Landau-Toeplitz theorems for slice regular functions over quaternions

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 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 $n\ge 2$. Finally, a $3$-diameter version of the Landau-Toeplitz Theorem is proved using the notion of slice $3$-diameter.

20 pages

References in corpus (3)

• A Bloch-Landau Theorem for slice regular functions
• The Bohr Theorem for slice regular functions
• On a theorem of Landau and Toeplitz

Cited by in corpus (8)

• Twistor interpretation of slice regular functions
• Landau's theorem for slice regular functions on the quaternionic unit ball
• Slice starlike functions over quaternions
• Carathéodory theorems for Slice Regular Functions
• On Some Geometric Properties of Slice Regular Functions of a Quaternion Variable
• On geometric aspects of quaternionic and octonionic slice regular functions
• Julia theory for slice regular functions
• Boundary Julia theory for slice regular functions