CUBO A Mathematical Journal Vol.11, No¯ 01, (1–20). March 2009 A Generalization of Wiman and Valiron’s theory to the Clifford analysis setting D. Constales 1 Department of Mathematical Analysis, Ghent University, Building S-22, Galglaan 2, B-9000 Ghent, Belgium. email: dc@cage.UGent.be R. De Almeida 2 Departamento de Matemática, Universidade de Trás-os-Montes e Alto Douro, P-5000-911 Vila Real, Portugal. email: ralmeida@utad.pt and R.S. Kraußhar 3 Department of Mathematics, Section of Analysis, Katholieke Universiteit Leuven, Celestijnenlaan 200-B, B-3001 Leuven (Heverlee), Belgium. email: soeren.krausshar@wis.kuleuven.be ABSTRACT The classical notions of growth orders, maximum term and the central index provide powerful tools to study the asymptotic growth behavior of complex-analytic functions. 1Financial support from BOF/GOA 01GA0405 of Ghent University gratefully acknowledged. 2Partial supported by the R&D unit Matemática e Aplicações (UIMA) of the University of Aveiro, through the Portuguese Foundation for Science and Technology (FCT), co-financed by the European Community fund FEDER, gratefully acknowledged. 3Financial support from FWO project G.0335.08 gratefully acknowledged. 2 D. Constales, R. De Almeida and R.S. Kraußhar CUBO 11, 1 (2009) This leads to much insight into the structure of the solutions to many two dimensional partial differential equations that are related to boundary value problems from harmonic analysis in the plane. In this overview paper we show how the classical techniques and results from Wiman and Valiron can be extended to the Clifford analysis setting in order to treat successfully analogous higher dimensional problems. RESUMEN Las nociones clásicas de orden de crecimiento, término máximo y de índice central pro- porcionan herramientas poderosas para estudiar el comportamiento de crecimiento asin- tótico de funciones complejas analíticas. Esto nos revela la estructura de las soluciones de varias ecuaciones diferenciales parciales de dimensión dos que son relacionadas con problemas de valores en la frontera venidos de análisis armónico en el plano. Mostramos como las técnicas clásicas y resultados de Wiman y Valiron pueden ser extendidas al contexto de análisis de Clifford para tratar con éxito problemas análogos de dimensión grande. Key words and phrases: monogenic functions, growth orders, growth type, maximum term, central index, Valiron’s inequalities, asymptotic growth, partial differential equations. Math. Subj. Class.: 30G35, 30D15. 1 Introduction The study of the asymptotic growth behavior of holomorphic and meromorphic functions in one and several complex variables is one of the central topics in complex analysis. This line of investigation started with early works of E. Lindelöf [22], A. Pringsheim [24], A. Wiman [26] and G. Valiron [25] and had its major breakthrough in the 1920s by works of R. Nevanlinna [23] and his school. Their results turned out to be very useful in the study of complex partial differential equations, see e.g. [20, 21] and elsewhere. This provides a strong motivation to also develop analogous methods for other function classes and in higher dimensions. One natural higher dimensional generalization of complex analysis is Clifford analysis. In this context one considers Clifford algebra valued solutions of the generalized Cauchy-Riemann system Df := ∂f ∂x0 + n∑ i=1 ∂f ∂xi ei = 0. (1) Solutions to this system are often called monogenic or Clifford holomorphic. Many classical theorems from complex analysis, such as for instance the Cauchy integral formula, the residue CUBO 11, 1 (2009) A Generalization of Wiman and Valiron’s theory to ... 3 theorem, Laurent expansion theorems, etc. carry over to the higher dimensional context using this operator, see for instance [8, 5, 7]. Nevertheless, as far as we know, questions concerning possible generalizations of Wiman-Valiron theory remained untouched for a long time. In [1] M.A. Abul-Ez and the first author introduced the notion of the growth order and the type for a particular subclass of entire Clifford holomorphic functions. See also the follow-up papers [2, 3]. In our recent papers [8, 9, 11, 13] we developed the basics for a generalized Wiman-Valiron theory for general entire monogenic functions and for monogenic Taylor series of finite convergence radius. We also managed to extend these techniques to the context of more general systems of partial differential equations, such as higher dimensional iterated Cauchy-Riemann systems [6, 7] and to polynomial Cauchy-Riemann systems equations with complex coefficients [10]. In this paper we give a concise overview over our results concerning the entire monogenic case. We show how the notions of growth orders, growth type, maximum term and the central index can be reasonably generalized to the Clifford analysis context. We exhibit how these tools can be applied to get insight in the asymptotics of related function classes and in the structure of solutions to related higher dimensional partial differential equations. This line of investigation should be regarded as a starting point to develop analogous methods for larger classes of functions that are in kernels of elliptic differential operators. We hope to get more insight in the structure of the solutions to larger classes of higher dimensional partial differential equations. 2 Preliminaries We begin by introducing the basic notions and concepts. For detailed information about Clifford algebras and their function theory we refer for example to [8, 1] and [7]. 2.1 Clifford algebras By {e1,e2, . . . ,en} we denote the canonical basis of the Euclidean vector space R n. The attached real Clifford algebra Cl0n is the free algebra generated by R n modulo the relation x 2 = −‖x‖2e0, where x ∈ Rn and e0 is the neutral element with respect to multiplication of the Clifford algebra Cl0n. In the Clifford algebra Cl0n the following multiplication rules hold eiej + ejei = −2δije0, i,j = 1, · · · ,n, where δij is the Kronecker symbol. A basis for the Clifford algebra Cl0n is given by the set {eA : A ⊆ {1, · · · ,n}} with eA = el1el2 · · ·elr , where 1 ≤ l1 < · · · < lr ≤ n, e∅ = e0 = 1. Each a ∈ Cl0n can be written in the form a = ∑ A aAeA with aA ∈ R. Two examples of real Clifford algebras are the complex number field C and the Hamiltonian skew field H. 4 D. Constales, R. De Almeida and R.S. Kraußhar CUBO 11, 1 (2009) The conjugation anti-automorphism in the Clifford algebra Cl0n is defined by a = ∑ A aAeA, where eA = elrelr−1 · · ·el1 and ej = −ej for j = 1, · · · ,n, e0 = e0 = 1. The linear subspace span R {1,e1, · · · ,en} = R ⊕ R n ⊂ Cl0n is the so-called space of paravectors z = x0 + x1e1 + x2e2 + · · · + xnen which we simply identify with R n+1. The term x0 =: Sc(z) is called the scalar part of the paravector z and x := x1e1 + · · · + xnen =: V ec(z) its vector part. A scalar product between two Clifford numbers a,b ∈ Cl0n is defined by 〈a,b〉 := Sc(ab) and the Clifford norm of an arbitrary a = ∑ A aAeA is ‖a‖ = ( ∑ A |aA| 2 ) 1/2. Any paravector z ∈ Rn+1\{0} has an inverse element in Rn+1 given by z−1 = z/‖z‖2. In order to present the calculations in a more compact form, the following notations will be used, where m = (m1, . . . ,mn) ∈ N n 0 is an n-dimensional multi-index: x m := x m1 1 · · ·xmn n , m! := m1! · · ·mn!, |m| := m1 + · · · + mn. By τ(i) we denote the multi-index (m1, . . . ,mn) with mj = δij for 1 ≤ j ≤ n. 2.2 Clifford analysis One way to generalize complex function theory to higher dimensional hypercomplex spaces is offered by the Riemann approach which considers Clifford algebra valued functions defined in Rn+1 that are annihilated by the generalized Cauchy-Riemann operator D := ∂ ∂x0 + n∑ i=1 ei ∂ ∂xi . (2) If U ⊂ Rn+1 is an open set, then a real differentiable function f : U → Cl0n is called left (right) monogenic or Clifford holomorphic at a point z ∈ U if Df(z) = 0 (or fD(z) = 0). Functions that are left monogenic in the whole space are also called left entire. The notion of left (right) monogenicity in Rn+1 provides indeed a powerful generalization of the concept of complex analyticity to Clifford analysis. Many classical theorems from complex analysis could be generalized to higher dimensions by this approach, we refer e.g. to [8]. One important tool is the generalized Cauchy integral formula. Let us denote by An+1 the n-dimensional surface “area” of the (n + 1)-dimensional unit ball, and by q0(z) = z ‖z‖n+1 the Cauchy kernel function. Then every function f that is left monogenic in a neighbourhood of the closure D of a domain D satisfies f(z) = 1 An+1 ∫ ∂D q0(z − w) dσ(w) f(w), (3) CUBO 11, 1 (2009) A Generalization of Wiman and Valiron’s theory to ... 5 where dσ(w) is the paravector-valued outer normal surface measure, i.e., dσ(w) = n∑ j=0 (−1)jejdw0 ∧ · · · ∧ d̂wj ∧ · · · ∧ dwn with d̂wj = dw0 ∧ · · · ∧ dwi−1 ∧ dwi+1 ∧ · · · ∧ dwn. It is important to mention that the set of left (right) monogenic functions forms only a Clifford right (left) module for n > 1. In contrast to complex analysis, the ordinary powers of the hypercomplex variables are not null-solutions to the generalized Cauchy-Riemann system. In Clifford analysis these are substituted by the Fueter polynomials. These are defined by Pm(z) = 1 |m|! ∑ π∈perm(m) zπ(m1) · · ·zπ(mn) where perm(m) is the set of all permutations of the sequence (m1, . . . ,mn) and zi := xi −x0ei for i = 1, . . . ,n and P0(z) := 1. In this paper we prefer to work with the slightly modified Fueter polynomials Vm(z) := m!Pm(z) (4) which turns out to be more convenient in our calculations. These polynomials play the analogous role of the complex power functions in the Taylor series representation of a monogenic function. More precisely, if f is a left monogenic function in a ball ‖z‖ < R, then for all ‖z‖ ≤ r with 0 < r < R f(z) = +∞∑ |m|=0 Vm(z)am, where the elements am are Clifford numbers which — as a consequence of Cauchy’s integral formula (3) — are uniquely defined by am = 1 m!An+1 ∫ ‖z‖ 0 satisfying lim inf r→∞ M(r,f) r|s| = L < ∞, (9) then f(z) = |s|∑ |m|=0 Vm(z)am. In order to characterize larger classes of monogenic functions by their asymptotic growth behavior it turned out to be convenient to introduce growth orders for monogenic functions [1, 3, 13]. For convenience we recall its definition. First we need, cf. e.g. [20]: CUBO 11, 1 (2009) A Generalization of Wiman and Valiron’s theory to ... 7 Definition 1. Let α ≥ 0. Then the plus logarithm is defined by log + (α) := max{0, log(α)}. (10) In the same way as in the planar case (see [20]) one introduces in the Clifford analysis setting (see also [2, 13]): Definition 2. (Order of growth) Let f : Rn+1 → Cl0n be an entire function. Then ρ(f) = ρ := lim sup r→∞ log + (log + M(r,f)) log(r) , 0 ≤ ρ ≤ ∞ (11) is called the order of growth of the function f. We further introduce λ(f) = λ : lim inf r→∞ log + (log + M(r,f)) log(r) , 0 ≤ λ ≤ ∞ (12) as the inferior order of growth of f. If ρ = λ, then we say that f is a function of regular growth. If ρ > λ then f has irregular growth. To get a finer classification of the growth behavior within the set of monogenic functions that have the same growth order, one further introduces the growth type of a monogenic function as follows, cf. [9]. Definition 3. For an entire monogenic function f : Rn+1 → Cl0n of order ρ (0 < ρ < ∞) the (growth) type is defined by τ(f) = τ := lim sup r→∞ log + M(r,f) rρ . Let us start with discussing some particular examples. In [13] we have proved that the following higher dimensional generalizations of the exponential function all have growth order equal to 1: (i) The monogenic plane wave exponential function from [1] defined for m ∈ Rn \ {0} by f1(m,z) := (|m| + im)e −|m|x0ei, (ii) The monogenic generalization exponential function from [8] f2(z) = exp(x0,x1, . . . ,xn) = e x1+···+xn cos(x0 √ n) − ex1+···+xn 1 √ n (e1 + · · · + en) sin(x0 √ n) (iii) The quaternion-valued 3-fold periodic exponential function from [17] given by f3(z) := Exp0(z) + e1Exp1(z) + e2Exp2(z) + e3Exp3(z) 8 D. Constales, R. De Almeida and R.S. Kraußhar CUBO 11, 1 (2009) where Exp0(z) = e x0 (cos( x1 √ 3 ) cos( x2 √ 3 ) cos( x3 √ 3 ) − sin( x1 √ 3 ) sin( x2 √ 3 ) sin( x3 √ 3 )) Exp1(z) = e x0 √ 3 3 (sin( x1 √ 3 ) cos( x2 √ 3 ) cos( x3 √ 3 ) + cos( x1 √ 3 ) sin( x2 √ 3 ) sin( x3 √ 3 )) Exp2(z) = e x0 √ 3 3 (cos( x1 √ 3 ) sin( x2 √ 3 ) cos( x3 √ 3 ) + sin( x1 √ 3 ) cos( x2 √ 3 ) sin( x3 √ 3 )) Exp3(z) = e x0 √ 3 3 (sin( x1 √ 3 ) sin( x2 √ 3 ) cos( x3 √ 3 ) + cos( x1 √ 3 ) cos( x2 √ 3 ) sin( x3 √ 3 )). However, not all of these higher dimensional analogues of the exponential function turn out to be of the same type. For the first and the second example we can determine the value of M(r,f) exactly. We obtain that M(r,f1) = ‖|m| + im‖e |m|r, thus τ(f1) = |m|. For f2 we obtain that ‖f2(z)‖ = e x1+···+xn which implies that M(r,f2)e nr and therefore τ(f2) = n. For the third example we are able to establish a useful lower and upper bound estimate for the maximum modulus. By a direct calculation we obtain that √ 3 3 er ≤ M(r,f3) ≤ e r so that τ(f3) = 1. When |m| = 1, f1 and f3 thus share the same growth order and growth type. After having discussed some concrete examples, let us now turn to the more general framework. As a consequence of Cauchy’s integral formula we can establish, cf. [13]: Theorem 2. Let f be a left entire function in Rn+1. By fi we denote the function fi := ∂ ∂xi f and Mi(r) := max‖z‖=r { ‖fi(z)‖ } where r > 0 and i ∈ {0, . . . ,n}. Then ρ(f) = ρ′(f) and λ(f) = λ′(f), where ρ′(f) := lim sup r→∞ log + (log + (M′(r))) log(r) λ′(f) := lim inf r→∞ log + (log + (M′(r))) log(r) , for M′(r) := max i=0,1,...,n {Mi(r)}. Proof. We consider an arbitrary rectifiable curve from the origin to z. Then f(z) = f(0) + 1∫ 0 n∑ i=0 xi fi(tz)dt. CUBO 11, 1 (2009) A Generalization of Wiman and Valiron’s theory to ... 9 For z ∈ Rn+1 with ‖z‖ = r we get ‖f(z)‖ ≤ ‖f(0)‖ + r n∑ i=0 Mi(r) ≤ ‖f(0)‖ + r(n + 1)M′(r). Therefore M(r) ≤ ‖f(0)‖ + r(n + 1)M′(r). Applying some properties of log+ we obtain that log + (M(r,f)) ≤ log+(‖f(0)‖) + log+(r(n + 1)) + log+(M′(r)) + log(2). This in turn leads to ρ(f) ≤ ρ′(f) and λ(f) ≤ λ′(f). To show the inequality in the other direction, we apply on fi Cauchy’s integral formula: fi(z) = 1 An+1 ∫ ‖ζ−z‖=R−r qτ(i)(ζ − z)dσ(ζ)f(ζ). (13) Applying the estimate (6) to (13) we hence obtain ‖fi(z)‖ ≤ 1 An+1 ∫ ‖ζ−z‖=R−r n (R − r)n+1 M(R)dS from which we then infer that Mi(r) ≤ n (R − r) M(R,f). In particular, for M′(r) := max i=0,1,...,n {Mi(r)} we have M′(r) ≤ n (R − r) M(R,f). (14) Replacing R = 2r into (14) yields: M′(r) ≤ n r M(2r,f). Thus, log + M′(r) ≤ log+ M(2r,f) + log+ ( n r ) . For what follows we may assume without loss of generality that r > n. Hence, log + M′(r) ≤ log+ M(2r,f). Furthermore, log + (log + M′(r)) log(r) ≤ log + (log + M(2r,f)) log(r) = log + (log + M(2r,f)) log(2r) log + (2r) log(r) 10 D. Constales, R. De Almeida and R.S. Kraußhar CUBO 11, 1 (2009) Thus, we have log + (log + M′(r)) log(r) ≤ log + (log + M(2r,f)) log(2r) ( log 2 log(r) + 1 ) from which we can infer directly that ρ(f) ≥ ρ′(f) and λ(f) ≥ λ′(f). After having computed the growth order ρ(f) resp. λ(f) of a monogenic function f, we know the maximal value of the growth order of all partial derivatives. Notice that Cauchy’s integral formula was an important ingredient in the proof of this state- ment. To establish these types of results in a more general framework it is thus indeed important to work in classes of functions that are in the kernel of a differential operator that satisfy a Cauchy type integral formula. See also our paper [7] where we treated more general solutions to higher dimensional iterated Cauchy-Riemann and Dirac operators. The class of monogenic functions actu- ally provides us with the canonical and easiest example of a function class which satisfies a Cauchy type integral formula. 4 Generalizations of some theorems from Valiron to Clifford analysis In this section we present some generalizations of some classical theorems from G. Valiron to the Clifford analysis setting. To this end we first define the maximum term and central index which are associated to the Taylor series of a monogenic function. Let us consider a left entire function f(z) = +∞∑ |l|=0 Vl(z)al. Let r > 0 be a fixed real. If f is transcendental, i.e. infinitely many al 6= 0, then lim |l|→∞ ‖al‖r |l| = 0. The following expression thus is well-defined: Definition 4. (Maximum term) Let f : Rn+1 → Cl0n be a left entire function with the Taylor series representation f(z) +∞∑ |l|=0 Vl(z)al. Furthermore, let r > 0 be a fixed real. Then the associated maximum term is defined by µ(r) := µ(r,f) := max |l|≥0 {‖al‖r |l|}. (15) CUBO 11, 1 (2009) A Generalization of Wiman and Valiron’s theory to ... 11 We further introduce Definition 5. (Central indices) Let f(z) = +∞∑ |l|=p Vl(z)al be a left entire function. For r > 0 the index (or the indices) m with maximal length |m| with µ(r)‖am‖r |m| is (are) called central index (indices) which shall be denoted by ν(r) = ν(r,f) = m. By ν(0) we denote the indices l which satisfy |l| = p. The following theorem proved in [13], providing us with a direct generalization of Valiron theorem, states a relation between the maximum term, central index and the maximum modulus. Theorem 3. If f : Rn+1 → Cl0n is a left entire function, then for all 0 < r < R M(r) ≤ µ(r) [ |ν(R)|(1 + |ν(R)|)n−1 + ( R R − r ) n ] . (16) Proof. The function f is assumed to be left entire. Thus, it can be represented by f(z) = +∞∑ |l|=0 Vl(z)al where infinitely many al 6= 0, since f is transcendental. From the maximum modulus theorem for monogenic functions we infer that for 0 < r < R: M(r) ≤ +∞∑ |l|=0 ‖al‖r |l| = |ν(R)|−1∑ |l|=0 ‖al‖r |l| + +∞∑ |l|=|ν(R)| ‖al‖r |l| ≤ |ν(R)|−1∑ |l|=0 µ(r) + +∞∑ |l|=|ν(R)| ‖al‖r |l|. (17) In view of |ν(R)|−1∑ |l|=0 1 = ∑ |l|=0 1 + ∑ |l|=1 1 + · · · + ∑ |l|=|ν(R)|−1 1 = 1 + ((n − 1) + 1)! (n − 1)!1! + · · · + [(n − 1) + (|ν(R) − 1)]! (n − 1)!(|ν(R)| − 1)! ≤ |ν(R)| [ [(n − 1) + |ν(R)| − 1]! (n − 1)!(|ν(R)| − 1)! ] where we use that for all n ≥ 1 the inequality (n − 1 + k)! (n − 1)!k! ≤ (n − 1 + (k + 1))! (n − 1)!(k + 1)! 12 D. Constales, R. De Almeida and R.S. Kraußhar CUBO 11, 1 (2009) holds, which itself can be verified by a straightforward induction over k. Further, |ν(R)| [ [(n − 1) + |ν(R)| − 1]! (n − 1)!(|ν(R)| − 1)! ] = |ν(R)| [ (|ν(R)| + n − 2)(|ν(R)| + n − 3) · · · (|ν(R)| + 1)|ν(R)| (n − 1)! ] = |ν(R)| [ |ν(R)| + n − 2 n − 1 · |ν(R)| + n − 3 n − 2 · · · · · |ν(R)| + 1 2 · |ν(R)| 1 ] ≤ |ν(R)| [ (1 + |ν(R)| n − 1︸ ︷︷ ︸ ≤1+|ν(R)| )(1 + |ν(R)| n − 2︸ ︷︷ ︸ ≤1+|ν(R)| ) · · · · · (1 + |ν(R)| 1︸ ︷︷ ︸ =1+|ν(R)| ) ] ≤ |ν(R)| [ (1 + |ν(R)|)n−1 ] . Inserting these results into (17) leads to M(r) ≤ µ(r)|ν(R)| [ (1 + |ν(R)|)n−1 ] + +∞∑ |l|=|ν(R)| ‖al‖r |l| ‖aν(r)‖r |ν(r)|R|l+ν(R)| ‖aν(r)‖r |ν(R)|R|l+ν(R)| = µ(r)|ν(R)| [ (1 + |ν(R)|)n−1 ] + µ(r) +∞∑ |l|=|ν(R)| ‖al‖R |l|R|ν(R)|r|l| ‖aν(R)‖R |ν(R)|R|l|r|ν(R)| ≤ µ(r)|ν(R)| [ (1 + |ν(R)|)n−1 ] + µ(r) +∞∑ |l|=|ν(R)| ( r R )|l|−|ν(R)| = µ(r) [ |ν(R)| [ (1 + |ν(R)|)n−1 ] + ( R R − r ) n ] . G. Valiron has also proved that an entire complex-analytic function f of finite order shows the asymptotic behavior log(M(r,f)) ≈ log(M′(r)) where M′ is the maximum modulus of the deriva- tive. The classical proof is based on the fact that one has the relation µ(r) ≤ M(r,f) for a complex-analytic function in one complex variable. In the framework of working with Clifford algebra valued monogenic Taylor series which are built with the Fueter polynomials, we have a more complicated upper bound estimate of the form µ(r) ≤ n(n + 1) · · · (n + |ν(r)| − 1) ν(r)! M(r,f) for a central index ν(r). This is a consequence of the higher dimensional Cauchy’s inequality. Notice that this is a sharp upper bound, cf. [5]. Adapting the classical methods based on Cauchy’s inequality to the higher dimensional case provides us only with a weaker result in the Clifford analysis setting. In [13] we proved that CUBO 11, 1 (2009) A Generalization of Wiman and Valiron’s theory to ... 13 Proposition 1. For a left entire function f : Rn+1 → Cl0n of order ρ and inferior order λ set ρ1 := lim sup r→∞ log + log + µ(r) log(r) , ρ2 := lim sup r→∞ log + |ν(r)| log(r) , (18) and λ1 := lim inf r→∞ log + log + µ(r) log(r) , λ2 : lim inf r→∞ log + |ν(r)| log(r) . (19) Then ρ ≤ ρ1 = ρ2 and λ ≤ λ1 = λ2. Remark: In the two-dimensional complex case where we have µ(r) ≤ M(r) these methods allow one to establish the stronger result ρ = ρ1 = ρ2 and λ = λ1 = λ2, as shown for instance in [20, Theorem 4.5]. With this proposition we may establish the following theorem. It provides us with a weaker analogy of Valiron’s asymptotic result on the growth of the logarithm of the derivative of a given analytic function: Theorem 4. If f : Rn+1 → Cl0n is left entire with ρ2(f) < ∞, then lim sup r→∞ log Mi(r) log µ(r) ≤ 1 (20) where Mi(r) := max ‖z‖=r {∥∥∥ ∂ ∂xi f(z) ∥∥∥ } for i = 1, . . . ,n. 5 The growth behavior and the Taylor coefficients of a mono- genic function In general it is difficult to determine the precise value of the maximum modulus. In many cases it is even complicated to just get a useful estimate on M(r,f) from below. In this section we present an explicit relation between the Taylor coefficients and the growth order and the type of an entire monogenic function. This allows us to compute the growth type directly on the knowledge of the Taylor coefficients without any knowledge on the maximum modulus of the function. Notice that Taylor series actually are a natural method to construct and to define entire monogenic functions. Recall, that the product of two monogenic functions is not monogenic anymore in general. Hence it is natural to construct entire monogenic functions in an additive way, for instance by its Taylor series. The following two theorems provide us with higher dimensional generalizations in the Clifford analysis setting of two theorems proved by Lindelöf and Pringsheim for complex analytic functions. In [8] we established 14 D. Constales, R. De Almeida and R.S. Kraußhar CUBO 11, 1 (2009) Theorem 5. For an entire monogenic function f : Rn+1 → Cl0n, with Taylor series representation f(z) = ∑ +∞ |m|=0 Vm(z)am let Π = lim sup |m|→+∞ |m| log |m| − log ‖ 1 c(n,m) am‖ . (21) Then we have ρ(f) = Π. Remark: In the cases where ‖am‖ = 0 one puts lim sup |m|→+∞ |m| log |m| − log ‖ 1 c(n,m) am‖ := 0. The following theorem, proved in [9], also relates the growth type with the Taylor coefficients of an entire monogenic function: Theorem 6. Let f : Rn+1 → Cl0n be an entire monogenic function with Taylor series expansion f(z) = ∑ +∞ |m|=0 Vm(z)am with order ρ (0 < ρ < +∞) and Θ = lim sup |m|→+∞ |m| ( ‖am‖ ) ρ |m| . (22) Then Θ = τeρ, where τ is the type of f. In turn, Theorem 6 allows us to construct immediately examples of entire monogenic Taylor series of non-zero finite growth order ρ of any arbitrary real growth type 0 ≤ τ ≤ +∞. Recalling from [9], we start with Proposition 2. Suppose that f : Rn+1 → Cl0n is an entire monogenic function. If ρ(f) = 0, then τ(f) = ∞ or f is a constant. Proof. If ρ(f) = 0, then τ(f) = lim sup r→+∞ log + M(r,f). If τ(f) 6= ∞, then lim sup r→+∞ M(r,f) = eτ, which implies that ‖f(z)‖ ≤ eτ for all z ∈ Rn+1. As a consequence of Theorem 1, f must be a constant. Example: Consider P(z) to be an arbitrary left monogenic polynomial, i.e. there exist Clifford numbers am ∈ Cl0n and N ∈ N0 such that P(z) N∑ |m|=0 Vm(z)am. From [13, Theorem 3.1] we know that ‖P(z)‖ ≤ ( (n − 1 + N)! (n − 1)!N! + ε ) ‖aN‖r N, (23) CUBO 11, 1 (2009) A Generalization of Wiman and Valiron’s theory to ... 15 where N is the index of length N for which ‖aN‖ ≥ ‖am‖ for all |m| = N, with an arbitrarily small ε > 0 for r sufficiently large. Thus, it follows with C(N) : ( (n−1+N)! (n−1)!N! + ε ) ‖aN‖ that lim r→∞ log + (log + (M(r,P)) log(r) ≤ lim r→∞ log + (log + (C(N)rN) log(r) 0. Thus, all monogenic polynomials satisfy ρ(P) = λ(P) = 0, like in the complex case. In view of Proposition 2 the growth type τ equals +∞. More generally, we could establish, cf. [9]: Proposition 3. Let 0 < δ < +∞ and 0 < λ < +∞ be arbitrary real numbers. Then f(z) = +∞∑ |m|=1 c(n, m)|m|− |m| δ Vm ( ( λeδ nδ ) 1 δ z ) (24) is an entire monogenic function of growth order ρ = δ and growth type τ = λ. Proof. By applying Hadamard’s formula, one may directly conclude that the convergence ra- dius of (24) is +∞. Since the Fueter polynomials Vm are homogeneous polynomials of total degree |m|, f can directly be rewritten in the form f(z) = +∞∑ |m|=1 Vm(z)am with am = c(n, m)|m| − |m| δ ( λeδ n δ ) |m| δ . According to Theorem 5, the growth order of f therefore equals ρ(f) = lim sup |m|→+∞ |m| log |m| − log ‖ 1 c(n,m) am‖ = lim sup |m|→+∞ |m| log |m| − log ∣∣∣|m|− |m| δ ( λeδ n δ ) |m| δ ∣∣∣ = lim sup |m|→+∞ δ log |m| log |m| − log(λeδ nδ ) = δ. By Theorem 6 we indeed furthermore obtain that τ(f) = 1 eδ lim sup |m|→+∞ c(n, m) δ |m| λeδ nδ = λ nδ lim sup M→+∞ max |m|=M c(n, m) δ M = λ nδ lim sup M→+∞ [ (n + M − 1)! (n − 1)!(M n )!n ] δ M = λ nδ lim sup M→+∞ [ 1 [ (n − 1)! ] δ M ( (n + M − 1)n+M−1+ 1 2 e−(n+M−1) ) δ M (( M n )M n + 1 2 e− M n )nδ M = λ nδ lim sup M→+∞ ( n + M − 1 M n ) δ λ. 16 D. Constales, R. De Almeida and R.S. Kraußhar CUBO 11, 1 (2009) By analogous calculations one can further show that Proposition 4. Let 0 < ρ < ∞. The functions g(z) = +∞∑ |m|=2 c(n, m) [ log |m| |m| ] |m| ρ Vm(z) h(z) = +∞∑ |m|=2 c(n, m) [ 1 |m| log |m| ] |m| ρ Vm(z) are entire monogenic functions of growth order ρ and τ(g) = +∞ and τ(h) = 0. 6 Applications to partial differential equations In this section we show how the notions of the maximum term and the central indices can be applied to obtain some information on the structure of the solutions of certain class of higher dimensional partial differential equations. To proceed in this direction it turns out to be useful to first establish a relation between the asymptotic behavior of the maximum term of a monogenic function and that of their iterated radial derivatives. In [13] we established: Theorem 7. Let f : Rn+1 → Cl0n be a left entire function. Then for all k ∈ N holds asymptotically 1 |ν(r)|k ∥∥∥[Ek]f(z) − f(z) ∥∥∥ ≤ Cµ(r)|ν(r)|− 1 2 +ε, r 6∈ F (25) where E := n∑ i=0 xi ∂ ∂xi is the Euler operator on Rn+1, C is a real positive constant and F is a set of finite logarithmical measure. As a direct consequence of Theorem 7 one obtains Proposition 5. Let 0 < δ < 1 2 . We assume that ‖z‖ = r and that r be sufficiently large. Suppose further that the relation ‖f(z)‖ > µ(r)|ν(r)|− 1 2 +δ (26) is satisfied for all those z that belong to a neighborhood Vz0 of a point z0 in which we have ‖z0‖ = r and ‖f(z0)‖ = max ‖z‖=r {‖f(z)‖}. Then for all k ∈ N holds asymptotically 1 |ν(r)|k [Ek]f(z) − f(z) = o(1)f(z), r 6∈ F, (27) where F is again a set of finite logarithmical measure. CUBO 11, 1 (2009) A Generalization of Wiman and Valiron’s theory to ... 17 Remark: This statement provides us with an analogy in the context of Clifford analysis of the classical result [20, Theorem 21.3] which states that entire complex-analytic functions that satisfy ‖f(z)‖ > M(r,f)[ν(r)]− 1 4 +δ have the asymptotic behavior f(m)(z) = ( ν(r) z ) m (1 + o(1))f(z). In the Clifford analysis setting one thus obtains a similar asymptotic result when substituting the complex operator z d dz by the higher dimensional Euler operator E. With these tools in hand we can study the structure of the solutions to some classes of partial differential equations. As a concrete example we present the following special case of an unpublished result from [12]: Theorem 8. Let f be an entire monogenic function of finite order ρ2 < ∞. Let ‖z‖ = r and assume that r is sufficiently large. Suppose further that the relation ‖f(z)‖ > µ(r)|ν(r)|− 1 2 +δ, r 6∈ F is satisfied for all those z that belong to a neighborhood Vz0 of a point z0 in which we have ‖z0‖ = r and ‖f(z0)‖ = max ‖z‖=r {‖f(z)‖}. Let Mj[f] = aj k∏ i=0 (Ei(f))ni, where aj are polynomials of degree j, and Mj[f] has degree γMj = k∑ i=0 ni and weight ΓMj = k∑ i=0 ini. Let Q[f] = s∑ j=0 Mj[f] be of degree γQ and weight ΓQ. If γQγM0 then the differential equation Q[f] = 0 has no transcen- dental entire solutions. Proof. If Q[f] = 0, then M0[f] = − s∑ j=1 Mj[f]. >From the definition of Mj it follows that a0 [ k∏ i=0 (Ei(f))ni ] M0 = − s∑ j=1 [ aj k∏ i=0 (Ei(f))ni ] Mj . Applying Proposition 5, we obtain that ‖a0‖|ν(r)| ΓM0 ‖f(z)‖γM0 ≤ s∑ j=1 ( ‖aj‖|ν(r)| ΓMj ‖f(z)‖ γMj ) . 18 D. Constales, R. De Almeida and R.S. Kraußhar CUBO 11, 1 (2009) Since a0 is a non zero constant and aj are polynomials of degree j, taking the maximum over the norm, and applying (23) leads to |ν(r)|ΓM0 M(r,f)γM0 ≤ |ν(r)|ΓQM(r,f)γQ−1 s∑ j=1 max ‖z‖r ‖aj‖ ‖a0‖ ≤ |ν(r)|ΓQM(r,f)γQ−1rα. (28) Therefore, in view of γQ = γM0 one has M(r,f) ≤ |ν(r)|ΓQ−ΓM0 rα. (29) For ΓQ − ΓM0 < 0 it follows lim inf r→∞ M(r,f) rα ≤ lim inf r→∞ |ν(r)|ΓQ−ΓM0 = 0 which implies that f is a polynomial, as a consequence of Theorem 1. Let us now consider the case where ΓQ − ΓM0 > 0. Since ρ2 < ∞, we have that |ν(r)| < r ρ2+ǫ for ǫ > 0. Therefore, there exists a β > (ΓQ − ΓM0 )(ρ2 + ǫ) such that lim inf r→∞ M(r,f) rβ+α ≤ lim inf r→∞ |ν(r)|ΓQ−ΓM0 rβ ≤ lim inf r→∞ r(ΓQ−ΓM0 )(ρ2+ǫ)−β = 0 which implies that f is a polynomial, as a consequence of Theorem 1. Concluding remarks: One can apply these techniques to obtain analogous statements for much more general classes of partial differential equations. 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