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A Burns-Krantz type theorem for Blaschke products

Annika Moucha

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.

A Burns-Krantz type theorem for Blaschke products

Abstract

Let 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 near some boundary point that forces 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.

Paper Structure

This paper contains 8 sections, 13 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Maximal Blaschke products (MBP)
  3. Hyperbolic geometry, angular derivative and Stolz regions
  4. Some facts from hyperbolic geometry
  5. Angular derivative
  6. Injectivity on (ends of) Stolz regions
  7. A local Julia type inequality
  8. Proof of Theorem \ref{['thm:sequenceBurnsKrantzforMBP']}

Key Result

Theorem A

Let $f:\mathbb D\to\mathbb D$ be a holomorphic function and $\xi\in\partial\mathbb D$. If then $f(z)=z$ for all $z\in\mathbb D$.

Theorems & Definitions (24)

  • Theorem A: Burns-Krantz (1994); see burnsRigidityHolomorphicMappings1994
  • Theorem B: Baracco-Zaitsev-Zampieri (2006); see baraccoBurnsKrantzTypeTheorem2006
  • Theorem C: Chelst (2001); see chelstGeneralizedSchwarzLemma2001
  • Theorem 1.1
  • Corollary 1.2
  • Theorem D: Bracci-Kraus-Roth (2023), see bracciNewSchwarzPickLemma2023
  • Theorem E: Kraus (2013); see krausCriticalSetsBounded2013. Kraus-Roth (2013); see krausMaximalBlaschkeProducts2013
  • Definition 2.1: Maximal Blaschke products
  • Remark 2.2
  • Lemma 3.1
  • ...and 14 more