Slice Regular Functions on Regular Quadratic Cones of Real Alternative Algebras
The theory of slice regular functions is a natural generalization of that of holomorphic functions of one complex variable to the setting of quaternions, octonions, paravectors in Clifford algebras, and more generally quadratic cones of real alternative a
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Slice Regular Functions on Regular Quadratic Cones of Real Alternative Algebras Guangbin Ren, Xieping Wang and Zhenghua Xu Abstract. The theory of slice regular functions is a natural generalization of that of holomorphic functions of one complex variable to the setting of quaternions, octonions, paravectors in Clifford algebras, and more generally quadratic cones of real alternative algebras, in virtue of a slight modification of a well-known Fueter construction. In this paper, we focus on slice regular functions on the so-called regular quadratic cones, which are generally smaller than quadratic cones introduced by Ghiloni–Perotti and turn out to be the appropriate sets on which some nice properties of slice regular functions can be considered, including particularly the growth and distortion theorems for slice regular extensions of univalent holomorphic functions on the unit disc D ⊂ C, the Erd˝ os–Lax inequality and the Turan inequality for a subclass of slice regular polynomials with all the coefficients in a same complex plane. It is noteworthy that the notion of regular quadratic cones also provides additionally an effective approach to unifying the theory of slice regular functions on quaternions, octonions, and paravectors in Clifford algebras. Mathematics Subject Classification (2010). 30G35, 32A30. Keywords. Slice regular functions; real alternative algebras; growth and distortion theorems; Erd˝ os–Lax inequality; Turan inequality.
1. Introduction A theory of slice regular functions on the so-called quadratic cones of real alternative algebras was recently initiated by Ghiloni and Perotti [24]. It contains and generalizes the theory of slice regular functions introduced initially by Gentili and Struppa in [20,21] for quaternions H, and subsequently by Colombo, Sabadini and Struppa [7,8] in the real Clifford algebra Rn , and later also by Gentili and Struppa in [22] for octonions O. This new slice regular theory involves a notion of slice regularity, which goes back to a work of Cullen [10] and is significantly different from This work was supported by Chinese NSF grant No. 11371337.
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that of Cauchy–Fueter (see [2] for example). It also has elegant applications to the functional calculus for noncommutative operators [9], Schur analysis [1], and the construction and classification of orthogonal complex structures on dense open subsets of R4 [18]. The strategy proposed by Ghiloni and Perotti [24] is motivated by a wellknown Fueter construction, which provides an effective way to generate quaternionic regular functions (in the sense of Cauchy–Fueter) starting from complex holomorphic functions (cf. [15,40]) and has been generalized by Sce [37], Qian [31], and Sommen [39] to the setting of Clifford algebras. Many variants have been given since then in Clifford analysis [6, 13, 30, 32] and Dunkl–Clifford analysis [14]. The approach introduced by Ghiloni and Perotti in [24] for an alternative algebra A over R makes use of the complexified algebra A ⊗R C, denoted by AC . It turns out that for eac
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