Bohr theorems for slice regular functions over octonions

2020 ◽  
pp. 1-16
Author(s):  
Zhenghua Xu
Keyword(s):  
Unit Ball ◽  
Division Algebras ◽  
Coefficient Estimate ◽  
Coefficient Estimates ◽  
Covering Theorem ◽  
Slice Regular Functions ◽  
Regular Functions ◽  
Carathéodory Class

In this paper, we mainly investigate two versions of the Bohr theorem for slice regular functions over the largest alternative division algebras of octonions $\mathbb {O}$ . To this end, we establish the coefficient estimates for self-maps of the unit ball of $\mathbb {O}$ and the Carathéodory class in this setting. As a further application of the coefficient estimate, the 1/2-covering theorem is also proven for slice regular functions with convex image.

scholarly journals Further properties of the Bergman spaces of slice regular functions

2015 ◽  
Vol 15 (4) ◽  
Author(s):  
Fabrizio Colombo ◽  
J. Oscar González-Cervantes ◽  
Irene Sabadini
Keyword(s):  
Unit Ball ◽  
Half Space ◽  
Half Plane ◽  
Bergman Kernel ◽  
Bergman Spaces ◽  
Reflection Principle ◽  
Slice Regular Functions ◽  
Regular Functions ◽  
Complex Half Plane ◽  
Schwarz Reflection Principle

AbstractWe continue the study of Bergman theory for the class of slice regular functions. In the slice regular setting there are two possibilities to introduce the Bergman spaces, that are called of the first and of the second kind. In this paperwe mainly consider the Bergman theory of the second kind, by providing an explicit description of the Bergman kernel in the case of the unit ball and of the half space. In the case of the unit ball, we study the Bergman-Sce transform. We also show that the two Bergman theories can be compared only if suitableweights are taken into account. Finally,we use the Schwarz reflection principle to relate the Bergman kernel with its values on a complex half plane.

scholarly journals Landau’s theorem for slice regular functions on the quaternionic unit ball

2017 ◽  
Vol 28 (03) ◽  
pp. 1750017 ◽  
Author(s):  
Cinzia Bisi ◽  
Caterina Stoppato
Keyword(s):  
Unit Ball ◽  
Unit Disk ◽  
Analogous Result ◽  
Open Unit ◽  
Open Unit Ball ◽  
Slice Regular Functions ◽  
New Approach ◽  
The Real ◽  
Regular Functions ◽  
Real Algebra

During the development of the theory of slice regular functions over the real algebra of quaternions [Formula: see text] in the last decade, some natural questions arose about slice regular functions on the open unit ball [Formula: see text] in [Formula: see text]. This work establishes several new results in this context. Along with some useful estimates for slice regular self-maps of [Formula: see text] fixing the origin, it establishes two variants of the quaternionic Schwarz–Pick lemma, specialized to maps [Formula: see text] that are not injective. These results allow a full generalization to quaternions of two theorems proven by Landau for holomorphic self-maps [Formula: see text] of the complex unit disk with [Formula: see text]. Landau had computed, in terms of [Formula: see text], a radius [Formula: see text] such that [Formula: see text] is injective at least in the disk [Formula: see text] and such that the inclusion [Formula: see text] holds. The analogous result proven here for slice regular functions [Formula: see text] allows a new approach to the study of Bloch–Landau-type properties of slice regular functions [Formula: see text].

scholarly journals Division algebras of slice functions

2019 ◽  
Vol 150 (4) ◽  
pp. 2055-2082 ◽  
Author(s):  
Riccardo Ghiloni ◽  
Alessandro Perotti ◽  
Caterina Stoppato
Keyword(s):  
Maximum Modulus ◽  
Division Algebras ◽  
Weierstrass Theorem ◽  
Minimum Modulus ◽  
Slice Regular Functions ◽  
Open Mapping Theorem ◽  
Zero Sets ◽  
Regular Functions ◽  
Finite Dimensional

AbstractThis work studies slice functions over finite-dimensional division algebras. Their zero sets are studied in detail along with their multiplicative inverses, for which some unexpected phenomena are discovered. The results are applied to prove some useful properties of the subclass of slice regular functions, previously known only over quaternions. Firstly, they are applied to derive from the maximum modulus principle a version of the minimum modulus principle, which is in turn applied to prove the open mapping theorem. Secondly, they are applied to prove, in the context of the classification of singularities, the counterpart of the Casorati-Weierstrass theorem.

scholarly journals Quaternionic Fock space on slice hyperholomorphic functions

Filomat ◽  
2020 ◽  
Vol 34 (4) ◽  
pp. 1197-1207
Author(s):  
Sanjay Kumar ◽  
S.D. Sharma ◽  
Khalid Manzoor
Keyword(s):  
Unit Ball ◽  
Recent Work ◽  
Fock Space ◽  
Fock Spaces ◽  
Slice Regular Functions ◽  
Regular Functions ◽  
Slice Hyperholomorphic Functions ◽  
Hyperholomorphic Functions

In this paper, we define the quaternionic Fock spaces F p? of entire slice hyperholomorphic functions in a quaternionic unit ball B in H: We also study growth estimates and various results of entire slice regular functions in these spaces. The work of this paper is motivated by the recent work of [5] and [26].

scholarly journals Spherical Coefficients of Slice Regular Functions

2021 ◽  
Vol 76 (4) ◽  
Author(s):  
Amedeo Altavilla
Keyword(s):  
Differential Equations ◽  
Regular Function ◽  
Countable Family ◽  
Slice Regular Functions ◽  
Regular Functions ◽  
Spherical Expansion ◽  
Derivatives Of

AbstractGiven a quaternionic slice regular function f, we give a direct and effective way to compute the coefficients of its spherical expansion at any point. Such coefficients are obtained in terms of spherical and slice derivatives of the function itself. Afterwards, we compare the coefficients of f with those of its slice derivative $$\partial _{c}f$$ ∂ c f obtaining a countable family of differential equations satisfied by any slice regular function. The results are proved in all details and are accompanied to several examples. For some of the results, we also give alternative proofs.

scholarly journals A Phragmén - Lindelöf principle for slice regular functions

2011 ◽  
Vol 18 (4) ◽  
pp. 749-759 ◽  
Author(s):  
Graziano Gentili ◽  
Caterina Stoppato ◽  
Daniele C. Struppa
Keyword(s):  
Slice Regular Functions ◽  
Regular Functions ◽  
Lindelöf Principle
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