On the Growth Order and Growth Type of Entire Functions of Several Complex Matrices

Abul-Ez, M.; Abd-Elmageed, H.; Hidan, M.; Abdalla, M.

doi:https://doi.org/10.1155/2020/4027529

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Research Article | Open Access

Volume 2020 | Article ID 4027529 | https://doi.org/10.1155/2020/4027529

On the Growth Order and Growth Type of Entire Functions of Several Complex Matrices

M. Abul-Ez,¹H. Abd-Elmageed,²M. Hidan,³and M. Abdalla^2,3

Academic Editor: Ismat Beg

Received11 Jun 2019

Accepted03 Sept 2019

Published17 Feb 2020

Abstract

In this paper, we establish an explicit relation between the growth of the maximum modulus and the Taylor coefficients of entire functions in several complex matrix variables (FSCMVs) in hyperspherical regions. The obtained formulas enable us to compute the growth order and the growth type of some higher dimensional generalizations of the exponential, trigonometric, and some special FSCMVs which are analytic in some extended hyperspherical domains. Furthermore, a result concerning linear substitution of the mode of increase of FSCMVs is given.

1. Introduction

The study of the asymptotic mode of increase behavior of entire functions in one and several complex variables is one of the classical central topics in complex analysis. Basic tools to study the mode of increase behavior of holomorphic functions are entities such as growth orders, the growth type, the maximum term, and the central index. Several developments made in this direction started with the early works of Lindelöf [1], Pringsheim [2], Valiron [3], Shah [4, 5], and Gol’dberg [6]. Their results turned out to be very useful in the study of partial differential equations and elsewhere. Generalization to higher dimensions has been given by many other authors (see, e.g., [7–10]) where they have contributed to the study of the order and type of entire functions of several complex variables. As Clifford analysis offers another possibility of generalizing complex function theory to higher dimensions, many authors introduced a study on the mode of increase of entire monogenic functions (see [11–16]). In recent years, the theory of matrix functions has grown significantly, with new applications appearing and the literature expanding at the last rate (see [12, 17]). From this point of view, many authors have dealt with different problems concerning functions in several complex matrix variables (see [18–20]). However, a number of central questions concerning the asymptotic growth behavior of functions in several complex matrix variables are still open. Recently, Kishka et al. [18] obtained the order and type of entire functions of two complex matrices in complete Reinhardt domains. The present paper is devoted to investigate the mode of increase of entire functions in several complex matrix variables in hyperspherical regions independent of the scalar entire functions of several complex variables associated with them (e.g., [16, 17]). The obtained results are applicable by illustrative examples. We end our study with an interesting result on linear substitution of entire functions in several complex matrix variables in hyperspherical regions.

Following Nassif [21], we give some notations and associated properties in the framework of several complex variables.

Assume and belonging to are -dimensional multi-indices, andwhere and are nonnegative numbers. For the -dimensional space, we write and , and for , we setwhere is an open spherical region of radius r and is its closure.

Consider the function , which is regular in ; then, (see [19, 21])

The maximum modulus of is denoted by

Note that (2) leads to

Cauchy’s inequality for functions of several complex variables was introduced by Nassif in the form (see [21])whereand on the assumption that , whenever .

The number in (7) is considered to be a generalization to the number in the two-complex variable case (cf. [22]), where

Now, the radius of convergence of the power series (3) is defined in the open sphere by

Then, the function is an entire function if . The growth order of the function is given in [21] in the form

If , we can prove using the same way as in the single complex variable case (see [23–25]) that the growth type θ of can be given in the form

2. Functions of Several Complex Matrix Variables (FSCMVs)

We shall cite some preliminaries and notations which are essential to establish our main results (see [20]).

2.1. Preliminaries and Notations

Let be the space of matrices whose entries are complex numbers. Let . The multiplication law of matrices takes the formand in general,where the summation includes all symbols independently, from 1 to N. Following [20], suppose that . We shall use the following notation:which means that a matrix A whose each of its elements have been taken to be moduli of the elements.

If a matrix B has positive elements which are greater than the elements of the matrix , we have . In other words, this inequality is equivalent to the following system of inequalities:

Also referring to [20], the notation shall mean that a matrix in which all its elements are equal to the number d and determine its positive integral powers as follows:

Thus, , and generally for positive integral powers we have

2.2. Convergence Property of FSCMVs

In the light of Section 2.1, we discuss convergence property of a power series of several complex matrix variables in hyperspherical regions by the convergence of a power series of several complex variables without any restrictions on the coefficients.

Let be commutative matrices in , and then the function of several complex matrices can be written as a power series in the form

Since and , Z may enjoy the same notation given in Section 2.1.

Thus, we write

Hence, , where

Let us investigate the convergence of series (20) in a domain which is a subset of the space determined by the following inequalities:

For this purpose, let

The convergence of this series guarantees the convergence of series (20), and in this case, series (20) will be absolutely convergent.

To show the sufficient condition for the absolute convergence of series (20), suppose that the scalar function of several complex variables associated with the matrix function in (18) is an analytic function in the region , where is the radius of convergence of this series, N is the common order of our matrices, and R is a positive number. Sinceusing similar procedure in deriving (6), we can deduce the following inequality for the coefficients of series (23), taking into account the common order of matrices . Thus,where

We obviously have

Thus, using (24) and (26) in (22), it follows thatthat is, the power series in (18) will be absolutely convergent. We thus have the following theorem.

Theorem 1. If the radius of convergence of the series in (23) is equal to , then series (18) will be absolutely convergent for all matrices situated in the neighborhood of domain (21).

3. Mode of Increase of FSCMVs

In this section, we prove two main results which provide the mode of increase of the entire FSCMVs. These results then enable us to compute the growth order and the growth type of some entire FSCMVs.

Let be an entire function of several complex matrix variables of common order N with Taylor expansion:and the maximum modulus

Therefore, Cauchy’s inequality for the matrix function can be given in the form

For the description of the asymptotic growth behavior of the maximum modulus of entire matrix functions, we write the following.

Definition 1. Let be an entire function of several complex matrices. Then, the order of growth of the maximum modulus of an entire function of several complex matrices is described byNow, we are going to prove our main result which provides us with a generalization to the context of entire FSCMVs of the famous theorem of Lindelöf and Pringsheim in terms of the relation between the growth order and the Taylor coefficients.

Theorem 2. For an entire function in several complex matrix variables with a Taylor series representation of form (28), letThen, .

Proof. We first show that . For , this inequality is trivial. So, let us assume that in what follows. In view of (32), there exist infinitely many withwhere b is a real constant to be chosen such that with an if . In the case where , one can take for b any arbitrary positive real number. We then haveThe coefficients of a matrix Taylor series (28) satisfy Cauchy’s inequality in the formThus, we may infer thatFrom relation (33), we find thatNow, let in (37), and we get for this arbitrarily large r the following inequality:This permits us to conclude thatfor arbitrary . For , one gets .
For the case we see the following.
In the case where , there is nothing to prove. So, let us assume without loss of generality that . Since is an entire matrix function, we have that . Because of this property and in view of (32), one can find that, for all , such that for all multi-indices with ,For all these multi-indices with , it holds thatTherefore,that is,where is a positive real constant. Choose the number such thatand then fix the positive integer ε such thatThus, Summarizing, we have obtainedwhere , and are constants. Making such that for ε tends to zero, and then we arrive at the desired estimate , and the theorem is hereby deduced.
To get a finer classification of the growth behavior within the set of entire matrix functions that have the same growth order, one further introduces the growth type of an entire matrix functions as follows.

Definition 2. For any entire function of several complex matrices of order , the growth type τ is given byThe following is the second main result which provides us with a generalization of the famous Pringsheim–Lindelöf result on the relation between the growth type and the Taylor coefficients (see, e.g., [1, 2]) in the framework of matrix function theory.

Theorem 3. If is an entire matrix function of finite growth order and growth type , thenTo prove this theorem, we need the following lemma.

Lemma 1. Let be a function in several complex matrix variables of common order N which have a Taylor series expansion in (28). Suppose there are numbers and an integer such that for all . Then, is an entire matrix function, and given any , there is a number such that

Proof. Since , for all these multi-indices with , it holds thatHence, and is an entire matrix function.Furthermore,if multi-indices with . Choosing , which is so large that , and if , thenprovided multi-indices with .
We now establish an upper bound for :However,The maximum is achieved for all these multi-indices with ; hence,Therefore, if ,Given any , there is a number such that the expression in brackets is less than provided that . Hence,

Now Lemma 1 can be used to determine Theorem 3 as follows.

Proof of Theorem 3.2. Suppose τ is finite. Then, given any , there is a number such that , for . According to Cauchy’s inequality in (30), we haveThe minimum value of occurs for ; thus,for all these multi-indices with and .
Rewriting,Hence,Since π is an arbitrary number exceeding ,where the right-hand side is clearly finite. Now let be any number exceeding the right-hand side of (49). Then, there is a number such thatApplying Lemma 1 with and , given such thatTherefore, and because of the choice of ,Thus, the result is derived. Also, if the right-hand side of (49) is finite so is τ, and if τ is infinite, so is the right-hand side of (49).

3.1. Basic Applications

Theorem 2 and Theorem 3 allow us to evaluate the exact value of the growth order and the growth type for some elementary matrix generalizations of the classical exponential, trigonometric, and other special functions. To illustrate this, let us afford here some important examples.

Example 1. Letbe an entire function of several complex matrix variables all of which are of common order N in ; are positive numbers, and then the growth order and the growth type .

Proof. Definition 1 and Definition 2 tell us thatThus,This leads to

Remark 1. One could expect to get entire function of several complex matrices of growth order and growth type just by taking

Example 2. The generalized cosine function of several complex matrices , all of which are of common order N in , given bywhere have growth order and growth type .

Proof. First, we note that the power series in (72) has indeed an infinite radius of convergence in ; thus, the essentialconverges for all in .
Applying Theorem 2 and Theorem 3 lead immediately toAccording to Stirling’s formula in the form(see [13] and [14]) we get , and if is sufficiently large we haveThis leads to

Example 3. Similarly, one can show that the generalized sine function of several complex matrices all of which are of common order N in bywhere also have growth order and growth type .

Example 4. We can use the same construction principle to get examples of the rational growth order and growth type. This elementary example is, for instance, the generalized Bessel matrix function of order 0 of several complex matrices all of which are of common order N in defined byand then using Theorem 2 and Theorem 3, we have its growth order and growth type .

Example 5. According to Theorem 2 and Theorem 3. All matrix power serieshave growth order and growth type .

4. Linear Substitution for FSCMs

Let be an entire matrix function of growth order and growth type , where are any constant numbers. It is required in this section to establish and as follows.

If , are several complex matrices situated in the neighbourhood of the origin given by (21), then

The matrices X + A are situated in the neighbourhood of the origin given by

Thus,

Now, suppose that is any number, then there is a positive number such that, applying (48) and (83), one gets

That is,

On the other hand, if , then , and thus .

Considering the growth type, we see from (84) and (48) that

Since ϵ can be chosen arbitrarily as near to τ as we please, we infer that . In the usual way, if , we infer that , and hence . Thus, the following theorem is determined.

Theorem 4. The entire matrix function is of the same growth order and growth type as the matrix function .
Now, we consider the matrix function and are an entire matrix function of growth order ρ and growth type τ. Then, we obtained the following result by modicum calculations based on Theorem 4 and relations (31) and (48).

Theorem 5. Let be an entire matrix function of positive order ρ and type τ, and let be the real constant. Then, the matrix function is an entire matrix function of the growth order ρ and the growth type [26].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All the authors contributed equally and significantly to writing this article. All the authors read and approved the final manuscript.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through the Research Groups Program under Grant no. R.G.P.1/153/40.

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Copyright © 2020 M. Abul-Ez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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