Commentarii Mathematici Helvetici

Volume 76, Issue 2, 2001, pp. 183–199
DOI: 10.1007/PL00000377

Published online: 2001-06-30

On Nehari disks and the inner radius

L. Miller-Van Wieren[1]

(1) University of Texas at Austin, USA

Let D be a simply connected plane domain and B the unit disk. The inner radius of D, $\sigma (D)$, is defined by $\sigma ({\rm D}) = {\rm sup}\left\{a: a \geq 0, \Vert{\rm S}_{f}\Vert_{\rm D} \le a\,{\rm implies}\,f\, {\rm is\,univalent\,in\,D} \right\}$. Here Sf is the Schwarzian derivative of f, $\rho_{\rm D}$ the hyperbolic density on D and $\Vert{\rm S}_{f}\Vert_{\rm D} = {\rm sup}_{z \in {\rm D}}\mid {\rm S}_{f}(z)\mid \rho_{\rm D}^{-2} (z)$. Domains for which the value of $\sigma {\rm (D)}$ 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 $\sigma {\rm (D)} = 2 - \Vert{\rm S}_h \Vert_{\rm B}$, where h maps B conformally onto D. Because of the importance of this property for computing $\sigma {\rm(D)}$, we say that D is a Nehari disk if $\sigma {\rm (D)} = 2 - \Vert{\rm S}_h \Vert_{\rm B}$ 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.

Keywords: Nehari disk, inner radius, univalence criteria

Miller-Van Wieren L.: On Nehari disks and the inner radius. Comment. Math. Helv. 76 (2001), 183-199. doi: 10.1007/PL00000377