On quasiconformal selfmappings of the unit disk and elliptic PDE in the plane

We prove the following theorem: if $w$ is a quasiconformal mapping of the unit disk onto itself satisfying elliptic partial differential inequality $|L[w]|\le \mathcal{B}|\nabla w|^2+\Gamma$, then $w$ is Lipschitz continuous. This {result} extends some recent results, where instead of an elliptic differential operator is {only} considered {the} Laplace operator.