Lipschitz continuity of quasiconformal mappings and of the solutions to second order elliptic PDE with respect to the distance ratio metric

The main aim of this paper is to study the Lipschitz continuity of certain $(K,K^{\mathrm{\prime }})$-quasiconformal mappings with respect to the distance ratio metric, and the Lipschitz continuity of the solution of a quasilinear differential equation with respect to the distance ratio metric.