Journal of Indian Acad. Math.  
ISSN: 0970-5120  
Vol. 48, No. 1 (2026) pp. 77–92.  
ON THE INTEGRAL REPRESENTATION OF  
RELATIVE GROWTH INDICATORS OF ENTIRE  
ALGEBROIDAL FUNCTIONS OF HIGHER  
INDEX  
Sanjib Kumar Datta  
Abstract. In this paper, we study the relative growth indicators of bicomplex valued  
entire algebroidal functions, their integral representation and some related results. A few  
examples are provided to justify the methodology of the work as carried out in the paper.  
Keywords: Bicomplex valued entire function, Relative order, Relative lower order, Idem-  
potent representation.  
2010 AMS Subject Classification: 30G35, 30D20, 32A30.  
1. Introduction, Definitions and Notations  
The order and lower order of an entire function f which is generally used in compu-  
tational purposes are classical in complex analysis. L. Bernal.[1] and [2], introduced the  
relative order (respectively relative lower order) between two entire functions to avoid  
comparing growth just with exp z. Extending the notion of relative order (respectively  
relative lower order) Ruiz et. al [9] introduced the relative (p, q)-th order (respectively  
relative lower (p, q)-th order) where p and q are any two positive integers. Now We  
extend these definitions to bicomplex and multicomplex valued entire functions. To com-  
pare the growth of bicomplex and multicomplex valued entire functions having the same  
relative (p, q)-th order or relative lower (p, q)-th order, We introduce the definition of rel-  
ative (p, q)-th type and relative (p, q)-th weak type of bicomplex and multicomplex entire  
functions with respect to bicomplex and multicomplex entire algebroidal functions and  
established their respective integral representations. We also investigate the equivalence  
of the computational definitions and their corresponding integral representations of the  
relative growth indicators as stated above in case of bicomplex and multicomplex valued  
entire algebroidal functions.  
1.1. The Bicomplex Number System {cf. [7]}. A bicomplex number z can be rep-  
resented in the form  
z = x1 + i1x2 + i2x3 + (i1i2) x4 = (x1 + i1x2) + i2(x3 + i1x4) = z1 + i2 z2,  
where each xk (k = 1, . . . , 4) is a real number, i2 = i2 = −1 and i1i2 = i2i1. We set  
1
2
j = i1i2, so that j2 = +1. Here z1 = x1 + i1x2 and z2 = x3 + i1x4 lie in the complex plane  
C.  
77  
78  
Sanjib Kumar Datta  
We denote the collection of real numbers by C0, the usual complex numbers by C1, and  
the bicomplex numbers by C2.  
1.2. Algebraic Structure of C2 {cf.  
[7]}. We equip the bicomplex field C2 with  
addition and scalar multiplication operations and a norm as follows.  
Addition and Scalar Multiplication: For z = (x1 +i1x2 +i2x3 +i1i2x4) and w = (y1 +i1y2 +  
i2y3 + i1i2y4) in C2, define  
z ⊕ w = (x1 + y1) + i1 (x2 + y2) + i2 (x3 + y3) + i1i2 (x4 + y4).  
Given a scalar a ∈ C1, scalar multiplication is  
a ⊙ z = a x1 + i1 a x2 + i2 a x3 + i1i2 a x4.  
With these operations, (C2, ⊕, ⊙) is a linear space.  
Norm and Completeness: Introduce the norm as  
ꢁ
ꢀ
ꢀ
x21 + x22 + x23 + x24.  
ꢀ
ꢀ
x1 + i1x2 + i2x3 + i1i2x4  
=
4
Observe that C2 can be identified with C0 via  
x1 + i1x2 + i2x3 + i1i2x4 ←→ (x1, x2, x3, x4).  
Equipping C0 with the standard Euclidean norm makes it a complete normed linear space  
4
or a Banach space. Because the norm on C2 coincides with this Euclidean norm under  
ꢂ
ꢃ
the above correspondence, C2, ⊕, ⊙, ꢀ · ꢀ is itself a Banach space.  
Bicomplex Multiplication: Define the bilinear product ⊗ : C2 × C2 → C2 by  
z ⊗ w = (x1y1 − x2y2 − x3y3 + x4y4)  
+ i1 (x1y2 + x2y1 − x3y4 − x4y3)  
+ i2 (x1y3 + x3y1 − x2y4 − x4y2)  
+ i1i2 (x1y4 + x4y1 + x2y3 + x3y2).  
Banach Algebra Property: One can verifies for all a ∈ C1, z, w ∈ C2:  
√
ꢀa ⊙ zꢀ = |a| ꢀzꢀ,  
ꢀz ⊗ wꢀ ≤  
2 ꢀzꢀ ꢀwꢀ.  
Therefore, (C2, ⊕, ⊙, ⊗, ꢀ · ꢀ) is a commutative Banach algebra.  
1.3. Idempotent Representation of Bicomplex Numbers {cf. [7]}. The bicomplex  
number system C2 admits four idempotent elements:  
1 + i1i2  
1 − i1i2  
0,  
1,  
,
.
2
2
Among these, the two nontrivial idempotents are denoted by  
1 + i1i2  
1 − i1i2  
e1 =  
and e2 =  
,
2
2
which satisfy the following relations  
e2 = e1,  
e2 = e2,  
e1e2 = e2e1 = 0,  
e1 + e2 = 1.  
1
2
The elements e1 and e2 are called orthogonal idempotents due to these properties.  
ON THE INTEGRAL REPRESENTATION OF RELATIVE GROWTH INDICATORS  
79  
Every bicomplex number z = z1+i2z2 ∈ C2, where z1, z2 ∈ C1, can be uniquely expressed  
as  
z = (z1 − i1z2)e1 + (z1 + i1z2)e2 = ξ1e1 + ξ2e2,  
where ξ1, ξ2 ∈ C1. This decomposition is known as the idempotent representation of z.  
A bicomplex number z = z1 + i2z2 ∈ C2 is said to be non-singular if |z2 + z2| = 0, and  
1
2
singular otherwise. We denote the set of all singular elements in C2 by θ2.  
1.4. Bicomplex Holomorphic Functions {cf. [7]}. Let  
f : Ω ⊂ C2 −→ C2  
be a bicomplex valued function. We say f is differentiable at ω0 ∈ Ω if the limit  
f(ω0 + h) − f(ω0)  
f′(ω0) = lim  
h→0  
h
h∈/θ2  
exists, where  
h = h0 + i1h1 + i2h2 + i1i2h3  
is required to be invertible in C2 (i.e. h ∈/ θ2).  
If f is differentiable at every point of Ω, it is called bicomplex valued holomorphic  
function. Writing  
ω = z1 + i2z2,  
f(ω) = g1(z1, z2) + i2 g2(z1, z2),  
with z1, z2 ∈ C1, one can show that f is holomorphic when both g1 and g2 are holomorphic  
in C1 and satisfy the Cauchy-Riemann equations  
∂g1  
∂g2  
∂g1  
∂g2  
=
,
= −  
.
∂z1  
∂z2  
∂z2  
∂z1  
In this case, the derivative takes the form  
∂g1  
∂g2  
f′(ω) =  
+ i2  
.
∂z2  
∂z1  
1.5. Bicomplex Entire Functions and Growth Orders {cf. [7]}. A function f :  
C2 → C2 is called bicomplex valued entire function if it is holomorphic on the entire  
bicomplex plane C2.  
Idempotent Representation of Bicomplex Valued Holomorphic Functions:  
Let fi’s be bicomplex valued entire functions ∀i = 0, 1, 2, · · · , k. Then fi can be written  
as  
fi(z1 + i2z2) = f1,i(z1 − i1z2)e1 + f2,i(z1 + i1z2)e2,  
Where idempotent components f1,i and f2,i are all complex valued entire functions ∀i =  
0, 1, 2, · · · , k and e1, e2 are idempotent elements of C2.  
Definition 1.1 (Order of a Bicomplex Valued Entire Function [3]). Mf (r) denotes the  
j,i  
maximal modulus of complex valued function fj,i on the circle |z| = r and defined by  
ꢄ
ꢄ
ꢄ
ꢄ
Mf (r) = max fj,i(z)  
j,i  
|z|=r  
80  
Sanjib Kumar Datta  
The order of fj,i in the classical sense is defined by  
log[2] Mf (r)  
j,i  
ρf  
= lim sup  
,
j = 1, 2.  
j,i  
log r  
r→∞  
Then the order of the bicomplex valued entire function fi is defined by  
ρ(fi) = max{ρf , ρf }.  
1,i  
2,i  
We now introduce the notions of relative order and relative lower order for bicomplex  
valued entire functions.  
Bernal {[1],[2]} defined the relative order of a complex valued entire function f with  
respect to another complex valued entire function g as  
ρg(f) = inf {µ > 0 : Mg(r) < Mf (rµ) ∀ r > r0(µ) > 0} ,  
where Mf (r) = max{|f(z)| : |z| = r} denotes the maximum modulus of f on the circle of  
radius r. Equivalently, it can be expressed as  
log[2] Mf−1Mg(r)  
ρg(f) = lim sup  
.
log r  
r→∞  
By replacing the lim sup with lim inf, one obtains the definition of the relative lower order,  
denoted by λg(f), which captures the lower growth behavior of f relative to g.  
Now, consider two bicomplex valued entire functions f and g, represented in their  
idempotent forms as  
f(z1 + i2z2) = f1(z1 − i1z2)e1 + f2(z1 + i1z2)e2,  
g(z1 + i2z2) = g1(z1 − i1z2)e1 + g2(z1 + i1z2)e2,  
where idempotent components f1, f2, g1, g2 are complex valued entire functions in C1.  
The relative order of the bicomplex valued entire function f with respect to g is defined  
as  
ρg(f) = max {ρg (f1), ρg (f2)} ,  
1
2
and the relative lower order is given by  
λg(f) = max {λg (f1), λg (f2)} .  
1
2
Furthermore, if the relative order and relative lower order of a bicomplex valued entire  
function f with respect to g are equal, then f is said to exhibit regular relative growth  
with respect to g.  
Example 1.2. Let f(z1 + i2z2) = exp(z1 + i2z2) and g(z1 + i2z2) = exp(7(z1 + i2z2)) be  
two bicomplex valued entire functions. Then,  
f(z1 + i2z2) = f1(z1 − i1z2)e1 + f2(z1 + i1z2)e2  
g(z1 + i2z2) = g1(z1 − i1z2)e1 + g2(z1 + i1z2)e2  
and  
where f1, f2, g1, g2 are complex valued entire functions.  
Then for ꢀz1 + i2z2ꢀ = r we can write,  
|z1 − i1z2| = |z1 + i1z2| = r.  
ON THE INTEGRAL REPRESENTATION OF RELATIVE GROWTH INDICATORS  
81  
Mf (r) = er,  
and Mg (r) = e7r.  
1
1
1
7
So, Mg−1(Mf (r)) = log Mf (r).  
1
1
1
Then one can easily verify that ρg (f1) = ρg (f2) = 1  
1
2
Therefore, ρg(f) = max{ρg (f1), ρg (f2)} = 1.  
1
2
Example 1.3. If f(z) = a + bz + cz2 and g(z) = exp(2z) where a, b, c, z ∈ C2, then one  
can easily check that ρg(f) = 0.  
1.6. Bicomplex Valued Entire Algebroidal Function.  
Definition 1.4. Let F be a k-valued function defined by the following irreducible equation  
fkFk + fk−1  
F
k−1 + fk−2  
F
k−2 + · · · + f0 = 0  
where fk = 0 and all fi (i = 0, 1, 2, . . . ., k) are bicomplex valued entire functions having  
no common zeros.  
If at least one of the fi (i = 0, 1, 2, . . . ., k) is transcendental then F is called a k-valued  
algebroidal function of bicomplex numbers. Further, if fk ≡ 1 then F is called a k-valued  
entire algebroidal function of bicomplex numbers.  
1.7. Relative (p, q)-th order and relative (p, q)-th lower order of a bicomplex  
valued entire functions with respect to another bicomplex valued entire func-  
tion.  
Definition 1.5. Let us define the relative (p, q) -th order and relative (p, q) -th lower  
order of bicomplex valued entire functions fi by  
ꢅ
ꢆ
ꢆ
ρ(Gp,q) (fi) = max ρ(p,q) (f1,i) , ρ(p,q) (f2,i)  
∀i = 0, 1, 2, · · · , k.  
G
G
and  
ꢅ
λ(Gp,q) (fi) = max λ(p,q) (f1,i) , λ(p,q) (f2,i)  
∀i = 0, 1, 2, · · · , k.  
G
G
where idempotent components f1,i and f2,i are all complex valued entire functions ∀i =  
0, 1, 2, · · · , k, with p and q being any two positive integers and  
log[p] MG−1Mf (r)  
log[p] MG−1Mf (r)  
ρ(Gp,q) (f) = lim sup  
and λ(Gp,q) (f) = lim inf  
,
log[q] r  
log[q] r  
r→∞  
r→∞  
for any two complex valued entire functions f and G.  
Definition 1.6. Let fi’s (0 ≤ i ≤ k − 1) be bicomplex valued entire functions with index-  
pair (m1, q) and G be any bicomplex valued entire algebroidal function with index-pair  
(m2, p) where m1 = m2 = m and p, q, m are all positive integers such that m ≥ max {p, q} .  
The relative (p, q) -th type of bicomplex valued entire functions fi with respect to the  
bicomplex valued entire algebroidal function G having finite positive relative (p, q) -th  
ꢇ
ꢈ
order ρ(Gp,q) (fi) 0 < ρ(p,q) (fi) < ∞ is defined as :  
G
ꢅ
ꢆ
σG(p,q) (fi) = max σ(p,q) (f1,i) , σ(p,q) (f2,i)  
∀i = 0, 1, 2, · · · , k.  
G
G
82  
Sanjib Kumar Datta  
where  
ꢊ
ꢇ
ꢈꢋ  
ꢉ
ꢌ
(p,q)(fj,i)  
φ > 0 : Mf (r) < MG exp[p−1] φ(log[q−1] r)ρ  
G
σG(p,q) (fj,i) = inf  
j,i  
for all r > r0 (φ) > 0  
log[p−1] MG−1Mf (r)  
j,i  
= lim sup  
ρ(Gp,q)(fj,i)  
ꢇ
ꢈ
r→∞  
log[q−1]  
r
∀j = 1, 2.  
and idempotent components f1,i and f2,i are all complex valued entire functions ∀i =  
0, 1, 2, · · · , k  
Definition 1.7. Let fi’s (i = 0, 1, 2, . . . ., k −1) have finite positive relative (p, q)-th lower  
ꢇ
ꢈ
order λ(Gp,q) (fi) a < λ(p,q) (fi) < ∞ with respect to G. The relative (p, q) -th weak type  
G
of fi with respect to G is denoted by τ(p,q) (fi ) and defined as :  
G
ꢅ
ꢆ
τG(p,q) (fi) = max τ(p,q) (f1,i) , τ(p,q) (f2,i)  
∀i = 0, 1, 2, · · · , k.  
G
G
where  
log[p−1] MG−1Mf (r)  
τG(p,q) (fj,i) = lim inf  
j,i  
λ(Gp,q)(fj,i)  
ꢇ
ꢈ
r→∞  
log[q−1]  
r
∀j = 1, 2.  
and fi, G, p, q, f1,i & f2,i are defined in a similar way as in Definition 1.6.  
Definition 1.8. Let fi’s (i = 0, 1, 2, . . . ., k − 1) have finite positive relative (p, q)-th order  
ꢇ
ꢈ
ρ(Gp,q) (fi) a < ρ(p,q) (fi) < ∞ . with respect to G. Then the relative (p, q)-th lower type  
G
of fi with respect to G is denoted by σ(p,q) (fi ) and defined as :  
G
ꢅ
ꢆ
σ(Gp,q) (fi) = max σ(p,q) (f1,i) , σ(p,q) (f2,i)  
∀i = 0, 1, 2, · · · , k.  
G
G
where  
log[p−1] MG−1Mf (r)  
σ(Gp,q) (fj,i) = lim inf  
j,i  
ρ(Gp,q)(fj,i)  
ꢇ
ꢈ
r→∞  
log[q−1]  
r
∀j = 1, 2.  
and fi, G, p, q, f1,i & f2,i are defined in a similar way as in Definition 1.6.  
Definition 1.9. Let fi’s (i = 0, 1, 2, . . . ., k − 1) have finite positive relative (p, q)-th  
ꢇ
ꢈ
lower order λ(Gp,q) (fi) a < λ(p,q) (fi) < ∞ with respect to G . Then the growth indicator  
G
τ(Gp,q) (fi) of fi with respect to G is defined as :  
ꢅ
ꢆ
τ(Gp,q) (fi) = max τ(p,q) (f1,i) , τ(p,q) (f2,i)  
∀i = 0, 1, 2, · · · , k.  
G
G
ON THE INTEGRAL REPRESENTATION OF RELATIVE GROWTH INDICATORS  
83  
where  
log[p−1] MG−1Mf (r)  
τ(Gp,q) (fj,i) = lim sup  
j,i  
λ(Gp,q)(fj,i)  
ꢇ
ꢈ
r→∞  
log[q−1]  
r
∀j = 1, 2.  
and fi, G, p, q, f1,i & f2,i are defined in a similar way as in Definition 1.6.  
2. Lemma  
In this section we need the following lemma which is needed in sequel.  
∞
log[p−2] MG−1Mfi (r)  
ꢍ
Lemma 2.1. Let the integral  
ꢀ
ꢁ
ꢂꢃt+1 dr (r0 > 0) converges where 0 < A <  
A
exp log[q−1]  
r
(
)
r0  
∞. Then  
log[p−2] MG−1Mf (r)  
i
lim  
= 0  
ꢎ
ꢏ
ꢐꢑ  
t
ꢇ
ꢈ
r→∞  
A
exp  
log[q−1]  
r
where f is a complex valued entire function and G is a complex valued entire algebroidal  
function.  
∞
log[p−2] MG−1Mfi (r)  
ꢍ
Proof. Since the integral  
ꢀ
ꢁ
ꢂꢃt+1 dr (r0 > 0) converges, then  
A
exp log[q−1]  
r
(
)
r0  
∞
ꢒ
log[p−2] MG−1Mf (r)  
i
t+1 dr < ε, if r0 > R (ε) .  
ꢎ
ꢏ
ꢐꢑ  
ꢇ
ꢈ
A
log[q−1]  
r
r0  
exp  
Therefore,  
ꢁ
ꢂ
A
exp log[q−1] r0  
+r0  
(
)
ꢒ
log[p−2] MG−1Mf (r)  
i
t+1 dr < ε .  
ꢎ
ꢏ
ꢐꢑ  
ꢇ
ꢈ
A
exp  
log[q−1]  
r
r0  
Since log[p−2] MG−1ꢁMf (r) incꢂreases with r, so  
i
A
exp log[q−1] r0  
+r0  
(
)
ꢒ
log[p−2] MG−1Mf (r)  
i
t+1 dr ≥  
ꢎ
ꢏ
ꢐꢑ  
ꢇ
ꢈ
A
log[q−1]  
r
r0  
exp  
ꢎ
ꢏ
ꢐꢑ  
ꢇ
ꢈ
log[p−2] MG−1Mf (r0)  
A
· exp  
log[q−1] r0  
.
i
ꢎ
ꢏ
ꢐꢑ  
t+1  
ꢇ
ꢈ
A
exp  
log[q−1] r0  
84  
Sanjib Kumar Datta  
i.e., for all sufficiently large values of r,  
ꢁ
ꢂ
A
exp log[q−1] r0  
+r0  
(
)
ꢒ
log[p−2] MG−1Mf (r)  
i
t+1 dr ≥  
ꢎ
ꢏ
ꢐꢑ  
ꢇ
ꢈ
A
exp  
log[q−1]  
r
r0  
log[p−2] MG−1Mf (r0)  
i
,
ꢎ
ꢏ
ꢐꢑ  
t
ꢇ
ꢈ
A
exp  
log[q−1] r0  
so that  
log[p−2] MG−1Mf (r0)  
i
< ε if r0 > R (ε) .  
ꢎ
ꢏ
ꢐꢑ  
t
ꢇ
ꢈ
A
exp  
log[q−1] r0  
log[p−2] MG−1Mf (r)  
i
i.e., lim  
= 0.  
ꢎ
ꢏ
ꢐꢑ  
t
ꢇ
ꢈ
A
r→∞  
exp  
log[q−1]  
r
This proves the lemma.  
□
Remark 2.2. Under the flavour of Lemma 2.1 Definition 1.6 can be viewed in the fol-  
lowing way.  
The relative (p, q)-th type σG(p,q) (fi) of fi with respect to G is defined as: The maximum  
between the following two integrals  
∞
ꢒ
log[p−2] MG−1Mf (r)  
1,i  
t+1 dr (r0 > 0)  
ꢓ
ꢔ
ꢕꢖ  
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ
r0  
exp  
log[q−1]  
r
converges for t > σG(p,q) (f1,i) and diverges for t < σ(p,q) (f1,i),  
G
and  
∞
ꢒ
log[p−2] MG−1Mf (r)  
2,i  
t+1 dr (r0 > 0)  
ꢓ
ꢔ
ꢕꢖ  
ρ(Gp,q)(f2,i)  
ꢇ
ꢈ
r0  
exp  
log[q−1]  
r
converges for t > σG(p,q) (f2,i) and diverges for t < σ(p,q) (f2,i),  
G
where fi, G, p, q, f1,i, f2,i and ρ(Gp,q) (fi) are defined in a similar way as in Definition 1.6.  
Remark 2.3. Under the flavour of Lemma 2.1 Definition 1.7 can be viewed in the fol-  
lowing way.  
The relative (p, q)-th weak type τG(p,q) (fi) of fi with respect to G is defined as: The  
maximum between the following two integrals  
∞
ꢒ
log[p−2] MG−1Mf (r)  
1,i  
t+1 dr (r0 > 0)  
ꢓ
ꢔ
ꢕꢖ  
λ(Gp,q)(f1,i)  
ꢇ
ꢈ
r0  
exp  
log[q−1]  
r
ON THE INTEGRAL REPRESENTATION OF RELATIVE GROWTH INDICATORS  
85  
converges for t > τG(p,q) (f1,i) and diverges for t < τ(p,q) (f1,i),  
G
and  
∞
ꢒ
log[p−2] MG−1Mf (r)  
2,i  
t+1 dr (r0 > 0)  
ꢓ
ꢔ
ꢕꢖ  
λ(Gp,q)(f2,i)  
ꢇ
ꢈ
r0  
exp  
log[q−1]  
r
converges for t > τG(p,q) (f2,i) and diverges for t < τ(p,q) (f2,i),  
G
where fi, G, p, q, f1,i, f2,i and λ(Gp,q) (fi) are defined in a similar way as in Definition 1.7.  
Remark 2.4. Under the flavour of Lemma 2.1 Definition 1.8 can be viewed in the fol-  
lowing way.  
The relative (p, q)-th lower type σ(Gp,q) (fi) of fi with respect to G is defined as: The  
maximum between the following two integrals  
∞
ꢒ
log[p−2] MG−1Mf (r)  
1,i  
t+1 dr (r0 > 0)  
ꢓ
ꢔ
ꢕꢖ  
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ
r0  
exp  
log[q−1]  
r
converges for t > σ(Gp,q) (f1,i) and diverges for t < σ(p,q) (f1,i),  
G
and  
∞
ꢒ
log[p−2] MG−1Mf (r)  
2,i  
t+1 dr (r0 > 0)  
ꢓ
ꢔ
ꢕꢖ  
ρ(Gp,q)(f2,i)  
ꢇ
ꢈ
r0  
exp  
log[q−1]  
r
converges for t > σ(Gp,q) (f2,i) and diverges for t < σ(p,q) (f2,i),  
G
where fi, G, p, q, f1,i, f2,i and ρ(Gp,q) (fi) are defined in a similar way as in Definition 1.8.  
Remark 2.5. Under the flavour of Lemma 2.1 Definition 1.9 can be viewed in the fol-  
lowing way.  
The growth indicator τ(Gp,q) (fi) of fi with respect to G is defined as: The maximum  
between the following two integrals  
∞
ꢒ
log[p−2] MG−1Mf (r)  
1,i  
t+1 dr (r0 > 0)  
ꢓ
ꢔ
ꢕꢖ  
λ(Gp,q)(f1,i)  
ꢇ
ꢈ
r0  
exp  
log[q−1]  
r
converges for t > τ(Gp,q) (f1,i) and diverges for t < τ(p,q) (f1,i),  
G
and  
∞
ꢒ
log[p−2] MG−1Mf (r)  
2,i  
t+1 dr (r0 > 0)  
ꢓ
ꢔ
ꢕꢖ  
λ(Gp,q)(f2,i)  
ꢇ
ꢈ
r0  
exp  
log[q−1]  
r
converges for t > τ(Gp,q) (f2,i) and diverges for t < τ(p,q) (f2,i) .  
G
where fi, G, p, q, f1,i, f2,i and λ(Gp,q) (fi) are defined in a similar way as in Definition 1.9.  
86  
Sanjib Kumar Datta  
3. Theorems  
Theorem 3.1. Let fi’s (i = 0, 1, 2, .....k − 1) be bicomplex valued entire functions having  
ꢇ
ꢈ
finite positive relative (p, q) -th order ρ(Gp,q) (fi ) 0 < ρ(p,q) (fi) < ∞ and relative (p, q)  
G
-th type σG(p,q) (fi ) with respect to a bicomplex valued entire algebroidal function G, where  
p and q are any two positive integers. Then Definition 1.6 and Remark 2.2 are equivalent.  
Proof. Let us consider fi’s (i = 0, 1, 2, .....k − 1) be bicomplex valued entire functions and  
ꢇ
ꢈ
G be a bicomplex valued entire algebroidal function such that ρ(Gp,q) (fi ) 0 < ρ(p,q) (fi ) < ∞  
G
exists for any two positive integers p and q.  
Case I. σG(p,q) (fi ) = ∞.  
Definition1.6 ⇒ Remark 2.2.  
As σG(p,q) (fi ) = ∞, then at least one of σ(p,q) (f1,i) and σ(p,q) (f2,i) is equal to ∞. Without  
G
G
loss of generality let σG(p,q) (f1,i) = ∞. From Definition 1.6 we have for arbitrary positive  
C and for a sequence of values of r tending to infinity that  
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ
log[p−1] MG−1Mf (r) > C · log[q−1]  
r
1,i  
ꢓ
ꢔ
ꢕꢖ  
C
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ
i.e., log[p−2] MG−1Mf (r) > exp  
log[q−1]  
r
.
(3.1)  
1,i  
∞
log[p−2] MG−1Mf1,i (r)  
ꢍ
ꢄ
ꢅ
ꢆꢇ  
If possible, let the integral  
Then by Lemma 2.1,  
C+1 dr (r0 > 0) be converges.  
(p,q)  
G
ρ
f
1,i  
(
)
exp log[q−1]  
r
r0  
(
)
log[p−2] MG−1Mf (r)  
1,i  
lim sup  
= 0 .  
ꢓ
ꢔ
ꢕꢖ  
ꢈ
C
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ
r→∞  
exp  
log[q−1]  
r
So for all sufficiently large values of r,  
ꢓ
ꢔ
ꢕꢖ  
C
ρ(Gp,q)(f1,i)  
ꢇ
log[p−2] MG−1Mf (r) < exp  
log[q−1]  
r
.
(3.2)  
1,i  
Therefore∞fromꢅEquations(3.1) and (3.2) , we arrive at a contradiction.  
log[p−2] MG−1Mf1,i (r)  
ꢍ
ꢄ
ꢆꢇ  
Hence  
C+1 dr (r0 > 0) diverges whenever C is finite, which is  
(p,q)  
G
ρ
f
1,i  
(
)
exp log[q−1]  
r
r0  
(
)
Remark 2.2.  
Remark 2.2 ⇒ Definition 1.6.  
Let C be aꢄny positive number. Since σG(p,q) (fi) = ∞, froꢄm Remark 2.2, at leasꢆtꢇone  
∞
∞
log[p−2] MG−1Mf1,i (r)  
log[p−2] MG−1Mf2,i (r)  
ꢍ
ꢍ
ꢅ
ꢆꢇ  
ꢅ
of the integrals  
C+1 dr (r0 > 0) and  
C+1 dr  
(p,q)  
G
(p,q)  
G
ρ
f
ρ
f
2,i  
(
)
(
)
1,i  
exp log[q−1]  
r
exp log[q−1]  
r
r0  
r0  
(
)
(
)
(r0 > 0) is divergent.  
ON THE INTEGRAL REPRESENTATION OF RELATIVE GROWTH INDICATORS  
87  
∞
log[p−2] MG−1Mf1,i (r)  
ꢍ
ꢄ
ꢅ
ꢆꢇ  
Let  
C+1 dr (r0 > 0) be divergent, then for arbitrary positive ε  
(p,q)  
G
ρ
f
1,i  
(
)
exp log[q−1]  
r
r0  
(
)
and for a sequence of values of r tending to infinity  
ꢓ
ꢔ
ꢕꢖ  
C−ε  
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ
log[p−2] MG−1Mf (r) > exp  
log[q−1]  
r
1,i  
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ
i.e., log[p−1] MG−1Mf (r) > (C − ε) log[q−1]  
r
,
1,i  
which implies that  
log[p−1] MG−1Mf (r)  
1,i  
lim sup  
≥ C − ε .  
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ
r→∞  
log[q−1]  
r
Since C > 0 is arbitrary, it follows that  
log[p−1] MG−1Mf (r)  
1,i  
lim sup  
= ∞ .  
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ
r→∞  
log[q−1]  
r
If we have considered the other integral to be divergent then in a similar way we will get  
log[p−1] MG−1Mf (r)  
2,i  
lim sup  
= ∞ .  
ρ(Gp,q)(f2,i)  
ꢇ
ꢈ
r→∞  
log[q−1]  
r
In either case σG(p,q) (fi) = ∞. Thus Definition 1.6 follows.  
Case II. 0 ≤ σG(p,q) (fi) < ∞.  
Definition 1.6 ⇒ Remark2.2.  
Subcase (A). 0 < σG(p,q) (fi) < ∞. Then both 0 < σ(p,q) (f1,i) , σ(p,q) (f2,i) < ∞.  
G
G
Let σG(p,q) (f1,i) > σ(p,q) (f2,i) Then according to Definition 1.6, for arbitrary positive ε and  
G
for all sufficiently large values of r, we obtain that  
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ ꢇ  
ꢈ
ꢈ
log[p−1] MG−1Mf (r) < σ(p,q) (f1,i) + ε  
log[q−1]  
r
1,i  
G
σG(p,q)(f1,i)+ε  
ꢓ
ꢔ
ꢔ
ꢕꢖ  
ρ(Gp,q)(f1,i)  
ꢇ
ꢇ
i.e., log[p−2] MG−1Mf (r) < exp  
log[q−1]  
r
r
1,i  
σG(p,q)(f1,i)+ε  
ꢓ
ꢕꢖ  
ρ(Gp,q)(f1,i)  
ꢈ
exp  
log[q−1]  
log[p−2] MG−1Mf (r)  
1,i  
i.e.,  
<
ꢓ
ꢔ
ꢕꢖ  
ꢓ
ꢔ
ꢇ
ꢕꢖ  
t
t
ρ(Gp,q)(f1,i)  
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ
ꢈ
exp  
log[q−1]  
r
exp  
log[q−1]  
r
log[p−2] MG−1Mf (r)  
1,i  
i.e.,  
<
ꢓ
ꢔ
ꢕꢖ  
t
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ
exp  
log[q−1]  
r
88  
Sanjib Kumar Datta  
1
.
t−σG(p,q)(f1,i)−ε  
ꢓ
ꢔ
ꢕꢖ  
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ
exp  
log[q−1]  
r
∞
log[p−2] MG−1Mf1,i (r)  
ꢍ
ꢄ
ꢅ
ꢆꢇ  
Therefore  
t+1 dr (r > 0) converges for t > σ(p,q) (f1,i) .  
0
G
(p,q)  
G
ρ
f
1,i  
(
)
exp log[q−1]  
r
r0  
(
)
Again by Definition 1.6, we obtain for a sequence of values of r tending to infinity that  
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ ꢇ  
ꢈ
log[p−1] MG−1Mf (r) > σ(p,q) (f1,i) − ε  
log[q−1]  
r
1,i  
G
σG(p,q)(f1,i)−ε  
ꢓ
ꢔ
ꢕꢖ  
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ
i.e., log[p−2] MG−1Mf (r) > exp  
log[q−1]  
r
.
(3.3)  
1,i  
So for t < σG(p,q) (f1,i), we get from Equation(3.3) that  
log[p−2] MG−1Mf (r)  
1
1,i  
ꢁ
ꢂ
>
.
ꢓ
ꢔ
ꢕꢖ  
t
t− σG(p,q)(f1,i)−ε  
ꢓ
ꢔ
ꢕꢖ  
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ
log[q−1]  
r
exp  
log[q−1]  
r
exp  
∞
log[p−2] MG−1Mf1,i (r)  
ꢍ
ꢄ
ꢅ
ꢆꢇ  
Therefore  
t+1 dr (r > 0) diverges for t < σ(p,q) (f1,i) .  
0
G
(p,q)  
G
ρ
f
1,i  
(
)
exp log[q−1]  
r
r0  
(
)
∞
log[p−2] MG−1Mf1,i (r)  
ꢍ
t+1 dr (r > 0) converges for t > σ(p,q) (f1,i) and di-  
ꢄ
ꢅ
ꢆꢇ  
Hence  
0
G
(p,q)  
G
ρ
f
1,i  
(
)
exp log[q−1]  
r
r0  
(
)
verges for t < σG(p,q) (f1,i).  
A similar result will arise if we take σG(p,q) (f1,i) < σ(p,q) (f2,i)  
G
Subcase (B). σG(p,q) (fi) = 0 both σ(p,q) (f1,i) = 0 and σ(p,q) (f2,i) = 0.  
G
G
Then Definition 1.6 gives for all sufficiently large values of r that  
log[p−1] MG−1Mf (r)  
j,i  
< ε  
∀j = 1, 2.  
ρ(Gp,q)(fj,i)  
ꢇ
ꢈ
log[q−1]  
r
∞
log[p−2] MG−1Mfj,i (r)  
ꢍ
ꢄ
ꢅ
ꢆꢇ  
Then as before we obtain that  
t+1 dr (r0 > 0) converges for t > 0  
(p,q)  
G
ρ
f
( )  
j,i  
exp log[q−1]  
r
r0  
(
)
and diverges for t < 0,  
∀j = 1, 2.  
Thus combining Subcase (A) and Subcase (B), Remark 2.2 follows.  
Remark 2.2 ⇒ Definition 1.6.  
From  
Remark  
2.2  
and  
for  
arbitrary  
positive  
ε,  
both  
integrals  
∞
ꢒ
log[p−2] MG−1Mf (r)  
1,i  
dr (r0 > 0)  
σG(p,q)(f1,i)+ε+1  
ꢓ
ꢔ
ꢕꢖ  
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ
r0  
exp  
log[q−1]  
r
ON THE INTEGRAL REPRESENTATION OF RELATIVE GROWTH INDICATORS  
89  
and  
∞
ꢒ
log[p−2] MG−1Mf (r)  
2,i  
dr (r0 > 0)  
σG(p,q)(f2,i)+ε+1  
ꢓ
ꢔ
ꢕꢖ  
ρ(Gp,q)(f2,i)  
ꢇ
ꢈ
r0  
exp  
log[q−1]  
r
converges. Then by Lemma 2.1, we get that  
log[p−2] MG−1Mf (r)  
1,i  
lim sup  
= 0 .  
σG(p,q)(f1,i)+ε  
ꢓ
ꢔ
ꢕꢖ  
r→∞  
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ
exp  
log[q−1]  
r
So we obtain all sufficiently large values of r that  
log[p−2] MG−1Mf (r)  
1,i  
< ε  
σG(p,q)(f1,i)+ε  
ꢓ
ꢔ
ꢕꢖ  
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ
exp  
log[q−1]  
r
σG(p,q)(f1,i)+ε  
ꢓ
ꢔ
ꢇ
ꢕꢖ  
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ
i.e., log[p−2] MG−1Mf (r) < ε · exp  
log[q−1]  
r
1,i  
ρ(Gp,q)(f1,i)  
ꢈ ꢇ  
ꢈ
i.e., log[p−1] MG−1Mf (r) < log ε + σ(p,q) (f1,i) + ε  
log[q−1]  
r
1,i  
G
log[p−1] MG−1Mf (r)  
≤ σG(p,q) (f1,i) + ε .  
1,i  
i.e., lim sup  
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ
r→∞  
log[q−1]  
r
Since ε (> 0) is arbitrary, it follows from above that  
log[p−1] MG−1Mf (r)  
lim sup  
≤ σG(p,q) (f1,i) .  
(3.4)  
1,i  
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ
r→∞  
log[q−1]  
r
On  
the  
other  
hand,  
the  
divergence  
of  
the  
integral  
∞
ꢒ
log[p−2] MG−1Mf (r)  
1,i  
dr (r0 > 0)  
σG(p,q)(f1,i)−ε+1  
ꢓ
ꢔ
ꢕꢖ  
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ
r0  
exp  
log[q−1]  
r
implies that there exists a sequence of values of r tending to infinity such that  
log[p−2] MG−1Mf (r)  
1
1,i  
>
ꢓ
ꢔ
ꢕꢖ  
σG(p,q)(f1,i)−ε+1  
1+ε  
ꢓ
ꢔ
ꢕꢖ  
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ
exp  
log[q−1]  
r
exp  
log[q−1]  
r
90  
Sanjib Kumar Datta  
σG(p,q)(f1,i)−2ε  
ꢓ
ꢔ
ꢕꢖ  
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ
i.e., log[p−2] MG−1Mf (r) > exp  
log[q−1]  
r
1,i  
ꢔ
ꢕ
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ ꢇ  
ꢈ
i.e., log[p−1] MG−1Mf (r) > σ(p,q) (f1,i) − 2ε  
log[q−1]  
r
1,i  
G
log[p−1] MG−1Mf (r)  
ꢇ
ꢈ
1,i  
i.e.,  
> σG(p,q) (f1,i) − 2ε  
.
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ
log[q−1]  
r
As ε (> 0) is arbitrary, it follows from the above that  
log[p−1] MG−1Mf (r)  
≥ σG(p,q) (f1,i) .  
(3.5)  
1,i  
lim sup  
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ
r→∞  
log[q−1]  
r
So from Equations(3.4) and (3.5) , we obtain that  
log[p−1] MG−1Mf (r)  
= σG(p,q) (f1,i) .  
1,i  
lim sup  
ρ(Gp,q)(f1,i)  
ꢇ
ꢈ
r→∞  
log[q−1]  
r
By similar calculations from the convergence and divergence of the integral  
∞
ꢒ
log[p−2] MG−1Mf (r)  
2,i  
dr (r0 > 0)  
σG(p,q)(f2,i)+ε+1  
ꢓ
ꢔ
ꢕꢖ  
ρ(Gp,q)(f2,i)  
ꢇ
ꢈ
r0  
exp  
log[q−1]  
r
we will arrive to  
log[p−1] MG−1Mf (r)  
= σG(p,q) (f2,i) .  
2,i  
lim sup  
ρ(Gp,q)(f2,i)  
ꢇ
ꢈ
r→∞  
log[q−1]  
r
Thus, Remark 2.2 follows.  
This proves the theorem.  
□
Theorem 3.2. Let fi’s (i = 0, 1, 2, .....k − 1) be bicomplex valued entire functions having  
ꢇ
ꢈ
finite positive relative (p, q) -th lower order λ(Gp,q) (fi ) 0 < λ(p,q) (fi) < ∞ and relative  
G
(p, q) -th weak type τG(p,q) (fi ) with respect to a bicomplex valued entire algebroidal function  
G, where p and q are any two positive integers. Then Definition 1.7 and Remark 2.3 are  
equivalent.  
Proof. We omit the proof of Theorem 3.2 as it can be carried similar to that of Theorem  
3.1, using Lemma 2.1 and idempotent decomposition of fi.  
□
Theorem 3.3. Let fi’s (i = 0, 1, 2, .....k − 1) be bicomplex valued entire functions having  
ꢇ
ꢈ
finite positive relative (p, q) -th order ρ(Gp,q) (fi ) 0 < ρ(p,q) (fi) < ∞ and relative (p, q)  
G
-th lower type σ(Gp,q) (fi ) with respect to a bicomplex valued entire algebroidal function G,  
where p and q are any two positive integers. Then Definition 1.8 and Remark 2.4 are  
equivalent.  
ON THE INTEGRAL REPRESENTATION OF RELATIVE GROWTH INDICATORS  
91  
Proof. The proof of the Theorem 3.3 is omitted because it is similar to that of Theorem  
3.1, using Lemma 2.1 and idempotent decomposition of fi.  
□
Theorem 3.4. Let fi’s (i = 0, 1, 2, .....k − 1) be bicomplex valued entire functions having  
ꢇ
ꢈ
finite positive relative (p, q) -th lower order λ(Gp,q) (fi ) 0 < λ(p,q) (fi) < ∞ and the growth  
G
indicator τ(Gp,q) (fi ) with respect to a bicomplex valued entire algebroidal function G, where  
p and q are any two positive integers. Then Definition 1.9 and Remark 2.5 are equivalent.  
Proof. Using Lemma 2.1 and idempotent decomposition of fi, the proof of Theorem 3.4  
can similarly be derived as the proof of Theorem 3.1.  
□
4. Future Prospect  
In the line of the works as carried out in this paper one may think of finding out  
the inter-relationships of relative growth indicators of multicomplex valued entire and  
meromorphic functions. As a consequence, the derivation of relevant results is still virgin  
and may be posed as an open problem to the future researchers of this area.  
5. Acknowledgement  
The 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.  
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Valued Entire Functions, South East Asian Journal of Mathematics and Mathematical Sciences  
19(1), 43-54, (2023).  
[4] W.R. Hamilton, On a New Species of Imaginary Quantities Connected with a Theory of Quaternions,  
Math. Proc. R. Ir. Acad. 2, 424-434, (1844).  
[5] M.E. Luna-Elizarrars, M. Shapiro, D.C. Struppa and A. Vajiac, Bicomplex Numbers and Their  
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92  
Sanjib Kumar Datta  
(Received, August 12, 2025)  
(Revised, September 17, 2025)  
Mathematics Department,  
University of Kalyani,  
P.O.: Kalyani, Dist: Nadia, PIN: 741235,  
West Bengal, India.  
Email: sanjibdatta05@gmail.com