Journal of Indian Acad. Math.

ISSN: 0970-5120

Vol. 48, No. 1 (2026) pp. 65–76.

ON MAXIMUM MODULUS THEOREM OF

MULTICOMPLEX VALUED FUNCTIONS

Debasmita Dutta^1∗, Sayan Jana², Sanjib Kumar Datta³and

Prosenjit Sen⁴

Abstract. In this paper, we consider the multicomplex space C_nwhich is a special

generalization of bicomplex space. We would like to derive the maximum modulus prin-

ciple and the minimum modulus principle in terms of multicomplex valued functions. In

addition, we also ﬁnd the analogue of Taylor’s series and Schwarz’s lemma in terms of

multicomplex valued functions.

Keywords: Multicomplex space, multicomplex function, holomorphic function, maxi-

mum modulus principle, minimum modulus principle.

2010 AMS Subject Classiﬁcation: 30D30, 30G35.

1. Introduction and Preliminaries

The theory of bicomplex numbers is a matter of active research for quite a long time

since seminal work as carried in [18] and [1] in search of special algebra. The algebra of

bicomplex numbers is widely used in the literature as it becomes a viable commutative

alternative [12] to the non-skew ﬁeld of quaternions introduced by Hamilton [10] (both

are four-dimensional and generalizations of complex numbers). Set of all real numbers R

and complex numbers C are denoted by C₀and C₁, respectively. The set of all bicomplex

numbers is denoted by C₂. Now we will discuss some basic deﬁnitions and preliminaries

of bicomplex analysis.

A bicomplex number is deﬁned as

ꢀ

ꢁ

z = x₁+ i₁x₂+ i₂x₃+ i₁i₂x₄= (x₁+ i₁x₂) + i₂x₃+ i₁x₄

= z₁+ i₂z₂,

(1)

where x_i, i = 1, 2, 3, 4, are real numbers with

i²₁= i²₂= −1,

i₁i₂= i₂i₁,

(i₁i₂)²= 1,

and z₁, z₂are complex numbers.

There are four idempotent elements in C₂, which are

1 + i₁i₂

1 − i₁i₂

0,

1,

,

.

2

Furthermore, for every z₁+ i₂z₂∈ C₂,

ꢂ

ꢃ

ꢂ

ꢃ

1 + i₁i₂

1 − i₁i₂

z₁+ i₂z₂= (z₁− i₁z₂)

+ (z₁+ i₁z₂)

.

2

Here, addition and scalar multiplication are deﬁned in the usual way. C₂is a linear

space with respect to addition and scalar multiplication.

65

66

Debasmita Dutta, Sayan Jana, Sanjib Kumar Datta and Prosenjit Sen

Here, ꢀ · ꢀ₂denotes the norm of elements in C₂and deﬁned as

2

2 1/2

ꢀz₁+ i₂z₂ꢀ₂= (|z₁| + |z₂| )

= (x²₁+ x²₂+ x²₃+ x²₄)^1/2

,

ꢂ

ꢃ

1/2

2

|z₁− i₁z₂| + |z₁+ i₁z₂|

ꢀz₁+ i₂z₂ꢀ₂=

.

2

With this norm, C₂is a normed linear space.

4

In addition, C₂embeds in C₀and C₀is complete. Therefore, C₂is complete and thus

a Banach space. For further study one can see [2], [5], [6], [7], [8] and [9].

2. Definitions

2.1. The Multicomplex Space C_n[15]. Now we deﬁne the space C_nfor n = 0, 1, 2, . . . .

C_nis not just a replacement of 2 by n; it is more than that. Many new relationships come

to light in C_nand many challenges arise in their treatment. An element ξ_n∈ C_ncan

be represented as a linear combination of elements in C₀, C₁, . . . , C_n−1. Corresponding to

each such representation of the elements in C_n−1there is a linear space in which C_ncan

be embedded.

A multicomplex number is deﬁned by

ξ_n= ξ_n−1,1+ i_nξ_n−1,2

,

where

ξ_n−1,1, ξ_n−1,2∈ C_n−1

,

(i_n)²= −1.

The addition is the operation on C_ndeﬁned by the function

⊕ : C_n× C_n−→ C_n

given by

ꢀ

ꢁ

ξ_n^′⊕ ξ_n^′′= (ξ_n^′_−1,1+ i_nξ_n^′_−1,2) + (ξ^′′

+ i_nξ_n^′′_−1,2) = (ξ_n^′_−1,1+ ξ_n^′′_−1,1) + i_nξ_n^′_−1,2+ ξ^′′

.

n−1,1

n−1,2

Multiplication is the operation on C_ndeﬁned by the function

⊙ : C_n× C_n−→ C_n

given by

ꢀ

ꢁ

ꢀ

ꢁ

′′

ξ_n^′⊙ξ_n^′′= ξ_n^′_−1,1

ξ

−ξ_n^′_−1,2

ξ

+i_nξ_n^′_−1,1

ξ

+ξ_n^′_−1,2

ξ

,

where (i_n)²= −1.

n−1,1

n−1,2

n−1,1

The norm on C_nis deﬁned by the the function

2

ꢀ · ꢀ_n: C_n−→ R_≥0,

ꢀξ_nꢀ_n= ꢀξ_n−1,1ꢀ_n−1+ ꢀξ_n−1,2ꢀ_n−1

.

So, the system (C_n, ⊕, ⊙) is a normed linear space and C_nis isomorphic and isometric

2ⁿ

with C₀.

ON MAXIMUM MODULUS THEOREM OF MULTICOMPLEX VALUED FUNCTIONS

67

2.2. Idempotent elements in C_n[15]. We know that in C₂the four elements

1 + i₁i₂

1 − i₁i₂

0,

1,

and

2

are idempotent, and any element of C₂can be written as a linear combination of these

idempotent elements.

Similarly, it is easy to verify that the following are idempotent elements in C_n.

1 + i₁i₂1 − i₁i₂1 + i₁i₃1 − i₁i₃1 + i₂i₃1 − i₂i₃

1 + i_{n−1 n}

i

1 − i_{n−1 n}

i

0,

1,

,

, . . . ,

,

.

2

For convenience in notation, deﬁne the symbols

1 + i_ri_s

1 − i_ri_s

e(i_ri_s) :=

,

e(−i_ri_s) :=

,

where r, s ∈ N.

2

2.3. Idempotent representation[15]. Let ξ be an element in C_nand let

ξ = ξ₁+ i_nξ₂

with ξ₁, ξ₂∈ C_n−1

.

Then

ꢀ

ꢁ ꢀ

ꢁ

ꢀ

ꢁ ꢀ

ꢁ

ξ = ξ₁− i_n−1ξ₂e i_{n−1 n}

i

+ ξ₁+ i_n−1ξ₂e − i_n−1i_n.

Let ξ ∈ C_nand let

ξ = ξ₁+ i_nξ₂,

where ξ₁, ξ₂∈ C_n−1. Then

ꢂ

ꢃ

1/2

2

ꢀξ₁− i_n−1ξ₂ꢀ_n−1+ ꢀξ₁+ i_n−1ξ₂ꢀ_n−1

ꢀξꢀ_n=

.

2

2.4. Singular Elements[15]. An element ξ ∈ C_nis called non-singular if and only if

there exists a unique η ∈ C_nsuch that ξη = 1, otherwise ξ is singular. The set of all

singular elements in C_nis denoted by

Θ_n,

n = 0, 1, 2, . . . .

Another way, the element ξ = ξ₁+i_nξ₂∈ C_nis non-singular if and only if both ξ₁−i_n−1ξ₂

and ξ₁+ i_n−1ξ₂are non-singular in C_n−1. The element ξ is singular if and only if at least

one of the elements ξ₁− i_n−1ξ₂or ξ₁+ i_n−1ξ₂is singular in C_n−1

.

2.5. Neighborhoods in C_n[15]. Suppose that

X₁= {ξ₁− i_n−1ξ₂: ξ₁, ξ₂∈ C_n−1},

X₂= {ξ₁+ i_n−1ξ₂: ξ₁, ξ₂∈ C_n−1}.

Then X(⊆ C_n) can be written as

X = {ξ₁+ i_nξ₂: ξ₁− i_n−1ξ₂∈ X₁, ξ₁+ i_n−1ξ₂∈ X₂}.

Then neighborhood N(a + i_nb, ε) ⊆ X is deﬁned as

N(a + i_nb, ε) = {ξ₁+ i_nξ₂∈ C_n: ꢀ(ξ₁+ i_nξ₂) − (a + i_nb)ꢀ_n< ε} .

Alternatively, it can be written in the form

{ξ₁+ i_nξ₂∈ C_n: ꢀ(ξ₁− i_n−1ξ₂) − (a − i_n−1b)ꢀ_n−1< ε, ꢀ(ξ₁+ i_n−1ξ₂) − (a + i_n−1b)ꢀ_n−1< ε} .

68

Debasmita Dutta, Sayan Jana, Sanjib Kumar Datta and Prosenjit Sen

2.6. Holomorphic Functions in C_n[15]. If a function f : X −→ C_n, X ⊆ C_n, n ≥ 1,

satisﬁes the strong Stolz condition at ξ₀∈ X, i.e. for all n ≥ 1, there exists a constant

α (which depends on ξ₀) in C_nand a function r(f; ξ₀, ξ) deﬁned in a neighborhood of ξ₀

and with values in C_n, such that:

(i) f(ξ) − f(ξ₀) = α(ξ − ξ₀) + r(f; ξ₀, ξ)(ξ − ξ₀)

(ii) lim r(f; ξ₀, ξ) = 0

ξ→ξ₀

Then f is diﬀerentiable at ξ₀. If f is diﬀerentiable in a neighborhood of each point in

domain X(⊆ C_n), then f is called a holomorphic multicomplex-valued function in X.

2.7. Idempotent Representation of Holomorphic Multicomplex Valued Functions[15].

Let X be a domain in C_n, n ≥ 1, and let f be a holomorphic function in C_nthen there

exists holomorphic functions

f₁: X₁→ C_n−1

and f₂: X₂→ C_n−1

where X₁, X₂⊆ C_n−1

,

such that

f(ξ₁+ i_nξ₂) = f₁(ξ₁− i_n−1ξ₂)e(i_n−1i_n) + f₂(ξ₁+ i_n−1ξ₂)e(−i_n−1i_n).

2.8. Power Series in C_n[15]. Let γ_k, k = 0, 1, . . ., be constants in C_n, and let ζ and ζ₀

denote elements in C_n. Then a power series about ζ₀in C_nis an inﬁnite series of the form

∞

ꢄ

γ_k(ζ − ζ₀)^k.

(2)

k=0

The study of the convergence of (2) is based on the idempotent representation of the

inﬁnite series. In order to state the relevant theorem, it is necessary to represent (2) in

the following notation:

∞

ꢄ

(c_k+ i_nd_k)[(ζ₁+ i_nζ₂) − (a₁+ i_na₂)]^k.

(3)

k=0

Now by Idempotent representation,

c_k+ i_nd_k= (c_k− i_n−1d_k)e(i_n−1i_n) + (c_k+ i_n−1d_k)e(−i_n−1i_n),

(4)

(5)

(6)

ζ₁+ i_nζ₂= (ζ₁− i_n−1ζ₂)e(i_n−1i_n) + (ζ₁+ i_n−1ζ₂)e(−i_n−1i_n),

a₁+ i_na₂= (a₁− i_n−1a₂)e(i_n−1i_n) + (a₁+ i_n−1a₂)e(−i_n−1i_n).

Then the inﬁnite series in (2) has the following idempotent representation:

∞

ꢄ

(c_k− i_n−1d_k)[(ζ₁− i_n−1ζ₂) − (a₁− i_n−1a₂)]^ke(i_n−1i_n)

(7)

(8)

k=0

∞

ꢄ

+

(c_k+ i_n−1d_k)[(ζ₁+ i_n−1ζ₂) − (a₁+ i_n−1a₂)]^ke(−i_n−1i_n).

k=0

ON MAXIMUM MODULUS THEOREM OF MULTICOMPLEX VALUED FUNCTIONS

69

If the inﬁnite series (3) converges at ζ₁+ i_nζ₂, n ≥ 2, then the inﬁnite series in (7) and

(8) converge at ζ₁− i_n−1ζ₂and ζ₁+ i_n−1ζ₂, respectively, and (3) equals the expressions in

(7) and (8).

If the series in (7) and (8) converge at ζ₁− i_n−1ζ₂and ζ₁+ i_n−1ζ₂, respectively, then

the series in (3) converges at ζ₁+ i_nζ₂and equals the expression in (7) and (8).

3. Main Theorems

Theorem 3.1. Let X be a domain in C_n, n ≥ 2, and let the function

f : X −→ C_n

be analytic in X. Then for each ξ₀∈ X there is a neighborhood of ξ₀in which

∞

k

ꢄ

D_ξf(ξ₀)

f(ξ) =

(ξ − ξ₀)^k.

k!

k=0

Proof. Since f is analytic in X, there exist holomorphic functions

f₁: X₁→ C_n−1

and f₂: X₂→ C_n−1

where X₁, X₂⊆ C_n−1

,

such that

f(ξ) = f(ξ₁+ i_nξ₂) = f₁(ξ₁− i_n−1ξ₂)e(i_n−1i_n) + f₂(ξ₁+ i_n−1ξ₂)e(−i_n−1i_n).

(9)

Where ξ₁+i_nξ₂is an arbitrary point in X and ξ₁−i_n−1ξ₂and ξ₁+i_n−1ξ₂are corresponding

points in X₁and X₂, respectively. We prove this result by induction on n.

For n = 2, the result is true for bicomplex valued diﬀerentiable functions. Let us assume

that the result is true for n − 1.

Let ξ₀= ξ⁰+ i_nξ⁰be a point in X.

1

2

Then ξ⁰+ i_nξ⁰= (ξ⁰− i_n−1ξ⁰)e(i_n−1i_n) + (ξ⁰+ i_n−1ξ⁰)e(−i_n−1i_n).

1

2

1

2

1

2

Since the functions f₁, f₂are holomorphic in C_n−1, by the induction hypothesis they

can be represented by power series in neighborhoods of ξ⁰− i_n−1ξ⁰and ξ⁰+ i_n−1ξ⁰. Thus

1

2

1

2

there exist r₁, r₂> 0 such that

∞

k

1

f₁(ξ⁰− i_n−1ξ⁰)

ꢄ

D_{ξ −i}

1

2

_n−1ξ₂

k

f₁(ξ₁− i_n−1ξ₂) =

{(ξ₁− i_n−1ξ₂) − (ξ⁰− i_n−1ξ⁰)} ,

1

2

k!

k=0

ꢀ(ξ₁− i_n−1ξ₂) − (ξ⁰− i_n−1ξ⁰)ꢀ_n−1< r₁

(10)

(11)

1

2

and

∞

k

1

ꢄ

D_{ξ +i}

f₂(ξ⁰+ i_n−1ξ⁰)

1

2

_n−1ξ₂

k

f₂(ξ₁+ i_n−1ξ₂) =

{(ξ₁+ i_n−1ξ₂) − (ξ⁰+ i_n−1ξ⁰)} ,

1

2

k!

k=0

ꢀ(ξ₁+ i_n−1ξ₂) − (ξ⁰+ i_n−1ξ⁰)ꢀ_n−1< r₂.

1

2

70

Debasmita Dutta, Sayan Jana, Sanjib Kumar Datta and Prosenjit Sen

Substituting (10) and (11) into (9) we get,

∞

k

1

ꢄ

D_{ξ −i}

f₁(ξ⁰− i_n−1ξ⁰)

1

2

_n−1ξ₂

k

f(ξ) =

{(ξ₁− i_n−1ξ₂) − (ξ⁰− i_n−1ξ⁰)} e(i_n−1i_n)

1

2

k!

k=0

∞

k

D_{ξ +i}

1

f₂(ξ⁰+ i_n−1ξ⁰)

ꢄ

1

2

_n−1ξ₂

k

+

{(ξ₁+ i_n−1ξ₂) − (ξ⁰+ i_n−1ξ⁰)} e(−i_n−1i_n).

(12)

1

2

k!

k=0

We know that [15]

D_ξ^k_−i

f₁(ξ⁰− i_n−1ξ⁰)e(i_n−1i_n) + D^k

f₂(ξ⁰+ i_n−1ξ⁰)e(−i_n−1i_n) = D^kf(ξ₀) (13)

_n−1ξ₂

1

2

ξ₁+i_n−1ξ₂

1

2

ξ

1

k

{(ξ₁− i_n−1ξ₂) − (ξ⁰− i_n−1ξ⁰)} e(i_n−1i_n)

1

2

+{(ξ₁+ i_n−1ξ₂) − (ξ⁰+ i_n−1ξ⁰)} e(−i_n−1i_n) = (ξ − ξ₀)^k

(14)

k

1

2

Therefore

D_ξ^k_−i

f₁(ξ⁰− i_n−1ξ⁰)

1

2

_n−1ξ₂

1

k

{(ξ₁− i_n−1ξ₂) − (ξ⁰− i_n−1ξ⁰)} e(i_n−1i_n)

1

2

k!

D_ξ^k_+i

f₂(ξ⁰+ i_n−1ξ⁰)

1

2

_n−1ξ₂

1

k

+

{(ξ₁+ i_n−1ξ₂) − (ξ⁰+ i_n−1ξ⁰)} e(−i_n−1i_n)

1

2

k!

D^kf(ξ₀)

ξ

=

(ξ − ξ₀)^k.

(15)

k!

Substituting (13),(14) and (15) into (12) we get,

∞

k

ꢄ

D_ξf(ξ₀)

f(ξ) =

(ξ − ξ₀)^k,

k!

k=0

and this will converge for all ξ = ξ₁+ i_nξ₂such that

ꢀ(ξ₁− i_n−1ξ₂) − (ξ⁰− i_n−1ξ⁰)ꢀ_n−1< r₁,

1

2

ꢀ(ξ₁+ i_n−1ξ₂) − (ξ⁰+ i_n−1ξ⁰)ꢀ_n−1< r₂.

1

2

□

Remark 3.2. This theorem is known as Taylor’s series for a holomorphic multicomplex-

valued function.

Example 3.3. Let f : C_n→ C_nbe a function such that

f(ξ₁+ i_nξ₂) = e^{ξ +i ξ}= e^{(ξ −i}

^{ξ )}e(i_n−1i_n) + e^{(ξ +i}

^{ξ )}e(−i_n−1i_n)

1

n

2

1

n−1

2

1

n−1

2

∞

(ξ₁− i_n−1ξ₂)^k

(ξ₁+ i_n−1ξ₂)^k

ꢄ

=

e(i_n−1i_n) +

e(−i_n−1i_n)

k!

k=0

∞

k=0

ꢄ

1

{(ξ₁− i_n−1ξ₂)^ke(i_n−1i_n) + (ξ₁+ i_n−1ξ₂)^ke(−i_n−1i_n)}

k!

k=0

∞

k

ꢄ

(ξ₁+ i_nξ₂)

k!

k=0

This series is convergent for all (ξ₁+ i_nξ₂) ∈ C_n. This is a power series of e^{ξ +i ξ}

.

1

n

2

ON MAXIMUM MODULUS THEOREM OF MULTICOMPLEX VALUED FUNCTIONS

71

Theorem 3.4 (Analogue to Maximum-Modulus Principle). Let X ⊆ C_nbe a bounded

domain and a function f : X −→ C_nbe holomorphic in X. Then ꢀf(ξ)ꢀ_ncan not attain

maximum in X unless f(ξ) is constant.

Proof. We prove it by induction on n.

For n = 2, the result is true for bicomplex valued holomorphic functions.

Let the result be true for n − 1.

Since f is holomorphic i.e. analytic in X then there exists holomorphic functions

f₁: X₁→ C_n−1

and f₂: X₂→ C_n−1

where X₁, X₂⊆ C_n−1

,

such that

f(ξ) = f(ξ₁+ i_nξ₂) = f₁(ξ₁− i_n−1ξ₂)e(i_n−1i_n) + f₂(ξ₁+ i_n−1ξ₂)e(−i_n−1i_n).

Where ξ₁+i_nξ₂is an arbitrary point in X and ξ₁−i_n−1ξ₂and ξ₁+i_n−1ξ₂are corresponding

points in X₁and X₂, respectively.

Let f(ξ) attains maximum at ξ₀= ξ⁰+ i_nξ⁰∈ X.

1

2

Now ξ⁰+ i_nξ⁰= (ξ⁰− i_n−1ξ⁰)e(i_n−1i_n) + (ξ⁰+ i_n−1ξ⁰)e(−i_n−1i_n).

1

2

1

2

1

2

Therefore for all ξ₁+ i_nξ₂∈ X

ꢀf(ξ₁+ i_nξ₂)ꢀ_n≤ ꢀf(ξ⁰+ i_nξ⁰)ꢀ_n

1

2

ꢂ

ꢃ

1/2

2

ꢀf₁(ξ₁− i_n−1ξ₂)ꢀ_n−1+ ꢀf₂(ξ₁+ i_n−1ξ₂)ꢀ_n−1

=⇒

2

ꢂ

ꢃ

1/2

2

ꢀf₁(ξ⁰− i_n−1ξ⁰)ꢀ

+ ꢀf₂(ξ⁰+ i_n−1ξ⁰)ꢀ

1

2

n−1

1

2

n−1

≤

2

=⇒ ꢀf₁(ξ₁− i_n−1ξ₂)ꢀ_n−1+ ꢀf₂(ξ₁+ i_n−1ξ₂)ꢀ_n−1

≤ ꢀf₁(ξ⁰− i_n−1ξ⁰)ꢀ

+ ꢀf₂(ξ⁰+ i_n−1ξ⁰)ꢀ

2

1

2

n−1

1

2

n−1

∀{(ξ₁− i_n−1ξ₂)e(i_n−1i_n) + (ξ₁+ i_n−1ξ₂)e(−i_n−1i_n)} ∈ X.

Hence,

ꢀf₁(ξ₁− i_n−1ξ₂)ꢀ_n−1≤ ꢀf₁(ξ⁰− i_n−1ξ⁰)ꢀ_n−1

1

2

for all ꢀ(ξ₁− i_n−1ξ₂) − (ξ⁰− i_n−1ξ⁰)ꢀ_n−1< r where (ξ₁− i_n−1ξ₂) ∈ X₁.

1

2

Similarly,

ꢀf₂(ξ₁+ i_n−1ξ₂)ꢀ_n−1≤ ꢀf₂(ξ⁰+ i_n−1ξ⁰)ꢀ_n−1

1

2

for all ꢀ(ξ₁+ i_n−1ξ₂) − (ξ⁰+ i_n−1ξ⁰)ꢀ_n−1< r where (ξ₁+ i_n−1ξ₂) ∈ X₂.

1

2

So by induction hypothesis we can say that f₁(ξ₁− i_n−1ξ₂) and f₂(ξ₁+ i_n−1ξ₂) both

are constant functions. Hence, f(ξ) is constant.

□

Remark 3.5. The following example ensures Theorem 3.4.

72

Debasmita Dutta, Sayan Jana, Sanjib Kumar Datta and Prosenjit Sen

Example 3.6. Let f : C_n→ C_nbe such that

f(ξ) = ξ²

To ﬁnd max_ξ∈Gꢀf(ξ)ꢀ_n, with G = {ξ ∈ C_n: ꢀξ − 1 − i_nꢀ_n< 1}

To ﬁnd the maximum value of f(ξ), ﬁrst we ﬁnd the maximum value of f in C₁(the

complex plane), where n = 1.

On |ξ − 1 − i| = 1,

ξ = 1 + i + e^iθ,

(1 + cos θ) + i(1 + sin θ),

0 ≤ θ ≤ 2π

|ξ²| = |ξ| = (1 + cos θ)²+ (1 + sin θ)²= 3 + 2(cos θ + sin θ)

2

√

It is clear that the maximum value of |ξ²| is 3 + 2 2 in C₁.

Then we ﬁnd the maximum value of f in C₂(the bicomplex plane), where n = 2.

Let ξ ∈ C₂, then it can be written in idempotent form as

ξ = (ξ₁− i₁ξ₂)e₁+ (ξ₁+ i₁ξ₂)e₂

where ξ₁, ξ₂∈ C₁

1 + i₁i₂

1 − i₁i₂

and e₁=

,

e₂=

.

2

√

2

√

|ξ₁− i₁ξ₂| + |ξ₁+ i₁ξ₂|

(3 + 2 2) + (3 + 2 2)

(ꢀξꢀ₂)²=

≤

= 3 + 2 2

2

√

∴ Maximum value of (ꢀξꢀ₂)²is 3 + 2 2 in C₂.

Next we ﬁnd the maximum value of f in C₃, i.e. here n = 3.

Let ξ ∈ C₃, then it can be written as

ξ = (ξ₁− i₂ξ₂)e(i₂i₃) + (ξ₁+ i₂ξ₂)e(−i₂i₃)

where ξ₁− i₂ξ₂∈ C₂,

ξ₁+ i₂ξ₂∈ C₂

1 + i₂i₃

1 − i₂i₃

and e(i₂i₃) =

,

e(−i₂i₃) =

.

2

√

(ꢀξ₁− i₂ξ₂ꢀ₂)²+ (ꢀξ₁+ i₂ξ₂ꢀ₂)²

(3 + 2 2) + (3 + 2 2)

2

√

(ꢀξꢀ₃)²=

≤

= 3 + 2 2

2

√

∴ Maximum value of (ꢀξꢀ₃)²is 3 + 2 2 in C₃.

√

In a similar way, we can say that the maximum value of (ꢀξꢀ_n)²is 3 + 2 2 in C_n.

Remark 3.7. The bounded condition of the domain in the Maximum Modulus Theorem

is impossible to be dropped, as shown by the following example.

Let

G = {ξ = ξ₁+ i_nξ₂∈ C_n: where ξ₁, ξ₂∈ C_nand ꢀξ₂ꢀ > 0} .

Deﬁne the function f : C_n→ C_nby

f(ξ) = exp(−i_nξ)

and then f is analytic on G.

Now, for ξ ∈ ∂G

ꢀf(ξ)ꢀ = ꢀexp(−i_nξ₁)ꢀ = 1

ON MAXIMUM MODULUS THEOREM OF MULTICOMPLEX VALUED FUNCTIONS

73

But for ξ = ξ₁+ i_nξ₂∈ G,

ꢀf(ξ)ꢀ = ꢀ exp(ξ₂)ꢀ → ∞ as ꢀξ₂ꢀ → ∞

Thus, the Maximum Modulus Theorem fails on unbounded regions.

Theorem 3.8 (Analogue of the Minimum-Modulus Theorem). Let X ⊆ C_nbe a bounded

domain and a function f : X −→ C_nbe holomorphic in X. Then ꢀf(ξ)ꢀ_ncan not attain

minimum in X unless f(ξ) is constant.

Proof. We prove it by induction on n.

For n = 2, the result is true for bicomplex valued holomorphic functions.

Let the result be true for n − 1.

Since f is holomorphic i.e. analytic in X then there exists holomorphic functions

f₁: X₁→ C_n−1

and f₂: X₂→ C_n−1

where X₁, X₂⊆ C_n−1

,

such that

f(ξ) = f(ξ₁+ i_nξ₂) = f₁(ξ₁− i_n−1ξ₂)e(i_n−1i_n) + f₂(ξ₁+ i_n−1ξ₂)e(−i_n−1i_n).

Where ξ₁+i_nξ₂is an arbitrary point in X and ξ₁−i_n−1ξ₂and ξ₁+i_n−1ξ₂are corresponding

points in X₁and X₂, respectively.

Let f(ξ) attains minimum at ξ₀= ξ⁰+ i_nξ⁰∈ X.

1

2

Now ξ⁰+ i_nξ⁰= (ξ⁰− i_n−1ξ⁰)e(i_n−1i_n) + (ξ⁰+ i_n−1ξ⁰)e(−i_n−1i_n).

1

2

1

2

1

2

Therefore for all ξ₁+ i_nξ₂∈ X

ꢀf(ξ₁+ i_nξ₂)ꢀ_n≤ ꢀf(ξ⁰+ i_nξ⁰)ꢀ_n

1

2

ꢂ

ꢃ

1/2

2

ꢀf₁(ξ₁− i_n−1ξ₂)ꢀ_n−1+ ꢀf₂(ξ₁+ i_n−1ξ₂)ꢀ_n−1

=⇒

2

ꢂ

ꢃ

1/2

2

ꢀf₁(ξ⁰− i_n−1ξ⁰)ꢀ

+ ꢀf₂(ξ⁰+ i_n−1ξ⁰)ꢀ

1

2

n−1

1

2

n−1

≥

2

=⇒ ꢀf₁(ξ₁− i_n−1ξ₂)ꢀ_n−1+ ꢀf₂(ξ₁+ i_n−1ξ₂)ꢀ_n−1

≥ ꢀf₁(ξ⁰− i_n−1ξ⁰)ꢀ

+ ꢀf₂(ξ⁰+ i_n−1ξ⁰)ꢀ

2

1

2

n−1

1

2

n−1

∀{(ξ₁− i_n−1ξ₂)e(i_n−1i_n) + (ξ₁+ i_n−1ξ₂)e(−i_n−1i_n)} ∈ X.

Hence,

ꢀf₁(ξ₁− i_n−1ξ₂)ꢀ_n−1≥ ꢀf₁(ξ⁰− i_n−1ξ⁰)ꢀ_n−1

1

2

for all ꢀ(ξ₁− i_n−1ξ₂) − (ξ⁰− i_n−1ξ⁰)ꢀ_n−1< r where (ξ₁− i_n−1ξ₂) ∈ X₁.

1

2

Similarly,

ꢀf₂(ξ₁+ i_n−1ξ₂)ꢀ_n−1≥ ꢀf₂(ξ⁰+ i_n−1ξ⁰)ꢀ_n−1

1

2

for all ꢀ(ξ₁+ i_n−1ξ₂) − (ξ⁰+ i_n−1ξ⁰)ꢀ_n−1< r where (ξ₁+ i_n−1ξ₂) ∈ X₂.

1

2

74

Debasmita Dutta, Sayan Jana, Sanjib Kumar Datta and Prosenjit Sen

So by induction hypothesis we can say that f₁(ξ₁− i_n−1ξ₂) and f₂(ξ₁+ i_n−1ξ₂) both

are constant functions. Hence, f(ξ) is constant.

□

Theorem 3.9 (Analogue of Schwarz’s Lemma). Let X ⊆ C_nbe a domain and the function

f : X → C_nbe analytic in a disk ꢀzꢀ_n< R(R > 0), having zero of order n at the origin

and let ꢀf(z)ꢀ_n≤ M for all z ∈ X. Then

(ꢀzꢀ_n)ⁿ

ꢀf(z)ꢀ_n≤M

for all z ∈ X

(16)

Rⁿ

Mn!

Rⁿ

and ꢀDⁿf(0)ꢀ_n≤

.

(17)

Proof. From Taylor’s series expansion of f in C_nand using the fact that f has a zero of

multiplicity n at z = 0, we have

Dⁿf(0)

Dⁿ⁺¹f(0)

f(z) =

zⁿ+

z

ⁿ⁺¹+ · · · ,

ꢀzꢀ_n< R.

n!

(n + 1)!

For z = 0, (1) is trivially true.

Let z = 0, then

f(z)

zⁿ

Dⁿf(0)

Dⁿ⁺¹f(0)

=

+

z + · · · ,

ꢀzꢀ_n< R.

n!

(n + 1)!

The inﬁnite series on the right-hand side of the above equation converges for ꢀzꢀ < R,

and if we deﬁne

ꢅ

f(z)

zⁿ

Dⁿf(0)

=

(18)

n!

z=0

f(z)

then

is analytic in disk X.

zⁿ

Let C be the circle ꢀzꢀ_n= ρ, where ρ < R. Then by the maximum modulus principle

in C_n,

ꢆ

f(z)

zⁿ

M

ρⁿ

≤

,

∀z with ꢀzꢀ_n≤ ρ.

n

Since the above inequality is true for all ρ < R, letting ρ → R, we get

ꢆ

f(z)

zⁿ

M

Rⁿ

≤

n

i.e.,

(ꢀzꢀ_n)ⁿ

ꢀf(z)ꢀ_n≤ M

.

Rⁿ

Now, (17) follows from (16) and in view of (18).

□

4. Conclusion

Introduction of quaternions which is Hamilton’s innovative approach is aimed to de-

scribe physical rotations in four dimensional space using mathematical framework. Build-

ing upon the complex number concept, bicomplex numbers extend into four dimensions

for ﬁnding applications in abstract algebra and theoretical physics. The multicomplex

space, which can be viewed as a special generalization of bicomplex space has implication

in various ﬁelds like topology, functional analysis, is now an active area of researchers.

ON MAXIMUM MODULUS THEOREM OF MULTICOMPLEX VALUED FUNCTIONS

75

5. Acknowledgement

The third author sincerely acknowledges the ﬁnancial support rendered by DST-FIST

2025-2026 running at the Department of Mathematics, University of Kalyani, P.O.: Kalyani,

Dist: Nadia, PIN: 741235, West Bengal, India.

REFERENCES

[1] Alpay D., Luna-Elizarras M. E., Shapiro M. and Struppa D. C.: Basics of functional analysis with bicomplex

scalars and bicomplex Schur analysis, Springer, 2013.

[2] Beg I., Datta S. K., Pal D., Fixed point in bicomplex valued metric spaces, Int. J. Nonlinear Anal. Appl.,

12(2), 2021, 717-727.

[3] Charak K. S., Rochon D. and Sharma N., Normal families of bicomplex meromorphic functions, Ann. Pol.

Math., 103(3), 2012, 303-317.

[4] Conway J. B., Functions of one complex variable, Second edition, Springer International Student Edition,

Narosa Publishing House, 2002.

[5] Datta D., Dey S., Sarkar S. and Datta S. K., Derivation of some consequences of Taylor’s Theorem - a

bicomplexial approach, Jn˜˜an˜abha, 51(2), 2021, 261-273.

[6] Datta S. K., Pal D., Sarkar R. and Manna A., Common ﬁxed point results for contractive type mappings in

bicomplex valued metric spaces, Applied Analysis and Optimization, 4(3), 2020, 323-338.

[7] Datta S. K., Pal D., Sarkar R. and Manna A., On a Common Fixed Point Theorem in Bicomplex Valued

b-Metric Space, Montes Taurus J. Pure Appl. Math., 3(3), 2021, 358-366.

[8] Datta S. K., Pal D., Sarkar R. and Saha J., Some Common Fixed Point Theorems for Contracting Mappings

in Bicomplex Valued b-Metric Spaces, Bull. Cal. Math. Soc., 112(4), 2020, 329-354.

[9] Datta S. K., Sarkar R., Biswas N. and Sk M., Common Fixed Point Theorems in Bicomplex Valued Metric

Spaces Under Both Rational Type Contrction Mappings Satisfying E. A. Property and Intimate Mappings,

South East Asian J. of Mathematics and Mathematical Sciences, 17(1), 2021, 347-360.

[10] Hamilton W. R., On a new species of imaginary quantities connected with a theory of quaternions, Math.

Proc. R. Ir. Acad., 2, 1844, 424-434.

[11] Kumar A., Kumar P., and Dixit P., Maximum and minimum modulus principle for bicomplex holomorphic

functions, Int. J. Eng. Sci. Technol., 3(2), 2011, 1484-1491.

[12] Luna-Elizarras M. E., Prez-Ragaldo C. O. and Shapiro M., On the Laurent series for bicomplex holomorphic

functions, Complex Var. Elliptic Equ., 62(2), 2017, 1266-1286.

[13] Luna-Elizarras M. E., Shapiro M., Struppa D. C. and Vajiac A., Bicomplex numbers and their elementary

functions, Cubo, 14 (2), 2013, 61-80.

[14] Luna-Elizarras M. E., Shapiro M., Struppa D. C. and Vajiac A., Bicomplex Holomorphic Functions: The

Algebra, Geometry and Analysis of Bicomplex numbers, Birkhuser, 2015.

[15] Price G. B., An introduction to multicomplex spaces and functions, Marcel Dekker Inc., New York, 1991.

[16] Riley J. D., Contribution to the theory of functions of bicomplex variable, Tokyo J. Math., 2, 1953, 132-165.

[17] Rochon D., A bicomplex Riemann zeta function, Tokyo J. Math., 27(2), 2004, 357-369.

[18] Segre C., Le rappresentazioni reali delle forme complesse e gli enti iperalgebrici, Math. Ann., 40, 1892,

413-467.

[19] Spampinato N., Sulla rappresentazioni delle funzioni di variabili bicomplessa total mente derivabili, Ann.

Mat. Pura Appl., 14 (4), 1936, 305-325.

[20] Titchmarsh E. C., The theory of functions, 2nd ed. Oxford University Press, Oxford, 1968.

[21] Wilson R., Hadamard multiplication of integral function, J. Lond. Math. Soc., 32, 1957, 421-429.

(Received, May 12, 2025)

(Revised, June 9, 2025)

^1∗Department of Mathematics,

Lady Brabourne College, P-1/2 Suhrawardy Avenue,

Beniapukur, Dist: Kolkata, PIN: 700017,

West Bengal, India

Email: debasmita.dut@gmail.com

76

Debasmita Dutta, Sayan Jana, Sanjib Kumar Datta and Prosenjit Sen

²Department of Mathematics,

Vivekananda College(Thakurpukur), 269 Diamond Harbour Road,

Thakurpukur, Dist: Kolkata, PIN: 700063, West Bengal, India

Email: sayanjana72926@gmail.com

³Department of Mathematics,

University of Kalyani, P.O.: Kalyani, Dist: Nadia,

PIN: 741235, West Bengal, India

Email: sanjibdatta05@gmail.com

⁴Department of Mathematics,

OmDayal Group of Institutions,

Uluberia, Howrah - 711316, India

Email: prosenjitsen83@gmail.com