Commentarii Mathematici Helvetici


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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.┬ÂThis 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