Generalized Growth of Special Monogenic Functions

Abstract

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 Kraußhar [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 only if its Taylor’s series near zero has the form (see [5], pp. 1259)

Definition 2. Let be a special monogenic function defined on a neighborhood of the closed ball . Then, where is the maximum modulus of (see [5], pp. 1260).

Definition 3. Let be a special monogenic function whose Taylor’s series representation is given by (9). Then, for the maximum term of this special monogenic function is given by (see [5], pp. 1260) Also the index with maximal length for which maximum term is achieved is called the central index and is denoted by (see [5], pp. 1260)

Definition 4. Let be a special monogenic function whose Taylor’s series representation is given by (9). Then, the order and lower order of are defined as (see [5], pp. 1263)

Definition 5. Let be a special monogenic function whose Taylor’s series representation is given by (9). Then, the type and lower type of special monogenic function having nonzero finite generalized order are defined as (see [5], pp. 1270) For generalization of the classical characterizations of growth of entire functions, Seremeta [6] introduced the concept of the generalized order and generalized type with the help of general growth functions as follows.
Let denote the class of functions satisfying the following conditions:(i) is defined on and is positive, strictly increasing, and differentiable, and it tends to as (ii),  for every function such that as .
Let denote the class of functions satisfying conditions (i) and (i), (ii)for every ; that is, is slowly increasing.
Following Srivastava and Kumar [7] and Kumar and Bala ([8–10]), here we give definitions of generalized order, generalized lower order, generalized type, and generalized lower type of special monogenic functions. For special monogenic function and functions , we define the generalized order and generalized lower order of as If in above equation we put and , then we get definitions of order and lower order as defined by Abul-Ez and De Almeida (see [5], pp. 1263). Hence, their definitions of order and lower order are special cases of our definitions.
Further, for , we define the generalized type and generalized lower type of special monogenic function having nonzero finite generalized order as If in above equation we put , , and , then we get definitions of type and lower type as defined by Abul-Ez and De Almeida (see [5], pp. 1270). Hence, their definitions of type and lower type are special cases of our definitions.
Abul-Ez and De Almeida [5] have obtained the characterizations of order, lower order, type, and lower type of special monogenic functions in terms of their Taylor’s series coefficients. In the present paper we have obtained the characterizations of generalized order, generalized lower order, generalized type and generalized lower type of special monogenic functions in terms of their Taylor’s series coefficients. The results obtained by Abul-Ez and De Almeida [5] are special cases of our results.

3. Main Results

We now prove the following.

Theorem 6. Let be a special monogenic function whose Taylor’s series representation is given by (9). If and , then the generalized order of is given as

Proof. Write Now, first we prove that . The coefficients of a monogenic Taylor’s series satisfy Cauchy’s inequality; that is, Also from (15), for arbitrary and all , we have Now, from inequality (19), we get Since , (see [11], pp. 148) so the above inequality reduces to Putting in the above inequality, we get, for all large values of , or or or Since ,  . Hence, proceeding to limits as , we get Since is arbitrarily small, so finally we get Now, we will prove that . If , then there is nothing to prove. So let us assume that . Therefore, for a given there exists such that, for all multi-indices with , we have or Now, from the property of maximum modulus (see [11], pp. 148), we have or On the lines of the proof of the theorem given by Srivastava and Kumar (see [7], Theorem 2.1, pp. 666), we get Combining this with inequality (28), we get (17). Hence, Theorem 6 is proved.

Next, we prove the following.

Theorem 7. Let be a special monogenic function whose Taylor’s series representation is given by (9). Also let and ; then the generalized type of is given as

Proof. Write Now, first we prove that . From (16), for arbitrary and all , we have where . Now, using (19), we get Now, as in the proof of Theorem 6, here this inequality reduces to Putting , we get or or Now, proceeding to limits as , we get Since is arbitrarily small, so finally we get Now, we will prove that . If , then there is nothing to prove. So let us assume that . Therefore, for a given there exists such that, for all multi-indices with , we have or Now, from the property of maximum modulus (see [11], pp. 148), we have or On the lines of the proof of the theorem given by Srivastava and Kumar (see [7], Theorem 2.2, pp. 670), we get Combining this with (43), we get (34). Hence, Theorem 7 is proved.

Next, we have the following.

Theorem 8. Let be a special monogenic function whose Taylor’s series representation is given by (9). If and then the generalized lower order of satisfies Further, if is a nondecreasing function of , then equality holds in (49).

Proof. The proof of the above theorem follows on the lines of the proof of Theorem 6 and [7] Theorem 2.4 (pp. 674). Hence, we omit the proof.

Next, we have the following.

Theorem 9. Let be a special monogenic function whose Taylor’s series representation is given by (9). Also let and ; then the generalized lower type of satisfies Further, if is a nondecreasing function of , then equality holds in (51).

Proof. The proof of the above theorem follows on the lines of the proof of Theorem 7 and [7] Theorem 2.4 (pp. 674). Hence, we omit the proof.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author is very thankful to the referee for the valuable comments and observations which helped in improving the paper.