On mappings related to the gradient of the conformal radius

We establish a criterion for the gradient ?R(D, z) of the conformal radius of a convex domain D to be conformal: the boundary ?D must be a circle. We obtain estimates for the coefficients K(r) for the K(r)-quasiconformal mappings ?R(D, z), D(r) ? D, 0 < r < 1, and supplement the results of Avkhadiev and Wirths concerning the structure of the boundary under diffeomorphic mappings of the domain D.