The Douady-Earle extension of quasihomographies

Quasihomography is a useful notion to represent a sense-preserving automorphism of the unit circle T which admits a quasiconformal extension to the unit disc. For K ≥ 1 let ${\displaystyle A_T\left(K\right)}$ denote the family of all K-quasihomographies of T. With any ${\displaystyle f\in A_T\left(K\right)}$ we associate the Douady-Earle extension ${\displaystyle E_f}$ and give an explicit and asymptotically sharp estimate of the ${\displaystyle L_{\infty }}$ norm of the complex dilatation of ${\displaystyle E_f}$.