Quasiconformal harmonic mappings and the curvature of the boundary (Q333852)

Let \(w\) be a harmonic diffeomorphism of the unit disk onto a convex Jordan domain of the form NEWLINE\[NEWLINEw(re^{i\tau})=P[F](re^{i\tau})=\int_0^{2\pi} P(r, t-\tau)F(e^{it})\, dt,NEWLINE\]NEWLINE where \(P (r,t)\) is the Poisson kernel, \(F\) is a homeomorphism of the unit circle onto a Jordan curve \(\gamma\). The Rado-Chouquet-Kneser theorem [\textit{T. Radó}, Jahresber. Dtsch. Math.-Ver. 35, 123--124 (1926; JFM 52.0498.03); \textit{G. Choquet}, Bull. Sci. Math., II. Sér. 69, 156--165 (1945; Zbl 0063.00851)] asserts that if \(\gamma\) is convex and \(F\) is a homeomorphism, then \(w=P[F]\) is a diffeomorphism.NEWLINENEWLINEOne of the main results of the paper is the following theorem.NEWLINENEWLINE{Theorem}. Let \(w\) be a harmonic diffeomorphism of the unit disk onto a convex Jordan domain \(D=\mathrm{int\,} \gamma \in C^2\) of the form \(w(re^{i\tau})=P[F](re^{i\tau})\). If \(F\) is \((m,M)\)-bi-Lipschitz, i.e., \(m=\min _{\tau \in [0,2\pi]} |F'(\tau)|\) and \(M=\max _{\tau \in [0,2\pi]} |F'(\tau)|\), then NEWLINE\[NEWLINE \varkappa_0 m^3 \leq J_w(e^{i\tau})\leq \varkappa_1 M^3,NEWLINE\]NEWLINE where \(J_w\) is the Jacobian, \(\varkappa_0 \) and \(\varkappa_1\) are the minimal and maximal curvature of \(\gamma\), respectively.NEWLINENEWLINEUsing these estimates and related results, some asymptotically sharp estimates for the constant of quasiconformality for harmonic diffeomorphisms of the unit disk onto a convex Jordan domain are found.