Angular derivatives of quasiconformal harmonic maps on the Poincaré disk

Abstract

Let \(S^1\) be the boundary of the open unit disk \(\mathbb {D}\) and let h be a quasisymmetric homeomorphism from the unit circle \(S^1\) onto itself. Let H be the quasiconformal harmonic extension of h to \(\mathbb {D}\) with respect to the Poincare metric. In this paper, it is shown that, if \(h'(\zeta )=\alpha \ne 0\) at \(\zeta \) in \(S^1\), then when \(z\rightarrow \zeta \) in \(\mathbb {D}\) non-tangentially, $$\begin{aligned} \lim _{z\rightarrow \zeta }\frac{H(z)-H(\zeta )}{z-\zeta }=\alpha \end{aligned}$$ and the complex derivatives \(H_z(z)\) and \(H_{\bar{z}}(z)\) approach \(\alpha \) and 0 respectively, i.e., H has an angular derivative \(\alpha \) at \(\zeta \); conversely, if H has a non-tangential derivative \(\alpha \ne 0\) at \(\zeta \), then \(h'(\zeta )=\alpha \) and hence H has an angular derivative \(\alpha \) at \(\zeta \).