On extremality and unique extremality of Teichmüller mappings (Q1899648)

Let \(D\) be the open unit disk and \(z_1, \dots, z_m\) be points of \(\partial D\) such that excising an arbitrary neighborhood \(D_i\) of each \(z_i\) from \(D\) it results in a region of finite \(\varphi\)-area. Let \(\varphi\) be a holomorphic quadratic differential in \(D\) and satisfy the following conditions: \[ \left |{\sqrt {\varphi (z)} \over \log^{t_i} (z_i - z)} - {\alpha_i \over z_i - z} \right |= O(1), \quad z \to z_i, \] where \(\{\alpha_i\}\) are complex numbers and \(\{t_i\}\) are real numbers. The main result of this paper is that the Teichmüller mapping of \(D\) associated with \(\varphi\) is uniquely extremal if and only if \(t_i \leq 1/2\), \(i = 1, \dots, m\). This result is an improvement of a result of \textit{G. C. Sethares} [Comment. Math. Helv. 43, 98-119 (1968; Zbl 0162.38002)].