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.