Asymptotic behaviour of the Conformal Representation of a Jordan domain with a small hole, and relative capacity

As it is well-known, given a plane Jordan domain with a hole, there exists a unique properly normalized holomorphic homeomorphism of a circular annulus with inner radius depending on the domain and with outer radius one, onto the domain itself. We analyze the behaviour of the inner radius and of the holomorphic homeomorphism, as the hole shrinks to a point. Since the relative capacity of the small hole is related to the inner radius, we deduce information on the limiting behaviour of the capacity.