On Mappings Related to the Gradient of the Conformal Radius

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