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Multipoint Schwarz-Pick Lemma for the quaternionic case

Cinzia Bisi, Davide Cordella

Abstract

Following ideas by Beardon, Minda and Baribeau, Rivard, Wegert in the context of the complex Schwarz-Pick Lemma, we use iterated hyperbolic difference quotients to prove a quaternionic multipoint Schwarz-Pick Lemma, in the context of the theory of slice regular functions. As applications, we obtain quaternionic Dieudonné and Goluzin estimates. Finally, an algorithm for the construction of (Nevanlinna-Pick) interpolating slice regular functions with real nodes is provided as a byproduct of the quaternionic multipoint Schwarz-Pick Lemma.

Multipoint Schwarz-Pick Lemma for the quaternionic case

Abstract

Following ideas by Beardon, Minda and Baribeau, Rivard, Wegert in the context of the complex Schwarz-Pick Lemma, we use iterated hyperbolic difference quotients to prove a quaternionic multipoint Schwarz-Pick Lemma, in the context of the theory of slice regular functions. As applications, we obtain quaternionic Dieudonné and Goluzin estimates. Finally, an algorithm for the construction of (Nevanlinna-Pick) interpolating slice regular functions with real nodes is provided as a byproduct of the quaternionic multipoint Schwarz-Pick Lemma.
Paper Structure (16 sections, 39 theorems, 144 equations)

This paper contains 16 sections, 39 theorems, 144 equations.

Table of Contents

  1. Preliminaries
  2. Slice regular functions
  3. Regular linear fractional and Möbius transformations
  4. Regular Blaschke products
  5. The hyperbolic difference quotient and the three-points Schwarz--Pick Lemma
  6. The quaternionic Schwarz--Pick Lemma
  7. Three-points quaternionic Schwarz--Pick Lemma
  8. Some consequences. Estimates in the case of f(0)=0
  9. Distortion estimates in terms of the hyperbolic derivative
  10. Distortion estimates for the class Bα
  11. Iterated hyperbolic difference quotients and the multipoint Schwarz--Pick Lemma
  12. On Nevanlinna--Pick interpolation problem
  13. Two points Nevanlinna--Pick interpolation
  14. Nevanlinna--Pick interpolation with three points
  15. Nevanlinna--Pick interpolation with more than three points
  16. ...and 1 more sections

Key Result

Lemma 1

Let $f:\mathbb D\longrightarrow\mathbb D$ be a holomorphic function. Let $z_0\in\mathbb D$. Then for any $z\in\mathbb D$ it is moreover Inequalities are strict for $z\neq z_0$, unless $f$ is an automorphism of the disk.

Theorems & Definitions (84)

  • Lemma 1: Schwarz--Pick Lemma
  • Theorem 2: BeardonMinda04
  • Theorem 3
  • Theorem 4
  • Definition 1.1
  • Lemma 1.2: Splitting Lemma, librospringer2
  • Definition 1.3
  • Example 1.4
  • Remark 1.5
  • Definition 1.6
  • ...and 74 more