Mappings connected with the gradient of conformal radius

In this paper we prove the following conformality criterion for the gradient of conformal radius $\nabla R(D,z)$ of a convex domain $D$: the boundary $\partial D$ has to be a circumference. We calculate coefficients $K(r)$ for $K(r)$-quasiconformal mappings $\nabla R(D(r),z)$, $D(r)\subset D$, $0<r<1$, and complete the results obtained by F. G. Avkhadiev and K.-J. Wirths for the structure of boundary elements of quasiconformal mappings of a domain $D$.