EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5929
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

The Relationship of Borel Distribution and Horadam
Polynomials Leads to Analytical Bi-Univalent Functions

Omar Alnajar1,∗, Omar S Khabour2, Ala Amourah3,4, Maslina Darus1

1 Department of Mathematical Sciences, Faculty of Science and Technology,
Universiti Kebangsaan Malaysia, Bangi 43600, Malaysia
2 Department of Curricula and Methods of Teaching Mathematics Education Program,
Faculty of Education Sciences, The University of Jordan, Amman 11942, Jordan
3 Mathematics Education Program, Faculty of Education and Arts, Sohar University,
Sohar 3111, Oman
4 Applied Science Research Center, Applied Science Private University, Amman, Jordan

Abstract. The Borel distribution is a practical and applicable model for a wide range of real-world
applications. Using the Borel distribution as a foundation, we create a novel subclass of analytic
bi-univalent functions in this study. We employ these functions, which involve the ultraspherical
polynomials, to create our new subclass. For functions that fall within the constructed class,
we investigate alternative estimations of the Maclaurin coefficients and solve the Fekete-Szego
functional problem.

2020 Mathematics Subject Classifications: 30C45

Key Words and Phrases: Borel distribution, Bi-univalent functions, Analytic Functions, Fekete-
Szegö problem

1. Preliminaries

In many branches of mathematics and physics, particularly in the study of differential
equations and approximation theory, orthogonal polynomials are a class of mathemat-
ical functions that appear. There are a collection of polynomials that are orthogonal
to a particular weight function across a specified range. This indicates that unless the
polynomials are equal, the outcome of multiplying the polynomials by one another and
integrating across the interval is zero.

Orthogonal polynomials come in a variety of families, each with a unique weight func-
tion and interval of orthogonality. The Legendre polynomials, Chebyshev polynomials,

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5929

Email addresses: P117246@siswa.ukm.edu.my (O. Alnajar),
AAmourah@su.edu.om (A. Amourah), o.khabour@ju.edu.jo (O. Khabour), 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 (2) (2025), 5929 2 of 12

Hermite polynomials, and Jacobi polynomials are a few of the most well-known families.
Each of these families has unique characteristics and uses, see [1–6].

Numerous areas of physics and mathematics, such as numerical analysis, probability
theory, and quantum mechanics, all heavily rely on orthogonal polynomials. For instance,
these polynomials can be applied to the numerical computation of integrals, the solution
of differential equations, and the investigation of the behavior of random variables. Let A
denote the class of functions f that takes the form:

f(φ) = φ+ k2φ
2 + k3φ

3 + · · · , (φ ∈ B), (1)

that are analytic in the disk B = {φ ∈ C : |φ| < 1}. Also, we represent by S the
subclass of A comprising functions of the Eq. (1) which are also univalent in B.

Geometric function theory can benefit greatly from the powerful tools that differential
subordination of analytical functions provides. Miller and Mocanu [7] introduced the first
differential subordination problem, additionally, see [8]. The majority of the developments
in the field are compiled in Miller and Mocanu’s book [9].

Every mathematical function f ∈ S has an inverse f−1, which is defined by

f−1(f(φ)) = φ (φ ∈ B)

and

w = f(f−1(w)) (|w| < r0(f); r0(f) ≥
1

4
)

where

g(w) = f−1(w) = w − k2w
2 + (−k3 + 2k22)w

3 − (k4 + 5k32 − 5k3k2)w
4 + · · · . (2)

A function is said to have the property of being bi-univalent in B if both f(φ) and f−1(φ)
have the property of being univalent in B.

Let us refer to the group of bi-univalent functions in B as Σ, which is defined by the
Eq. (1). Some examples from the class Σ are as follows:

φ

1− φ
, log

1

1− φ
.

However, Σ does not contain the well-known Koebe function. Other examples of
functions that are typical in B include the following:

2φ− φ2

2
and

φ

1− φ2
.

Furthermore, it is not a part of Σ. In class Σ and its subclasses, look for intriguing
functions ([10]-[11], [12]-[13]).

Also, in [14–20], estimates were made but not sharp for the first two coefficients |k2|
and |k3| in the Taylor-Maclaurin series expansion (1). These developments were motivated
by the groundbreaking work of Srivastava et al [21].



O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5929 3 of 12

In 2009, Horzum and Kocer published their related to Horadam polynomials hm(d),
[22]. The recurrence relation, which can be seen in the following sentence, gives us these
polynomials to work with, that

hm(d) = ϑdhm−1(d) + lhm−2(d), (m ∈ N \ {1, 2}), (3)

with

h1(d) = a, h2(d) = td and h3(d) = ϑtd2 + ϑl, (4)

assuming that a, t, ϑ, and l are real constants.

Remark 1. Special examples of the Horadam polynomials.
i) If a = t = ϑ = l = 1, the Fibonacci polynomials sequence is obtained

Fm(d) = dFm−1(d) + Fm−2(d); F1(d) = 1, F2(d) = d.

ii) If a = 2, t = ϑ = l = 1, the Lucas polynomials sequence is obtained

Lm−1(d) = dLm−2(d) + Lm−3(d); L0(d) = 2, L1(d) = d.

iii) a = 1, t = ϑ = 2, l = −1, the Chebyshev polynomials of second kind sequence is
obtained

Um−1(d) = 2dUm−2(d)− Um−3(d); U0(d) = 1, U1(d) = 2d.

iv) If a = t = 1, ϑ = 2, l = −1, the Chebyshev polynomials of first kind sequence is
obtained

Tm−1(d) = 2dTm−2(d)− Tm−3(d); T0(d) = 1, T1(d) = d.

v) If a = l = 1, t = ϑ = 2, the Pell polynomials sequence is obtained

Pm(d) = 2dPm−1(d) + Pm−2(d); P1(d) = 1, P2(d) = 2d.

vi) If a = t = ϑ = 2, l = 1, the Pell-Lucas polynomials sequence is obtained

Qm−1(d) = 2dQm−2(d) +Qm−3(d); Q0(d) = 2, Q1(d) = 2d.

In general, the Horadam polynomials have a lot of interesting and useful mathematical
properties, and they play a big role in a wide range of math, engineering, and physics
applications. Many studies, have looked into the properties of these polynomials, both
theoretical and practical.

The following expression serves as an example of how to generate the Horadam poly-
nomials hm(d):



O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5929 4 of 12

ψ(d, φ) =
∞∑

m=1

hm(d)φn−1 =
a+ (t− aϑ)dφ

1− ϑdφ− lφ2
. (5)

In recent years, a great number of studies have investigated significant aspects of geo-
metric function theory. These studies have focused on topics such as coefficient estimates,
inclusion relations, and propirtses of the classes. These investigations have made use of
a wide variety of probability distributions, such as the Poisson, Pascal, and many others
(see, [23–25]).

If it is conceivable for d to take on the values 1, 2, 3, ..., and so on with the stated
probability, then it is said that a discrete random variable, which is indicated by X,
should have a Borel distribution.

e−v

1!
,
2ve−2v

2!
,
9v2e−3v

3!
, ..., (6)

accordingly, in which cases they are referred to as the parameters. Hence

P (d = ∂) =
(v∂)∂−1e−v∂

∂!
, ∂ = 1, 2, 3, ....

Finally, we give a power series with Borel distribution coefficients.

F(v, φ) = φ+

∞∑
m=2

(v (m− 1))m−2 e−v(m−1)

(m− 1)!
φm, φ ∈ B. (7)

Take into account the convolution-defined linear operator Pγ : A → A.

Pγf(φ) = F(v, φ) ∗ f(φ) = φ+
∞∑

m=2

(v (m− 1))m−2 e−v(m−1)

(m− 1)!
kmφ

m, φ ∈ B. (8)

Too many scholars to count have studied the relationship between bi-univalent func-
tions and orthogonal polynomials recently, but some of the ones worth mentioning are
[26–35]. To the best of our knowledge, we have not been able to locate any previous work
in the literature that deals with bi-univalent functions for subordinate Horadam polynomi-
als that use the Borel distribution. We derive bounds for the |k2| and |k3| Taylor-Maclaurin
coefficients and describe a new subclass of Σ involving the Borel distribution connected
to Horadam polynomials. In addition, we address the Fekete-Szegö functional difficulties
for this new category of functions.

2. Delimitations of the class ϱtΣ(d, ϑ, l, γ)

This section starts off by providing a definition for the new subclass ϱtΣ(d, ϑ, l, γ), which
is related to the Borel distribution series.



O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5929 5 of 12

Definition 1. In the event that the subordinations listed below are satisfied, a function
denoted by (1) is considered to be a member of the class ϱtΣ(d, ϑ, l, γ):

(1− γ)
φPγf

′(φ)

Pγf(φ)
+ γ

(
1 +

φPγf
′′(φ)

Pγf ′(φ)

)
≺ ψ(d, φ) + 1− a (9)

and

(1− γ)
wPγf

′(w)

Pγf(w)
+ γ

(
1 +

wPγf
′′(w)

Pγf ′(w)
≺
)
ψ(d,w) + 1− a, (10)

where d ∈ R, and (2) describes the function g = f−1.

Example 1. ϱtΣ(d, ϑ, l, 0) = ϱtΣ(d, ϑ, l), is the class of functions f that is given by (1) and
satisfies the following condition: This holds true with regard to γ = 0.

φPγf
′(φ)

Pγf(φ)
≺ ψ(d, φ) + 1− a (11)

and
wPγf

′(w)

Pγf(w)
≺ ψ(d,w) + 1− a, (12)

where d ∈ R, and (2) describes the function g = f−1.

Example 2. ϱtΣ(d, ϑ, l, 1) = ϱtΣ(d, ϑ, l), is the class of functions f that is given by (1) and
satisfies the following condition: This holds true with regard to γ = 1..

1 +
φPγf

′′(φ)

Pγf ′(φ)
≺ ψ(d, φ) + 1− a (13)

and

1 +
wPγf

′′(w)

Pγf ′(w)
≺ ψ(d,w) + 1− a, (14)

where d ∈ R, and (2) describes the function g = f−1.
We will start by giving the estimated coefficients for class ϱtΣ(d, ϑ, l, γ) from Definition

1.

Theorem 1. Recognize that class ϱtΣ(d, ϑ, l) is a member of the function f ∈ Σ defined by
reference (1) . Then

|k2| ≤
|td|

√
t |d|√∣∣∣2 (1 + 2γ) ve−2v (td)2 − (1 + γ)2 e−2v (ϑtd2 + al)

∣∣∣ ,
and

|k3| ≤
t2d2

(1 + γ)2 e−2v
+

t|d|
2 (1 + 2γ) ve−2v

.



O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5929 6 of 12

Proof. Let f ∈ ϱtΣ(d, ϑ, l, γ). From Definition 1, we can write

(1− γ)
φPγf

′(φ)

Pλf(φ)
+ γ

(
1 +

φPλf
′′(φ)

Pλf ′(φ)

)
= ψ(d,κ(φ)) + 1− a (15)

and

(1− γ)
wPγf

′(w)

Pγf(w)
+ γ

(
1 +

wPγf
′′(w)

Pγf ′(w)

)
= ψ(d, τ(w)) + 1− a, (16)

the point at which the analytical functions κ and τ assume the form

κ(φ) = b1φ+ b2φ
2 + b3φ

3 + · · · , (φ ∈ B)

and
τ(w) = i1w + i2w

2 + i3w
3 + · · · , (w ∈ B),

such that κ(0) = τ(0) = 0 and |κ(φ)| < 1, |τ(w)| < 1 for all φ,w ∈ B.
From the equalities (15) and (16), it is what we get

(1−γ)φPγf
′(φ)

Pγf(φ)
+γ

(
1 +

φPγf
′′(φ)

Pγf ′(φ)

)
= 1+h2(d)b1φ+

[
h2(d)b2 + h3(d)b

2
1

]
φ2+ · · · (17)

and

(1−γ)wPγf
′(w)

Pγf(w)
+γ

(
1 +

wPγf
′′(w)

Pγf ′(w)

)
= 1+h2(d)i1w+

[
h2(d)i2 + h3(d)i

2
1

]
w2+· · · . (18)

It is common knowledge that if

|κ(φ)| =
∣∣b1φ+ b2φ

2 + b3φ
3 + · · ·

∣∣ < 1, (φ ∈ B)

and
|τ(w)| =

∣∣i1w + i2w
2 + i3w

3 + · · ·
∣∣ < 1, (w ∈ B),

then
|bj | ≤ 1 and |ij | ≤ 1 for all j ∈ N. (19)

When we compare the relevant coefficients in (17) and (18), we get the following:

(1 + γ) e−vk2 = h2(d)b1, (20)

2 (1 + 2γ) ve−2vk3 = h2(d)b2 + h3(d)b
2
1, (21)

− (1 + γ) e−vk2 = h2(d)i1, (22)

and
2 (1 + 2γ) ve−2v

(
2k22 − k3

)
= h2(d)i2 + h3(d)i

2
1. (23)



O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5929 7 of 12

According to(20) and (22),
b1 = −i1 (24)

and
2 (1 + γ)2 e−2vk22 = [h2(d)]

2 (b21 + i21
)
. (25)

When we combine (21) and (23), we obtain

4 (1 + 2γ) ve−2vk22 = h2(d) (b2 + i2) + h3(d)
(
b21 + i21

)
. (26)

We can find out what it is by changing the value of
(
b21 + i21

)
in (25) onto the right

side of (26) .

2

(
2 (1 + 2γ) v − (1 + γ)2

h3(d)

[h2(d)]
2

)
e−2vk22

= h2(d) (b2 + i2) . (27)

Moreover computations using (4), and (27), we find that

|k2| ≤
td
√
td√∣∣∣2 (1 + 2γ) ve−2v (td)2 − (1 + γ)2 e−2v (ϑtd2 + al)

∣∣∣ .
In addition to this, the result that we get when we take away (23) from (21) is.

4 (1 + 2γ) ve−2v
(
k3 − k22

)
= h2(d) (b2 − i2) + h3(d)

(
b21 − i21

)
. (28)

So, if you take into account (24) and (25), the equation on (28) can be rewritten as

k3 =
[h2(d)]

2

2 (1 + γ)2 e−2v

(
b21 + i21

)
+

h2(d)

4 (1 + 2γ) ve−2v
(b2 − i2) .

So, using(4), we come to the conclusion that

|k3| ≤
t2d2

(1 + γ)2 e−2v
+

td

2 (1 + 2γ) ve−2v
.

.

An exact limit on the functional space
∣∣k3 − ηk22

∣∣ was obtained by Fekete and Szego
in 1933 [36] . This limit was specific to a univalent function f and η that belongs to the
interval [0, 1].

Using the values of k22 and k3, we prove the functional
∣∣k3 − ηk22

∣∣ for class functions
ϱtΣ(d, ϑ, l, γ).



O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5929 8 of 12

Theorem 2. Recognize that class ϱtΣ(d, ϑ, l, γ) is a member of the function f ∈ Σ defined
by reference (1) . Then

∣∣k3 − ηk22
∣∣ ≤


|td|

2(1+2γ)ve−2v ,

(td)3|1−η|
e−2v|[2(1+2γ)v[td]2−(1+λ)2(ϑtd2+al)]| ,

|η − 1| ≤ θ

|η − 1| ≥ θ,

where

θ =

∣∣∣∣∣1− 2 (1 + γ)2 e−2v
(
ϑtd2 + al

)
4 (1 + 2γ) t2d2ve−2v

∣∣∣∣∣ .
Proof. From (27) and (28)

k3 − ηk22

= (1− η)
[h2(d)]

3 (b2 + i2)

2e−2v
[
2v (1 + 2γ) [h2(d)]

2 − (1 + γ)2 h3(d)
]

+
h2(d)

4 (1 + 2γ) ve−2v
(b2 − i2)

= h2(d)

[
℧(η) +

1

4 (1 + 2γ) ve−2v

]
b2

+ h2(d)

[
℧(η)− 1

4 (1 + 2γ) ve−2v

]
i2,

where

℧(η) =
[h2(d)]

2 (1− η)

2e−2v
[
2v (1 + 2γ) [h2(d)]

2 − (1 + γ)2 h3(d)
] ,

Consequently, based on (4), we deduce that

∣∣k3 − ηk22
∣∣ ≤


2|h2(d)|

4(1+2γ)ve−2v

2 |h2(d)| |℧(η)|

|℧(η)| ≤ 1
4(1+2γ)ve−2v ,

|℧(η)| ≥ 1
4(1+2γ)ve−2v .

.

3. Corollaries

As a result of the theorems called 1 and 2, the following corollaries are true. These
corollaries generally correspond to the examples referred to as 1 and 2.

Corollary 1. Recognize that class ϱtΣ(d, ϑ, l) is a member of the function f ∈ Σ defined
by reference (1) . Then



O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5929 9 of 12

|k2| ≤
td
√
td√∣∣∣2ve−2v (td)2 − e−2v (ϑtd2 + al)

∣∣∣ ,
|k3| ≤

t2d2

e−2v
+

td

2ve−2v
.

and

∣∣k3 − ηk22
∣∣ ≤


|td|

2ve−2v ,

(td)3|1−η|
e−2v |[2vt2d2−(ϑtd2+al)]| ,

|η − 1| ≤
∣∣∣∣1− 2e−2v(ϑtd2+al)

4t2d2ve−2v

∣∣∣∣
|η − 1| ≥

∣∣∣∣1− 2e−2v(ϑtd2+al)
4t2d2ve−2v

∣∣∣∣ .
Corollary 2. Recognize that class ϱtΣ(d, ϑ, l) is a member of the function f ∈ Σ defined
by reference (1) . Then

|k2| ≤
td
√
td√∣∣∣6ve−2v (td)2 − 4e−2v (ϑtd2 + al)

∣∣∣ ,
|k3| ≤

t2d2

4e−2v
+

td

6ve−2v
.

and

∣∣k3 − ηk22
∣∣ ≤


|td|

6ve−2v ,

2(td)3|1−η|
e−2v |[6vt2d2−4(ϑtd2+al)]| ,

|η − 1| ≤
∣∣∣∣1− 8e−2v(ϑtd2+al)

12t2d2ve−2v

∣∣∣∣
|η − 1| ≥

∣∣∣∣1− 8e−2v(ϑtd2+al)
12t2d2ve−2v

∣∣∣∣ .
4. Conclusions

In this important study, we created a new category of normalised analytic and bi-
univalent functions that are closely related to the famous Borel distribution series ϱtΣ(d, ϑ, l
, γ). With our new way of thinking, we were able to find accurate values for the Taylor-
Maclaurin coefficients |k2| and |k3| and solve the hard Fekete-Szego functional problems.
We were able to figure out the results for subclasses ϱtΣ(d, ϑ, l, 1) and ϱ

t
Σ(d, ϑ, l, 0), which

are shown in Examples 1 and 2, by cleverly changing the parameters γ. They are connected
in a complicated way to the Borel series of distributions. Based on our ground-breaking use
of the Borel distribution series (8), future researchers will be able to use the extraordinary
Horadam polynomials associated with this distribution series to solve Fekete-Szego func-
tional problems and estimate Taylor-Maclaurin coefficients for new classes of bi-univalent
functions.



O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5929 10 of 12

References

[1] B. A. Frasin, T. Al-Hawary, and F. Yousef. Necessary and sufficient conditions for
hypergeometric functions to be in a subclass of analytic functions. Afrika Matematika,
30:223–230, 2019.

[2] A. Amourah, B. Frasin, J. Salah, and F. Yousef. Subfamilies of bi-univalent functions
associated with the imaginary error function and subordinate to jacobi polynomials.
Symmetry, 17(2):157, 2025.

[3] B. A. Frasin, T. Al-Hawary, F. Yousef, and I. Aldawish. On subclasses of analytic
functions associated with struve functions. Nonlinear Functional Analysis and Appli-
cations, 27(1):99–110, 2022.

[4] B. A. Frasin, F. Yousef, T. Al-Hawary, and I. Aldawish. Application of generalized
bessel functions to classes of analytic functions. Afrika Matematika, 32:431–439, 2021.

[5] T. Al-Hawary, I. Aldawish, B. A. Frasin, O. Alkam, and F. Yousef. Necessary and
sufficient conditions for normalized wright functions to be in certain classes of analytic
functions. Mathematics, 10(24):4693, 2022.

[6] M. Illafe, M. H. Mohd, F. Yousef, and S. Supramaniam. A subclass of bi-univalent
functions defined by a symmetric q-derivative operator and gegenbauer polynomials.
European Journal of Pure and Applied Mathematics, 17(4):2467–2480, 2024.

[7] S. S. Miller and P. T. Mocanu. Second order differential inequalities in the complex
plane. J. Math. Anal. Appl., 65:289–305, 1978.

[8] S. S. Miller and P. T. Mocanu. Differential subordinations and univalent functions.
Mich. Math. J., 28:157–172, 1981.

[9] S. S. Miller and P. T. Mocanu. Differential Subordinations. Theory and Applications.
Marcel Dekker, Inc., New York, NY, USA, 2000.

[10] B. A. Frasin and M. K. Aouf. New subclasses of bi-univalent functions. Appl. Math.
Lett., 24:1569–1573, 2011.

[11] G. Murugusundaramoorthy. Subclasses of starlike and convex functions involving
poisson distribution series. Afr. Mat., 28:1357–1366, 2017.

[12] F. Yousef, T. Al-Hawary, and G. Murugusundaramoorthy. Fekete-szegö functional
problems for some subclasses of bi-univalent functions defined by frasin differential
operator. Afrika Matematika, 30(3-4):495–503, 2019.

[13] F. Yousef, S. Alroud, and M. Illafe. New subclasses of analytic and bi-univalent
functions endowed with coefficient estimate problems. Anal. Math. Physics, 11:58,
2021.

[14] A. Amourah, B. A. Frasin, G. Murugusundaramoorthy, and T. Al-Hawary. Bi-
bazilevič functions of order ϑ + iδ associated with (p, q)-lucas polynomials. AIMS
Mathematics, 6(5):4296–4305, 2021.

[15] G. Murugusundaramoorthy, N. Magesh, and V. Prameela. Coefficient bounds for
certain subclasses of bi-univalent function. Abst. Appl. Anal., 2013:Article ID 573017,
3 pages, 2013.

[16] Z. Peng, G. Murugusundaramoorthy, and T. Janani. Coefficient estimate of bi-
univalent functions of complex order associated with the hohlov operator. J. Complex



O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5929 11 of 12

Analysis, 2014:Article ID 693908, 6 pages, 2014.
[17] H. M. Srivastava, Ş. Altınkaya, and S. Yalçın. Certain subclasses of bi-univalent

functions associated with the horadam polynomials. Iranian Journal of Science and
Technology, Transactions A: Science, 43(4):1873–1879, 2019.

[18] H. O. Guney, G. Murugusundaramoorthy, and J. Sokol. Subclasses of bi-univalent
functions related to shell-like curves connected with fibonacci numbers. Acta Univ.
Sapientiae, Math., 10(1):70–84, 2018.

[19] S. Altınkaya and S. Yalçın. On the (p, q)-lucas polynomial coefficient bounds of the
bi-univalent function class. Boletin de la Sociedad Matematica Mexicana, pages 1–9,
2018.

[20] S. Bulut. Coefficient estimates for a class of analytic and bi-univalent functions. Novi
Sad J. Math., 43(2):59–65, 2013.

[21] H. M. Srivastava, A. K. Mishra, and P. Gochhayat. Certain subclasses of analytic
and bi-univalent functions. Appl. Math. Lett., 23(10):1188–1192, 2010.

[22] T. Horzum and E. G. Kocer. On some properties of horadam polynomials. Int. Math.
Forum, 2009.

[23] A. K. Wanas and J. A. Khuttar. Applications of borel distribution series on analytic
functions. Earthline J. Math. Sci., 4:71–82, 2020.

[24] H. M. Srivastava, A. K. Wanas, and G. Murugusundaramoorthy. A certain family of
bi-univalent functions.

[25] M. Ghaffar et al. Khan. Applications of mittag-leffer type poisson distribution to a
subclass of analytic functions involving conic-type regions. J. Funct. Spaces, pages
Article ID 4343163, 9 pages, 2021.

[26] A. Amourah, B. A. Frasin, M. Ahmad, and F. Yousef. Exploiting the pascal distri-
bution series and gegenbauer polynomials to construct and study a new subclass of
analytic bi-univalent functions. Symmetry, 14(1):147, 2022.

[27] A. Amourah, B. A. Frasin, and T. Abdeljawad. Fekete-szegö inequality for analytic
and bi-univalent functions subordinate to gegenbauer polynomials. J. Funct. Spaces,
2021:Article ID 5574673, 7 pages, 2021.

[28] A. Amourah, O. Alnajar, M. Darus, A. Shdouh, and O. Ogilat. Estimates for the
coefficients of subclasses defined by the bell distribution of bi-univalent functions
subordinate to gegenbauer polynomials. Mathematics, 11(8):1799, 2023.

[29] V. Kumar, N. E. Cho, V. Ravichandran, and H. M. Srivastava. Sharp coefficient
bounds for starlike functions associated with the bell numbers. Mathematica Slovaca,
69(5):1053–1064, 2019.

[30] A. Amourah, O. Alnajar, J. Salah, and M. Darus. Geometric properties and neigh-
borhoods of certain subclass of analytic functions defined by using bell distribution.
Contemporary Mathematics, pages 5473–5481, 2024.

[31] O. Alnajar and M. Darus. Coefficient estimates for subclasses of bi-univalent func-
tions related to gegenbauer polynomials and an application of bell distribution. AIP
Conference Proceedings, 3150(1):AIP Publishing, September, 2024.

[32] O. Alnajar, O. Ogilat, A. Amourah, M. Darus, and M. S. Alatawi. The miller-ross
poisson distribution and its applications to certain classes of bi-univalent functions



O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5929 12 of 12

related to horadam polynomials. Heliyon, 10(7):Article ID e04334, 2024.
[33] O. Alnajar, A. Amourah, and M. Darus. The characteristics of inclusion pertaining to

univalent functions associated with bell distribution functions. Int. J. Open Problems
Complex Analysis, 15(2):46–61, 2023.

[34] O. Alnajar, A. Amourah, and M. Darus. Application of gegenbauer polynomials to
certain classes of bi-univalent functions of order ν+ iς. Korean Journal of Mathemat-
ics, 32(1):183–193, 2024.

[35] O. Alnajar, A. Amourah, J. Salah, and M. Darus. Fekete-szegö functional prob-
lem for analytic and bi-univalent functions subordinate to gegenbauer polynomials.
Contemporary Mathematics, pages 5731–5742, 2024.

[36] M. Fekete and G. Szegö. Eine bemerkung Ãber ungerade schlichte funktionen. J.
Lond. Math. Soc., 1(2):85–89, 1933.