Geometria Complessa e Geometria Differenziale
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G. Gentili - G. Sarfatti

Landau-Toeplitz theorems for slice regular functions over quaternions

created by stoppato on 09 Nov 2016
modified by sarfatti on 19 Feb 2020


Published Paper

Inserted: 9 nov 2016
Last Updated: 19 feb 2020

Journal: Pacific J. Math.
Volume: 265
Number: 2
Pages: 381-404
Year: 2013
Doi: 10.2140/pjm.2013.265.381

ArXiv: 1404.3120 PDF


The theory of slice regular functions of a quaternionic variable extends the notion of holomorphic function to the quaternionic setting. This theory, already rich of 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 then show that the Landau-Toeplitz inequalities hold in the case of the regular $n$-diameter, for all $n\geq 2$. Finally, a $3$-diameter version of the Landau-Toeplitz Theorem is proved using the notion of slice $3$-diameter.

Tags: FIRB2012-DGGFT

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