Boundary Schwarz lemma for holomorphic self-mappings of strongly pseudoconvex domains

TL;DR

This paper extends the boundary Schwarz lemma for holomorphic self-mappings from the unit ball to strongly pseudoconvex domains, using growth estimates of the Carathéodory metric and boundary behavior analysis.

Contribution

It generalizes existing results to broader domains and introduces a different proof technique based on growth estimates and boundary behavior of the Carathéodory metric.

Findings

Generalization of boundary Schwarz lemma to strongly pseudoconvex domains

Introduction of a new proof method involving growth estimates

Connection to boundary behavior of the Carathéodory metric

Abstract

In this paper, we generalize a recent work of Liu et al. from the open unit ball $\mathbb B^n$ to more general bounded strongly pseudoconvex domains with $C^2$ boundary. It turns out that part of the main result in this paper is in some certain sense just a part of results in a work of Bracci and Zaitsev. However, the proofs are significantly different: the argument in this paper involves a simple growth estimate for the Carath\'eodory metric near the boundary of $C^2$ domains and the well-known Graham's estimate on the boundary behavior of the Carath\'eodory metric on strongly pseudoconvex domains, while Bracci and Zaitsev use other arguments.