EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6108
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Hankel Determinant of Analytical Functions Closely
Tied to Bell Polynomials

Omar Alnajar1,∗, Khalid M. K. Alshammari2, Ala Amourah3, Maslina Darus1,∗

1 Department of Mathematical Sciences, Faculty of Science and Technology,
Universiti Kebangsaan Malaysia, Bangi 43600, Malaysia
2 Department of Mathematics, College of Sciences, Faculty of Science and Technology,
University of Ha’il, Ha’il 55425 , Saudi Arabia
3 Mathematics Education Program, Faculty of Education and Arts, Sohar University,
Sohar 311, Oman.

Abstract. Applying the state-of-the-art Bell polynomials to the open unit disk, a differential
operator ϑm

ξ,P is produced. In this paper, we shall provide a family of analytic functions related to

the differential operator indicated above. The upper bound for the nonlinear functional |a2a4−a23|,
otherwise known as the Hankel determinant, is our primary finding. Aside from using the Bell
polynomial, the differential operator is gained using the Hadamard product. Coefficient equating
and other fundamentals of classical calculus will be used in the primary finding of the upper bound.

2020 Mathematics Subject Classifications: 30C45

Key Words and Phrases: Univalent function, Hankel determinant, Bell polynomial, Inclusion
relation

1. preliminaries

Various branches within mathematics, such as complex analysis, differential geometry,
and mathematical physics, present opportunities for application in the geometric func-
tion theory [1, 2]. Additionally, it provides tools that can be utilized to understand and
characterize the geometry of intricate functions and their associated mappings.

Castellares et al. [3] studied the Bell polynomial, which is useful in many areas, such as
biology, physics, engineering, and finance. Researchers have successfully used it to model
the distribution of stock returns, describe noisy signals, and examine the functioning of
biological systems. The Bell curve is used in many areas of statistics, such as testing
hypotheses, finding confidence ranges, and performing regression analysis. It is also used

∗Corresponding author.
∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6108

Email addresses: P117246@siswa.ukm.edu.my (O. Alnajar), aamourah@su.edu.om (A. Amourah),
khmo.alshammari@uoh.edu.sa (K. Alshammari), maslina@ukm.edu.my (M. Darus)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6108 2 of 11

to describe complicated systems and make predictions based on real-world data, such
as in psychology, economics, and finance. The Bell polynomial, which was made to be
better than the Bell numbers [4], is defined by a generating function for a discrete random
variable X, which can be written in the following way:

P (X = n) =
ξnee

(−ξ2)+1

Fn

n!
; n = 1, 2, 3, ..., (1)

where Fn = 1
e

∞∑
b=0

bn

b! is the Bell numbers, n ≥ 1, and 0 < ξ ≤ 1.

The first few terms for Bell numbers are as follows:

F1 = 1, F2 = 2, F3 = 5, F4 = 15,F5 = 52.

Let us now present a new power series, the coefficients of which will represent the Bell
generating function.

L(ξ, z) = z +

∞∑
n=2

ξn−1ee
(−ξ2)+1

Fn

(n− 1)!
zn, z ∈ U. where 0 < ξ ≤ 1. (2)

Consequently, coefficients can be viewed as probabilities linked to the Bell polynomial.
It is possible to confirm the convergence of the previously mentioned series on the unit disk
U by applying the ratio test, a widely recognized and proven effective method. Recently,
Alnajar and Darus [5], Alnajar et al.,[6–8], Amourah et al., [9], and Illafe et al., [10]
employed Bell, Borel and Neutrosophic Poisson polynomials to address specific problems
related to complex analysis. The motivation behind this study is to look at the behavior
of the polynomials determined by their coefficient values.

Suppose f is defined on the open unit disk, and A represents the categorization of all
analytical functions. This is valid only if conditions U = {z ∈ C : |z| < 1} and f(0) = 0
and f ′(0)− 1 = 0 are satisfied. For each f ∈ A, we write the following Taylor series:

f(z) = z +
∞∑
n=2

anz
n, (z ∈ U, an ∈ C, n ∈ N := {1, 2, 3, . . . }). (3)

The examination of inclusion relationships among analytic functions within specific
special sets was a topic that was previously incorporated into the realm of geometric
function theory. This area held considerable interest for researchers. An investigation
conducted by Ruscheweyh [11] centered on the neighborhood and inclusion relationships
of univalent functions. Simultaneously, Srivastava et al., [12] delved into a comprehensive
study on all inclusion characteristics of multivalent functions. In recent times, scholars in
the field of geometric function theory have directed their focus towards diverse sub-classes
of univalent functions. Amourah et al., [13], Mahmood et al., [14], Amini et al., [15],
Jahangiri et al., [16], and Amini et al., [17] provide additional details that yield a more
in-depth understanding, see also [18–22].



O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6108 3 of 11

Using the symbol ϑξ we may express the linear operator, which is defined by the
Hadamard product, also known as convolution: A → A

ϑξf(z) = L(ξ, z) ∗ f(z) = z +
∞∑
n=2

ξn−1ee
(−ξ2)+1

Fn

(n− 1)!
anz

n, z ∈ U. (4)

we define the operator ϑm
ξ,P f(z) : A → A as

ϑ0
ξ,P f(z) = z +

∞∑
n=2

ξn−1ee
(−ξ2)+1

Fn

(n− 1)!
anz

n,

ϑ1
ξ,P f(z) = (1− P )ϑ0

ξ,P f(z) + Pz
(
ϑ0
ξ,P f(z)

)′
,

ϑ1
ξ,P f(z) = z +

∞∑
n=2

ξn−1ee
(−ξ2)+1

Fn

(n− 1)!
[1 + P (n− 1)] anz

n,

ϑ2
ξ,P f(z) = (1− P )ϑ1

ξ,P f(z) + Pz
(
ϑ1
ξ,P f(z)

)′
,

ϑ2
ξ,P f(z) = z +

∞∑
n=2

ξn−1ee
(−ξ2)+1

Fn

(n− 1)!
[1 + P (n− 1)]2 anz

n,

...

ϑm
ξ,P f(z) = z +

∞∑
n=2

ξn−1ee
(−ξ2)+1

Fn

(n− 1)!
[1 + P (n− 1)]m anz

n, (5)

m ∈ N ∪ {0}, P ≥ 0, 0 < ξ ≤ 1.

Pommerenke [23, 24] defined the Hankel determinant of f for r ≥ 1 and n ≥ 1 as

Hr(n) =

∣∣∣∣∣∣∣∣∣
an an+1 . . . an+r−1

an+1 an+2 . . . an+r
...

...
. . .

...
an+r−1 an+r . . . an+2(r−1)

∣∣∣∣∣∣∣∣∣ .

Many authors have also given this problem some thoughts. For example, Noor [25]
calculated the growth rate of Hr(n) as n → ∞ with a constrained boundary, Ehrenborg
[26] examined the Hankel determinant of exponential polynomials, and Layman [27] and
Panigrahi, and Murugusundaramoorthy [28] covered some of its characteristics. Mishra



O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6108 4 of 11

and Gochhayat [29] also investigated the Hankel determinant using fractional operators.
Furthermore, this study develops the classes utilised in [30–39] by incorporating the Bell
polynomial.

In the present work, we will examine the Hankel determinant when r = 2 and n = 2,
that is:

H2(2) =

∣∣∣∣ a2 a3
a3 a4

∣∣∣∣ = |a2a4 − a23|.

Bear in mind, H2(1) =

∣∣∣∣ a1 a2
a2 a3

∣∣∣∣ = |a1a3 − a22| is the well-known Fekete-Szego func-

tional for a1 = 1.
This study aims to find the upper bound for the functional |a2a4 − a23| of a function f

that belongs to the class Tk(m) defined as follows:

Definition 1. Let f be given by (3). It is said to satisfy the inequality if f ∈ Tk(m).

ℜ
{(

ϑm
ξ,P f(z)

)′}
> 0, z ∈ U. (6)

We begin by stating a few foundational lemmas that will be utilized in our proof.
Let B stand for the function class

B(z) = 1 + y1z + y2z
2 + y3z

3 + ··· = 1 +
∞∑
n=1

ynz
n, (7)

which are analytic in U and satisfy Re {B(z)} > 0 for any z ∈ U .

Lemma 1. [40] If y ∈ B, then |yn| ≤ 2, for each n ≥ 1.

Lemma 2. [41, 42] If y ∈ B, then

2y2 = y21 +
(
4− y21

)
x = y2 =

1

2
(y21 +

(
4− y21

)
x), (8)

for some x, |x| ≤ 1, and

4y3 = y31 + 2y1
(
4− y21

)
x− y1

(
4− y21

)
x2 + 2

(
4− y21

)
(1− |x2||)z =

y3 =
1

4
(y31 + 2y1

(
4− y21

)
x− y1

(
4− y21

)
x2 + 2

(
4− y21

)
(1− |x2|)z), (9)

for some z, |z| ≤ 1.



O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6108 5 of 11

2. Main Result

Our main result as follows:

Theorem 1. Let f ∈ Tk(m). Then∣∣a2a4 − a23
∣∣ ≤ 16

9
(
ξ4e2e

(−ξ2)+1F3 [1 + 2P ]2m
) .

Proof. Since f ∈ Tk(m), it follows from Eq (5) and Eq (7) that:(
ϑm
ξ,P f(z)

)′
= B(z),

we can write as follows:

1 + 2ξee
(−ξ2)+1

F2 [1 + P ]m a2z +
3

2

(
ξ2ee

(−ξ2)+1

F3 [1 + 2P ]m
)
a3z

2+

4

6

(
ξ3ee

(−ξ2)+1

F4 [1 + 3P ]m
)
a4z

3 = 1 + y1z + y2z
2 + y3z

3.

Through coefficient comparison, we obtain(
2ξee

(−ξ2)+1

F2 [1 + P ]m
)
a2 = y1,

3

2

(
ξ2ee

(−ξ2)+1

F3 [1 + 2P ]m
)
a3 = y2,

2

3

(
ξ3ee

(−ξ2)+1

F4 [1 + 3P ]m
)
a4 = y3.

Therefore,

a2 =
y1(

2ξee
(−ξ2)+1F2 [1 + P ]m

) , a3 =
2y2

3
(
ξ2ee

(−ξ2)+1F3 [1 + 2P ]m
) ,

a4 =
3y3

2
(
ξ3ee

(−ξ2)+1F4 [1 + 3P ]m
) .

Then combining a2a4 − a23 and take the magnitude based on Hankel determinant, also
by applying Lemma (2) and taking common factors we have the following:

∣∣a2a4 − a23
∣∣ = 1

ξee
(−ξ2)+1

∣∣∣∣∣ 3y1y3
4ξ2F2F4 [1 + P ]m [1 + 3P ]m

− 4y22

9ξ3ee
(−ξ2)+1F3 [1 + 2P ]2m

∣∣∣∣∣ . (10)

Given that the function B(z) belongs to the class B simultaneously, we can assume that
y1= y > 0 without losing generality. For ease of notation, we will use y1 = y(y ∈ [0, 2]).



O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6108 6 of 11

Next, by combining (8) with (9) and substitute in (10), we obtain the following:

∣∣a2a4 − a23
∣∣ = Q(h)

∣∣∣∣∣∣∣∣∣∣∣

27y4+54y2(4−y2)x−27y2(4−y2)x2

144

+
ξ2F2F4[1+P ]m[1+3P ]m

(
−16y4−32y2(4−y2)x−16(4−y2)

2
x2

)
144

(
ξ3ee

(−ξ2)+1
F3[1+2P ]2m

)
+

6y(4−y2)(1−|x2|)z
16

∣∣∣∣∣∣∣∣∣∣∣
where

Q(h) =
1(

ξee
(−ξ2)+1

)
(ξ2F2F4 [1 + P ]m [1 + 3P ]m)

.

When |k| is substituted with v and the triangle inequality is applied, we have

∣∣a2a4 − a23
∣∣ ≤ Q(h)



 27
144 − 16(ξ2F2F4[1+P ]m[1+3P ]m)

144

(
ξ3ee

(−ξ2)+1
F3[1+2P ]2m

)
 y4

+

 54
144 − 32(ξ2F2F4[1+P ]m[1+3P ]m)

144

(
ξ3ee

(−ξ2)+1
F3[1+2P ]2m

)
 y2(4− y2)v

+

27y2

144 +
16(4−y2)(ξ2F2F4[1+P ]m[1+3P ]m)

144

(
ξ3ee

(−ξ2)+1
F3[1+2P ]2m

)
 (4− y2)v2

+6y(4−y2)(1−v2)
16


(11)

= Q(h)



 27
144 − 16(ξ2F2F4[1+P ]m[1+3P ]m)

144

(
ξ3ee

(−ξ2)+1
F3[1+2P ]2m

)
 y4

+

 54
144 − 32(ξ2F2F4[1+P ]m[1+3P ]m)

144

(
ξ3ee

(−ξ2)+1
F3[1+2P ]2m

)
 y2(4− y2)v

+

27y2

144 +
16(4−y2)(ξ2F2F4[1+P ]m[1+3P ]m)

144

(
ξ3ee

(−ξ2)+1
F3[1+2P ]2m

) − 6y
16

 (4− y2)v2

+6y(4−y2)
16


= K(y, v) (12)

where 0 ≤ y ≤ 2 and 0 ≤ v ≤ 1. The function K(y, v) is then maximized on the closed
square [0, 2]× [0, 1]. Differentiating K(y, v) with respect to v, we get

dK

dv
= Q(h)



 54
144 − 32(ξ2F2F4[1+P ]m[1+3P ]m)

144

(
ξ3ee

(−ξ2)+1
F3[1+2P ]2m

)
 y2(4− y2)

+

3y(y−2)
8 +

16(4−y2)(ξ2F2F4[1+P ]m[1+3P ]m)

72

(
ξ3ee

(−ξ2)+1
F3[1+2P ]2m

)
 (4− y2)v

 .



O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6108 7 of 11

For 0 < v < 1, and for fixed y with 0 < y < 2, and
(ξ2F2F4[1+P ]m[1+3P ]m)(
ξ3ee

(−ξ2)+1
F3[1+2P ]2m

) < 27
16 , we

observe that dK
dv > 0. Consequently, a maximum of K(y, v) cannot exist inside the closed

square [0, 2] × [0, 1]. Additionally, for fixed y ∈ [0, 2], we have max
0≤v≤1

K(y, v) = K(y, 1) =

G(y).

G(y) = Q(h)



 27
144 − 16(ξ2F2F4[1+P ]m[1+3P ]m)

144

(
ξ3ee

(−ξ2)+1
F3[1+2P ]2m

)
 y4

+

 54
144 − 32(ξ2F2F4[1+P ]m[1+3P ]m)

144

(
ξ3ee

(−ξ2)+1
F3[1+2P ]2m

)
 y2(4− y2)

+

27y2

144 +
16(4−y2)(ξ2F2F4[1+P ]m[1+3P ]m)

144

(
ξ3ee

(−ξ2)+1
F3[1+2P ]2m

) − 6y
16

 (4− y2)

+6y(4−y2)
16


.

Next

G
′
(y) = Q(h)

3y(3− y2)

2
−

8y(4− y2)
(
ξ2F2F4 [1 + P ]m [1 + 3P ]m

)
9
(
ξ3ee

(−ξ2)+1F3 [1 + 2P ]2m
)

 = 0,

implies y = 0. Further, we observe that G′′(y) =

Q(h)

9
2 − 27y2

4 − 8(4−3y2)(ξ2F2F4[1+P ]m[1+3P ]m)

9

(
ξ3ee

(−ξ2)+1
F3[1+2P ]2m

)
 < 0, and

81
64 <

(ξ2F2F4[1+P ]m[1+3P ]m)(
ξ3ee

(−ξ2)+1
F3[1+2P ]2m

) < 27
16 .

We also note that G(y) > G(2). Thus, max
0≤y≤2

G(y) happens at y = 0, that is when v = 1

and y = 0 we obtain the bound of Eq. (11):∣∣a2a4 − a23
∣∣ ≤ 16

9
(
ξ4e2e

(−ξ2)+1F3 [1 + 2P ]2m
) .

The proof is completed.

Corollary 1. Let f ∈ Tk(0), for m = 0. Then∣∣a2a4 − a23
∣∣ ≤ 16

9
(
ξ4e2e

(−ξ2)+1F3

) .



O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6108 8 of 11

3. Conclusions

The bound of the second Hankel determinant is our main focus. It is established within
a class of univalent functions associated with a new operator linked to the Bell polynomial.
The result is not sharp and may be improved in future work.

Acknowledgements

We would like to thank Universiti Kebangsaan Malaysia for the support given at the
Universiti while doing research.

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