On Nehari disks and the inner radius

Abstract

Let D be a simply connected plane domain and B the unit disk. The inner radius of D, $\sigma (D)$ , is defined by $\sigma (\mathrm{D})=\sup \left\{a:a\ge 0,\Vert {\mathrm{S}}_f{\Vert }_{\mathrm{D}}\le a\mathrm{implies}f\mathrm{is}\mathrm{univalent}\mathrm{in}\mathrm{D}\right\}$ . Here Sf is the Schwarzian derivative of f, ${\rho }_{\mathrm{D}}$ the hyperbolic density on D and $\Vert {\mathrm{S}}_f{\Vert }_{\mathrm{D}}={\sup }_{z\in \mathrm{D}}\mid {\mathrm{S}}_f(z)\mid {\rho }_{\mathrm{D}}^{-2}(z)$ . Domains for which the value of $\sigma (\mathrm{D})$ is known include disks, angular sectors and regular polygons, as well as certain classes of rectangles and equiangular hexagons. All of the mentioned domains except non-convex angular sectors have an interesting property in common, namely that $\sigma (\mathrm{D})=2-\Vert {\mathrm{S}}_h{\Vert }_{\mathrm{B}}$ , where h maps B conformally onto D. Because of the importance of this property for computing $\sigma (\mathrm{D})$ , we say that D is a Nehari disk if $\sigma (\mathrm{D})=2-\Vert {\mathrm{S}}_h{\Vert }_{\mathrm{B}}$ holds.¶This paper is devoted to the problem of characterizing Nehari disks. We give a necessary and sufficient condition for a domain to be a Nehari disk provided it is a regulated domain with convex corners.