Mappings connected with the gradient of conformal radius

In this paper we prove the following conformity criterion for the gradient of conformal radius ∇R(D, z) of a convex domain D: the boundary ∂D has to be a circumference. We calculate coefficients K(r) for K(r)-quasiconformal mappings ∇R(D(r), z), D(r) ⊂ 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 the domain D.