# On Nehari disks and the inner radius

### L. Miller-Van Wieren

University of Texas at Austin, USA

## Abstract

Let D be a simply connected plane domain and B the unit disk. The inner radius of D, $σ(D)$ , is defined by $σ(D)=sup{a:a≥0,∥S_{f}∥_{D}≤aimpliesfisunivalentinD}$ . Here Sf is the Schwarzian derivative of f, $ρ_{D}$ the hyperbolic density on D and $∥S_{f}∥_{D}=sup_{z∈D}∣S_{f}(z)∣ρ_{D}(z)$ . Domains for which the value of $σ(D)$ 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 $σ(D)=2−∥S_{h}∥_{B}$ , where h maps B conformally onto D. Because of the importance of this property for computing $σ(D)$ , we say that D is a Nehari disk if $σ(D)=2−∥S_{h}∥_{B}$ 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