Quasiconformal extension of harmonic mappings in the plane (Q2860887)

The authors consider sense-preserving harmonic mappings \(f: \mathbb{D} \to \mathbb{C}\) with the canonical representation \(f= h+ \overline{g}\) where \(f\) and \(g\) are analytic functions of the unit disk \(\mathbb{D}\) and \(g(0) =0\). For a locally univalent \(f\) the (second) complex dilatation of \(f\) is defined as \(\omega=g'/h'\). In another paper [J. Geom. Anal. 25, No. 1, 64--91 (2015; Zbl 1308.31001)] the authors introduced the pre-Schwarzian derivative of a locally univalent harmonic mapping \(f\) as NEWLINE\[NEWLINE P_f= \frac{h''}{h'}-\frac{\overline{\omega} \omega'}{1- |\omega|^2} NEWLINE\]NEWLINE and they proved that NEWLINE\[NEWLINE |P_f (z)|(1-|z|^2) + |\omega^*(z)| \leq 1\,, \quad \omega^*(z)= \frac{\omega'(z)(1-|z|^2)}{1-|\omega(z)|^2}\,, NEWLINE\]NEWLINE for all \(|z|<1\), implies that \(f\) is univalent in \(\mathbb{D}\).NEWLINENEWLINEHere the authors prove the following theorem which is analogous to a result of \textit{J. Becker} [J. Reine Angew. Math. 255, 23--43 (1972; Zbl 0239.30015)] and \textit{L. V. Ahlfors} [in: Discontin. Groups Riemann Surf., Proc. 1973 Conf. Univ. Maryland, 23--29 (1974; Zbl 0324.30034)].NEWLINENEWLINETheorem. Let \(f=h+ \overline{g}\) be a sense-preserving harmonic mapping in the unit disk. If for all \(z \in \mathbb{D}\) NEWLINE\[NEWLINE |P_f (z)|(1-|z|^2) + |\omega^*(z)| \leq k<1, NEWLINE\]NEWLINE then \(f\) has a continuous injective extension \(\tilde{f} \) to \(\overline{\mathbb{D}}\). Moreover, the function NEWLINE\[NEWLINEF(z)=\begin{cases} \tilde{f}(z) & |z|\leq 1,\\ f(1/\overline{z})+ U(1/\overline{z}) & |z|>1, \end{cases}NEWLINE\]NEWLINE is a homeomorphic extension of \(f\) to the whole plane onto itself. The function \(U\) is defined as follows NEWLINE\[NEWLINE U(z) = \frac{h'(z)(1-|z|^2)}{\overline{z}} + \frac{\overline{g'(z)} (1-|z|^2)}{z}\,,\quad z\in \mathbb{D}\setminus \{0 \}. NEWLINE\]