Slice Regular Functions on Regular Quadratic Cones of Real Alternative Algebras

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 $\mathbb{D}\subset \mathbb{C}$, the Erdős–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.