On certain quasiconformal and elliptic mappings

Let $\bar{\mathbb{D}}$ be the closure of the unit disk $\mathbb{D}$ in the complex plane $\mathbb{C}$ and $g$ be a continuous function in $\bar{\mathbb{D}}$. In this paper, we discuss some characterizations of elliptic mappings $f$ satisfying the Poisson’s equation $\mathrm{\Delta }f=g$ in $\mathbb{D}$, and then establish some sharp distortion theorems on elliptic mappings with the finite perimeter and the finite radial length, respectively. The obtained results are the extension of the corresponding classical results.