Generalized Growth of Special Monogenic Functions

We study the generalized growth of special monogenic functions. The characterizations of generalized order, generalized lower order, generalized type, and generalized lower type of special monogenic functions have been obtained in terms of their Taylor’s series coefficients. 1. Introduction Clifford analysis offers the possibility of generalizing complex function theory to higher dimensions. It considers Clifford algebra valued functions that are defined in open subsets of for arbitrary finite and that are solutions of higher-dimensional Cauchy-Riemann systems. These are often called Clifford holomorphic or monogenic functions. In order to make calculations more concise, we use the following notations, where is -dimensional multi-index and : Following Almeida and Krau？har [1] and Constales et al. [2, 3], we give some definitions and associated properties. By we denote the canonical basis of the Euclidean vector space . The associated real Clifford algebra is the free algebra generated by modulo , where is the neutral element with respect to multiplication of the Clifford algebra . In the Clifford algebra , the following multiplication rule holds: where is Kronecker symbol. A basis for Clifford algebra is given by the set with , where , . Each can be written in the form with . The conjugation in Clifford algebra is defined by , where and for , . The linear subspace is the so-called space of paravectors which we simply identify with . Here, is scalar part and is vector part of paravector . The Clifford norm of an arbitrary is given by Also, for , we have . Each paravector has an inverse element in which can be represented in the form . In order to make calculations more concise, we use the following notation: The generalized Cauchy-Riemann operator in is given by If is an open set, then a function is called left (right) monogenic at a point if ( ). The functions which are left (right) monogenic in the whole space are called left (right) entire monogenic functions. Following Abul-Ez and Constales [4], we consider the class of monogenic polynomials of degree , defined as Let be -dimensional surface area of -dimensional unit ball and let be -dimensional sphere. Then, the class of monogenic polynomials described in (6) satisfies (see [5], pp. 1259) Also following Abul-Ez and De Almeida [5], we have 2. Preliminaries Now following Abul-Ez and De Almeida [5], we give some definitions which will be used in the next section. Definition 1. Let be a connected open subset of containing the origin and let be monogenic in . Then, is called special monogenic in , if and