On Nehari disks and the inner radius (Q5943647)

Let \(D\) be a simply connected domain in the complex plane \(\mathbb C\) and \(B\) the unit disc in \(\mathbb C\). For \(f\) analytic in \(D\), \(S_{f}(z)\) denotes the Schwarzian derivative \((f''/f')'(z)-{1\over 2}(f''/f')^2(z)\). Let \(\rho_{D}\) denote the hyperbolic density on \(D\) and for a locally univalent function in \(D\), we define the hyperbolic norm of \(S_{f}\) with respect to \(D\) by \(\|S_f\|= \sup_{z\in D}|S_f(z)|\rho_{D}^{-2}(z)\). We define the inner radius of \(D\), denoted by \(\sigma(D)\), by \(\sigma(D)=\sup \{a: a\geq 0, \|S_{f}\|_{D}\leq a\) implies \(f\) is univalent in \(D\}\). It is well known that \(\sigma(B)=2\). Later Lehtinen showed that \(\sigma(D)\leq 2 \) for all simply connected domains in the extended complex plane \(\overline{\mathbb C}\). In addition domains for which the value of \(\sigma(D)\) is known include disks, angular sectors, and regular polygons, as well as certain classes of rectangles and equiangular hexagons. The computations of \(\sigma(D)\) for some domains can be based on the understanding of the behaviour of the Riemann mapping \(h(z)\) which carries B conformally onto \(D\). All of the above mentioned domains except non-convex angular sectors have an interesting property in common, namely \(\sigma(D)=2-\|S_{h}\|_{B}\), where \(h\) maps \(B\) conformally onto \(D\). We say \(D\) a Nehari disk if \(\sigma(D) =2-\|S_{h}\|_{B}\) holds. In this paper the author establishes a necessary and sufficient condition for a domain to be a Nehari disk, provided it is a regulated domain with convex corners. More precisely, one of the main results is the following: ``Suppose that \(D\) is a regulated domain with convex corners and that \(h\) maps \(B\) conformally onto \(D\). Then, \(D\) is a Nehari disk if and only if \(\|S_h\|_B = \limsup_{|z|\rightarrow 1}|S_h(z)|(1-|z|^2)^2\).'' The proof of this nice result relies on several preliminary facts and two lemmas that describe the behaviour of the Schwarzian derivative of the mapping \(h\) near the boundary \(\partial B\).