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

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. *e 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.


Introduction
e 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].
eir 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][8][9][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][12][13][14][15][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][19][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. e 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]). e 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.
Consider the function f(z), which is regular in S r ; then, (see [19,21]) e maximum modulus of f(z) 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]) where and 1 ≤ σ m ≤ ( ) [m] on the assumption that m m s /2 s � 1, whenever m s � 0. e number σ m in (7) is considered to be a generalization to the number σ h,k 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 S r by en, the function f(z) is an entire function if R f � ∞. e growth order of the function f(z) is given in [21]  ln σ m / a m .
If 0 < 9 < ∞, we can prove using the same way as in the single complex variable case (see [23][24][25]) that the growth type θ of f(z) can be given in the form . (11)

Functions of Several Complex Matrix Variables (FSCMVs)
We shall cite some preliminaries and notations which are essential to establish our main results (see [20]).

Preliminaries and Notations.
Let M N (C) be the space of N × N matrices whose entries are complex numbers. Let A ∈ M N (C), A � (a uv ), u, v � 1, 2, . . . , N. e multiplication law of matrices takes the form and in general, where the summation includes all symbols j v 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 |A|, we have |A| < B. In other words, this inequality is equivalent to the following system of N × N inequalities: Also referring to [20], the notation ‖d‖ shall mean that a matrix in which all its elements are equal to the number d and determine its positive integral powers as follows: us, ‖d‖ 2 � ‖Nd‖ 2 , and generally for positive integral powers we have

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 X � ([x s;ij ]); s � 1, 2, . . . , k be commutative matrices in M N (C), and then the function F(X) of several complex matrices can be written as a power series in the form 1≤i,j≤N and Z ∈ M N (C), Z may enjoy the same notation given in Section 2.1.
us, we write Let us investigate the convergence of series (20) in a domain which is a subset of the space M N (C) determined by the following inequalities: For this purpose, let e 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 F(z) � [n]�0 a n z n of several complex variables associated with the matrix function in (18) is an analytic function in the region S NR , where NR is the radius of convergence of this series, N is the common order of our matrices, and R is a positive number. Since a n z n , using 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 N. us, where We obviously have us, using (24) and (26) in (22), it follows that that is, the power series in (18) will be absolutely convergent. We thus have the following theorem. (23) is equal to NR, then series (18) will be absolutely convergent for all matrices situated in the neighborhood of domain (21).

Mode of Increase of FSCMVs
In this section, we prove two main results which provide the mode of increase of the entire FSCMVs. ese results then enable us to compute the growth order and the growth type of some entire FSCMVs. Let F(X) be an entire function of several complex matrix variables of common order N with Taylor expansion: and the maximum modulus erefore, Cauchy's inequality for the matrix function F(X) can be given in the form

Journal of Function Spaces
For the description of the asymptotic growth behavior of the maximum modulus of entire matrix functions, we write the following. Definition 1. Let F(X) be an entire function of several complex matrices. en, the order of growth of the maximum modulus of an entire function of several complex matrices is described by Now, 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), let
Proof. We first show that ρ(F) ≥ Ω. For Ω � 0, this inequality is trivial. So, let us assume that 0 < Ω ≤ +∞ in what follows. In view of (32), there exist infinitely many n ∈ N k 0 with [n]ln ([n]) ≥ − b ln a n K n , where b is a real constant to be chosen such that b � Ω − ε > 0 with an ε > 0 if Ω < ∞. In the case where Ω � ∞, one can take for b any arbitrary positive real number. We then have e coefficients of a matrix Taylor series (28) satisfy Cauchy's inequality in the form a n ≤ NK n r − [n] M F; S r .
For the case Ω ≥ ρ(F) we see the following.
In the case where Ω � +∞, there is nothing to prove. So, let us assume without loss of generality that 0 ≤ Ω < ∞. Since F(X) is an entire matrix function, we have that lim [n]⟶∞ |a n | � 0. Because of this property and in view of (32), one can find that, for all ε > 0, κ ∈ N such that for all multi-indices [n] with [n] ≥ κ, 0 ≤ [n]ln ([n]) − ln a n /K n ≤ Ω + ε. (Nr) [n] a n σ n , that is, where L 1 is a positive real constant. Choose the number r 0 > 1 such that and then fix the positive integer ε such that κ < ε ≤ 2r 0 Ω+ε < ε + 1; r > r 0 , Summarizing, we have obtained where L 2 , L 3 , and L 4 are constants. Making r ⟶ ∞ such that ρ(F) ≤ Ω + ε for ε tends to zero, and then we arrive at the desired estimate ρ(F) ≤ Ω, 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 F(X) of order ρ(0 < ρ < ∞), the growth type τ is given by e 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. .

Lemma 1. Let F(X) be a function in several complex matrix variables of common order N which have a Taylor series expansion in
Proof. Since (|a n |/σ n ) < (eαβ/N α [n]) [n]/α , for all these multi-indices with [n] > c, it holds that Hence, (N [n] |a n |/σ n ) 1/[n] ⟶ 0, [n] ⟶ ∞, and F(X) is an entire matrix function.Furthermore, (Nr) [n] a n σ n

1/[n]
< eαβ Given any ε > 0, there is a number R � R(ε) > R 1 such that the expression in brackets is less than exp εr α { } provided that r > R. Hence, M F; S r < exp (β + ϵ)r α for all r > R.
(58) □ Now Lemma 1 can be used to determine eorem 3 as follows.

(62)
Since π is an arbitrary number exceeding τ, where the right-hand side is clearly finite. Now let π 1 be any number exceeding the right-hand side of (49). en, there is a number c � c(π 1 ) > 0 such that Applying Lemma 1 with β � π 1 and α � ρ, given such that M F; S r < exp π 1 + ϵ r ρ , for all r > R.
us, 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).
be an entire function of several complex matrix variables X � (X 1 , X 2 , . . . , X k ) all of which are of common order N in S r ; b s are positive numbers, and then the growth order ρ(F) � 1 and the growth type τ � k Proof. Definition 1 and Definition 2 tell us that ρ � lim sup r⟶∞ ln ln M F; S r ln r , us, is leads to □ Remark 1. One could expect to get entire function of several complex matrices of growth order p(p ∈ N) and growth type (k Example 2. e generalized cosine function of several complex matrices X � (X 1 , X 2 , . . . , X k ), all of which are of common order N in S r , given by where 2n � (2n 1 , 2n 2 , . . . , 2n k ) have growth order ρ � 1 and growth type τ � Nk.
Example 4. We can use the same construction principle to get examples of the rational growth order and growth type. is elementary example is, for instance, the generalized Bessel matrix function of order 0 of several complex matrices X � (X 1 , X 2 , . . . , X k ) all of which are of common order N in S r defined by and then using eorem 2 and eorem 3, we have its growth order ρ � (1/2) and growth type τ � 0.
Example 5. According to eorem 2 and eorem 3. All matrix power series Journal of Function Spaces