analytically across the boundary $\partial \Omega_1$, except maybe at the "corners". My question is the following :
Is it true that F' has an analytic square root in $\Omega_1$, i.e. F'=G^2 where $G$ is analytic in $\Omega_1$ and continuous in $\overline{\Omega_1}$, except maybe at the "corners" of $\partial \Omega_1$?
Any reference on this and biholomorphic mappings between multiply connected domains is welcome.
Thank you, Malik