Slice Regular Functions on Regular Quadratic Cones of Real Alternative Algebras

Ren, Guangbin; Wang, Xieping; Xu, Zhenghua

doi:10.1007/978-3-319-42529-0_13

Cited by 16 publications

(8 citation statements)

References 4 publications

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“…The counterpart of the convex combination identity (3.11) in Lemma 3.2 also holds for slice regular functions defined on octonions or more general real alternative algebras under the extra assumption that f preserves at least one slice. This can be verified similarly as in the proof of Proposition 3.1, see [29,32] for details.…”

Section: I I ∧ J I(i ∧ J)supporting

confidence: 70%

Growth and distortion theorems for slice monogenic functions

Ren

Wang

2017

Pacific J. Math.

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Abstract. The sharp growth and distortion theorems are established for slice monogenic extensions of univalent functions on the unit disc D ⊂ C in the setting of Clifford algebras, based on a new convex combination identity. The analogous results are also valid in the quaternionic setting for slice regular functions and we can even prove the Koebe type one-quarter theorem in this case. Our growth and distortion theorems for slice regular (slice monogenic) extensions to higher dimensions of univalent holomorphic functions hold without extra geometric assumptions, in contrast to the setting of several complex variables in which the growth and distortion theorems fail in general and hold only for some subclasses with the starlike or convex assumption.

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Section: I I ∧ J I(i ∧ J)supporting

confidence: 70%

Growth and distortion theorems for slice monogenic functions

Ren

Wang

2017

Pacific J. Math.

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“…It is easy to verify that every hypercomplex subspace of is, in particular, a strong regular quadratic cone in , as defined in [35]. Lemma 1.…”

Section: -Rmentioning

confidence: 99%

Cauchy-Riemann operators and local slice analysis over real alternative algebras

Perotti¹

2022

Preprint

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A. We prove some formulas relating Cauchy-Riemann operators defined on hypercomplex subspaces of an alternative *-algebra to a differential operator associated with the concept of slice-regularity and to the spherical Dirac operator. These results in particular allow to introduce a definition of locally slice-regular function and open the path for local slice analysis. Since Cauchy-Riemann operators factor the corresponding Laplacian operators, the proven formulas let us also obtain several results about the harmonicity and polyharmonicity properties of slice-regular functions.

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“…This technique makes it possible to extend some properties of holomorphic functions in one complex variable to the high dimensional and non-commutative case of quaternions. It has found significant applications especially in operator theory [3,9,10], differential geometry [14], geometric function theory [26,27] and it can be generalized to other higher dimensional settings like Clifford algebras [7,8] and real alternative algebras [18,19,20,28].…”

Section: Introductionmentioning

confidence: 99%

Slice Dirac operator over octonions

2020

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The slice Dirac operator over octonions is a slice counterpart of the Dirac operator over quaternions. It involves a new theory of stem functions, which is the extension from the commutative O(1) case to the non-commutative O(3) case. For functions in the kernel of the slice Dirac operator over octonions, we establish the representation formula, the Cauchy integral formula (and, more in general, the Cauchy-Pompeiu formula), and the Taylor as well as the Laurent series expansion formulas.

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Slice Regular Functions on Regular Quadratic Cones of Real Alternative Algebras

Cited by 16 publications

References 4 publications

Growth and distortion theorems for slice monogenic functions

Growth and distortion theorems for slice monogenic functions

Cauchy-Riemann operators and local slice analysis over real alternative algebras

Slice Dirac operator over octonions

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