On mappings related to the gradient of the conformal radius

Abstract

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. © Pleiades Publishing, Ltd., 2010.