Asymptotic behaviour of the conformal representation of a Jordan domain with a small hole in Schauder spaces

We consider a suitably normalized Riemann map $g{[\mbox{\boldmath$\zeta$}]}$ of the plane annulus $\mathbb A\left(r\left[\mbox{\boldmath$\zeta$}\right],1\right)\equiv \left\{z\in\mathbb C:\, r\left[\mbox{\boldmath$\zeta$}\right]<|z|<1\right\}$ to the plane annular domain $\mathbb A[\mbox{\boldmath$\zeta$}]$ enclosed by the pair of Jordan curves $\mbox{\boldmath$\zeta$}\equiv (\zeta^{i}, \zeta^{o})$. Here $\zeta^{i}$ is of the form $w+\epsilon\xi$, where $w$ is a point in the domain enclosed by the external curve $\zeta^{o}$, and $\xi$ is a curve enclosing $0$, and $\epsilon>0$ is a real parameter. We analyze the behaviour of the corresponding $g{[\mbox{\boldmath$\zeta$}]}$ as $\epsilon$ tends to $0$. More precisely, we show that the nonlinear operator which takes the quadruple $(w,\epsilon,\xi,\zeta^{o})$ to the corresponding triple of functions $\left(r^{-1}[\mbox{\boldmath$\zeta$}]g{[\mbox{\boldmath$\zeta$}]}^{(-1)}\circ\zeta^{i}, g{[\mbox{\boldmath$\zeta$}]}^{(-1)}\circ\zeta^{o},\epsilon^{-1}r[\mbox{\boldmath$\zeta$}] \right)$ can be continued real analytically around a singular quadruple $(w,0,\xi,\zeta^{o})$ corresponding to an annular domain with an interior degenerate curve. As a corollary, one can deduce information on the behaviour of the relative capacity of the domain enclosed by $\zeta^{i}=w+\epsilon\xi$ with respect to that enclosed by $\zeta^{o}$ as $\epsilon$ tends to $0$.