Generalized Growth of Special Monogenic Functions

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 Rn for arbitrary finite n ∈ N and that are solutions of higherdimensionalCauchy-Riemann systems.These are often called Clifford holomorphic or monogenic functions. In order to make calculations more concise, we use the following notations, where m = (m 1 , . . . ,m n ) ∈ Nn 0 is ndimensional multi-index and x ∈ R:


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 R  for arbitrary finite  ∈ N and that are solutions of higherdimensional 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 m = ( 1 , . . .,  ) ∈ N  0 is dimensional multi-index and x ∈ R  : (1) Following Almeida and Kraußhar [1] and Constales et al. [2,3], we give some definitions and associated properties.
By { 1 ,  2 , . . .,   } we denote the canonical basis of the Euclidean vector space R  .The associated real Clifford algebra Cl 0 is the free algebra generated by R  modulo x 2 = −||x|| 2  0 , where  0 is the neutral element with respect to multiplication of the Clifford algebra Cl 0 .In the Clifford algebra Cl 0 , the following multiplication rule holds: where   is Kronecker symbol.A basis for Clifford algebra Cl 0 is given by the set {  :  ⊆ {1, 2, . . ., }} with Each  ∈ Cl 0 can be written in the form  = ∑      with   ∈ R.
Also, for  ∈ Cl 0 , we have ‖‖ ≤ 2 /2 ‖‖‖‖.Each paravector  ∈ R +1 \ {0} has an inverse element in R +1 which can be represented in the form  −1 = /|||| 2 .In order to make calculations more concise, we use the following notation: The generalized Cauchy-Riemann operator in R +1 is given by If  ⊆ R +1 is an open set, then a function  :  → Cl 0 is called left (right) monogenic at a point  ∈  if () = 0 (() = 0).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  m of degree |m|, defined as Let   be -dimensional surface area of  + 1-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 max

Preliminaries
Now following Abul-Ez and De Almeida [5], we give some definitions which will be used in the next section.

Definition 1.
Let Ω be a connected open subset of R +1 containing the origin and let () be monogenic in Ω.Then, () is called special monogenic in Ω, if and only if its Taylor's series near zero has the form (see [5], pp.1259) m be a special monogenic function defined on a neighborhood of the closed ball (0, ).Then, where (, ) = max ‖‖= ‖()‖ is the maximum modulus of () (see [5], pp.1260).
Definition 3. Let  : R +1 → Cl 0 be a special monogenic function whose Taylor's series representation is given by (9).Then, for  > 0 the maximum term of this special monogenic function is given by (see [5], pp.1260) Also the index m with maximal length |m| for which maximum term is achieved is called the central index and is denoted by (see [5], pp.1260) Definition 4. Let  : R +1 → Cl 0 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  : R +1 → Cl 0 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  0 denote the class of functions ℎ() satisfying the following conditions: (i) ℎ() is defined on [, ∞) and is positive, strictly increasing, and differentiable, and it tends to ∞ as  → ∞, for every function () such that () → ∞ as  → ∞.
Let Λ denote the class of functions ℎ() satisfying conditions (i) and (i) lim  → ∞ (ℎ()/ℎ()) = 1, (ii) for every  > 0; that is, ℎ() is slowly increasing.Following Srivastava and Kumar [7] and ), 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 () ∈ Λ, () ∈  0 , we define the generalized order (, , ) and generalized lower order (, , ) of () as If in above equation we put () = log  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 (),  −1 (), () ∈  0 , we define the generalized type (, , , ) and generalized lower type of special monogenic function () having nonzero finite generalized order as  (, , , )  (, , , ) = lim 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.

Main Results
We now prove the following.Theorem 6.Let  : R +1 →  0 be a special monogenic function whose Taylor's series representation is given by (9).If () ∈ Λ and () ∈  0 , 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  > 0 and all  >  0 (), we have Now, from inequality (19), we get Since (1/√ m ) ≤ 1, (see [11] Since () ∈  0 , (1+) ≃ ().Hence, proceeding to limits as |m| → ∞, we get Since  > 0 is arbitrarily small, so finally we get Now, we will prove that  ≥ .If  = ∞, then there is nothing to prove.So let us assume that 0 ≤  < ∞.Therefore, for a given  > 0 there exists  0 ∈ N such that, for all multi-indices m with |m| >  0 , 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  : R +1 →  0 be a special monogenic function whose Taylor's series representation is given by (9).Also let (), (), () ∈  0 and 0 <  < ∞; then the generalized type  of () is given as Now, first we prove that  ≥ .From ( 16), for arbitrary  > 0 and all  >  0 (), we have where  =  + .Now, using (19), we get Now, as in the proof of Theorem 6, here this inequality reduces to Putting or or Now, proceeding to limits as |m| → ∞, we get Since  > 0 is arbitrarily small, so finally we get Now, we will prove that  ≥ .If  = ∞, then there is nothing to prove.So let us assume that 0 ≤  < ∞.Therefore, for a given  > 0 there exists  0 ∈ N such that, for all multi-indices m with |m| >  0 , we have or Now, from the property of maximum modulus (see [11], pp.148), we have 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.