Landau–Toeplitz theorems for slice regular functions over quaternions

Graziano Gentili and Giulia Sarfatti

Vol. 265 (2013), No. 2, 381–404

DOI: 10.2140/pjm.2013.265.381

Abstract

The theory of slice regular functions of a quaternionic variable extends the notion of holomorphic function to the quaternionic setting. This theory, already rich in results, is sometimes surprisingly different from the theory of holomorphic functions of a complex variable; however, several fundamental results in the two environments are similar even if their proofs for the case of quaternions need new technical tools.

In this paper we prove the Landau–Toeplitz theorem for slice regular functions in a formulation that involves an appropriate notion of regular 2-diameter. We show that the Landau–Toeplitz inequalities hold in the case of the regular n-diameter for all n ≥ 2. Finally, a 3-diameter version of the Landau–Toeplitz theorem is proved using the notion of slice 3-diameter.

Keywords

functions of hypercomplex variables, geometric theory of regular functions of a quaternionic variable, Schwarz lemma and generalizations

Mathematical Subject Classification 2010

Primary: 30G35

Secondary: 30C80

Milestones

Received: 28 June 2011

Revised: 14 April 2013

Accepted: 3 July 2013

Published: 28 August 2013