On the Growth Orders and Types of Biregular Functions

Abstract

One of the main aims of Clifford analysis is to study the growth properties of regular functions. Biregular functions are a well-known generalization of regular functions. In this paper, the growth orders and types of biregular functions are studied. First, generalized growth orders and types of biregular functions are defined in the context of Clifford analysis. Then, using the methods of Wiman and Valiron, generalized Lindelöf–Pringsheim theorems are proved, which show the relationship between growth orders, growth types, and Taylor series. These connections allow us to calculate the growth order and determine the type of biregular functions.

1. Introduction

Clifford analysis is the theory of regular functions generalizing complex function theory to higher dimensions [1,2]. This analysis is widely used in mathematics physics and engineering [3,4]. Regular functions of Clifford analysis are solutions of higher-dimensional Cauchy–Riemann systems and are defined in $R^n$ and take values in real Clifford algebra. The biregular functions are functions of two variables of arbitrary dimension, which are left regular in one variable and right regular in the other. The early research on the theory of biregular functions originates from Brakex and Pincket (see [5,6,7,8,9]). In [7], the Cauchy integral formula for regular functions is extended to the case of biregular functions. Furthermore, applying the Cauchy formula, the Taylor series expansion for biregular functions in the open balls is obtained. The Hartogs extension phenomenon and the Hartogs theorem are also studied in [8]. Then, by the method of integral equations and the Schauder fixed-point theorem, the nonlinear boundary value problems for biregular functions are studied in [10,11]. In this paper, the asymptotic behavior of the growth of biregular functions is investigated by using a Taylor series.

The study of the growth properties of holomorphic and meromorphic functions is one of the central topics of complex analysis. Research on the topic started with the early works of Lindelöf [12] and Pringsheim [13], and a short time later, continued by Wiman [14] and Valiron [15] in the early twentieth century. Wiman–Valiron methods are widely used to study the growth behavior of solutions to complex differential equations. The growth properties of meromorphic functions are studied by Wiman–Valiron methods [16]. The Wiman–Valiron theory for the Askey–Wilson operator is studied in [17]. The connection between the growth of coefficients and the growth of solutions in the case of the unit disc is established in [18]. Using Wiman–Valiron methods and complex analytic methods, meromorphic solutions of some third-order differential equations are studied in [19]. The growth properties of solutions of complex differential equations with entire coefficients having a multiply-connected Fatou component are studied in [20]. Growth properties of solutions of complex differential equations with entire coefficients of finite $(\alpha ,\beta ,\gamma )$-order are investigated in [21]. Some characteristics of the order of growth for transcendental meromorphic functions are studied in [22]. In recent years, these methods have been used for some classes of functions in Clifford analysis. The growth properties of null-solutions to polynomial Cauchy–Riemann equations with constant complex coefficients are considered in [23]. The growth orders and types of hyperholomorphic functions and slice hyperholomorphic functions are studied in [24]. The growth of axially monogenic functions are studied by the generalized Hadamard product [25]. Applying Cauchy inequality, the relation between the asymptotic growth behavior of left-regular functions and the coefficients of its power series expansion is investigated in [26]. The proximate order of entire monogenic functions and their applications are studied in [27]. Inspired by their works, we consider the growth orders and types of biregular functions in Clifford analysis.

The aim of this paper is to study the growth behavior of biregular functions by Wiman and Valiron methods. This paper is structured as follows. Section 2 briefly introduces the fundamental concepts of biregular functions in Clifford analysis and presents the Cauchy integral formula and the Taylor series for biregular functions. In Section 3, the growth order of biregular functions is defined. Then, using the Cauchy integral formula and the Taylor series for biregular functions, a connection is established between the order of growth and the Taylor series, which allows one to calculate the order of growth without the maximum values of the function. That is, a generalized Lindelöf–Prinsheim theorem regarding the order of growth is obtained. In order to obtain a more precise classification of the growth behavior of biregular functions, the growth type of biregular functions is discussed in Section 4. The generalized Lindelöf–Pringsheim theorem for biregular functions with respect to the growth type is proved. This theorem implies the connection between the growth type and Taylor series. It follows that one may calculate the growth type without the maximum values of a biregular function.

2. Preliminaries

For a detailed study of biregular functions of Clifford analysis, readers are referred to [5,6,7,8,9].

Let $\Omega$ be an open subset of $R^{m+1}\times R^{k+1},1<m,k\le n.$ Let $C{l}_n$ be the real Clifford algebra constructed over $R^n$ with orthonormal basis $(e_1,\cdots ,e_n).$ The functions are defined in $\Omega$ and take their values in $C{l}_n.$ The functions of two variables in Clifford analysis are given by

$$f:\Omega \to C{l}_n$$

$$(x,y)=(x_0,x_1,\cdots ,x_m;y_0,y_1,\cdots ,y_k)\to f(x,y)=\sum _A{f}_A(x,y)e_A,$$

where $A=(h_1,h_2,\cdots ,h_r)\in P\{1,2,\cdots ,n\},1\le h_1<h_2<\cdots <h_r\le n,$ $f_A$ are real-valued. Here, $e_i{e}_j+e_j{e}_i=-2{\delta }_{i,j},i,j=1,\cdots ,n,$ where ${\delta }_{i,j}$ is the Kronecker symbol.

For any $a\in C{l}_n,$ we have

$$a=\sum _A{a}_A{e}_A,\parallel a\parallel ={\left(\sum _A|a_A{|}^2\right)}^{{\displaystyle \frac{1}{2}}},$$

where $\parallel a\parallel$ is the norm of $a.$

We also introduce the following notations for the intersections parallel to the $x-$ and $y-$ space:

$$U_y=\{x\in R^{m+1}:(x,y)\in \Omega \},yfixedin{R}^{k+1}.$$

$$V_x=\{y\in R^{k+1}:(x,y)\in \Omega \},xfixedin{R}^{m+1}.$$

Next, the biregular function is defined as follows.

Definition 1 

([5]). A function f is called biregular, if

$$\left\{\begin{array}{c}{\displaystyle D_{x}f=\sum _{i=0}^m{e}_i{\partial }_{x_i}f=0,f\in C^1(U_y;C{l}_n),}\hfill \\ {\displaystyle f{D}_y=\sum _{j=0}^k{e}_j{\partial }_{y_j}f=0,f\in C^1(V_x;C{l}_n).}\hfill \end{array}\right.$$

Lemma 1 

([7]). Let $S_m\times S_k\subset \Omega .$ Let $S_m$ (resp. $S_k$) be an $(m+1)-$ (resp. $(k+1)-$) dimensional, compact, differentiable and oriented manifold. Let $S_m^{\circ }=S_m\setminus \partial S_m,S_k^{\circ }=S_k\setminus \partial S_k.$ If f is biregular in $\Omega ,$ then for each $(x,y)\in S_m^{\circ }\times S_k^{\circ },$

$$f(x,y)={\int }_{\partial S_m\times \partial S_k}{E}_m(u-x)d{\sigma }_{u}f(u,v)d{\sigma }_v{E}_l(v-y),$$

where the Cauchy kernels are given by

$$E_m\left(x\right)={\displaystyle \frac{1}{{\omega }_{m+1}}}{\displaystyle \frac{\overline{x}}{{\left|x\right|}^{m+1}}},E_k\left(y\right)={\displaystyle \frac{1}{{\omega }_{k+1}}}{\displaystyle \frac{\overline{y}}{{\left|y\right|}^{k+1}}}.$$

Let us take $e_0=1.$ Then,

$$x=\sum _{i=0}^m{x}_i{e}_i,y=\sum _{j=0}^k{x}_j{e}_j,\overline{x}=\sum _{i=0}^m{x}_i\overline{e_i},\overline{y}=\sum _{j=0}^k{y}_j\overline{e_j},$$

where $\overline{e_i}=-e_i,$ $\left|x\right|$ is the Euclidean norm of x and ${\omega }_{m+1}={\displaystyle \frac{2{\pi }^{\frac{m+1}{2}}}{\Gamma \left(\frac{m+1}{2}\right)}}$ is the area of the $(m+1)$-dimensional unit sphere. These kernels satisfy the equalities

$$D_x{E}_m=E_m{D}_x=\delta ,D_y{E}_k=E_k{D}_y=\delta ,$$

where δ is the Dirac measure at the origin.

To make the calculations more clear, we use the following notation

$${\left(l\right)}_m=(l_1,\cdots ,l_m),{\left(l\right)}_m!=l_1!\cdots l_m!,|{\left(l\right)}_m|=|l_1+\cdots +l_m|.$$

$${\left(h\right)}_k=(h_1,\cdots ,l_k),{\left(h\right)}_k!=h_1!\cdots h_k!,|{\left(h\right)}_k|=|h_1+\cdots +h_k|.$$

Lemma 2 

([7]). If the function f is biregular in balls $B_m^{\circ }(0,R)\times B_k^{\circ }(0,R),$ then in this region, f may be expanded uniquely into the normally convergent Taylor series

$$f(x,y)=\sum _{{|\left(l\right)}_m|=0}^{+\infty }\sum _{{|\left(h\right)}_k|=0}^{+\infty }{V}_{{\left(l\right)}_m}\left(x\right){\lambda }_{{\left(l\right)}_m{\left(h\right)}_k}{W}_{{\left(h\right)}_k}\left(y\right),$$

where

$$V_{{\left(l\right)}_m}\left(x\right)={\displaystyle \frac{{\left(l\right)}_m!}{{|\left(l\right)}_m|!}}\sum _{\pi \in perm\left({\left(l\right)}_m\right)}{\xi }_{\pi \left(l_1\right)}\cdots {\xi }_{\pi \left(l_m\right)},W_{{\left(h\right)}_k}\left(y\right)={\displaystyle \frac{{\left(h\right)}_k!}{{|\left(h\right)}_k|!}}\sum _{\pi \in perm\left({\left(h\right)}_k\right)}{\eta }_{\pi \left(h_1\right)}\cdots {\eta }_{\pi \left(h_k\right)},$$

and $perm\left({\left(l\right)}_m\right)$ (resp. $perm\left({\left(h\right)}_k\right)$) is the set of all permutations of the sequence $(l_1,\cdots ,l_m)$ (resp. $(h_1,\cdots ,h_k)$). The hypercomplex variables are given by

$${\xi }_s=x_s{e}_0-x_0{e}_s,s=1,\cdots ,m;{\eta }_t=y_t{e}_0-y_0{e}_t,t=1,\cdots ,k.$$

Here

$${\lambda }_{{\left(l\right)}_m{\left(h\right)}_k}={\int }_{\partial B_m\left(R_m\right)\times \partial B_k\left(R_k\right)}{X}_{{\left(l\right)}_m}\left(u\right)d{\sigma }_{u}f(u,v)d{\sigma }_v{Y}_{{\left(h\right)}_k}\left(v\right),$$

where $X_{{\left(l\right)}_m}\left(u\right)={(-1)}^m{\partial }_{x_{l_1}}\cdots {\partial }_{x_{l_m}}{E}_m\left(u\right)$ and $Y_{{\left(h\right)}_k}\left(v\right)={(-1)}^k{\partial }_{y_{h_1}}\cdots {\partial }_{y_{h_k}}{E}_k\left(v\right).$

3. The Growth Order of a Biregular Function

In this section, the generalized Lindelöf and Pringsheim theorem for biregular functions is proved. This theorem establishes a connection between the order of growth and the Taylor series. In [26], the order of left-regular functions is defined. Inspired by this work, we next define the growth order of biregular functions.

Definition 2. 

Let $f(x,y):R^{m+1}\times R^{k+1}\to C{l}_n$ be a biregular function. Then, the order of growth of the function f is defined by

$$\rho =\underset{R\to +\infty }{lim\; sup}{\displaystyle \frac{loglogM(R,f)}{logR}},$$

where

$${\displaystyle M(R,f)=\underset{\left|x\right|,\left|y\right|=R}{sup}\left\{\right|f(x,y)|,f\in C{l}_n,(x,y)\in R^{m+1}\times R^{k+1}\}.}$$

In this paper, we assume that $0<\rho <+\infty .$

Theorem 1. 

If f is biregular in $R^{m+1}\times R^{k+1},$ then the order of growth of the function f is given by

$$\rho =\underset{{|\left(l\right)}_m{|,|\left(h\right)}_k|\to +\infty }{lim\; sup}{\displaystyle \frac{{\left(\right|\left(l\right)}_m{|+|\left(h\right)}_k{\left|\right)log\left(\right|\left(l\right)}_m{|+|\left(h\right)}_k\left|\right)}{-log\parallel \frac{1}{c(m,{\left(l\right)}_m)}{\lambda }_{{\left(l\right)}_r{\left(h\right)}_k}\frac{1}{c(k,{\left(h\right)}_k)}\parallel }},$$

where

$$c(m,{\left(l\right)}_m)={\displaystyle \frac{m(m+1)\cdots (m+|{\left(l\right)}_m|-1)}{{\left(l\right)}_m!}},$$

$$c(k,{\left(h\right)}_k)={\displaystyle \frac{k(k+1)\cdots (k+|{\left(h\right)}_k|-1)}{{\left(h\right)}_k!}}.$$

Proof of Theorem 1. 

Let

$$\underset{{|\left(l\right)}_m{|,|\left(h\right)}_k|\to +\infty }{lim\; sup}{\displaystyle \frac{{\left(\right|\left(l\right)}_m{|+|\left(h\right)}_k{\left|\right)log\left(\right|\left(l\right)}_m{|+|\left(h\right)}_k\left|\right)}{-log\parallel \frac{1}{c(m,{\left(l\right)}_m)}{\lambda }_{{\left(l\right)}_r{\left(h\right)}_k}\frac{1}{c(k,{\left(h\right)}_k)}\parallel }}=L.$$

Then, we need to prove that $\rho \ge L,$ (resp. $\rho \le L$). The first inequality will be proved below. Without loss of generality, we assume that $0<L<+\infty .$

According to the definition of the supremum, there exists $\epsilon >0,$ such that

$${\displaystyle \frac{{\left(\right|\left(l\right)}_m{|+|\left(h\right)}_k{\left|\right)log\left(\right|\left(l\right)}_m{|+|\left(h\right)}_k\left|\right)}{-log\parallel \frac{1}{c(m,{\left(l\right)}_m)}{\lambda }_{{\left(l\right)}_r{\left(h\right)}_k}\frac{1}{c(k,{\left(h\right)}_k)}\parallel }}\ge L-\epsilon ,$$

where $L-\epsilon >0.$ This implies that

$$log∥{\displaystyle \frac{1}{c(m,{\left(l\right)}_m)}}{\lambda }_{{\left(l\right)}_m{\left(h\right)}_k}{\displaystyle \frac{1}{c(k,{\left(h\right)}_k)}}∥\ge -{\displaystyle \frac{{\left(\right|\left(l\right)}_m{|+|\left(h\right)}_k{\left|\right)log\left(\right|\left(l\right)}_m{|+|\left(h\right)}_k\left|\right)}{L-\epsilon }}.$$

By using Lemmas 1 and 2, we obtain

$$\begin{array}{ccc}\Vert {\lambda }_{{\left(l\right)}_m{\left(h\right)}_k}\Vert \hfill & =\hfill & {\displaystyle \left|{\int }_{\partial B_m\left(R_m\right)\times \partial B_k\left(R_k\right)}{X}_{{\left(l\right)}_m}\left(u\right)d{\sigma }_{u}f(u,v)d{\sigma }_v{Y}_{{\left(h\right)}_k}\left(v\right)\right|}\hfill \\ \\ & \le \hfill & {\displaystyle M(R,f){\int }_{\partial B_m\left(R_m\right)\times \partial B_k\left(R_k\right)}|{\partial }_{x_{l_1}}\cdots {\partial }_{x_{l_m}}{E}_m\left(u\right)\left|\right|{\partial }_{y_{h_1}}\cdots {\partial }_{y_{h_k}}{E}_k\left(v\right)|d{\sigma }_{u}d{\sigma }_v}\hfill \\ \\ & =\hfill & {\displaystyle \frac{m(m+1)\cdots (m+|{\left(l\right)}_m|-1)}{R^{{|\left(l\right)}_m|}}\frac{k(k+1)\cdots (k+|{\left(h\right)}_k|-1)}{R^{{|\left(h\right)}_k|}}M(R,f)}\hfill \\ \\ & =\hfill & {\displaystyle \frac{{m(m+1)\cdots (m+|\left(l\right)}_m{|-1)k(k+1)\cdots (k+|\left(h\right)}_k|-1)}{{\left(l\right)}_m!{\left(h\right)}_k!R^{{|\left(l\right)}_m{|+|\left(h\right)}_k|}}M(R,f).}\hfill \end{array}$$

The inequality leads us to deduce the conclusion that

$$M(R,f)\ge {\displaystyle \frac{{\left(l\right)}_m!{\left(h\right)}_k!\Vert {\lambda }_{{\left(l\right)}_m{\left(h\right)}_k}\Vert R^{{|\left(l\right)}_m{|+|\left(h\right)}_k|}}{{m(m+1)\cdots (m+|\left(l\right)}_m{|-1)k(k+1)\cdots (k+|\left(h\right)}_k|-1)}}.$$

The inequality implies that

$$\begin{array}{ccc}logM(R,f)\hfill & \ge \hfill & {\displaystyle log\left[\frac{{\left(l\right)}_m!{\left(h\right)}_k!\Vert {\lambda }_{{\left(l\right)}_m{\left(h\right)}_k}\Vert R^{{|\left(l\right)}_m{|+|\left(h\right)}_k|}}{{m(m+1)\cdots (m+|\left(l\right)}_m{|-1)k(k+1)\cdots (k+|\left(h\right)}_k|-1)}\right]}\hfill \\ \\ & =\hfill & {\displaystyle log\left[\frac{{\left(l\right)}_m!{\left(h\right)}_k!\Vert {\lambda }_{{\left(l\right)}_m{\left(h\right)}_k}\Vert }{{m(m+1)\cdots (m+|\left(l\right)}_m{|-1)k(k+1)\cdots (k+|\left(h\right)}_k|-1)}\right]+log{R}^{{|\left(l\right)}_m{|+|\left(h\right)}_k|}}\hfill \\ \\ & \ge \hfill & {\displaystyle \frac{{log\left(\right|\left(l\right)}_m{|+|\left(h\right)}_k{\left|\right)}^{{-|\left(l\right)}_m{|+|\left(h\right)}_k|}}{L-\epsilon }+log{R}^{{|\left(l\right)}_m{|+|\left(h\right)}_k|}}\hfill \\ \\ & \ge \hfill & {\displaystyle \frac{{log\left(\right|\left(l\right)}_m{|+|\left(h\right)}_k{\left|\right)}^{{-|\left(l\right)}_m{|+|\left(h\right)}_k|}}{e^{L-\epsilon }}+log{R}^{{|\left(l\right)}_m{|+|\left(h\right)}_k|}}\hfill \\ \\ & =\hfill & {\displaystyle log\left[{\left(\right|\left(l\right)}_m{|+|\left(h\right)}_k{\left|\right)}^{-\frac{{|\left(l\right)}_m{|+|\left(h\right)}_k|}{e^{L-\epsilon }}}{R}^{{|\left(l\right)}_m{|+|\left(h\right)}_k|}\right]}\hfill \\ \\ & \ge \hfill & {\displaystyle log\left[{\left(\frac{R}{{|\left(l\right)}_m{|+|\left(h\right)}_k|}\right)}^{\frac{{|\left(l\right)}_m{|+|\left(h\right)}_k|}{e^{L-\epsilon }}{e}^{L-\epsilon }}\right].}\hfill \end{array}$$

From Definition 2, it follows that

$$\begin{array}{ccc}\rho \hfill & \ge \hfill & {\displaystyle \underset{R\to +\infty }{lim\; sup}\frac{loglog{\left(\frac{R}{{|\left(l\right)}_m{|+|\left(h\right)}_k|}\right)}^{\frac{{|\left(l\right)}_m{|+|\left(h\right)}_k|}{R^{L-\epsilon }}{R}^{L-\epsilon }}}{logR}}\hfill \\ \\ & \ge \hfill & {\displaystyle \underset{R\to +\infty }{lim\; sup}\left[\frac{(L-\epsilon )logR}{logR}-\frac{{loglog\left(\right|\left(l\right)}_m{|+|\left(h\right)}_k{\left|\right)}^{{|\left(l\right)}_m{|+|\left(h\right)}_k|}}{logR}\right]}\hfill \\ \\ & =\hfill & {\displaystyle L-\epsilon ,}\hfill \end{array}$$

which implies that $\rho \ge L$ as $\epsilon \to 0.$

Now let us prove another the inequality $\rho \le L.$ We use Lemma 2, which states

$$f(x,y)=\sum _{{|\left(l\right)}_m|=0}^{+\infty }\sum _{{|\left(h\right)}_k|=0}^{+\infty }{V}_{{\left(l\right)}_m}\left(x\right){\lambda }_{{\left(l\right)}_m{\left(h\right)}_k}{W}_{{\left(h\right)}_k}\left(y\right).$$

It is easy to check that

$$|V_{{\left(l\right)}_m}\left(x\right)|\le R^{{|\left(l\right)}_m|},\left|W_{{\left(h\right)}_k}\left(y\right)\right|\le R^{{|\left(h\right)}_k|}.$$

Hence,

$$\begin{array}{ccc}M(R,f)\hfill & \le \hfill & {\displaystyle \sum _{{|\left(l\right)}_m|=0}^{+\infty }\sum _{{|\left(h\right)}_k|=0}^{+\infty }\Vert {\lambda }_{{\left(l\right)}_m{\left(h\right)}_k}\Vert R^{{|\left(l\right)}_m{|+|\left(h\right)}_k|}}\hfill \\ \\ & =\hfill & {\displaystyle \sum _{{|\left(l\right)}_m|=0}^{N_0}\sum _{{|\left(h\right)}_k|=0}^{N_0}\Vert {\lambda }_{{\left(l\right)}_m{\left(h\right)}_k}\Vert R^{{|\left(l\right)}_m{|+|\left(h\right)}_k|}+\sum _{{|\left(l\right)}_m|=N_0+1}^{+\infty }\sum _{{|\left(h\right)}_k|=N_0+1}^{+\infty }\Vert {\lambda }_{{\left(l\right)}_m{\left(h\right)}_k}\Vert R^{{|\left(l\right)}_m{|+|\left(h\right)}_k|}}\hfill \\ \\ & =\hfill & {\displaystyle I_1+I_2.}\hfill \\ \end{array}$$

For the first term of the above inequality, we have

$$\begin{array}{ccc}{I}_1\hfill & \le \hfill & {\displaystyle \sum _{{|\left(l\right)}_m|=0}^{N_0}\sum _{{|\left(h\right)}_k|=0}^{N_0}\Vert {\lambda }_{{\left(l\right)}_m{\left(h\right)}_k}\Vert R^{2N_0}\le C_1{R}^{2N_0},}\hfill \end{array}$$

where $C_1$ is a positive real constant.

According to the definition of the limits and the supremum, there exists $\epsilon >0,$ such that

$${\displaystyle \frac{{\left(\right|\left(l\right)}_m{|+|\left(h\right)}_k{\left|\right)log\left(\right|\left(l\right)}_m{|+|\left(h\right)}_k\left|\right)}{-log\parallel \frac{1}{c(m,{\left(l\right)}_m)}{\lambda }_{{\left(l\right)}_r{\left(h\right)}_k}\frac{1}{c(k,{\left(h\right)}_k)}\parallel }}\le L+\epsilon$$

with ${|\left(l\right)}_m|\ge N_0,\left|{\left(h\right)}_k\right|\ge N_0.$ This inequality leads us to deduce the conclusion that

$$\parallel {\lambda }_{{\left(l\right)}_r{\left(h\right)}_k}\parallel \le c(m,{\left(l\right)}_m)c(k,{\left(h\right)}_k){\left(\right|\left(l\right)}_m{|+|\left(h\right)}_k{\left|\right)}^{-{\displaystyle \frac{{|\left(l\right)}_m{|+|\left(h\right)}_k|}{L+\epsilon }}}.$$

Let us now evaluate $I_2:$

$$\begin{array}{ccc}{I}_2\hfill & \le \hfill & {\displaystyle \sum _{{|\left(l\right)}_m|=N_0+1}^{+\infty }\sum _{{|\left(h\right)}_k|=N_0+1}^{+\infty }c(m,{\left(l\right)}_m)c(k,{\left(h\right)}_k){\left(\right|\left(l\right)}_m{|+|\left(h\right)}_k{\left|\right)}^{-\frac{{|\left(l\right)}_m{|+|\left(h\right)}_k|}{L+\epsilon }}{R}^{{|\left(l\right)}_m{|+|\left(h\right)}_k|}}\hfill \\ \\ & \le \hfill & {\displaystyle \sum _{{|\left(l\right)}_m|=N_0+1}^{{\left[(m+1)R\right]}^{L+\epsilon }}\sum _{{|\left(h\right)}_k|=N_0+1}^{{\left[(k+1)R\right]}^{L+\epsilon }}c(m,{\left(l\right)}_m)c(k,{\left(h\right)}_k){\left(\right|\left(l\right)}_m{|+|\left(h\right)}_k{\left|\right)}^{-\frac{{|\left(l\right)}_m{|+|\left(h\right)}_k|}{L+\epsilon }}{R}^{{|\left(l\right)}_m{|+|\left(h\right)}_k|}}\hfill \\ \\ & +\hfill & {\displaystyle \sum _{{|\left(l\right)}_m{|=\left[(m+1)R\right]}^{L+\epsilon }}^{+\infty }\sum _{{|\left(h\right)}_k{|=\left[(k+1)R\right]}^{L+\epsilon }}^{+\infty }c(m,{\left(l\right)}_m)c(k,{\left(h\right)}_k){\left(\right|\left(l\right)}_m{|+|\left(h\right)}_k{\left|\right)}^{-\frac{{|\left(l\right)}_m{|+|\left(h\right)}_k|}{L+\epsilon }}{R}^{{|\left(l\right)}_m{|+|\left(h\right)}_k|}}\hfill \\ \\ & =\hfill & {\displaystyle I_{21}+I_{22}.}\hfill \end{array}$$

For $I_{21},$ we can write

$$\begin{array}{ccc}{I}_{21}\hfill & \le \hfill & {\displaystyle R^{{\left[(m+1)R\right]}^{L+\epsilon }+{\left[(k+1)R\right]}^{L+\epsilon }}\sum _{{|\left(l\right)}_m|=N_0+1}^{{\left[(m+1)R\right]}^{L+\epsilon }}\sum _{{|\left(h\right)}_k|=N_0+1}^{{\left[(k+1)R\right]}^{L+\epsilon }}c(m,{\left(l\right)}_m)c(k,{\left(h\right)}_k){\left(\right|\left(l\right)}_m{|+|\left(h\right)}_k{\left|\right)}^{-\frac{{|\left(l\right)}_m{|+|\left(h\right)}_k|}{L+\epsilon }}}\hfill \\ \\ & \le \hfill & {\displaystyle R^{{\left[(m+1)R\right]}^{L+\epsilon }+{\left[(k+1)R\right]}^{L+\epsilon }}\sum _{{|\left(l\right)}_m|=1}^{+\infty }\sum _{{|\left(h\right)}_k|=1}^{+\infty }c(m,{\left(l\right)}_m)c(k,{\left(h\right)}_k){\left(\right|\left(l\right)}_m{|+|\left(h\right)}_k{\left|\right)}^{-\frac{{|\left(l\right)}_m{|+|\left(h\right)}_k|}{L+\epsilon }}.}\hfill \end{array}$$

In order to evaluate $I_{21},$ we consider the sum in $I_{21}.$

$$\begin{array}{ccc}& & {\displaystyle \sum _{{|\left(l\right)}_m|=1}^{+\infty }\sum _{{|\left(h\right)}_k|=1}^{+\infty }c(m,{\left(l\right)}_m)c(k,{\left(h\right)}_k){\left(\right|\left(l\right)}_m{|+|\left(h\right)}_k{\left|\right)}^{-\frac{{|\left(l\right)}_m{|+|\left(h\right)}_k|}{L+\epsilon }}}\hfill \\ \\ & \le \hfill & {\displaystyle \sum _{{|\left(l\right)}_m|=1}^{+\infty }c(m,{\left(l\right)}_m){|\left(l\right)}_m{|}^{-\frac{{|\left(l\right)}_m|}{L+\epsilon }}\sum _{{|\left(h\right)}_k|=1}^{+\infty }c(k,{\left(h\right)}_k){\left|{\left(h\right)}_k\right|}^{-\frac{{|\left(h\right)}_k|}{L+\epsilon }}}\hfill \\ \\ & \le \hfill & {\displaystyle \sum _{p=1}^{+\infty }\sum _{{|\left(l\right)}_m|=p}c(m,{\left(l\right)}_m)p^{-\frac{p}{L+\epsilon }}\sum _{q=1}^{+\infty }\sum _{{|\left(h\right)}_k|=q}c(k,{\left(h\right)}_k)q^{-\frac{q}{L+\epsilon }}.}\hfill \end{array}$$

Let $f\left(t\right)={\displaystyle \frac{1}{{(1-nt)}^n}}.$ Then $f\left(t\right)$ only has an isolated pole at $t={\displaystyle \frac{1}{n}}.$ Since

$${\displaystyle \frac{1}{{(1-nt)}^n}}=\sum _{p=0}^{+\infty }\left(\sum _{{|\left(l\right)}_m|=p}c(n,{\left(l\right)}_m)\right)t^p,$$

the radius of convergence of the above series is ${\displaystyle \frac{1}{n}}.$ Thus,

$$\underset{p\to +\infty }{lim\; sup}{\left(\sum _{{|\left(l\right)}_m|=p}c(m,{\left(l\right)}_m)\right)}^{{\displaystyle \frac{1}{p}}}=n.$$

Furthermore, it follows that

$$\underset{p\to +\infty }{lim\; sup}{\left(\sum _{{|\left(l\right)}_m|=p}c(m,{\left(l\right)}_m)p^{-{\displaystyle \frac{p}{L+\epsilon }}}\right)}^{{\displaystyle \frac{1}{p}}}=0.$$

Using the Cauchy discriminant method, it is easy to see that the series

$$\sum _{p=1}^{+\infty }\sum _{{|\left(l\right)}_m|=p}c(m,{\left(l\right)}_m)p^{-{\displaystyle \frac{p}{L+\epsilon }}}$$

converges. Similarly, the series

$$\sum _{q=1}^{+\infty }\sum _{{|\left(h\right)}_k|=q}c(k,{\left(h\right)}_k)q^{-{\displaystyle \frac{q}{L+\epsilon }}}$$

converges.

The above inequalities imply that the series

$$\sum _{{|\left(l\right)}_m|=1}^{+\infty }\sum _{{|\left(h\right)}_k|=1}^{+\infty }c(m,{\left(l\right)}_m)c(k,{\left(h\right)}_k){\left(\right|\left(l\right)}_m{|+|\left(h\right)}_k{\left|\right)}^{-{\displaystyle \frac{{|\left(l\right)}_m{|+|\left(h\right)}_k|}{L+\epsilon }}}$$

converges.

Hence,

$$I_{21}\le C_2{R}^{{\left[(m+1)R\right]}^{L+\epsilon }+{\left[(k+1)R\right]}^{L+\epsilon }},$$

where $C_2$ is a positive real constant. Let us now consider the inequalities for $I_{22}$:

$$\begin{array}{ccc}{I}_{22}\hfill & \le \hfill & {\displaystyle \sum _{{|\left(l\right)}_m|=1}^{+\infty }\sum _{{|\left(h\right)}_k|=1}^{+\infty }c(m,{\left(l\right)}_m)c(k,{\left(h\right)}_k){({\left[(m+1)R\right]}^{L+\epsilon }+{\left[(k+1)R\right]}^{L+\epsilon })}^{-\frac{{|\left(l\right)}_m{|+|\left(h\right)}_k|}{L+\epsilon }}{R}^{{|\left(l\right)}_m{|+|\left(h\right)}_k|}}\hfill \\ \\ & =\hfill & {\displaystyle \sum _{{|\left(l\right)}_m|=1}^{+\infty }\sum _{{|\left(h\right)}_k|=1}^{+\infty }c(m,{\left(l\right)}_m)c(k,{\left(h\right)}_k){({\left[(m+1)\right]}^{L+\epsilon }+{\left[(k+1)\right]}^{L+\epsilon })}^{-\frac{{|\left(l\right)}_m{|+|\left(h\right)}_k|}{L+\epsilon }}}\hfill \\ \\ & \le \hfill & {\displaystyle \sum _{{|\left(l\right)}_m|=1}^{+\infty }c(m,{\left(l\right)}_m){(m+1)}^{{-|\left(l\right)}_m|}\sum _{{|\left(h\right)}_k|=1}^{+\infty }c(k,{\left(h\right)}_k){(k+1)}^{{-|\left(h\right)}_k|}}\hfill \\ \\ & =\hfill & {\displaystyle \sum _{p=1}^{+\infty }\sum _{{|\left(l\right)}_m|=p}c(m,{\left(l\right)}_m){(m+1)}^{-p}\sum _{q=1}^{+\infty }\sum _{{|\left(h\right)}_k|=q}c(k,{\left(h\right)}_k){(k+1)}^{-q}.}\hfill \end{array}$$

Note that

$$\underset{p\to +\infty }{lim\; sup}{\left(\sum _{{|\left(l\right)}_m|=p}c(m,{\left(l\right)}_m){(m+1)}^{-p}\right)}^{{\displaystyle \frac{1}{p}}}={\displaystyle \frac{m}{m+1}}<1.$$

Using the Cauchy discriminant method, it is easy to see that the series

$$\sum _{{|\left(l\right)}_m|=p}c(m,{\left(l\right)}_m){(m+1)}^{-p}$$

converges. Similarly, the series

$$\sum _{q=1}^{+\infty }\sum _{{|\left(h\right)}_k|=q}c(k,{\left(h\right)}_k){(k+1)}^{-q}$$

converges. It follows that $I_{22}\le C_3,$ where $C_3$ is a positive real constant.

Based on the inequalities above, it can be concluded that

$$M(R,f)\le C_1{R}^{2N_0}+C_2{R}^{{\left[(m+1)R\right]}^{L+\epsilon }+{\left[(k+1)R\right]}^{L+\epsilon }}+C_3\le C_4{R}^{R^{L+\epsilon }},$$

where $C_4$ is a positive real constant. Therefore, for $\epsilon >0,$

$$\begin{array}{c}{\displaystyle \rho =\underset{R\to +\infty }{lim\; sup}\frac{loglogM(R,f)}{logR}\le L+\epsilon .}\hfill \end{array}$$

If $\epsilon \to 0,$ then $\rho \le L.$ This completes the proof. □

As far as we know, it is very difficult to calculate the maximum value of functions. This means that it has become difficult to use the maximum value to calculate the growth order of functions. However, this theorem allows us to calculate the growth order of biregular functions with the help of Taylor series without the exact value of $M(R,f).$

4. The Growth Type of a Biregular Function

In [26], the type of left-regular functions is defined. Inspired by it, the definition of the growth type of biregular functions is given in the section. Then, by the definition, the relation between the growth type and the Taylor series of biregular functions is established. This allows us to obtain a finer classification of the growth behavior of biregular functions if the growth order of the function is given.

Definition 3. 

Let $f(x,y):R^{m+1}\times R^{k+1}\to C{l}_n$ be a biregular function. Then, the type of growth of the function f is defined as

$$\tau =\underset{R\to +\infty }{lim\; sup}{\displaystyle \frac{logM(R,f)}{R^{\rho }}},$$

where $0<\rho <+\infty .$

Theorem 2. 

Let $f(x,y):R^{m+1}\times R^{k+1}\to C{l}_n$ be a biregular function. If $0<\rho <+\infty ,$ then the growth type of a biregular function is given by

$$\tau ={\displaystyle \frac{1}{e\rho }}\underset{{|\left(l\right)}_m{|,|\left(h\right)}_k|\to +\infty }{lim\; sup}{\left(\right|\left(l\right)}_m{|+|\left(h\right)}_k\left|\right)\parallel {\lambda }_{{\left(l\right)}_m{\left(h\right)}_k}{\parallel }^{{\displaystyle \frac{\rho }{{|\left(l\right)}_m{|+|\left(h\right)}_k|}}},$$

where ρ is the order of $f.$

Proof of Theorem 2. 

Let

$$G=\underset{{|\left(l\right)}_m{|,|\left(h\right)}_k|\to +\infty }{lim\; sup}{\left(\right|\left(l\right)}_m{|+|\left(h\right)}_k\left|\right)\parallel {\lambda }_{{\left(l\right)}_m{\left(h\right)}_k}{\parallel }^{{\displaystyle \frac{\rho }{\left(\right|{\left(l\right)}_m|+|{\left(h\right)}_k\left|\right)}}}.$$

Then, by the definition of the supremum, we have

$${\left(\right|\left(l\right)}_m{|+|\left(h\right)}_k\left|\right)\parallel {\lambda }_{{\left(l\right)}_m{\left(h\right)}_k}{\parallel }^{{\displaystyle \frac{\rho }{\left(\right|{\left(l\right)}_m|+|{\left(h\right)}_k\left|\right)}}}\ge G-\epsilon$$

with $\epsilon >0.$ From Lemmas 1 and 2, it follows that

$$\begin{array}{ccc}M(R,f)\hfill & \ge \hfill & {\displaystyle \frac{{\left(l\right)}_m!{\left(h\right)}_k!\Vert {\lambda }_{{\left(l\right)}_m{\left(h\right)}_k}\Vert R^{{|\left(l\right)}_m{|+|\left(h\right)}_k|}}{{m(m+1)\cdots (m+|\left(l\right)}_m{|-1)k(k+1)\cdots (k+|\left(h\right)}_k|-1)}}\hfill \\ \\ & \ge \hfill & {\displaystyle \frac{{\left(l\right)}_m!{\left(h\right)}_k!{\left[\frac{(G-\epsilon )}{{|\left(l\right)}_m{|+|\left(h\right)}_k|}\right]}^{\frac{{|\left(l\right)}_m{|+|\left(h\right)}_k|}{\rho }}{R}^{{|\left(l\right)}_m{|+|\left(h\right)}_k|}}{{m(m+1)\cdots (m+|\left(l\right)}_m{|-1)k(k+1)\cdots (k+|\left(h\right)}_k|-1)}}\hfill \\ \\ & =\hfill & {\displaystyle \frac{{\left(l\right)}_m!{\left(h\right)}_k!{\left[\frac{(G-\epsilon )R^{\rho }}{{|\left(l\right)}_m{|+|\left(h\right)}_k|}\right]}^{\frac{{|\left(l\right)}_m{|+|\left(h\right)}_k|}{\rho }}}{{m(m+1)\cdots (m+|\left(l\right)}_m{|-1)k(k+1)\cdots (k+|\left(h\right)}_k|-1)}.}\hfill \end{array}$$

Since ${\left(l\right)}_m!\le \left|{\left(l\right)}_m\right|!$ and ${|\left(l\right)}_m|>0,$ we have

$${\displaystyle \frac{{\left(l\right)}_m!}{m(m+1)\cdots (m+|{\left(l\right)}_m|-1)}}={\displaystyle \frac{{\left(l\right)}_m!(m-1)!}{(m+|{\left(l\right)}_m|-1)!}}\le {\displaystyle \frac{{(m-1)!|\left(l\right)}_m|!}{(m+|{\left(l\right)}_m|-1)!}}.$$

Note that Stirling’s formula is given by

$$ln(m!)\approx mlnm-m+{\displaystyle \frac{1}{2}}ln\left(2\pi m\right),$$

which implies that

$$m!=e^{ln(m!)}\approx \sqrt{2\pi m}{m}^m{e}^{-m}.$$

Using the above equation, we have

$${\displaystyle \frac{(m+|{\left(l\right)}_m|-1)!}{{|\left(l\right)}_m|!}}\approx {\left({\displaystyle \frac{m-1}{{|\left(l\right)}_m|}}+1\right)}^{{|\left(l\right)}_m|+{\displaystyle \frac{1}{2}}}{(m+|\left(l\right)}_m{|-1)}^{m-1}{e}^{-(m-1)}.$$

Similarly, it follows that

$${\displaystyle \frac{(m+|{\left(h\right)}_k|-1)!}{{|\left(h\right)}_k|!}}\approx {\left({\displaystyle \frac{k-1}{{|\left(h\right)}_k|}}+1\right)}^{{|\left(h\right)}_k|+{\displaystyle \frac{1}{2}}}{(m+|\left(h\right)}_k{|-1)}^{k-1}{e}^{-(k-1)}.$$

Since R tends to approach positive infinity, R is arbitrarily large. This leads us to deduce the conclusion that there exists $t_1>0,t_2>0,$ such that

$${|\left(l\right)}_m|=t_1{\displaystyle \frac{(G-\epsilon )R^{\rho }}{e}},\left|{\left(h\right)}_k\right|=t_2{\displaystyle \frac{(G-\epsilon )R^{\rho }}{e}},$$

with $t_1+t_2=1.$ Now we focus on the inequalities for $M(R,f)$

$${\displaystyle M(R,f)\ge \frac{(m-1)!(k-1)!e^{\frac{(G-\epsilon )R^{\rho }}{e\rho }}}{{\left(\frac{m-1}{t_1\frac{(G-\epsilon )R^{\rho }}{e}}+1\right)}^{t_1\frac{(G-\epsilon )R^{\rho }}{e}+\frac{1}{2}}{\left(\frac{m+t_1\frac{(G-\epsilon )R^{\rho }}{e}-1}{e}\right)}^{m-1}{\left(\frac{k-1}{t_2\frac{(G-\epsilon )R^{\rho }}{e}}+1\right)}^{t_2\frac{(G-\epsilon )R^{\rho }}{e}+\frac{1}{2}}{\left(\frac{k+t_2\frac{(G-\epsilon )R^{\rho }}{e}-1}{e}\right)}^{k-1}}.}$$

It is easy to verify the validity of the following three equalities

$$\underset{R\to +\infty }{lim}{\left[{\left({\displaystyle \frac{m-1}{t_1\frac{(G-\epsilon )R^{\rho }}{e}}}+1\right)}^{t_1{\displaystyle \frac{(G-\epsilon )R^{\rho }}{e}}+{\displaystyle \frac{1}{2}}}{\left(m+t_1{\displaystyle \frac{(G-\epsilon )R^{\rho }}{e}}-1\right)}^{m-1}{e}^{-(m-1)}\right]}^{{\displaystyle \frac{1}{R^{\rho }}}}=1,$$

$$\underset{R\to +\infty }{lim}{\left[{\left({\displaystyle \frac{k-1}{t_2\frac{(G-\epsilon )R^{\rho }}{e}}}+1\right)}^{t_2{\displaystyle \frac{(G-\epsilon )R^{\rho }}{e}}+{\displaystyle \frac{1}{2}}}{\left(k+t_2{\displaystyle \frac{(G-\epsilon )R^{\rho }}{e}}-1\right)}^{k-1}{e}^{-(k-1)}\right]}^{{\displaystyle \frac{1}{R^{\rho }}}}=1,$$

and

$$\underset{R\to +\infty }{lim}{[(m-1)!(k-1)!]}^{{\displaystyle \frac{1}{R^{\rho }}}}=1.$$

From Definition 3, it follows that

$$\begin{array}{ccc}{\displaystyle \tau }\hfill & =\hfill & {\displaystyle \underset{R\to +\infty }{lim\; sup}\frac{logM(R,f)}{R^{\rho }}=\underset{R\to +\infty }{lim\; sup}{logM(R,f)}^{\frac{1}{R^{\rho }}}}\hfill \\ \\ & \ge \hfill & {\displaystyle \underset{R\to +\infty }{lim\; sup}\left(log\left(e^{\frac{(G-\epsilon )}{e\rho }}\right)\right)=\frac{G-\epsilon }{e\rho }.}\hfill \end{array}$$

If $\epsilon$ tends to zero, then $\tau \ge {\displaystyle \frac{G}{e\rho }}.$ Let us prove the opposite inequality below. The definition of the supremum implies that

$${\left(\right|\left(l\right)}_m{|+|\left(h\right)}_k\left|\right)\parallel {\lambda }_{{\left(l\right)}_m{\left(h\right)}_k}{\parallel }^{{\displaystyle \frac{\rho }{\left(\right|{\left(l\right)}_m|+|{\left(h\right)}_k\left|\right)}}}\le G+\epsilon ,$$

where $\epsilon >0.$ By using the inequality, it follows that

$$\begin{array}{ccc}M(R,f)\hfill & \le \hfill & {\displaystyle \sum _{{|\left(l\right)}_m|=0}^{+\infty }\sum _{{|\left(h\right)}_k|=0}^{+\infty }\Vert {\lambda }_{{\left(l\right)}_m{\left(h\right)}_k}\Vert R^{{|\left(l\right)}_m{|+|\left(h\right)}_k|}}\hfill \\ \\ & =\hfill & {\displaystyle \sum _{{|\left(l\right)}_m|=0}^{N_0}\sum _{{|\left(h\right)}_k|=0}^{N_0}\Vert {\lambda }_{{\left(l\right)}_m{\left(h\right)}_k}\Vert R^{{|\left(l\right)}_m{|+|\left(h\right)}_k|}+\sum _{{|\left(l\right)}_m|=N_0+1}^{+\infty }\sum _{{|\left(h\right)}_k|=N_0+1}^{+\infty }{\left[\frac{(G-\epsilon )R^{\rho }}{{|\left(l\right)}_m{|+|\left(h\right)}_k|}\right]}^{\frac{{|\left(l\right)}_m{|+|\left(h\right)}_k|}{\rho }}=I_3+I_4.}\hfill \end{array}$$

By a simple calculation, $I_3$ can be estimated as

$$\begin{array}{ccc}{I}_3\hfill & \le \hfill & {\displaystyle \sum _{{|\left(l\right)}_m|=0}^{N_0}\sum _{{|\left(h\right)}_k|=0}^{N_0}\Vert {\lambda }_{{\left(l\right)}_m{\left(h\right)}_k}\Vert R^{2N_0}\le C_5{R}^{2N_0},}\hfill \end{array}$$

where $C_5$ is a positive real constant. For $I_4$, we obtain

$$\begin{array}{ccc}{I}_4\hfill & =\hfill & {\displaystyle \sum _{{|\left(l\right)}_m|=N_0+1}^{+\infty }\sum _{{|\left(h\right)}_k|=N_0+1}^{+\infty }{\left[\frac{(G+\epsilon )R^{\rho }}{{|\left(l\right)}_m{|+|\left(h\right)}_k|}\right]}^{\frac{{|\left(l\right)}_m{|+|\left(h\right)}_k|}{\rho }}}\hfill \\ \\ & \le \hfill & {\displaystyle \sum _{{|\left(l\right)}_m|=N_0+1}^{(G+\epsilon )R^{\rho }}\sum _{{|\left(h\right)}_k|=N_0+1}^{(G+\epsilon )R^{\rho }}{e}^{\frac{(G+\epsilon )R^{\rho }}{e\rho }}+\sum _{{|\left(l\right)}_m|=(G+\epsilon )R^{\rho }}^{+\infty }\sum _{{|\left(h\right)}_k|=(G+\epsilon )R^{\rho }}^{+\infty }{\left[\frac{(G+\epsilon )R^{\rho }}{{|\left(l\right)}_m{|+|\left(h\right)}_k|}\right]}^{\frac{{|\left(l\right)}_m{|+|\left(h\right)}_k|}{\rho }}}\hfill \\ \\ & \le \hfill & {\displaystyle \sum _{{|\left(l\right)}_m|=1}^{(G+\epsilon )R^{\rho }}\sum _{{|\left(h\right)}_k|=1}^{(G+\epsilon )R^{\rho }}{e}^{\frac{(G+\epsilon )R^{\rho }}{e\rho }}+\sum _{{|\left(l\right)}_m|=(G+\epsilon )R^{\rho }}^{+\infty }\sum _{{|\left(h\right)}_k|=(G+\epsilon )R^{\rho }}^{+\infty }{\left[\frac{(G+\epsilon )R^{\rho }}{(G+\epsilon )R^{\rho }+(G+\epsilon )R^{\rho }}\right]}^{\frac{{|\left(l\right)}_m{|+|\left(h\right)}_k|}{\rho }}}\hfill \\ \\ & \le \hfill & {\displaystyle C_6{e}^{\frac{(G+\epsilon )R^{\rho }}{e\rho }}(G+\epsilon )R^{\rho }+\sum _{{|\left(l\right)}_m|=1}^{+\infty }\sum _{{|\left(h\right)}_k|=1}^{+\infty }{\left[\frac{1}{2}\right]}^{\frac{{|\left(l\right)}_m{|+|\left(h\right)}_k|}{\rho }}}\hfill \\ \\ & \le \hfill & {\displaystyle C_6{e}^{\frac{(G+\epsilon )R^{\rho }}{e\rho }}(G+\epsilon )R^{\rho }+C_7,}\hfill \end{array}$$

where $C_6,C_7$ are positive real constants. To sum up, one can see that

$$M(R,f)\le C_5{R}^{2N_0}+C_6{e}^{{\displaystyle \frac{(G+\epsilon )R^{\rho }}{e\rho }}}(G+\epsilon )R^{\rho }+C_7.$$

Note that

$$\underset{R\to +\infty }{lim}{\displaystyle \frac{log{R}^{2N_0}}{R^{\rho }}}=2N_0\underset{R\to +\infty }{lim}{\displaystyle \frac{logR}{R^{\rho }}}=0,\underset{R\to +\infty }{lim}{\displaystyle \frac{log\left[(G+\epsilon )R^{\rho }\right]}{R^{\rho }}}=0.$$

Thus, the growth type of the biregular function can be written as

$$\begin{array}{ccc}\tau \hfill & =\hfill & {\displaystyle \underset{R\to +\infty }{lim\; sup}\frac{logM(R,f)}{R^{\rho }}}\hfill \\ \\ & \le \hfill & {\displaystyle \underset{R\to +\infty }{lim\; sup}\frac{log[C_5{R}^{2N_0}+C_6{e}^{\frac{(G+\epsilon )R^{\rho }}{e\rho }}(G+\epsilon )R^{\rho }+C_7]}{R^{\rho }}}\hfill \\ \\ & =\hfill & {\displaystyle \underset{R\to +\infty }{lim\; sup}\frac{log{e}^{\frac{(G+\epsilon )R^{\rho }}{e\rho }}}{R^{\rho }}=\frac{G+\epsilon }{e\rho }.}\hfill \end{array}$$

If $\epsilon$ tends to zero, then $\tau \le {\displaystyle \frac{G}{e\rho }}.$ Therefore, the theorem is proved. □

In the following example, the applications of Theorems 1 and 2 are shown.

Example 1. 

Let $f(x,y):R^{m+1}\times R^{m+1}\to C{l}_n,1<m\le n.$ We take $f(x,y)=e^{x+y},$ then the exponential function $e^{x+y}$ is a biregular function. Applying Lemma 2, the Taylor series representation of the biregular functions can be written as

$$e^{x+y}=\sum _{{|\left(l\right)}_m|=0}^{+\infty }\sum _{{|\left(l\right)}_m|=0}^{+\infty }{\displaystyle \frac{1}{{\left(l\right)}_m!{\left(l\right)}_m!}}{V}_{{\left(l\right)}_m}\left(x\right)W_{{\left(l\right)}_m}\left(y\right).$$

By Theorem 1 and Stirling’s formula, we have

$$\begin{array}{ccc}\rho \hfill & =\hfill & {\displaystyle \underset{{|\left(l\right)}_m|\to +\infty }{lim\; sup}\frac{{\left(2\right|\left(l\right)}_m{\left|\right)log\left(2\right|\left(l\right)}_m\left|\right)}{-log\parallel \frac{1}{c(m,{\left(l\right)}_m)}\frac{1}{{\left(l\right)}_m!{\left(l\right)}_m!}\frac{1}{c(m,{\left(l\right)}_m)}\parallel }}\hfill \\ \\ & =\hfill & {\displaystyle \underset{{|\left(l\right)}_m|\to +\infty }{lim\; sup}\frac{{\left(2\right|\left(l\right)}_m{\left|\right)log\left(2\right|\left(l\right)}_m\left|\right)}{{log[m(m+1)\cdots (m+|\left(l\right)}_m{|-1)]}^2}}\hfill \\ \\ & =\hfill & {\displaystyle \underset{{|\left(l\right)}_m|\to +\infty }{lim\; sup}\frac{{\left(\right|\left(l\right)}_m{\left|\right)log\left(2\right|\left(l\right)}_m\left|\right)}{{(m+|\left(l\right)}_m{|-1)log(m+|\left(l\right)}_m|-1)}=1,}\hfill \end{array}$$

which is the order of growth of the function $f.$

Theorem 2 and Stirling’s formula lead us to deduce the conclusion that

$$\begin{array}{ccc}\tau \hfill & =\hfill & {\displaystyle \frac{1}{e\rho }\underset{{|\left(l\right)}_m|\to +\infty }{lim\; sup}{\left(2\right|\left(l\right)}_m\left|\right)\parallel {\lambda }_{{\left(l\right)}_m{\left(l\right)}_m}{\parallel }^{\frac{\rho }{{|2\left(l\right)}_m|}}}\hfill \\ \\ & =\hfill & {\displaystyle \frac{2}{e}\underset{{|\left(l\right)}_m|\to +\infty }{lim\; sup}\left|{\left(l\right)}_m\right|{\left|\frac{1}{{\left(l\right)}_m!{\left(l\right)}_m!}\right|}^{\frac{1}{{2|\left(l\right)}_m|}}}\hfill \\ \\ & =\hfill & {\displaystyle \frac{2}{e}\underset{{|\left(l\right)}_m|\to +\infty }{lim\; sup}{e}^{log{|\left(l\right)}_m|-log\frac{{|\left(l\right)}_m|}{m}+1-\cdots }=2m,}\hfill \end{array}$$

which is the type of growth of the function $f.$

5. Conclusions

The study of the asymptotic growth behavior of Clifford variable functions is one of the central topics in Clifford analysis. In this paper, we develop an explicit relation between the Taylor series and the growth order and type for biregular functions. This will indeed allow us to calculate the growth order and type of function on the Taylor series without taking into account the maximum values of the function. One may further study the asymptotic growth behavior of biregular functions and related functions. This is a starting part for the general Wiman–Valiron theory for Clifford-valued functions with several variables.