Quasi-invariant domain constants
\(\lambda_ \Omega(z)| dz|\) denote the hyperbolic metric with curvature \(-4\) on a hyperbolic region \(\Omega\) in the complex plane \(\mathbb{C}(\mathbb{C}\backslash\Omega\) contains at least two points). \(\delta_ \Omega(z)\) denotes the Euclidean distance from \(z\) to \(\partial\Omega\). A region \(\Omega\) is called uniformly perfect if the domain constant \(c(\Omega):= \inf\{\lambda_ \Omega(z)\delta_ \Omega(z): z\in\Omega\}\) is positiv. \textit{B. Osgood} [Ind. Univ. J. 31, 449-461 (1982; Zbl 0503.30014)] showed that if two hyperbolic regions \(\Omega\) and \(\Delta\) are conformally equivalent, then \(B^{-1}\leq c(\Omega)/c(\Delta)\leq B\) for a constant \(B\leq 6\). The authors improve this bound to \(2\leq B\leq 2.84\dots\) and conjecture that \(B= 2\). Further, the authors study four other domain constants and show that these can be used to characterize uniformly perfect hyperbolic regions, too.
scientific article; zbMATH DE number 203269
scientific article