On Nehari disks and the inner radius

  • L. Miller-Van Wieren

    University of Texas at Austin, USA


Let D be a simply connected plane domain and B the unit disk. The inner radius of D, σ(D)\sigma (D) , is defined by σ(D)=sup{a:a0,SfDaimpliesfisunivalentinD}\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, ρD\rho_{\rm D} the hyperbolic density on D and SfD=supzDSf(z)ρD2(z)\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 σ(D)\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 σ(D)=2ShB\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 σ(D)\sigma {\rm(D)} , we say that D is a Nehari disk if σ(D)=2ShB\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.

Cite this article

L. Miller-Van Wieren, On Nehari disks and the inner radius. Comment. Math. Helv. 76 (2001), no. 2, pp. 183–199

DOI 10.1007/PL00000377