One property of Schwarzian derivative (Q2725236)

Let \(D\) be a domain in the extended complex plane \(\overline{\mathbb{C}}\) with a quasiconformal boundary and \(A(D)\) be the set of analytic functions \(f: D \rightarrow \overline\mathbb{C}\). A functional \(I(f)\) is called admissible if there exists a positive number \(t_1\) such that \(f\in A(D)\) and \(I(f)\leq t_1\) implies \(f\) is univalent in \(D\). An admissible function \(I\) is called regular if there exists a positive \(t_n>\sup t_1\) such that \(f\in A(D)\) and \(I(f) \leq t_n\) implies \(f\) is no more than \(n\)-valent; otherwise \(I\) is called catastropic. The author has proved that the Schwarzian derivative \(S(f,z)=|(f''/f')'-(f''/f')^2/2|\) is catastropic in \(B_k=\{z\); \(|\arg z|< k\pi/2\}\) for \(0 < k < 2\).