Grunsky inequalities and Fredholm spectrum in general domains

We discuss the Fredholm spectrum for general domains and study its applications to conformal and quasi-conformal mappings. In particular, we establish an improvement of the Grunsky inequalities which is valid for general domains. This improvement constitutes an extension of a recent result of Schiffer concerning the sharpening of Grunsky inequalities for the unit disk by a factor smaller than 1, and which is the reciprocal of the least Fredholm eigenvalue of the smooth simply-connected image domain.