The third extremal problem (due to Mori) presented in the Ahlfors' lectures on quasiconformal mappings is the following:
Let $G$ be a doubly connected region in $\mathbb{C}$, and denote by $C_1$ the bounded component of $\mathbb{C}\backslash G$ and by $C_2$ the unbounded component. Find the largest value of the conformal module M(G) such that $diam(C_1\cap {|z|\leq1})\geq\lambda$ and $C_2$ contains the origin.
While the author presented the solution in p.37, I did not understand that in the beginning he said
open up the plane by $\zeta=\sqrt z$.
and what he meant by
$\hat{G}$ is the region between $C_{1}^{-}$ and $C_{1}^{+}$.
where he obtained the inequality $M(G)\leq \frac{1}{2}M(\hat{G})$.
It seems to me that this is related to the Riemann surface defined by the square root. However, I haven't got a right picture of how should this region lie on the Riemann surface. Could any one please explain Ahlfors' solution here?
Also, any recommend reading on how to understand and use of this notion of conformal module?
A lot of thanks!