Different aspects of the boundary value problem for quasiconformal mappings and Teichmüller spaces are expressed in a unified form by the use of the trace and extension operators. Moreover, some new results on harmonic and quasiconformal extensions are included.

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 D, if F(D) is a convex domain, then the inequality |G(z2)− G(z1)| < |H(z2) − H(z1)| holds for all distinct points z1, z2∈ D. Here H and G are holomorphic mappings in D determined by F = H + Ḡ, up to a constant function. We extend this inequality by replacing the unit disk by an arbitrary nonempty domain Ω in ℂ and improve it provided F...

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 ${A}_{T}\left(K\right)$ denote the family of all K-quasihomographies of T. With any $f\in {A}_{T}\left(K\right)$ we associate the Douady-Earle extension ${E}_{f}$ and give an explicit and asymptotically sharp estimate of the ${L}_{\infty}$ norm of the complex dilatation of ${E}_{f}$.

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 $F\left(\mathbb{D}\right)$ is a convex domain, then the inequality $|G\left({z}_{2}\right)-G\left({z}_{1}\right)|<|H\left({z}_{2}\right)-H\left({z}_{1}\right)|$ holds for all distinct points ${z}_{1},{z}_{2}\in \mathbb{D}$. Here $H$ and $G$ are holomorphic mappings in $\mathbb{D}$ determined by $F=H+\overline{G}$, up to a constant function. We extend this inequality by replacing the unit disk by an arbitrary nonempty domain $\Omega $ in $\u2102$ and improve it provided $F$ is additionally a quasiconformal mapping in $\Omega $.

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