On the images of horodisks under holomorphic self-maps of the unit disk

Abstract

Suppose that f is a holomorphic self map of the unit disk \({\mathbb{D}}\). Recently several monotonicity results related to the image of smaller disks under f have been proved. These results extend the classical Schwarz lemma in various ways. We prove analogous monotonicity results in the context of Julia’s boundary Schwarz lemma. A horodisk is a disk internally tangent to the unit circle. For positive \({\lambda}\), we denote by \({H_{\lambda}}\) the disk of radius \({\lambda/(1\,+\,\lambda)}\) centered at the point \({1/(1\,+\,\lambda)}\). This is a horodisk that touches the unit circle at the point 1. Suppose that f(1) = 1 (in the sense of radial limit) and denote by \({f^{\prime}(1)}\) the angular derivative. By Julia’s lemma \({f(H_{\lambda})\,\subset H_{{\lambda}f^{\prime}(1)}}\). Let \({\Psi_f(\lambda)\,=\,\inf\,\{\rho > 0 : f(H_{\lambda}) \subset H_\rho\}}\). We show that the function \({\Psi_f(\lambda)/\lambda}\) is a decreasing function of \({\lambda}\) and that \({\lim_{\lambda\,\to\,0+} \Psi_f(\lambda)/\lambda = f^\prime(1)}\). This result implies that the constant \({f^\prime(1)}\) in Julia’s lemma is the best possible.