A Burns-Krantz type theorem for Blaschke products

TL;DR

This paper extends Schwarz's lemma to boundary points for holomorphic functions, showing conditions under which such functions must be Blaschke products with specified critical points, using a Julia-type inequality.

Contribution

It introduces a boundary Schwarz lemma variant and characterizes when a holomorphic self-map of the disk is a Blaschke product with given critical points.

Findings

Established a boundary Schwarz lemma for holomorphic functions.

Provided conditions that force a function to be a Blaschke product.

Developed a Julia-type inequality based on Nehari's sharpening.

Abstract

Let $f$ be a holomorphic function mapping the open unit disk into itself. We establish a boundary version of Schwarz' lemma in the spirit of a result by Burns and Krantz and provide sufficient conditions on the local behaviour of $f$ near some boundary point that forces $f$ to be a Blaschke product with predescribed critical points. For the proof, a local Julia type inequality based on Nehari's sharpening of Schwarz' lemma is established.