Archiv der Mathematik

, Volume 102, Issue 1, pp 91–99

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

• Dimitrios Betsakos
Article

## 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.

30C80 30H10

## Keywords

Holomorphic function Horodisk Julia’s lemma Angular derivative Hyperbolic distance Schwarz–Pick lemma

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