EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5955
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

The Neutrosophic Poisson Distribution Applied to
Horadam Polynomial-Subordinate Bi-Univalent

Functions

Khalid M. K. Alshammari1, Omar Alnajar2,∗, Ala Amourah3,4

1 Department of Mathematics, College of Sciences, Faculty of Science and Technology,
University of Ha’il, Ha’il 55425 , Saudi Arabia
2 Department of Mathematics Sciences, Faculty of Science and Technology,
Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia
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. In the open unit disk, we introduce a new subclass of analytic and bi-univalent functions
described by the Neutrosophic Poisson distribution series. The estimates for the Fekete–Szegö type
and the Taylor coefficients. Then, using the Horadam polynomials, the inequalities corresponding
to the functions that belong to this recently discovered subclass are investigated. Along with
some corollaries, we also go over the implications of the findings presented in this essay. Numerous
branches of mathematics, science, and technology are anticipated to heavily rely on the neutrophilic
Poisson distribution.

2020 Mathematics Subject Classifications: 30C45

Key Words and Phrases: Horadam polynomials, bi-univalent functions, analytic functions,
Fekete-Szegö problem

1. Preliminaries

The orthogonality and completeness of Legendre polynomials, two of their defining
characteristics, make them well-suited for a wide range of applications. Legendre first
introduced these polynomials in 1784 [1]. They are utilised quite frequently in analysis,
where they are utilised in polynomial interpolation and approximation, in order to address
boundary value problems in analysis, such as the Laplace equation and the Schrödinger
equation. In addition, they are utilised in a variety of other contexts.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5955

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

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



K. Alshammari, O. Alnajar, A. Amourah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5955 2 of 12

In the field of physics, Legendre polynomials are put to use for a broad variety of tasks,
including the study of spherical harmonics and the development of gravitational potentials,
to name just a couple of examples. When it comes to analysing data, particularly in
the fields of statistics and signal processing, they can be used to perform regression and
smoothing.

A probabilistic model that handles ambiguous or incomplete data in a more flexible
manner is called the neutrosophic Poisson distribution. This model is an extension of the
traditional Poisson distribution that was developed many years ago. The neutrosophic
logic theory is a generalisation of classical logic that permits for the existence of inde-
terminate or unknown truth values. This distribution was first presented by Florentin
Smarandache in 1998 [2], as a part of the neutrosophic logic theory. There have been a
few researchers who have combined the Bi-Univalent distribution with this one; for an
example, see the citation provided by [3].

Numerous branches of mathematics, including geometry and complex analysis, use
a family of analytical functions called ”bi-univalent functions,” which are defined in the
complex plane. The two-univalent functions are the univalent functions’ generalization.
They are in charge of assigning a specific region to the complex plane’s unit disk.

Research on bi-univalent functions continues to be active and has contributed to the
development of a number of important theories and breakthroughs in complex analysis
and geometry, few to mention [4–8]. generally.

Assume that the class of functions f of the type is A.

f(z) = z +m2z
2 + · · · = z +

∞∑
k=2

mkz
k, (z ∈ h). (1)

on the disk h = {z ∈ C : |z| < 1}, which are analytical. Additionally, we refer to S as
the subclass of A composed of univalent functions of Eq. (1) in h.

The study of geometric function theory benefits greatly from the many strong tools that
are made possible by the differential subordination of analytical functions. Additionally,
see [9]. Miller and Mocanu introduced the first differential subordination problem in [10].
Miller and Mocanu book [11], which includes publication date references, compiles the
most of the field’s accomplishments.

The symbol for the inverse of each mathematical function f ∈ S is f−1.

f−1(f(z)) = z (z ∈ h)

and
w = f(f−1(w)) (|w| < r0(f); r0(f) ≥ 0.25)

where

g(w) = f−1(w) = w −m2w
2 + (−m3 + 2m2

2)w
3 − (m4 + 5m3

2 − 5m3m2)w
4 + · · · . (2)



K. Alshammari, O. Alnajar, A. Amourah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5955 3 of 12

In the case where both f(z) and f−1(z) are univalent in the set h, it can be concluded
that the function is bi-univalent in the set h. Both of these functions have the same
evaluation value, which is the reason why this is the case.

The class of bi-univalent functions in h that are defined by the equation (1) will be
denoted as Σ using the notation. Here are a few instances from section Σ:

z

1− z
, log

1

1− z
.

On the other hand, the well-known Koebe function is not included in the Σ. Here
are some more instances of standard functions that can be found in h:

2z − z2

2
and

z

1− z2

also don’t belong to Σ. To learn about interesting subclasses of functions in class Σ,
see ([12–20]).

Inspired by the groundbreaking work of Srivastava et al. [21], several subclasses of the
bi-univalent function class Σ were introduced, [22–30] obtained non-sharp estimates on
the first two coefficients |m2| and |m3| in the Taylor-Maclaurin series expansion (1).

The Horadam polynomials hn(x), which are determined by the following recurrence
relation, were examined by Horzum and Kocer in 2009. [31].

hn(x) = pxhn−1(x) + qhn−2(x), (n ∈ N \ {1, 2}), (3)

with

h1(x) = a, h2(x) = tx and h3(x) = ptx2 + aq, (4)

for some real constant a, t, p and q.

Remark 1. Instances of the Horadam polynomials that are particularly noteworthy.

i) If a = t = p = q = 1, the Fibonacci polynomials sequence is obtained

Fn(x) = xFn−1(x) + Fn−2(x); F1(x) = 1, F2(x) = x.

ii) If a = 2, t = p = q = 1, the Lucas polynomials sequence is obtained

Ln−1(x) = xLn−2(x) + Ln−3(x); L0(x) = 2, L1(x) = x.

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

Un−1(x) = 2xUn−2(x)− Un−3(x); U0(x) = 1, U1(x) = 2x.

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



K. Alshammari, O. Alnajar, A. Amourah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5955 4 of 12

Tn−1(x) = 2xTn−2(x)− Tn−3(x); T0(x) = 1, T1(x) = x.

v) If a = q = 1, t = p = 2, the Pell polynomials sequence is obtained

Pn(x) = 2xPn−1(x) + Pn−2(x); P1(x) = 1, P2(x) = 2x.

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

Qn−1(x) = 2xQn−2(x) +Qn−3(x); Q0(x) = 2, Q1(x) = 2x.

For more information, (see [32] and [31]).

Numerous disciplines in mathematics, physics, statistics, and engineering make use of
the Fibonacci, Lucas, Chebyshev, and families of orthogonal polynomials, as well as their
extensions. Numerous academic articles have been written.

The polynomials in question.

The following creates Horadam polynomials, hn(x):

Ω(x, z) =
∞∑
n=1

hn(x)z
n−1 =

a+ (t− ap)xz

1− pxz − qz2
. (5)

In recent years, a significant amount of research has concentrated on various essential
areas of geometric function theory. These studies looked at coefficient estimates, relation-
ships between different types, and what is needed to belong to certain groups (see [33–39]).
Various distributions, such as the Poisson, Pascal, Borel, and Mittag-Leffler-type Poisson
distributions, have been utilized in this research, see [40–43].

Neutrosophic theory was just recently, specifically in 1995, presented by Smarandache.
It is a novel area of philosophy that serves as a generalisation of both the fuzzy and the
intuitionistic fuzzy logic, see [44]. Even though the neutrosophic Poisson distribution of
a discrete variable X is a classic Poisson distribution of X, its parameter l is not precise
and can be specified with two or more elements. This type of distribution occurs most
frequently when l is an interval. Let

NP (X = d) =
(l)d

d!
e−l, d = 0, 1, 2, ..., (6)

where the expected value and the variance, denoted by the distribution parameter l

NL(X) = NV (X) = l ,

a neutrosophic statistical number is N = O + I; for more information, see [44] and the
sources therein.We are going to provide a new power series, and the coefficients of this
series will be the probabilities of the Neutrosophic Poisson Distribution



K. Alshammari, O. Alnajar, A. Amourah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5955 5 of 12

B(l, z) = z +
∞∑
k=2

(l)k−1e−l

(k − 1)!
zk, z ∈ h. (7)

Consider the linear operator Pm : A → A defined by the convolution

Pmf(z) = B(l, z) ∗ f(z) = z +
∞∑
k=2

(l)k−1e−l

(k − 1)!
mkz

k, z ∈ h. (8)

In this study, we derive bounds for the |m2| and |m3| Taylor-Maclaurin coefficients
and establish a new subclass involving the neutrosophic Poisson distribution associated
with Horadam polynomials. For functions in this new class, we also solve the Fekete-Szego
functional problems.

2. Class boundaries Gt
Σ(x, p, q, u, γ)

This section starts by defining the new subclass of Bell distribution series, Gt
Σ(x, p, q, u, γ).

Definition 1. A function f ∈ Σ given by (1) is considered to belong to the class Gt
Σ(x, p, q, u, γ)

if the following subordinations are satisfied:

1 +
1

u

[
(Pmf (z))′ + γz(Pmf (z))′′ − 1

]
≺ Ω(x, z) + 1− a (9)

and

1 +
1

u

[
(Pmf (w))′ + γz(Pmf (w))′′ − 1

]
≺ Ω(x,w) + 1− a, (10)

where γ ≥ 0, u ≥ 1, x ∈ R, and the function g = f−1 is given by (2).

Example 1. The following equation is obtained when γ = 0 and u = 1,The expression
Gt

Σ(x, p, q, 1, 0), where Gt
Σ(x, p, q) denotes the class of functions f ∈ Σ that are provided

by (1) and satisfy the subsequent condition:The expression Gt
Σ(x, p, q, 1, 0).

(Pmf(z))′ ≺ Ω(x, z) + 1− a (11)

and
(Pmf(w))′ ≺ Ω(x,w) + 1− a, (12)

where x ∈ R, and the function g = f−1 is given by (2).
First, we will provide some estimates for the coefficients that belong to the class

Gt
Σ(x, p, q, u, γ) which may be found in Definition 1.



K. Alshammari, O. Alnajar, A. Amourah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5955 6 of 12

Theorem 1. Consider f ∈ Σ to be a member of the class Gt
Σ(x, p, q, u, γ) as indicated by

(1). After that,

|m2| ≤
tx
√
2txu√[∣∣∣3u (1 + 2γ) (tx)2 − 8 (1 + γ)2 (ptx2 + aq) e−l

∣∣∣] l2e−l

,

and

|m3| ≤
t2x2u2

4 (1 + γ)2 l2e−2l
+

2txu

3 (1 + 2γ) l2e−l
.

Proof. Suppose that f ∈ Gt
Σ(x, p, q, u, γ). From 1 Definition, we can write

1 +
1

u

[
(Pmf (z))′ + γz(Pmf (z))′′ − 1

]
= Ω(x,κ(z)) + 1− a (13)

and

1 +
1

u

[
(Pmf (w))′ + γz(Pmf (w))′′ − 1

]
= Ω(x, τ(w)) + 1− a, (14)

where κ and τ , the analytical functions, have the form

κ(z) = c1z + c2z
2 + c3z

3 + · · · , (z ∈ h)

and
τ(w) = d1w + d2w

2 + d3w
3 + · · · , (w ∈ h),

such that κ(0) = τ(0) = 0 and |κ(z)| < 1, |τ(w)| < 1 for all z, w ∈ h.
From the equalities (13) and (14), it is what we get

1 +
1

u

[
(Pmf (z))′ + γz(Pmf (z))′′ − 1

]
= 1+ h2(x)c1z +

[
h2(x)c2 + h3(x)c

2
1

]
z2 + · · · (15)

and

1 +
1

u

[
(Pmf (w))′ + γz(Pmf (w))′′ − 1

]
= 1 + h2(x)d1w +

[
h2(x)d2 + h3(x)d

2
1

]
w2 + · · · .

(16)
It is well known that if

|κ(z)| =
∣∣c1z + c2z

2 + c3z
3 + · · ·

∣∣ < 1, (z ∈ h)

and
|τ(w)| =

∣∣d1w + d2w
2 + d3w

3 + · · ·
∣∣ < 1, (w ∈ h),

then
|cj | ≤ 1 and |dj | ≤ 1 for all j ∈ N. (17)



K. Alshammari, O. Alnajar, A. Amourah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5955 7 of 12

The following is the outcome of comparing the pertinent coefficients in (15) and (16)

2 (1 + γ)

u
le−lm2 = h2(x)c1, (18)

3 (1 + 2γ)

2u
l2e−lm3 = h2(x)c2 + h3(x)c

2
1, (19)

−2 (1 + γ)

u
le−lm2 = h2(x)d1, (20)

and
3 (1 + 2γ)

2u
l2e−l

[
2m2

2 −m3

]
= h2(x)d2 + h3(x)d

2
1. (21)

It follows from (18) and (20) that

c1 = −d1 (22)

and
8 (1 + γ)2

u2
l2e−2lm2

2 = [h2(x)]
2 (c21 + d21

)
. (23)

If we add (19) and (21), we get

3 (1 + 2γ)

u
l2e−lm2

2 = h2(x) (c2 + d2) + h3(x)
(
c21 + d21

)
. (24)

We determine that by replacing the value of
(
c21 + d21

)
from (23) in the right hand side

of (24)

[
3 (1 + 2γ)

u
l2e−l − 8 (1 + γ)2

u2
l2e−2l h3(x)

[h2(x)]
2

]
m2

2

= h2(x) (c2 + d2) . (25)

Moreover computations using (4), (17) and (25), we find that

|m2| ≤
tx
√
2txu√[∣∣∣3u (1 + 2γ) (tx)2 − 8 (1 + γ)2 (ptx2 + aq) e−l

∣∣∣] l2e−l

.

Moreover, if we subtract (21) from (19), we obtain

3 (1 + 2γ)

u
l2e−l

(
m3 −m2

2

)
= h2(x) (c2 − d2) + h3(x)

(
c21 − d21

)
. (26)



K. Alshammari, O. Alnajar, A. Amourah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5955 8 of 12

Then, in view of (22) and (23), Eq. (26) becomes

m3 =
[h2(x)]

2 u2

8 (1 + γ)2 l2e−2l

(
c21 + d21

)
+

h2(x)u

3 (1 + 2γ) l2e−l
(c2 − d2) .

Thus applying (4), we conclude that

|m3| ≤
t2x2u2

4 (1 + γ)2 l2e−2l
+

2txu

3 (1 + 2γ) l2e−l
.

.

1933 [45] was the year when Fekete and Szego succeeded in obtaining a precise
constraint for the functional

∣∣m3 − ηm2
2

∣∣.. This was carried out for a function with a
single value known as f and the interval η ∈ [0, 1].

We use the values of m2
2 and m3 to illustrate that the functional expression for class

functions Gt
Σ(x, p, q, u, γ) is

∣∣m3 − ηm2
2

∣∣. This is carried out to show that the difference
between the two numbers is the key to the answer.

Theorem 2. Allow f ∈ Σ to belong to the class Gt
Σ(x, p, q, u, γ) as given by (1). Then

∣∣m3 − ηm2
2

∣∣ ≤


2|tx|u
3(1+2γ)l2e−l ,

2(tx)3|1−η|u2

[|3u(1+2γ)[tx]2−8(1+γ)2(ptx2+aq)e−l|]l2e−l
,

|η − 1| ≤ ℘

|η − 1| ≥ ℘,

where

℘ = 1−

[∣∣∣8 (1 + γ)2
(
ptx2 + aq

)∣∣∣] e−l

3u [tx]2 (1 + 2γ)
.

Proof. From (25) and (26)

m3 − ηm2
2

= (1− η)
[h2(x)]

3 (c2 + d2)u
2[∣∣∣3u (1 + 2γ) [h2(x)]

2 − 8 (1 + γ)2 h3(x)e−l
∣∣∣] l2e−l

+
h2(x)u

3 (1 + 2γ) l2e−l
(c2 − d2)

= h2(x)

[
F(η) +

u

3 (1 + 2γ) l2e−l

]
c2

+ h2(x)

[
F(η)− u

3 (1 + 2γ) l2e−l

]
d2,

where

F(η) =
[h2(x)]

2 (1− η)u2[∣∣∣3u (1 + 2γ) [h2(x)]
2 − 8 (1 + γ)2 h3(x)e−l

∣∣∣] l2e−l
,



K. Alshammari, O. Alnajar, A. Amourah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5955 9 of 12

Then, in view of (4), we conclude that

∣∣m3 − ηm2
2

∣∣ ≤


2|h2(x)|u
3(1+2γ)l2e−l

2 |h2(x)| |F(η)|

|F(η)| ≤ u
3(1+2γ)l2e−l ,

|F(η)| ≥ u
3(1+2γ)l2e−l .

.

3. Corollary

The mentioned theorems 1 and 2 directly generate this corollary.

Corollary 1. Allow f ∈ Σ to belong to the class Gt
Σ(x, p, q, 1, 0) as given by (1). Then

|m2| ≤
tx
√
2tx√[∣∣∣3 (tx)2 − 8 (ptx2 + aq) e−l

∣∣∣] l2e−l

,

|m3| ≤
t2x2

4l2e−2l
+

2tx

3l2e−l
.

and

∣∣m3 − ηm2
2

∣∣ ≤


2|tx|
3l2e−l ,

2(tx)3|1−η|
[|3[tx]2−8(ptx2+aq)e−l|]l2e−l

,

|η − 1| ≤ Φ

|η − 1| ≥ Φ,

where

Φ =

∣∣∣∣∣1− 8
(
ptx2 + aq

)
e−l

3 [tx]2

∣∣∣∣∣ .
4. Conclusions

The coefficient problems associated with each of the new subclasses of the class of
bi-univalent functions have been introduced and investigated in this paper.

Gt
Σ(x, p, q, u, γ) in the open unit disk’s configuration h. Definition 1 thus provides a

description of these bi-univalent function classes. For each of these three classes of bi-
univalent functions, we have calculated the Taylor–Maclaurin coefficients |m2| and |m3| as
well as the estimates of the Fekete–Szego functional issues. We discovered that a number
of other novel findings would emerge after focussing on the factors that were crucial to
our main findings. You can utilise bi-univalent functions in this investigation by using a
altered form of the derivative operator proposed by Caputo.



K. Alshammari, O. Alnajar, A. Amourah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5955 10 of 12

References

[1] A. Legendre. Recherches sur l’attraction des sphéröıdes homogènes. Mémoires
Présentés par Divers Savants à l’Académie des Sciences de l’Institut de France,
10:411–434, 1785.

[2] R. Padilla. Smarandache algebraic structures. Bull. Pure Appl. Sci., 17:119–121,
1998.

[3] A. Alsoboh, A. Amourah, M. Darus, and R.I. Al Sharefeen. Applications of neu-
trosophic q-poisson distribution series for subclasses of analytic functions and bi-
univalent functions. Mathematics, 11:868, 2023.

[4] O. Alnajar, A. Amourah, and M. Darus. Application of gegenbauer polynomials to
certain classes of bi-univalent functions of order ν+ iς. Korean J. Math., 32:183–193,
2024.

[5] 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.

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

[7] O. Alnajar, A. Amourah, J. Salah, and M. Darus. Fekete-szegö functional problem for
analytic and bi-univalent functions subordinate to gegenbauer polynomials. Contemp.
Math., pages 5731–5742, 2024.

[8] Ala Amourah, Omar Alnajar, Maslina Darus, Ala Shdouh, and Osama Ogilat. Esti-
mates for the coefficients of subclasses defined by the bell distribution of bi-univalent
functions subordinate to gegenbauer polynomials. Mathematics, 11(8):1799, 2023.

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

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

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

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

[13] 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.

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

[15] A. Amourah, T. Al-Hawary, F. Yousef, and J. Salah. Collection of bi-univalent func-
tions using bell distribution associated with jacobi polynomials. Int. J. Neutrosophic
Sci., 25(1):228–238, 2025.

[16] T. Al-Hawary, A. Amourah, J. Salah, and F. Yousef. Two inclusive subfamilies of
bi-univalent functions. Int. J. Neutrosophic Sci., 24(4):315–323, 2024.



K. Alshammari, O. Alnajar, A. Amourah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5955 11 of 12

[17] 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.

[18] 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.

[19] 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.

[20] T. Al-Hawary, B.A. Frasin, and F. Yousef. Coefficients estimates for certain classes
of analytic functions of complex order. Afrika Matematika, 29:1265–1271, 2018.

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

[22] G. Murugusundaramoorthy, N. Magesh, and V. Prameela. Coefficient bounds for cer-
tain subclasses of bi-univalent functions. Abst. Appl. Anal., pages Article ID 573017,
3 pages, 2013.

[23] Z. Peng, G. Murugusundaramoorthy, and T. Janani. Coefficient estimate of bi-
univalent functions of complex order associated with the hohlov operator. J. Complex
Anal., pages Article ID 693908, 6 pages, 2014.

[24] H.M. Srivastava, Ş. Altınkaya, and S. Yalçın. Certain subclasses of bi-univalent func-
tions associated with the horadam polynomials. Iran. J. Sci. Technol. Trans. Sci.,
43:1873–1879, 2019.

[25] F. Yousef, T. Al-Hawary, and G. Murugusundaramoorthy. Fekete-szegö functional
problems for some subclasses of bi-univalent functions defined by frasin differential
operator. Afr. Mat., 30:495–503, 2019.

[26] B.A. Frasin, S.R. Swamy, and J. Nirmala. Some special families of holomorphic
and al-oboudi type bi-univalent functions related to k-fibonacci numbers involving
modified sigmoid activation function. Afr. Mat., 2020.

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

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

[29] Ş. Altınkaya and S. Yalçın. On the (p, q)-lucas polynomial coefficient bounds of the
bi-univalent function class. Bol. Soc. Mat. Mex., pages 1–9, 2018.

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

[31] T. Horzum and E.G. Kocer. On some properties of horadam polynomials. Int. Math.
Forum, 4(25), 2009.

[32] A.F. Horadam and J.M. Mahon. Pell and pell-lucas polynomials. Fibonacci Q.,
23:7–20, 1985.

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



K. Alshammari, O. Alnajar, A. Amourah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5955 12 of 12

[34] 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
related to horadam polynomials. Heliyon, page Article ID 693908, 2024.

[35] O. Alnajar and M. Darus. Coefficient estimates for subclasses of bi-univalent functions
related to gegenbauer polynomials and an application of bell distribution. In AIP
Conf. Proc., volume 3150. AIP Publishing, 2024.

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

[37] H.M. Srivastava, A.K. Wanas, and G. Murugusundaramoorthy. A certain family of
bi-univalent functions associated with the pascal distribution series based upon the
horadam polynomials. Surv. Math. Appl., 16:193–205, 2021.

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

[39] O. Alnajar, A. Amourah, and M. Darus. The characteristics of inclusion pertaining
to univalent functions associated with bell distribution functions. Int. J. Open Probl.
Complex Anal., 15:46–61, 2023.

[40] B. A. Frasin, T. Al-Hawary, and F. Yousef. Some properties of a linear operator involv-
ing generalized mittag-leffler function. Stud. Univ. Babes-Bolyai Math, 65(1):67–75,
2020.

[41] A. Hussen and A. Zeyani. Coefficients and fekete–szegö functional estimations of
bi-univalent subclasses based on gegenbauer polynomials. Mathematics, 11(13):2852,
2023.

[42] A. Hussen and M. Illafe. Coefficient bounds for a certain subclass of bi-univalent
functions associated with lucas-balancing polynomials. Mathematics, 11(24):4941,
2023.

[43] A. Hussen. An application of the mittag-leffler-type borel distribution and gegenbauer
polynomials on a certain subclass of bi-univalent functions. Heliyon, 10(10):e31469,
2024.

[44] R. Alhabib, M.M. Ranna, H. Farah, and A.A. Salama. Some neutrosophic probability
distributions. Neutrosophic Sets Syst., 22:30–38, 2018.

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