On a result by Clunie and Sheil-Small

In 1984 J. Clunie and T. Sheil-Small proved ([2, Corollary 5.8]) that for any complex-valued and sense-preserving injective harmonic mapping $F$ in the unit disk $\mathbb{D}$, if $F(\mathbb{D})$ is a convex domain, then the inequality $|G(z_2)-G(z_1)|<|H(z_2)-H(z_1)|$ holds for all distinct points $z_1,z_2\in \mathbb{D}$. Here $H$ and $G$ are holomorphic mappings in $\mathbb{D}$ determined by $F=H+\bar{G}$, up to a constant function. We extend this inequality by replacing the unit disk by an arbitrary nonempty domain $\mathrm{\Omega }$ in $\mathbb{C}$ and improve it provided $F$ is additionally a quasiconformal mapping in $\mathrm{\Omega }$.