CUBO A Mathematical Journal

Vol.11, No¯ 01, (73–100). March 2009

On mapping properties of monogenic functions

K. Gürlebeck and J. Morais

Institute of Mathematics / Physics,

Bauhaus-University,

Weimar, Germany.

email: guerlebe@fossi.uni-weimar.de

email: jmorais@mat.ua.pt

ABSTRACT

Main goal of this paper is to study the description of monogenic functions by their

geometric mapping properties. At first monogenic functions are studied as general

quasi-conformal mappings. Moreover, dilatations and distortions of these mappings

are estimated in terms of the hypercomplex derivative. Then pointwise estimates from

below and from above are given by using a generalized Bohr’s theorem and a Borel-

Carathéodory theorem for monogenic functions. Finally it will be shown that mono-

genic functions can be defined as mappings which map infinitesimal balls to special

ellipsoids.

RESUMEN

El principal objetivo de este artículo es estudiar la descripción de funciones monogéni-

cas a través de las propiedades geométricas de sus mapeos. Primero son estudiadas

funciones monogénicas como aplicaciones casi-conformes generales. Además, dilata-

ciones y distorciones de estas aplicaciones son estimadas en términos de la derivada

hipercompleja. Entonces estimativas puntuales por abajo y por arriba son dadas us-

ando un teorema de Bohr generalizado y un Teorema de Borel-Carathéodory para

funciones monogénicas. Finalmente es demostrado que funciones monogénicas pueden

ser definidas como mapeos que aplican bolas infinitesimales en elipsoides especiales.

Key words and phrases: monogenic functions, quasi-conformal mappings, geometric mapping

properties.



74 K. Gürlebeck and J. Morais CUBO
11, 1 (2009)

Math. Subj. Class.: 30G35.

1 Introduction

Quaternionic analysis provides us with a spatial analogue of the one-dimensional complex function

theory in the plane. Generalizing ideas from the complex case to the higher dimensional real

Euclidean space, quaternionic analysis is applied to construct solutions of some classes of partial

differential equations for vector-valued functions in three or four dimensions (see, e.g., [17, 18, 31,

32]), involving the Dirac operator or a generalized Cauchy-Riemann operator, respectively.

We are mainly interested in the study of spatial generalizations of holomorphic or anti-

holomorphic functions. The crucial fact of such functions (transformations) is that they describe

essentially mappings of the unit disk onto or into the unit disk transforming partial differential equa-

tions to other differential equations and moreover, they embrace Cauchy’s inequalities, maximum

modulus principle, Bohr’s Theorem, Schwarz Lemma, Hadamard Theorem and others. Since they

provide with the best description of the pointwise behavior from a given function, at first one has

to ask whether or not those results can be generalized to ”holomorphic” (resp. anti-holomorphic)

functions in higher dimensions (sections 3 and 4). In addition, to deal with these results it is also

necessary to study the mapping properties of such functions more detailed. It is already known,

as the ideas of [7] and [5] show, that these functions preserve geometrical properties like length,

distance and angles, while mapping domains to the ball. Therefore they are applied as well for the

transformation of differential equations. Our main task is to characterize such mappings which

map technically relevant domains to mathematically simple domains and to find out if such class

of functions leave the differential equations and/or some geometrical properties invariant.

At this point it is of interest to know that in contrast to the situation in the plane, the set of

conformal mappings is restricted only to the set of Möbius transformations. But the theory of gen-

eralized holomorphic functions (by historical reasons they are also called monogenic functions, cf.

[8]) does not cover the set of Möbius tranformations in Rn+1, and since the Möbius transformations

are not monogenic, one can only expect that monogenic functions represent certain quasi-conformal

mappings. On the other hand, the class of all quasi-conformal mappings is much bigger than the

class of monogenic functions. The question arises if monogenic functions correspond to a special

subclass of quasi-conformal mappings.

In the paper [28], the concept of monogenic-conformal mappings realized by functions in

R
n+1 and with values in the Clifford algebra Cl0,n was already considered. Together with the

geometric interpretation of the hypercomplex derivative (see [17]), dilatations and distortions of

these mappings can be estimated. Compared with the related work, the advantage of our approach

lies in the possibility to study the description of monogenic functions by their geometric mapping

properties. The local mapping properties of a monogenic function or of a real analytic function

are mainly determined by the behaviour of the linear part of their Taylor expansions. According

to this, in [23] the geometric behaviour of the linear part of a monogenic function was considered.



CUBO
11, 1 (2009)

On mapping properties of monogenic functions 75

As a consequence, it is shown that monogenic functions can be defined as mappings which map

infinitesimal balls to special ellipsoids and vice versa. Here we extend this result considering the

all series expansion of a function.

The text is based on recent publications of the authors [19, 20, 21, 22, 23] and contains new

material as well.

2 Preliminaries

Let H := {a = a0 + a1e1 + a2e2 + a3e3, ai ∈ R, i = 0, 1, 2, 3} be the algebra of the real quater-

nions, where the imaginary units ei (i = 1, 2, 3) are subject to the multiplication rules

e2
1

= e2
2

= e2
3

= −1,

e1e2 = e3 = −e2e1, e2e3 = e1 = −e3e2, e3e1 = e2 = −e1e3.

Through this paper we shall denote by Sc(a) := a0 the scalar part of a and by Vec(a) :=

a1e1 + a2e2 + a3e3 its vector part. Analogously to the complex case, the (quaternion-)conjugate

element of a is the quaternion

a := Sc(a) − Vec(a) = a0 − a1e1 − a2e2 − a3e3.

Also, we shall use the Euclidean norm |a|2 = aa =
(
a2
0

+ a2
1

+ a2
2

+ a2
3

)1/2
. The real vector

space R3 is to be embedded in the subset A := spanR{1, e1, e2} of H via the identification of each

element x = (x0,x) = (x0,x1,x2) ∈ R
3 with the paravector (also called reduced quaternion)

x := x0 + x = x0 + x1e1 + x2e2 ∈ A.

As a consequence, no distinction will be made between x as a point in R3 or its correspondent

reduced quaternion. Also, we emphasize that A is only a real vector space but not a subalgebra

of H. For more details on the real algebra of quaternions we refer e.g. to [8], [31], [18], [15].

Let now Ω be an open subset of R3 with piecewise smooth boundary. A quaternion-valued

function or, briefly, an H-valued function is a mapping f : Ω −→ H such that

f(x) =
3∑

i=0

fi(x)ei,

where e0 = 1 and the coordinate-functions fi (i = 0, 1, 2, 3) are real-valued in Ω. Properties such

as continuity, differentiability or integrability are ascribed coordinate-wisely.

For continuously real-differentiable functions f : Ω −→ H, the operator

D = ∂x0 + e1∂x1 + e2∂x2 (1)



76 K. Gürlebeck and J. Morais CUBO
11, 1 (2009)

is called generalized Cauchy-Riemann operator. The conjugate generalized Cauchy-Riemann op-

erator is defined by

D = ∂x0 − e1∂x1 − e2∂x2. (2)

A function f : Ω −→ H is called left (resp. right) monogenic in Ω if

Df = 0 in Ω (resp.,fD = 0 in Ω).

Remark 1. In general, left (resp. right) monogenic functions are not right (resp. left) mono-

genic. From now on, we refer only to left monogenic functions. For simplicity, we will call them

monogenic. However, all results achieved to left monogenic functions can easily be adapted to right

monogenic functions.

The generalized Cauchy-Riemann operator (1) and its conjugate (2) factorize the Laplace

operator in R3. In fact, it holds

∆3f = DDf = DDf,

which implies that any monogenic function is also a harmonic function. Analogously as in the

complex one-dimensional case 1
2
D̄ defines a derivative of monogenic functions. This was shown in

[16], where 1
2
D̄f was called hypercomplex derivative of f.

A monogenic function f : Ω −→ H with an identically vanishing hypercomplex derivative

(i.e. a function from ker D ∩ ker D̄) is called hyperholomorphic constant (see again [16]). It is

immediately clear that such function depends only on x1 and x2.

Additionally, we introduce the following notations: BR := BR(0) will denote the ball of radius

R in R3 centered at the origin, SR = ∂BR its boundary and dσR (resp. dVR) the Lebesgue measure

on SR (resp. BR). For simplicity, in the case R = 1 we omit R in the notations. We will also denote

by L2(SR; X; R) (resp. L2(BR; X; R)) the R-linear Hilbert space of square integrable functions on

SR (resp. BR) with values in X (X = R or A). In the case X = R we abbreviate L2(SR; R; R)

(resp. L2(BR; R; R)) briefly by L2(SR) (resp. L2(BR)). Also, the real-valued inner product in

L2(SR; A; R) (resp. L2(BR; A; R)) is given by

〈f,g〉
L2(SR;A;R) =

∫

SR

Sc(fg)dσR, (3)

respectively,

〈f,g〉
L2(BR;A;R) =

∫

BR

Sc(fg)dVR, (4)

for any f,g ∈ L2(SR; A; R) (resp. L2(BR; A; R)). Each homogeneous harmonic polynomial Pn of

degree n can be written in spherical coordinates as

Pn(x) = R
nPn(ω), ω ∈ SR, (5)



CUBO
11, 1 (2009)

On mapping properties of monogenic functions 77

its restriction, Pn(ω), to the boundary of the ball BR is called spherical harmonic
1 of degree n.

From (5), it is clear that a homogeneous polynomial is determined by its restriction to SR. Denoting

by Hn(SR) the space of real-valued spherical harmonics of degree n on SR, it is well-known (see

[2] and [29]) that

dim Hn(SR) = 2n + 1.

It is also known (see [2] and [29]) if n 6= m, the spaces Hn(SR) and Hm(SR) are orthogonal

in L2(SR).

Let us denote the homogeneous monogenic polynomials of degree n by Hn. In an analogous

way to the spherical harmonics, the restriction of Hn to the boundary of the ball BR is called

spherical monogenic2 of degree n. Now let M+(Ω; A; n) be the space of A-valued homogeneous

monogenic polynomials of degree n in Ω ⊂ R3. In [27], it is shown that the space M+(Ω; A; n)

has dimension 2n + 3. Later, this result was generalized for arbitrary higher dimensions by R.

Delanghe in [12].

Consider, for each n ∈ N0, a basis {H
ν

n
: ν1, ..., dim M+(Ω; A; n)} of M+(Ω; A; n). Since the

coordinates of Hν
n

are harmonic, for arbitrary n,k = 0, 1, ..., we have

‖Hν
n
‖2
L2(BR;A;R) =

R2n+3

2n + 3
‖Hν

n
‖2
L2(S;A;R). (6)

3 Homogeneous Monogenic Polynomials

In [9] and [10], a special R-linear complete orthonormal system of A-valued homogeneous mono-

genic polynomials defined in the unit ball of R3 is explicitly constructed. The main idea of this

construction is based on the already referred factorization of the Laplace operator. The authors

took a system of real-valued homogeneous harmonic polynomials and applied the D operator in

order to obtain a system of A-valued homogeneous monogenic polynomials.

For an easier description, we introduce spherical coordinates

x0 = r cos θ, x1 = r sin θ cos ϕ, x2 = r sin θ sin ϕ,

where 0 < r < ∞, 0 < θ ≤ π, 0 < ϕ ≤ 2π. Each point x = (x0,x1,x2) ∈ R
3 \ {0} admits a unique

representation x = rω, where r|x| and |ω| = 1. Therefore, ωi =
xi

r
for i = 0, 1, 2.

As described, the homogeneous monogenic polynomials (solid spherical monogenics)

{X0,†
n
, Xm,†

n
, Y m,†

n
: m = 1, ...,n + 1}, (7)

1This restriction is also called surface spherical harmonic by some authors (see [8]).
2Such restriction is also called surface inner spherical monogenic (see [8]).



78 K. Gürlebeck and J. Morais CUBO
11, 1 (2009)

formed by the extensions to the ball of {X0
n
,Xm

n
,Y m
n

: m = 1, ...,n + 1} are obtained by applying

the operator 1
2
D to the system of homogeneous harmonic polynomials

U
0,†
n+1

,U
m,†
n+1

,V
m,†
n+1

, m = 1, . . . ,n + 1,

with the notations

U
0,†
n+1

:= rn+1U0
n+1

= rn+1Pn+1(cos θ)

U
m,†
n+1

:= rn+1Um
n+1

= rn+1Pm
n+1

(cos θ) cos mϕ

V
m,†
n+1

:= rn+1V m
n+1

= rn+1Pm
n+1

(cos θ) sin mϕ, m = 1, ...,n + 1.

Hereby Pn+1 stands for the Legendre polynomial of degree n + 1 and the functions P
m

n+1
are

the associated Legendre functions3. The set {U0
n+1

,Um
n+1

,V m
n+1

: m = 1, . . . ,n + 1} denotes the

standard orthogonal basis of spherical harmonics of degree n + 1 in R3 (considered, e.g., in [34])

with respect to the inner product

〈f,g〉
L2(S)

=

∫

S

fg dσ.

Moreover, their norms are given by

‖ U0
n+1

‖L2(S) = 2

√
π

2n + 3

‖ Um
n+1

‖L2(S) = ‖ V
m

n+1
‖L2(S) =

√
2π

2n + 3

(n + 1 + m)!

(n + 1 − m)!
.

We begin by considering the following norm estimates already obtained in [9] which will be

used later on.

Proposition 1. For a given fixed n ∈ N0, the spherical monogenics X
0
n
, Xm

n
and Y m

n
(m =

1, . . . ,n + 1) are orthogonal to each other with respect to the inner product (3) and their norms are

given by

‖ X0
n
‖L2(S;A;R) =

√
π(n + 1)

‖ Xm
n

‖L2(S;A;R) = ‖ Y
m

n
‖L2(S;A;R) =

√
π

2
(n + 1)

(n + 1 + m)!

(n + 1 − m)!
.

Remark 2. A similar result can be obtained for the homogeneous monogenic polynomials (7) if

one takes into account relation (6).

3These functions were introduced in 1877 by Ferrers. For that reason, some authors (c.f. [34]) call them Ferrers
functions.



CUBO
11, 1 (2009)

On mapping properties of monogenic functions 79

In the second and third sections we will look more closely to the pointwise behavior of a given

function. For that reason in what follows we present pointwise estimates of our basis polynomials

(7), already obtained by the authors in [21].

Proposition 2. Let n ∈ N0. For the homogeneous monogenic polynomials (7) the following

estimates hold:

|X0,†
n

(x)| ≤
1

2
√
π

(n + 1)(2r)n ‖ X0
n
‖L2(S;A;R)

|Xm,†
n

(x)| ≤
1

2
√
π

(n + 1)(2r)n ‖ Xm
n

‖L2(S;A;R)

|Y m,†
n

(x)| ≤
1

2
√
π

(n + 1)(2r)n ‖ Y m
n

‖L2(S;A;R),

with m = 1, . . . ,n + 1.

An interesting point to note here is that the real part of these polynomials are again related

with the set {U0
n
,Um

n
,V m
n

: m = 1, . . . ,n}. These relations are given in the next theorem:

Theorem 1. Given a fixed n ∈ N0, we have the following relations:

Sc(X0
n
) =

(n + 1)

2
U0
n
(θ,ϕ)

Sc(Xm
n

) =
(n + m + 1)

2
Um
n

(θ,ϕ)

Sc(Y m
n

) =
(n + m + 1)

2
V m
n

(θ,ϕ),

for m = 1, . . . ,n.

Proof. We just prove the relation for the spherical harmonics Sc(Xl
n
) (l = 0, . . . ,n). The proof for

Sc(Y m
n

) (m = 1, . . . ,n) is similar. Taking results from [9], the real part of the spherical monogenics

Xl
n

(l = 0, ...,n) is given by

Sc(Xl
n
) = Al,n(θ) cos(lϕ)

with

Al,n(θ) =
1

2

(
sin

2
θ
d

dt

[
P l
n+1

(t)
]
t=cos θ

+ (n + 1) cos θP l
n+1

(cos θ)

)
.

It is well known that the Legendre polynomials, together with the associated Legendre func-

tions, satisfy in particular the recurrence formula

(1 − t2)(P l
n+1

(t))′ = (n + l + 1)P l
n
(t) − (n + 1)tP l

n+1
(t), (8)



80 K. Gürlebeck and J. Morais CUBO
11, 1 (2009)

for l = 0, . . . ,n+ 1. Now, making the change of variable cos θ = t in Al,n(θ) and using the previous

recurrence formula, it follows immediately that

Al,n(arccos t) =
1

2

[
(1 − t2)(P l

n+1
(t))′ + (n + 1)tP l

n+1
(t)
]

=
(n + l + 1)

2
P l
n
(t).

Making again a change of variable t = cos θ our statement is proved.

As a consequence, it turns out the result:

Theorem 2. For a fixed n ∈ N0, the spherical harmonics

{
Sc(X0

n
), Sc(Xm

n
), Sc(Y m

n
) : m = 1, . . . ,n

}

are orthogonal in L2(S).

With such relations together with the norms of the spherical harmonics, we are ready to

establish, as well, the L2-norms of Sc(X
0
n
), Sc(Xm

n
) and Sc(Y m

n
) (m = 1, ...,n).

Proposition 3. For a fixed n ∈ N0, the norms of the spherical harmonics Sc(X
0
n
), Sc(Xm

n
) and

Sc(Y m
n

) are given by

‖Sc(X0
n
)‖L2(S) = (n + 1)

√
π

2n + 1

and

‖Sc(Xm
n

)‖L2(S) = ‖Sc(Y
m

n
)‖L2(S)(n + 1 + m)

√
π

2

1

(2n + 1)

(n + m)!

(n − m)!
,

for m = 1, . . . ,n.

Remark 3. Using a different measure, we have established Theorem 2 and the previous proposition

already in [21].

For future use we need also the next results:

Proposition 4. Given a fixed n ∈ N0, the spherical harmonics

{
Sc(Xn+1

n
ei), Sc(Y

n+1
n

ei) : i = 1, 2
}

are orthogonal to each other with respect to the inner product (3) and their norms are given by

‖Sc(Xn+1
n

ei)‖L2(S) = ‖Sc(Y
n+1
n

ei)‖L2(S) =
1

2

√
π(n + 1)(2n + 2)!.



CUBO
11, 1 (2009)

On mapping properties of monogenic functions 81

The case i = 1 was already studied by the authors in [20]. For i = 2 the proof is similar.

Remark 4. The orthogonality is ensured if one takes into account the following representations

([9], Proposition 3.4.3)

Xn+1
n

= −Cn+1,n cos(nϕ)e1 + C
n+1,n

sin(nϕ)e2 (9)

Y n+1
n

= −Cn+1,n sin(nϕ)e1 − C
n+1,n

cos(nϕ)e2.

Since some of our further results are not only restricted to the unit ball, from now on we repre-

sent by X0,†,∗R
n

,Xm,†,∗R
n

,Y m,†,∗R
n

(m = 1, ...,n+1) the normalized basis functions X0,†
n
,Xm,†

n
,Y m,†
n

in L2(BR; A; R). Based on these functions, in [9] and [10] the following orthonormal basis is con-

structed, therein restricted to the unit ball.

Theorem 3. For each n, the set of 2n + 3 homogeneous monogenic polynomials
{
X0,†,∗R
n

,Xm,†,∗R
n

,Y m,†,∗R
n

, m1, ...,n + 1
}

(10)

forms an orthonormal basis in M+(BR; A; n) with respect to the inner product (4).

Remark 5. The estimates stated in Proposition 2 are still valid for this new system of polynomials

(10). In particular, taking into account relation (6) it follows:

|X0,†,∗R
n

(x)| =

√
2n + 3

R2n+3
|X0,†

n
(x)|

‖X
0,†
n ‖L2(S;A;R)

and moreover, from Proposition 2

|X0,†
n

(x)| =
∣∣∣RnX0,†n

(
x

R

)∣∣∣ ≤ Rn
1

2
√
π

(n + 1)2n
∣∣∣
x

R

∣∣∣
n

‖ X0
n
‖L2(S;A;R)

for 0 < |x| = r < R.

Theorem 3 makes it possible to define the Fourier expansion of a square integrable A-valued

monogenic function in L2(BR). Moreover, each monogenic function can be decomposed in an

orthogonal sum of a monogenic "main part" (g) of the function and a hyperholomorphic constant

(h). More precisely, it holds:

Lemma 1. A monogenic L2-function f : Ω ⊂ R
3 −→ A can be decomposed into

f = f(0) + g + h, (11)

where the functions g and h have Fourier series

g(x) =

∞∑

n=1

(
X0,†,∗R
n

(x)α0
n

+

n∑

m=1

[
Xm,†,∗R
n

(x)αm
n

+ Y m,†,∗R
n

(x)βm
n

]
)

h(x) =

∞∑

n=1

[
Xn+1,†,∗R
n

(x)αn+1
n

+ Y n+1,†,∗R
n

(x)βn+1
n

]
.



82 K. Gürlebeck and J. Morais CUBO
11, 1 (2009)

The associated Fourier coefficients α0
n
,αm

n
,βm
n

(m = 1, ...,n + 1) are real-valued.

4 Bohr’s Theorem for monogenic functions

During the last years the standard Bohr’s phenomena attracted a lot of attention. In 1914, H.

Bohr discovered that there exists a radius r ∈ (0, 1) such that if a power series of a holomorphic

function converges in the unit disk and its sum has a modulus less than 1, then for |z| < r the sum

of the absolute values of its terms is again less than 1. The significance of the theorem is that such

radius does not depend on the function.4 To be more precise, the classical Bohr’s Theorem says

that:

Theorem 4. [6] Let f be a bounded analytic function in the open unit disk, with Taylor expansion

f(z) =

∞∑

n=0

anz
n convergent in the unit disk and with modulus less than 1. Then

∞∑

n=0

|an|r
n < 1 for

0 ≤ r < 1
3
.

This result, known as Bohr’s inequality, is true for 0 < r < 1
3

and the constant 1
3

cannot

be improved, that is, the inequality fails for any r ≥ 1
3
. Originally, this theorem was proved

for 0 ≤ r < 1
6
, but soon improved to the sharp result by M. Riesz, I. Schur, and N. Wiener

independently. In Bohr’s paper [6] his own proof was published as well as a proof by Wiener based

on function theory methods. Later, S. Sidon gave a different proof [35], which was subsequently

rediscovered by M. Tomić [36].

Recently, multi-dimensional analogues and other generalizations of Bohr’s theorem are treated

by several mathematicians such as Aizenberg [1], Beneteau, Dahlner and Khavinson [3], Boas and

Khavinson [4], Dineen and Timoney [14], Paulsen, Popescu and Singh [30], and many others. In

several of these papers, the proof of Bohr’s inequality or of Bohr-type inequalities, respectively,

in the theory of holomorphic functions of one or n variables is based on the orthogonality of the

powers of the complex variable(s). To use similar ideas in the quaternionic case, it seems to be

natural to work with the Fourier expansion of monogenic functions. It is not so simple as in the

complex case to switch between the Taylor expansion and the Fourier series of a function. The

reason for this is that the Taylor expansion with respect to the Fueter variables does not give us

orthogonal summands.

It should be also remarked that in some papers (see, e.g., [30]) the idea is to work with the

Fourier coefficients of the boundary values of a holomorphic function. We prefer here to consider

(analogously to the original formulation of Bohr’s theorem) only monogenic functions in the ball.

It follows directly from the supposed boundedness of the monogenic functions that they are also

square integrable in the ball and therefore we can work with Fourier series there. The existence of

integrable boundary values needs additional assumptions.

4For the physicists, the notion "Bohr radius" is associated to Niels Bohr, the founder of the quantum theory and
winner of the Nobel Prize in Physics in 1922.



CUBO
11, 1 (2009)

On mapping properties of monogenic functions 83

In the remainder of this section, we collect generalizations and different modifications of this

theorem (see [19, 21]) and we show that the result can be extended to the all class of monogenic

functions with |f(x)| < 1 in B. In [19], the first version of a quaternionic Bohr type theorem was

obtained, therein restricted to the case of functions with f(0) = 0.

Theorem 5. [19] Let f be a square integrable A-valued monogenic function with f(0) = 0 and

|f(x)| < 1 in B and let

∞∑

n=1

[
X0,†,∗
n

α0
n

+

n+1∑

m=1

(
Xm,†,∗
n

αm
n

+ Y m,†,∗
n

βm
n

)
]

be its Fourier expansion. Then

∞∑

n=1

∣∣∣X0,†,∗n α0n +
n+1∑

m=1

(
Xm,†,∗
n

αm
n

+ Y m,†,∗
n

βm
n

)∣∣∣ < 1

holds in the ball {x : |x| < 0.047}.

This result is adapted very well to the complex situation. The absolute value is taken from

all summands of the same degree n. In the complex case this is also a first important step. All

the considered functions with f(0) = 0 are orthogonal to the constants. This is used later on

to estimate all Fourier coefficients of a general holomorphic function with |f(z)| ≤ 1 by the first

Fourier coefficient (see, e.g., [30]).

However, it is important to remark that in the quaternionic context, the set of ”constants” is

much bigger. Hereby constants are also monogenic functions which have an identically vanishing

hypercomplex derivative. Then it is immediately clear that the constant function and all monogenic

functions which depend only on x1 and x2 are the so called hyperholomorphic constants. Moreover,

if we, as in this paper, consider only A-valued functions then a non-trivial hyperholomorphic

constant must have values in span
R
{e1, e2}. With these observations it seems to be natural to

study at first the class of functions which are orthogonal to the non-trivial hyperholomorphic

constants in L2(B; A; R) with |f(x)| < 1 in B (see [21]). This approach is also supported by

the fact that in Lemma 1 it is shown that each monogenic function can be decomposed in an

orthogonal sum of a monogenic ”main part” of the function and a hyperholomorphic constant.

Remind that an orthonormal basis of the subspace of hyperholomorphic constants is given by the

set {Xn+1,†
n

,Y n+1,†
n

}∞
n=0

.

The Fourier representation in the hypothesis of the next Theorem describes the general form

of these main parts. The non-trivial hyperholomorphic constants in the decomposition do not

influence the real part of the function at the origin because their image lies in span
R
{e1, e2}.

Theorem 6. [21] Let f be an A-valued monogenic function such that f(x) − f(0) is orthogonal

to the hyperholomorphic constants with respect to the inner product (4) with |f(x)| < 1 in B and



84 K. Gürlebeck and J. Morais CUBO
11, 1 (2009)

let

∞∑

n=0

[
X0,†,∗
n

α0
n

+

n∑

m=1

(
Xm,†,∗
n

αm
n

+ Y m,†,∗
n

βm
n

)
]

be its Fourier expansion. Then

∞∑

n=0

[
|X0,†,∗

n
||α0

n
| +

n∑

m=1

(
|Xm,†,∗

n
||αm

n
| + |Y m,†,∗

n
||βm

n
|
)
]
< 1

holds in the ball of radius r, with 0 ≤ r < 0.004.

Proof. We give only the main ideas of the proof. For more details see [21]. At first it is important

to note that in the previous series the sum which contains the variable m runs now only from 1 to

n. This fact expresses the supposed orthogonality to the hyperholomorphic constants Xn+1,†
n

and

Y n+1,†
n

. Since the basis polynomials are homogeneous, the value of f at the origin is

f(0) =

∞∑

n=0

(
X0,†,∗
n

(0)α0
n

+

n∑

m=1

[
Xm,†,∗
n

(0)αm
n

+ Y m,†,∗
n

(0)βm
n

]
)

=
1

2

√
3

π
α0

0
∈ R.

Without loss of generality we assume that f(0) is positive (otherwise we work with −f). Since

the associated Fourier coefficients are real-valued, the real part of f is given by

Sc(f) =
∞∑

n=0

{
Sc(X0,†,∗

n
)α0
n

+

n∑

m=1

[
Sc(Xm,†,∗

n
)αm
n

+ Sc(Y m,†,∗
n

)βm
n

]
}
.

Basically, the main idea of the proof is to find relations between the general Fourier coefficients

and the coefficient of the zeroth term, i.e., α0
0
. Multiplying both sides of the equation

Sc(1 − f) = 1 − Sc(f) (12)

by the solid spherical harmonics {Sc(X
0,†,∗
k

), Sc(X
p,†,∗
k

), Sc(Y
p,†,∗
k

) : p = 1, ...,k}, integrating over

the ball and applying the modulus we get the following relations:

|α0
k
| ≤ max

B

|X
0,†,∗
k

|
2
√

π

3

‖Sc(X
0,†,∗
k

)‖2
L2(B)

(
2

√
π

3
− α0

0

)

|α
p

k
| ≤ max

B

|X
p,†,∗
k

|
2
√

π

3

‖Sc(X
p,†,∗
k

)‖2
L2(B)

(
2

√
π

3
− α0

0

)

|β
p

k
| ≤ max

B

|Y
p,†,∗
k

|
2
√

π

3

‖Sc(Y
p,†,∗
k

)‖2
L2(B)

(
2

√
π

3
− α0

0

)
, p = 1, ...,k.



CUBO
11, 1 (2009)

On mapping properties of monogenic functions 85

With some calculations, applying Propositions 1-3 we arrive at

1

2

√
3

π
α0

0
+

∞∑

n=1

[
|X0,†,∗

n
||α0

n
| +

n∑

m=1

(
|Xm,†,∗

n
||αm

n
| + |Y m,†,∗

n
||βm

n
|
)
]

≤
1

2

√
3

π
α0

0
+

1
√

3π

(
2

√
π

3
− α0

0

) ∞∑

n=1

(4r)n(n + 1)4(2n + 3).

The principal significance is that

1

2

√
3

π
α0

0
+

∞∑

n=1

[
|X0,†,∗

n
||α0

n
| +

n∑

m=1

(
|Xm,†,∗

n
||αm

n
| + |Y m,†,∗

n
||βm

n
|
)
]
< 1

if 2
3

∞∑

n=1

(4r)n(n + 1)4(2n + 3) < 1. We see that the last series converges for r < 1
4
, and therefore,

the inequality is satisfied for 0 ≤ r < 0.004.

As we have seen, the set {Xn+1,†
n

,Y n+1,†
n

} belonging to h play a special role. In order to

extend the previous result, next we present some important properties of this function and/or of

its coordinates. For simplicity we restrict ourselves to the unit ball.

Lemma 2. The hyperholomorphic constant h can be written as

h = h1e1 + h2e2,

where its coordinates have Fourier series

h1(x) =

∞∑

n=1

(
[Xn+1,†,∗

n
(x)]1α

n+1
n

+ [Y n+1,†,∗
n

(x)]1β
n+1
n

)

h2(x) =

∞∑

n=1

(
[Xn+1,†,∗

n
(x)]2α

n+1
n

+ [Y n+1,†,∗
n

(x)]2β
n+1
n

)
.

Moreover, the following properties hold:

Proposition 5. The harmonic functions h1 and h2 are orthogonal with respect to the inner product

(3).

Proof. Because relation (6) we just need to ensure the orthogonality in L2(S). For technical

reasons, we rewrite the function h as follows

h(x) =

∞∑

n=1

√
2n + 3 rn

(
Xn+1,∗
n

(θ,ϕ)αn+1
n

+ Y n+1,∗
n

(θ,ϕ)βn+1
n

)
.



86 K. Gürlebeck and J. Morais CUBO
11, 1 (2009)

By definition of the real-valued inner product in L2(S), using representation (9) and the

previous expression we have

< h1,h2 >L2(S)=

∞∑

n=1

∞∑

n′=1

√
2n + 3 rn

‖Xn+1n ‖L2(S;A;R)

√
2n′ + 3 rn

′

‖Xn
′+1

n′
‖L2(S;A;R)

∫

S

An(θ,ϕ)Bn′ (θ,ϕ)dσ

where

An(θ,ϕ) = −C
n+1,n

(θ)
(
cos(nϕ)αn+1

n
+ sin(nϕ)βn+1

n

)
= An(θ)An(ϕ)

Bn′ (θ,ϕ) = C
n
′
+1,n

′

(θ)
(
sin(n′ϕ)αn

′
+1

n′
− cos(n′ϕ)βn

′
+1

n′

)
= Bn′ (θ)Bn′ (ϕ).

If the degrees of the summands in the series of h1 and h2 are different (n 6= n
′
) then the

orthogonality is ensured. Therefore it is only necessary to prove that the previous integral vanishes

for n = n′. And since
∫

S

An(θ,ϕ)Bn(θ,ϕ)dσ =

∫
π

0

An(θ)Bn(θ) sin θdθ

∫
2π

0

An(ϕ)Bn(ϕ)dϕ,

it is enough to prove that the second integral on the right-hand side is zero. Moreover,

∫
2π

0

An(ϕ)Bn(ϕ)dϕ =
[
(αn+1
n

)
2 − (βn+1

n
)
2
]∫ 2π

0

cos(nϕ) sin(nϕ)dϕ

+ αn+1
n

βn+1
n

(∫
2π

0

sin
2
(nϕ)dϕ −

∫
2π

0

cos
2
(nϕ)dϕ

)

= 0.

Proposition 6. The harmonic functions h1 and h2 verify the following relation:

sup
B

|h1(x)| = sup
B

|h2(x)|.

The proof follows immediately from representation (9).

We are thus led to the following generalization of Theorem 6.

Theorem 7. Let f be an A-valued monogenic function with |f(x)| < 1 in B and let

∞∑

n=0

[
X0,†,∗
n

α0
n

+

n+1∑

m=1

(
Xm,†,∗
n

αm
n

+ Y m,†,∗
n

βm
n

)
]

be its Fourier expansion. Then

∞∑

n=0

[
|X0,†,∗

n
||α0

n
| +

n+1∑

m=1

(
|Xm,†,∗

n
||αm

n
| + |Y m,†,∗

n
||βm

n
|
)
]
< 1

holds in the ball of radius r, with 0 ≤ r < 0.004.



CUBO
11, 1 (2009)

On mapping properties of monogenic functions 87

Proof. Considering f written as in (11) (Lemma 1)

f(x) = f(0) + g(x) + h(x),

with f(0) = g(0) + h(0). The study of the function g was already considered in Theorem 6. We

showed that

|g(x)| ≤
∞∑

n=1

[
|X0,†,∗

n
||α0

n
| +

n∑

m=1

(
|Xm,†,∗

n
||αm

n
| + |Y m,†,∗

n
||βm

n
|
)
]

≤
1

√
3π

(
2

√
π

3
− |α0

0
|

) ∞∑

n=1

(4r)n(n + 1)4(2n + 3).

We consider now the function h written as Fourier series

h(x) =

∞∑

n=1

(
Xn+1,†,∗
n

(x)αn+1
n

+ Y n+1,†,∗
n

(x)βn+1
n

)
.

The proof for the function h follows the same idea as the previous one. According to its Fourier

expansion, in this case we should find relations with α1
0

and/or β1
0
. However, it is important to

note that the function h has no real part, and therefore, it is not possible to apply straight the

previous idea. Because h lies in span
R
{e1, e2}, the interesting point is that multiplying h at right

by e1 (resp., by e2) the real part is different of zero, standing then for −h1 (resp., −h2). Moreover,

since there are two different Fourier coefficients in the zeroth term, it is natural to ask: which

coefficient should be compared, α1
0

or β1
0
? Multiplying the function h at right by e1 we obtain

h̃(0) + h̃(x) := he1(0) + he1(x)

= (X
1,†,∗
0

e1)α
1
0

+ (Y
1,†,∗
0

e1)β
1
0

+

∞∑

n=1

[
(Xn+1,†,∗

n
e1)(x)α

n+1
n

+ (Y n+1,†,∗
n

e1)(x)β
n+1
n

]

where

(X
1,†,∗
0

e1)α
1
0

+ (Y
1,†,∗
0

e1)β
1
0

=
1

2
√
π
α1

0
+

1

2
√
π
β1

0
e3.

For this case, taking into account the previous assumption, it is obviously that we should

consider for zeroth term the coefficient α1
0
. In a similar way, considering a new function

˜̃
h := he2,

β1
0

should be surely considered.

Having disposed of this preliminary step let us return to the proof. It follows easily multiplying

h at right by e1

Sc(h̃) = Sc(he1)

=

∞∑

n=1

[
Sc(Xn+1,†,∗

n
e1)α

n+1
n

+ Sc(Y n+1,†,∗
n

e1)β
n+1
n

]
.



88 K. Gürlebeck and J. Morais CUBO
11, 1 (2009)

Taking into account Proposition 6, for 0 < δ2 ≤ 1 we multiply both sides of the equation

Sc

(
δ2

2
− h̃

)
=
δ2

2
− Sc(h̃)

by the orthonormal solid spherical harmonics Sc(X
k+1,†,∗
k

e1) ( resp. Sc(Y
k+1,†,∗
k

e1)), integrating

over the ball and taking the modulus it follows

|αk+1
k

| ≤ max
B

|X
k+1,†,∗
k

|
2
√

π

3

‖Sc(X
k+1,†,∗
k

)‖2
L2(B)

(√
π

3
δ2 − |α

1
0
|

)

|βk+1
k

| ≤ max
B

|X
k+1,†,∗
k

|
2
√

π

3

‖Sc(X
k+1,†,∗
k

)‖2
L2(B)

(√
π

3
δ2 − |α

1
0
|

)

Now with some calculations, using Propositions 1-3 and applying the Maximum modulus

principle, the previous inequalities can be rewritten as follows

|αk+1
k

| ≤
2
√

3
2
k
√

2k + 3(k + 1)

(√
π

3
δ2 − |α

1
0
|

)

|βk+1
k

| ≤
2
√

3
2
k
√

2k + 3(k + 1)

(√
π

3
δ2 − |α

1
0
|

)
.

In a similar way, from the study of the function
˜̃
h we obtain

|αk+1
k

| ≤
2
√

3
2
k
√

2k + 3(k + 1)

(√
π

3
δ2 − |β

1
0
|

)

|βk+1
k

| ≤
2
√

3
2
k
√

2k + 3(k + 1)

(√
π

3
δ2 − |β

1
0
|

)
.

Using the previous inequalities we end with

|h(x)| ≤

∞∑

n=1

(
|Xn+1,†,∗

n
||αn+1

n
| + |Y n+1,†,∗

n
||βn+1

n
|
)

≤
1

√
3π

(
2

√
π

3
δ2 − |α

1
0
| − |β1

0
|

) ∞∑

n=1

(4r)n(n + 1)2(2n + 3).

Finally, we obtain

|f(x)| ≤
1

2

√
3

π
|α0

0
− α1

0
e1 − β

1
0
e2|

+
1

√
3π

(
2

√
π

3
(1 + δ2) − |α

0
0
| − |α1

0
| − |β1

0
|

) ∞∑

n=1

(4r)n(n + 1)4(2n + 3),

and since |f(x)| < 1 it is clear that

1
√

3π

(
2
√

π

3
(1 + δ2) − |α

0
0
| − |α1

0
| − |β1

0
|

2
√

π

3
− |α0

0
| − |α1

0
| − |β1

0
|

) ∞∑

n=1

(4r)n(n + 1)4(2n + 3) < 1.



CUBO
11, 1 (2009)

On mapping properties of monogenic functions 89

Of crucial importance is the fact that the coefficient

2
√

π

3
− |α0

0
| − |α1

0
| − |β1

0
|

2
√

π

3
(1 + δ2) − |α

0
0
| − |α1

0
| − |β1

0
|

is bounded from above by 1. A simple calculation shows that the last series converges for r < 1
4
,

and therefore, the inequality is satisfied for 0 ≤ r < 0.004.

This shows that such a radius exists in the three-dimensional Euclidean ball. It has to be

studied how the estimate for the Bohr radius can be improved.

5 Real part Theorems for monogenic functions

In the remainder of this section, we refer to the results as "real part theorems" in honor to the first

assertion of such a kind, the classical (improved) Hadamard’s real part theorem (1892). His work

on functions of a complex variable was one of the first to examine the general theory of analytic

functions. Since then, the acceptance of his work is worldwide. Looking back to all of these years

one can say that time has shown that his topic has a wide range of applications. Some important

indicators for such a development is that based on it, many recent results with strong applications

are still coming out. Moreover, they provide with the best description of the pointwise behavior of

analytic functions from a given space. A lot of results and extended list of references concerning

these and other fundamental inequalities, as well as their applications, can be found in the book

by Kresin and Maz’ya (see [33]).

In the complex case, Hadamard’s real part theorem contains only the modulus of the function

in the left-hand side of an inequality and bounds the growth of a function by the growth of its real

part. More precisely, for r < R the inequality

|f(z) − f(0)| ≤
Cr

R − r
sup
|ξ|≤R

|Re (f(ξ) − f(0)) |, (13)

holds for analytic functions on a closed disk of radius R centered at the origin. Such an inequality

appeared first in 1892 (see [24]) with C = 4. Later, Borel and Carathéodory found the sharp

constant C = 2. As a matter of this fact, a more general estimate for |f(z)| with f(0) 6= 0 was

noticed by Carathéodory (see Landau [25, 26])

|f(z)| ≤
2r

R − r
sup
|ξ|≤R

|Ref(ξ)| +
R + r

R − r
|f(0)|. (14)

Having in mind the type of inequalities we want to prove, we will restrict ourselves to the

three-dimensional case. The frequent use of quaternionic analysis in the study of three-dimensional



90 K. Gürlebeck and J. Morais CUBO
11, 1 (2009)

problems motivate us to consider functions defined in R3 and with values in the reduced quater-

nions. It is also known that already in the four-dimensional case (quaternion-valued functions)

there are a lot of non-trivial monogenic functions with vanishing scalar part. For such functions

we cannot get the result as desired.

In [20, 22] it is shown that it is possible to generalize Borel-Carathéodory and Hadamard’s real

part theorems to monogenic functions, therein restricted to the unit ball in the Euclidean space

R
3. In this section we generalize these results for an arbitrary ball of radius R, analogously to the

complex case.

Remark 6. Several proofs of Bohr’s inequality are based on estimating all Fourier coefficients

by the first one. We will observe that the proof of both theorems follows from this idea. Here we

consider relations between the Fourier coefficients of the function and the Fourier coefficients of

its real part.

The referred relations come with the next lemma:

Lemma 3. Let f be a square integrable A-valued monogenic function and n ∈ N0. Then the

Fourier coefficients of f are given by

α0
n

=
‖X0,†

n
‖L2(BR;A;R)

‖Sc(X
0,†
n )‖

2

L2(BR)

∫

BR

Sc(f)Sc(X0,†
n

)dVR

αm
n

=
‖Xm,†

n
‖L2(BR;A;R)

‖Sc(X
m,†
n )‖

2

L2(BR)

∫

BR

Sc(f)Sc(Xm,†
n

)dVR

βm
n

=
‖Y m,†

n
‖L2(BR;A;R)

‖Sc(Y
m,†
n )‖

2

L2(BR)

∫

BR

Sc(f)Sc(Y m,†
n

)dVR, m = 1, ...,n

αn+1
n

=
‖Xn+1,†

n
‖L2(BR;A;R)

‖Sc(X
n+1,†
n e1)‖

2

L2(BR)

∫

BR

Sc(he1)Sc(X
n+1,†
n

e1)dVR

βn+1
n

=
‖Y n+1,†

n
‖L2(BR;A;R)

‖Sc(Y
n+1,†
n e1)‖

2

L2(BR)

∫

BR

Sc(he1)Sc(Y
n+1,†
n

e1)dVR.

Originally the Fourier coefficients are defined by the inner product of the function f and

elements of the space M+(R3; A,n). Now as we can see these coefficients, up to a factor, are also

associated with the scalar part of f.

Proof. We give only some ideas of the proof. For more details see [20]. According to Lemma 1, f



CUBO
11, 1 (2009)

On mapping properties of monogenic functions 91

can be written as Fourier series respecting the decomposition (11)

f(x) = f(0) +

∞∑

n=1

(
X0,†,∗R
n

(x)α0
n

+

n∑

m=1

[
Xm,†,∗R
n

(x)αm
n

+ Y m,†,∗R
n

(x)βm
n

]
)

︸ ︷︷ ︸
=g

+

∞∑

n=1

[
Xn+1,†,∗R
n

(x)αn+1
n

+ Y n+1,†,∗R
n

(x)βn+1
n

]

︸ ︷︷ ︸
=h

.

We will present the proof only for the coefficients α0
n

of g. The remaining coefficients αm
n

and

βm
n

(m = 1, ...,n) are obtained in a similar way. As described, we aim to compare each Fourier

coefficient α0
n

with Sc(f). We have seen several times before that

Sc(f) =

∞∑

n=0

(
Sc(X0,†,∗R

n
)α0
n

+

n∑

m=1

[
Sc(Xm,†,∗R

n
)αm
n

+ Sc(Y m,†,∗R
n

)βm
n

]
)
.

Multiplying both sides of the previous expression by the solid spherical harmonics {Sc(X
0,†,∗R
k

), Sc(X
p,†,∗R
k

), Sc(

p = 1, ...,k} (k ≥ 1) and integrating over the ball we get the desired relations.

For the study of the coefficients αn+1
n

and βn+1
n

we multiply the equation

Sc(he1) =

∞∑

n=0

[
Sc(Xn+1,†,∗R

n
e1)α

n+1
n

+ Sc(Y n+1,†,∗R
n

e1)β
n+1
n

]

by the orthogonal solid spherical harmonics Sc(X
k+1,†,∗R
k

e1) (resp.

Sc(Y
k+1,†,∗R
k

e1)) (k ≥ 1) and integrating over the ball carries our results.

Lemma 4. Let f be a square integrable A-valued monogenic function. For each n ∈ N0, the

Fourier coefficients satisfy the inequalities

|α0
n
| ≤ 2

√
π

3

√
R3

‖X0,†
n

‖L2(BR;A;R)

‖Sc(X
0,†
n )‖L2(BR)

sup
|ξ|≤R

|Sc (f(ξ)) |

|αm
n
| ≤ 2

√
π

3

√
R3

‖Xm,†
n

‖L2(BR;A;R)

‖Sc(X
m,†
n )‖L2(BR)

sup
|ξ|≤R

|Sc (f(ξ)) |

|βm
n
| ≤ 2

√
π

3

√
R3

‖Y m,†
n

‖L2(BR;A;R)

‖Sc(Y
m,†
n )‖L2(BR)

sup
|ξ|≤R

|Sc (f(ξ)) |, m = 1, ...,n

|αn+1
n

| ≤ 2

√
π

3

√
R3

‖Xn+1,†
n

‖L2(BR;A;R)

‖Sc(X
n+1,†
n e1)‖L2(BR)

sup
|ξ|≤R

|Sc (he1(ξ)) |

|βn+1
n

| ≤ 2

√
π

3

√
R3

‖Y n+1,†
n

‖L2(BR;A;R)

‖Sc(Y
n+1,†
n e1)‖L2(BR)

sup
|ξ|≤R

|Sc (he1(ξ)) |.



92 K. Gürlebeck and J. Morais CUBO
11, 1 (2009)

The previous inequalities are basic results to prove the following theorem:

Theorem 8 (Real-Part Theorem). Let f be a square integrable A-valued monogenic function in

BR. Then, for 0 ≤ r <
R

2
we have the inequality

|f|r ≤ |f(0)| +

√
2

3

8r

(R − 2r)3

(
A1(r,R) sup

|ξ|≤R
|Sc (f(ξ)) |

+ A2(r,R) sup
|ξ|≤R

|Sc (he1(ξ)) |

)
,

where

|f|r = max
|x|=r

|f(x)|

and

A1(r,R) =
8r2(2R − r)

R − 2r
+ 6R2

A2(r,R) = 4r
2 − 6rR + 3R2.

Proof. Considering f written as in (11) and taking into account the maximum modulus principle

we have

|f|r ≤ |f(0)| + |g|R + |h|R.

Let us start with the study of the function g. Using the previous lemma it follows that

|g|R ≤ 2

√
π

3

√
R3 sup

|ξ|≤R
|Sc (f(ξ)) |

∞∑

n=1

[
|X0,†,∗R

n
|
‖X0

n
‖L2(BR;A;R)

‖Sc(X0
n
)‖L2(BR)

+

n∑

m=1

(
|Xm,†,∗R

n
|
‖Xm,†

n
‖L2(BR;A;R)

‖Sc(X
m,†
n )‖L2(BR)

+ |Y m,†,∗R
n

|
‖Y m,†

n
‖L2(BR;A;R)

‖Sc(Y
m,†
n )‖L2(BR)

)]
.

Applying Proposition 2 and taking into account Remark 5 it follows

|g|R ≤ 2

√
2

3
sup
|ξ|≤R

|Sc (f(ξ)) |

∞∑

n=1

(
2r

R

)
n

(n + 1)2(n + 2).

In the same way, we can study the function h.

|h|R = |h̃|R ≤ 2

√
2

3
sup
|ξ|≤R

|Sc (he1(ξ)) |
∞∑

n=1

(
2r

R

)
n

(n + 1)(n + 2).

Finally, we obtain

|f|r ≤ |f(0)| + 2

√
2

3
sup
|ξ|≤R

|Sc (f(ξ)) |

∞∑

n=1

(
2r

R

)
n

(n + 1)2(n + 2)

+ 2

√
2

3
sup
|ξ|≤R

|Sc (he1(ξ)) |

∞∑

n=1

(
2r

R

)
n

(n + 1)(n + 2).



CUBO
11, 1 (2009)

On mapping properties of monogenic functions 93

Now, note that the last series are convergent for 0 ≤ r < R
2
.

The previous theorem states that a monogenic L2-function f : Ω ⊂ R
3 −→ A is bounded by a

combination of its real part and one of its other components. This result is a more general estimate

but, in fact, it is not a complete analogy to the complex case. Therein, an analytic function is

only bounded by its real part. However, restricting ourselves to the class of functions which are

orthogonal to the subspace of the non-trivial hyperholomorphic constants in L2(BR; A; R), we get

a stronger result:

Corollary 1. Let f̃ be a square integrable A-valued monogenic function in BR orthogonal to the

non-trivial hyperholomorphic constants with respect to the inner product (3). Then, for 0 ≤ r < R
2

we have the following inequality:

|f̃|r ≤ |f̃(0)| +
8rA(r,R)

(R − 2r)4
sup
|ξ|≤R

|Sc(f̃(ξ))|

where

A(r,R) = 8r2(2R − r) + 6R2(R − 2r).

Remark 7. Replacing f̃(x) by f(x) − f(0) in the resulting relation from the previous corollary,

we arrive at

|f(x) − f(0)|r ≤
8rA(r,R)

(R − 2r)4
sup
|ξ|≤R

|Sc (f(ξ) − f(0)) |

with A(r,R) as in the previous theorem, which is a refinement of Hadamard’s real part theorem.

We observe that in the constants of our estimates, the factor 1/(R − 2r) occurs and not R −r

as it could be expected from the complex case. As we will see in the next section, to explain this

is because monogenic functions do not map balls to balls in the small but balls to ellipsoids.

6 First applications

As in the case of holomorphic functions in the complex plane we have to ask if also the growth of

the derivative (here of the hypercomplex derivative) can be bounded by the growth of the function.

If this is possible then, consequently, the behaviour of the derivative can be estimated by the scalar

part of the monogenic function. The following theorem gives a first result.

Theorem 9. Let f be a square integrable A-valued monogenic function in BR. Then, for 0 ≤ r <
R

2

we have the following inequality:
∣∣∣∣(

1

2
D)f(x)

∣∣∣∣
r

≤
8
√

3

R2(2R2 + 7rR + 2r2)

(R − 2r)5
sup
|ξ|≤R

|Sc (f(ξ)) |.



94 K. Gürlebeck and J. Morais CUBO
11, 1 (2009)

Proof. We consider f written as in (11) (Lemma 1). Since the referred series is convergent in

L2, it converges uniformly to f in each compact subset of BR. Also the series of all partial

derivatives converges uniformly to the corresponding partial derivatives of f in compact subsets of

BR. Applying the hypercomplex derivative
1

2
D term by term to the series, it follows formally

(
1

2
D)f =

∞∑

n=1

[
(
1

2
D)X0,†,∗R

n
α0
n

+

n∑

m=1

(
(
1

2
D)Xm,†,∗R

n
αm
n

+ (
1

2
D)Y m,†,∗R

n
βm
n

)

+ (
1

2
D)Xn+1,†,∗R

n
αn+1
n

+ (
1

2
D)Y n+1,†,∗R

n
βn+1
n

]

The proof follows from the idea applied in Theorem 8 with the estimates of the Fourier

coefficients from Lemma 4 and taking into account the following equalities for the homogeneous

monogenic polynomials (7) and their derivatives:

(
1

2
D)Xl,†

n
= (n + l + 1)X

l,†
n−1

(
1

2
D)Y m,†

n
= (n + m + 1)Y

m,†
n−1,

for l = 0, ...,n and m = 1, ...,n.

7 M-conformal mappings

The concept of monogenic-conformal mappings described by paravector-valued real differentiable

functions in Ω ⊂ Rn+1 and with values in the Clifford algebra Cl0,n (in the Cauchy-Riemann sense)

was introduced by Malonek in [28]. Let z∗ ∈ S be a fixed point and {Sm} a regular sequence of

subdomains which is shrinking to z∗ if m tends to infinity and whereby z∗ belongs to all Sm. In

[28] it is shown that a function F realizes locally in the neighborhood of a fixed point z = z∗ a left

M-conformal mapping if and only if F is left monogenic and its left derivative is different from

zero.

This result is described by the limit of a ”quotient” of a 2-form (surface area) and a 3-form

(volume). However, the geometric properties of such a result are not directly visible. Here we show

that the description of monogenic functions can be now formulated easily by accessible geometric

mapping properties.

For simplicity in what follows we focus our attention on the case of the Dirac operator. Let

R
0,3 be the real vector space R3 endowed with a quadratic form of signature (0, 3) and let (ε0,ε1,ε2)

be an associated orthonormal basis for R0,3. Then R0,3 generates the Clifford algebra R0,3 which

is a real linear associative algebra of dimension 23 and with identity 1. The multiplication in R0,3

is given according to the multiplication rules

ε2
i

= −1, i = 0, 1, 2

εiεj + εjεi = 0, i 6= j, 0 ≤ i,j ≤ 2.



CUBO
11, 1 (2009)

On mapping properties of monogenic functions 95

We introduce the Dirac operator

∂ = ε0∂x0 + ε1∂x1 + ε2∂x2. (15)

We are mainly interested in the case of vector-valued functions F : Ω ⊂ R3 −→ R0,3 defined

as

F(x) = F0(x)ε0 + F1(x)ε1 + F2(x)ε2, (16)

where its coordinates Fi (i = 0, 1, 2) are real-valued functions defined in Ω. Continuously real-

differentiable functions F : Ω −→ R0,3 which satisfy

∂F = 0 ⇐⇒





∂x0F0 + ∂x1F1 + ∂x2F2 = 0

∂x0F1 − ∂x1F0 = 0

∂x0F2 − ∂x2F0 = 0

∂x1F2 − ∂x2F1 = 0

, (17)

are said to be (left) monogenic in Ω. Moreover, as ∂2 = −∆, where ∆ is the Laplace operator in

R
3, (left) monogenic functions in Ω are also harmonic in Ω.

Let F : Ω ⊂ R3 −→ R0,3 be an arbitrary real-differentiable function. Clearly then,

F(x) := F(0) + f(x) + R(x)

being f its linear part and R stands for the rest (degree n ≥ 2). Under the above hypotheses, we

claim that f is a general linear function given as in (16), where the coordinates fi (i = 0, 1, 2) are

given by

f0(x) = a0x0 + a1x1 + a2x2

f1(x) = b0x0 + b1x1 + b2x2 (18)

f2(x) = c0x0 + c1x1 + c2x2.

Let us denote by E the ellipsoid generated by the quadratic form

E := {(x0,x1,x2) :
x2

0

α2
+
x2

1

β2
+
x2

2

γ2
= 1}, (19)

where α, β and γ are the lengths of the semi-axes. The next theorem characterizes the local

mapping properties of the linear part f of an arbitrary function F .

Theorem 10. Let f be a linear function. Then, the function f is monogenic if and only if it maps

a ball to an ellipsoid centered at the origin with the property that the reciprocal of the length of one

semi-axis is equal to the sum of the reciprocals of the lengths of the other two semi-axes.



96 K. Gürlebeck and J. Morais CUBO
11, 1 (2009)

Proof. For simplicity we just prove the sufficient condition. The necessary condition can be found

in [23]. Suppose that there exists a linear analytic function f which maps the unit ball B to an

arbitrarily oriented ellipsoid ε with the referred property. We rotate this ellipsoid transforming it

to an ellipsoid ε∗ such that the directions of its semi-axes y(0), y(1), y(2) coincide with the directions

of the standard coordinate system (y0,y1,y2). Such an ellipsoid is given by

ε∗ : {(y0,y1,y2) :
y2
0

α2
+
y2
1

β2
+
y2
2

γ2
≤ 1}.

Let f̃ be the function whose image represents ε∗. We denote by D̃ the associated matrix to f̃

D̃ =




ã0 ã1 ã2

b̃0 b̃1 b̃2

c̃0 c̃1 c̃2


 .

We remind that ε∗ preserves the orientation, and therefore, it holds the property D̃y(i) = λy(i)

(i = 0, 1, 2). It is easily seen that D̃ is a diagonal matrix and moreover, its elements satisfy the

equation ã0 + b̃1 + c̃2 = 0. In this case the associated function is given by

f̃(x) = −(b̃1 + c̃2)x0e0 + b̃1x1e1 + c̃2x2e2.

It is easy to check that this function is monogenic (with respect to the Dirac operator). Now,

as it was described before, we apply a rotation R to ε∗ in order to obtain ε. Roughly speaking, for

the rotation

R =




r1,1 r1,2 r1,3

r2,1 r2,2 r2,3

r3,1 r3,2 r3,3




such that RTR = I = RRT , we have then that RD̃RT := A. Note that the original function f is

now represented by the symmetric matrix A, in fact AX = f(x) for X = (x0 x1 x2)
T , being its

coordinates given by

f0(x) = (ã0r
2
1,1

+ b̃1r
2
1,2

+ c̃2r
2
1,3

)x0 + (ã0r1,1r2,1 + b̃1r1,2r2,2 + c̃2r1,3r2,3)x1

+ (ã0r1,1r3,1 + b̃1r1,2r3,2 + c̃2r1,3r3,3)x2

f1(x) = (ã0r1,1r2,1 + b̃1r1,2r2,2 + c̃2r1,3r2,3)x0 + (ã0r
2
2,1

+ b̃1r
2
2,2

+ c̃2r
2
2,3

)x1

+ (ã0r2,1r3,1 + b̃1r2,2r3,2 + c̃2r2,3r3,3)x2

f2(x) = (ã0r1,1r3,1 + b̃1r1,2r3,2 + c̃2r1,3r3,3)x0

+ (ã0r2,1r3,1 + b̃1r2,2r3,2 + c̃2r2,3r3,3)x1 + (ã0r
2
3,1

+ b̃1r
2
3,2

+ c̃2r
2
3,3

)x2.

It is easy to check, as desired, that the function f is monogenic.



CUBO
11, 1 (2009)

On mapping properties of monogenic functions 97

Remark 8. By a simple linear transformation of variables the previous theorem can be extended

to a (small) ball with radius R.

Remark 9. The composition of a linear function with a translation allows to extend the result to

an ellipsoid centered at an arbitrary point x̃. This composition preserves the monogenicity.

Next one has to show that Theorem 10 can be generalized to arbitrary real-analytic functions

which have the described local mapping properties. A monogenic function with non-vanishing

linear part will map in the small balls to the special class of ellipsoids. Non-vanishing linear part

means that all directional first derivatives of the function are different from zero. Equivalently

this can be characterized by the Jacobian determinant. For details and further relations to the

hypercomplex derivative see the paper [11].

Theorem 11. Let F be a real-analytic function. Then, the function F is monogenic if and only

if it maps locally a ball to an ellipsoid with the property that the reciprocal of the length of one

semi-axis is equal to the sum of the reciprocals of the lengths of the other two semi-axes.

Proof. If the function is monogenic then there is almost nothing to prove. The local mapping

properties at a point x are determined by the linear part of the Taylor expansion at x. Theorem

10 leads to the stated result.

If a real-analytic function has the supposed local mapping properties then we have to expand

the function at x in a real Taylor series. Applying ideas from [15], Chapter II, paragraph 5.2.2

we can show after a longer calculation which we will skip here that the linear part of the Taylor

expansion satisfies at each point of the domain the Dirac equation and so the function must be

monogenic.

Monogenic functions as null solutions of the Dirac operator can be mapped to monogenic (or

anti-monogenic) functions in the sense of satisfied Cauchy-Riemann equations (for details see, e.g.,

[13]). This transformation is an isometry and so it becomes clear that the here discussed mapping

properties of monogenic functions remain true under this transformation.

The result of Theorem 11 allows to describe the monogenic functions as a special class of

quasi-conformal mappings. If we visualize quasi-conformal mappings in R3 by points, given by

the lengths of the semi-axes of the associated ellipsoids, then the monogenic functions (with non-

vanishing Jacobian determinant) can be seen as a two-dimensional manifold in R3.

Acknowledgments: Financial support from Foundation for Science and Technology (FCT)

via the PHD/grant SFRH/BD/19174/2004.



98 K. Gürlebeck and J. Morais CUBO
11, 1 (2009)

Received: April 2008. Revised: August 2008.

References

[1] L. Aizenberg, Multidimensional analogues of Bohr’s theorem on power series. Proc. Amer.

Math. Soc. 128, 1147-1155 (2000).

[2] S. Axler, P. Bourdon, and W. Ramey, Harmonic Function Theory. Springer-Verlag, New

York (1992).

[3] C. Beneteau, A. Dahlner and D. Khavinson, Remarks on the Bohr Theorem. Computa-

tional Methods and Function Theory. Volume 4, Number 1, 1-19 (2004).

[4] H.P. Boas and D. Khavinson, Bohr’s Power Series Theorem in several variables. Proceedings

of the American Mathematical Society. Volume 125, Number 10, 2975-2979 (1997).

[5] S. Bock , M. I. Falcão, K. Gürlebeck and H. Malonek, A 3-Dimensional Bergman

Kernel Method with Applications to Rectangular Domains, Journal of computational and applied

mathematics 189, 67-79 (2006).

[6] H. Bohr, A theorem concerning power series. Proc. London Math. Soc. (2) 13, 1-5 (1914).

[7] B. Boone, Bergmankern en conforme albeelding: een excursie in drie dimensies. Proefschrift,

Universiteit Gent Faculteit van de Wetenshappen (1991-1992).

[8] F. Brackx, R. Delanghe, and F. Sommen, Clifford Analysis. Pitman Publishing, Boston-

London-Melbourne, 1982.

[9] I. Cação, Constructive Approximation by Monogenic polynomials. Ph.D. thesis, Universidade

de Aveiro, Departamento de Matemática, Dissertation, 2004.

[10] I. Cação, K. Gürlebeck and S. Bock, On Derivatives of Spherical Monogenics. Complex

Var. Elliptic Equ. 51, No. 8-11, 847-869 (2006).

[11] P. Cerejeiras K. Gürlebeck, U. Kähler, H. Malonek, A quaternionic Beltrami-Type

equation and the existence of local homeomorphic solutions, ZAA 20 (2001) 1, 17-34.

[12] R. Delanghe, On a class of inner spherical monogenics and their primitives, to appear.

[13] R. Delanghe, On homogeneous polynomial solutions of the Riesz system and their harmonic

potentials. Complex Variables and Elliptic Equations 52, 1047-1062 (2007)

[14] S. Dineen and R. M. Timoney, On a problem of H. Bohr. Bull. Soc. Roy. Sci. Liège, 60

401-404 (1991).

[15] K. Gürlebeck, K. Habetha, W. Sprößig, Holomorphic Functions in the Plane and n-

dimensional Space, Birkhäuser Verlag, Basel - Boston - Berlin, 2008.



CUBO
11, 1 (2009)

On mapping properties of monogenic functions 99

[16] K. Gürlebeck, H.R. Malonek, A hypercomplex derivative of monogenic functions in Rn+1

and its applications, ”Complex Variables”, Vol. 39, 199-228 (1999).

[17] K. Gürlebeck, W. Sprössig, Quaternionic Analysis and Elliptic Boundary Value Problems.

Akademieverlag Berlin, Math. Research 56 (1989).

[18] K. Gürlebeck, W. Sprössig, Quaternionic Calculus for Engineers and Physicists. John

Wiley and Sons, Chichester (1997).

[19] K. Gürlebeck and J. Morais, Bohr’s Theorem for monogenic functions, AIP Conf. Proc.,

936, 750-753 (2007).

[20] K. Gürlebeck, J. Morais and P. Cerejeiras, Borel-Carathéodory Type Theorem for

monogenic functions. Accepted in Complex Analysis and Operator Theory.

[21] K. Gürlebeck and J. Morais, Bohr’s Theorem for Monogenic Power Series. Submitted to

Computational Methods and Function Theory.

[22] K. Gürlebeck and J. Morais, Hadamard’s real part theorem for monogenic functions, to

appear in AIP Conf. Proc. (2008)

[23] K. Gürlebeck and J. Morais, Geometric characterization of M-conformal mappings, sub-

mitted.

[24] J. Hadamard, Sur les fonctions entières de la forme eG(X). C.R. Acad. Sci., 114, 1053-1055

(1892).

[25] E. Landau, Über den Picardschen Satz. Vierteljahrschr. Naturforsch. Gesell. Zürich 51 (1906),

252-318.

[26] E. Landau, Beiträge zur analytischen Zahlentheorie. Rend. Circ. Mat. Palermo 26 (1908),

169-302.

[27] H. Leutwiller, Quaternionic analysis in R3 versus its hyperbolic modification, in: Brackx,

F., Chisholm, J.S.R. and Souček, V. (ed.). NATO Science Series II. Mathematics, Physics and

Chemistry, vol. 25, Kluwer Academic Publishers, Dordrecht, Boston, London, 2001.

[28] H. Malonek, Monogenic functions and M-conformal mappings. Clifford Analysis and its

applications ed. F. Brackx et al., Kluwer (2001).

[29] C. Müller, Spherical Harmonics. Lecture Notes in Mathematics, 17. Springer-Verlag, Berlin,

1966.

[30] V.I. Paulsen, G. Popescu and D. Singh, On Bohr’s Inequality. London Mathematical

Society, volume 85, 493-512 (2002).



100 K. Gürlebeck and J. Morais CUBO
11, 1 (2009)

[31] V. V. Kravchenko, M. V. Shapiro, Integral Representations for Spatial Models of Mathe-

matical Physics. Research Notes in Mathematics, Pitman Advanced Publishing Program, London

(1996).

[32] V.V. Kravchenko, Applied quaternionic analysis. Research and Exposition in Mathematics.

28. Lemgo: Heldermann Verlag (2003).

[33] G. Kresin and V. Maz’ya, Sharp Real-Part Theorems - A Unified Approach. Lecture Notes

in Mathematics, Vol. 1903, Springer 2007.

[34] G. Sansone, Orthogonal Functions: Pure and Applied Mathematics. Vol. IX. Interscience

Publishers, New York, 1959.

[35] S. Sidon, Über einen Satz von Herrn Bohr. Math. Z. 26, 731-732 (1927).

[36] M. Tomić, Sur un théorème de H. Bohr. Math. Scand. 11, 103-106. MR 31:316 (1962).