Properties of Nehari Disks

• Published in: Sarajevo journal of mathematics Vol. 4; no. 1; pp. 49 - 59
• Main Author: Miller-Van Wieren, Leila
• Format: Journal Article
• Language: English
• Published: 11.06.2024
• ISSN: 1840-0655; 2233-1964
• DOI: 10.5644/SJM.04.1.05

Let $D$ be a simply connected plane domain and let $B$ be the unit disk. The inner radius of $D$, $\sigma (D)$ is defined by $\sigma (D) = \sup \{a:a \geq 0,||S_f||_{D} \leq a$ implies $f$ is univalent in $D$ . Here $S_f$ is the Schwarzian derivative of $f$, $ \rho_{D}$ the hyperbolic density on $D$ and $ || S_{f} ||_{D} = \sup_{z \in D} |S_{f} (z)|\rho_{D}^{-2} (z) $. Domains for which the value of $ \sigma (D)$ is known include disks, angular sectors and regular polygons as well as certain classes of rectangles and equiangular hexagons. When the inner radius for the above-mentioned domains, except non convex angular sectors, is computed it is seen that $ \sigma (D) = 2- || S_h ||_{B}$, where $ h:B \longrightarrow D $ is the Riemann mapping and $B$ the unit disk, a fact that yields a convenient method for computing $\sigma(D)$. We introduce the name Nehari disks for domains with the above property. In this paper we generalize some results by Gehring, Pommerenke, Ahlfors and Minda that were proved in the unit disk, to analogous results for Nehari disks.   2000 Mathematics Subject Classification. Primary 30C55; Secondary 30C20