EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 7036 ISSN 1307-5543 – ejpam.com Published by New York Business Global A New Subclass of Bi-Univalent Functions Involving Bell and Meixner-Pollaczek Polynomials Omar Alnajar1, Ala Amourah2,3,∗, Abdullah Alsoboh4, Omar S. Khabour5, Mohammed Mattar Al Hatmi4,∗, Tala Sasa6 1 Department of Mathematics, Faculty of Science and Technology, Irbid National University, P.O. Box: 2600 Irbid 21110, Jordan 2 Mathematics Education Program, Faculty of Education and Arts, Sohar University, Sohar 311, Oman 3 Jadara University Research Center, Jadara University, Jordan 4 College of Applied and Health Sciences, A’Sharqiyah University, P.O. Box 42, Post Code 400, Ibra, Sultanate of Oman 5 Department of Curricula and Methods of Teaching Mathematics Education Program, Faculty of Education Sciences, The University of Jordan, Amman 11942, Jordan 6 Department of Mathematics, Faculty of Science, Applied Science Private University, Amman, Jordan Abstract. In this work, we present a novel subclass of bi-univalent functions defined by Meixner- Pollaczek and Bell polynomials. Deriving coefficient estimates is the primary focus, especially for the second and third Taylor-Maclaurin coefficients, a2 and a3. Fekete-Szegö functional inequalities related to these subclasses are also examined. By extending and generalizing current subclasses, the proposed class offers fresh perspectives on the geometric and analytic characteristics of bi- univalent functions. Our findings demonstrate the theoretical originality and possible uses of orthogonal-polynomial-based function classes. 2020 Mathematics Subject Classifications: 30C45 Key Words and Phrases: Bell polynomials, Fekete-Szegö problem functions, bi-univalent func- tions, analytic functions, Meixner-Pollaczek polynomials 1. Preliminaries When considering a particular weight function across a specified interval, orthogonal polynomials are a particular kind of polynomial that meets a specific orthogonality cri- terion. Numerous branches of mathematics, including approximation theory, numerical ∗Corresponding author. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7036 Email addresses: o.alnjar@inu.edu.jo (O. Alnajar), AAmourah@su.edu.om (A. Amourah), abdullah.alsoboh@asu.edu.om (A. Alsoboh), o.khabour@ju.edu.jo (O. Khabour), mohammed.alhatmi@asu.edu.om (M. M. Al Hatmi), t_sasa@asu.edu.jo (T. Sasa) 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 (4) (2025), 7036 2 of 19 analysis, and mathematical physics, have devoted a significant amount of time and energy to the study of these polynomials. The fact that they constitute a basis for the space of square-integrable functions with respect to the weight function is one of the most impor- tant characteristics of these functions. Consequently, this makes it possible to express and approximate functions in an effective manner by utilizing polynomial expansions. Several well-known families of orthogonal polynomials are available, such as Legendre polynomials, Chebyshev polynomials, Meixner-Pollaczek polynomials, and Jacobi polynomials. Each of these families has its unique weight function and orthogonality features, specially designed to cater to particular applications (see [1, 2] for more information). Building upon these classical families, several subclasses of bi-univalent functions have been constructed using orthogonal polynomials such as Chebyshev, Gegenbauer, and Horadam polynomials. In the present work, we extend these developments by introducing a new subclass defined through Bell and Meixner–Pollaczek polynomials. This construction not only general- izes the previous subclasses, but also establishes new analytical connections among these polynomial families within the framework of geometric function theory. As a result of their orthogonality property in connection to a certain weight function on the real line, the mathematicians Wolfgang Meixner and Erwin Pollaczek got a lot of attention. The study of stochastic processes, such as random walks and queuing systems, frequently benefits from the application of Meixner-Pollaczek polynomials. In the con- text of differential equations or difference equations with a discrete spectrum, they are frequently utilized as solutions. As a result of their links to special functions, such as hypergeometric functions and q-series, these polynomials have been the subject of a sig- nificant amount of research. They also possess intriguing combinatorial properties. When it comes to examination of the spectral properties of differential operators and analysis of probabilistic models, Meixner-Pollaczek polynomials are crucial instruments due to their versatility and analytical properties, as stated in studies ( [3], [4]). A represents the class of all analytic functions f defined on the disk U = {z ∈ C : |z| < 1}. These functions are normalized by the constraints f(0) = 0 and f ′(0) = 1. As a result, every f that is a member of the mathematical category A possesses a Taylor-Maclaurin series expansion of the form f(z) = z + ∞∑ n=2 anz n, (z ∈ U). (1) In addition, denote by S the collection of all functions f that belong to A and are univalent in U. In the discipline of geometric function theory, the robust tools that differential subordi- nation of analytic functions provides have the potential to make substantial contributions to the field’s overall advancement. Miller and Mocanu [5] were the ones who initially presented the differential subordination problem, and more references can be found in [6]. The book written by Miller and Mocanu cite5aa provides a detailed documentation of the developments that have taken place in this particular field, including the publishing dates. Each and every function f that belongs to S is known to have an inverse f−1 that is O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7036 3 of 19 defined by the following equation: f−1(f(z)) = z (z ∈ U) and f−1(f(w)) = w (|w| < r0(f); r0(f) ≥ 1 4 ) where f−1(w) = w − a2w 2 + (2a22 − a3)w 3 − (5a32 − 5a2a3 + a4)w 4 + · · · . (2) Assuming that both f(z) and f−1(z) are univalent in U, a function is considered bi- univalent in U. Given that equation (1) defines the class of bi-univalent functions in U, let Σ be assigned the role of representing this class. The class Σ contains a number of different implementations of functions z 1− z , log 1 1− z , log √ 1 + z 1− z . It is important to note that the well-known Koebe function is not included in the set Σ. In addition, there are other examples of functions in U that are well-known, specifically the following: 2z − z2 2 and z 1− z2 are also not members of Σ. An exact upper limit for functional ηa22 − a3, where η is a real number (0 ≤ η ≤ 1), applied to a univalent function f , was established in 1933 by Fekete and Szegö [7]. Find the best bounds for this functional across all compact families of functions f belonging to A, regardless of the complex value of η. 2. Both Bell polynomials and Meixner-Pollaczek polynomials are represented here In 2018 Castellares et al. presented Bell polynomials [8], which is an appropriate polynomial for count data that exhibit over-dispersion. The Bell polynomials are an advance in comparison to the Bell numbers, as stated in the references [9, 10]. The expression for the probability density function of a discrete random variable X, which is based on the Bell distribution, is as follows: λ(X = n) = ℸnee (−ℸ2)+1 Υn n! ; n = 1, 2, 3, · · · . (3) where Υn = 1 e ∞∑ k=0 kn k! are the Bell numbers, n ≥ 1, and ℸ > 0. O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7036 4 of 19 Example of Bell numbers are Υ2 = 2,Υ3 = 5,Υ4 = 15 and Υ5 = 52. Next, we will show a novel power series with coefficients that accurately reflect the probability connected to the Bell polynomials, Υ(ℸ, z) = z + ∞∑ n=2 ℸn−1Υn (n− 1)! eℸ2−1 zn, (z ∈ U) , (4) where ℸ > 0. Following that, we look at the linear operator Φℸ : A → A: The Hadamard product, often known as convolution, is defined. Φℸf(z) = Υ(ℸ, z) ∗ f(z) = z + ∞∑ n=2 ℸn−1e1−ℸ2 Υn (n− 1)! anz n, (z ∈ U) , = z + 2ℸ eℸ2−1 a2z 2 + 5ℸ2 2eℸ2−1 a3z 3 + 15ℸ3 3!eℸ2−1 a4z 4 + · · · . (5) The Meixner–Pollaczek polynomials λ(⅁)n (x; ℓ) (see [11]) of a real variable x as coeffi- cients of ϱ⅁(q̃, ℓ; z) = 1 (1− zeiℓ) ⅁−iq̃ (1− zeiℓ) ⅁+iq̃ = ∞∑ n=0 λ(⅁)n (q̃; ℓ)zn, (6) where λ(⅁)n (q̃; ℓ) = (2⅁)n n! einℓ ( e2iℓ e2iℓ − 1 )n 2F1 ( −n,⅁+ iq̃ 2⅁ 1− 1 e2iℓ ) , (7) are orthogonal with respect to the continuous weight; ω(x; ℓ) = ∣∣Γ(⅁+ iq̃) ∣∣2e(2ℓ−π)q̃, (8) For n ∈ N, ⅁ > 0, and 0 < ℓ < π in the interval (−∞,∞), observe that the complex Gamma function in Equation (7) has the form [12], ∣∣Γ(⅁+ iq̃) ∣∣2 = Γ(⅁+ iq̃)Γ(⅁− iq̃). Special cases: 1) lim ℓ→π 2 λ (α+1 2 ) n (−q̃2ℓ ; ℓ) is called Laguerre polynomial Lαn(x). 2) lim ⅁→∞ n!⅁ −n 2 λ (⅁) n (−q̃ √ ⅁−⅁ cos ℓ sin ℓ ; ℓ) is called Hermite polynomial Hn(x). O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7036 5 of 19 By means of a three-term recurrence relation, the Meixner-Pollaczek polynomials can be represented. λ(⅁)n (q̃; ℓ) = ( q̃ + α(⅁,ℓ) n ) λ (⅁) n−1(q̃; ℓ)− C(⅁,ℓ) n λ (⅁) n−2(q̃; ℓ), (9) where α(⅁,ℓ) n := ⅁+ n− 1 tan ℓ ; and C(⅁,ℓ) n := (n− 1)(2⅁+ n− 2) 4 sin2 ℓ , (10) with λ (⅁) −1 (q̃) = 0, λ(⅁)0 (q̃) = 1 and α (⅁,π 2 ) n = lim ℓ→π 2 α (⅁,ℓ) n = 0. The initial polynomials can be constructed λ(⅁)n (x; δ) using Equation (13) as described below (for further reference, see [13]). λ (⅁) 0 (q̃; δ) = 1 λ (⅁) 1 (q̃; δ) = q̃ + δ⅁ λ (⅁) 2 (q̃; δ) = q̃2 + ( δ⅁+ ⅁+ 1 ) q̃ − 2δ2⅁+ δ⅁2 + δ⅁− 2⅁ (11) Subclasses of bi-univalent functions that are connected with orthogonal polynomials have recently drawn the attention of a group of scholars who have begun their investi- gation. Estimates for the coefficients that initially are associated with these functions have been determined. Nevertheless, the difficulty in identifying accurate boundaries for coefficients |an|, (n = 3, 4, 5, · · · ) has not yet been resolved, as has been mentioned in a number of sources [14–46]. Many researchers have used a variety of probability distributions, including the Pascal, Poisson, and Borel distributions, to examine certain subclasses of analytic functions (for an example, see [30, 47, 48]) and other applications can be found in [49–55]). The primary purpose of this research is to analyze the characteristics of bi-univalent functions in a new class from a mathematical perspective. The following definitions serve as the starting point for the investigation. O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7036 6 of 19 Figure 1: Hierarchical relationship among Bell functions, Meixner–Pollaczek class, and the bi-univalent class. 3. Definition and Examples In this section, a novel subclass of bi-univalent functions within the unit disk will be defined and investigated. This will be accomplished by using the subordination principle. The Bell polynomials and subordination through Meixner-Pollaczek polynomials will be used to create this new class. Definition 1. The function f ∈ Σ, indicated by (1), belongs to the class GΣ ( ℸ, ξ,m, ψ, ϱ⅁(q̃, ℓ; z) ) , if the conditions in subsequent subordinations are fulfilled. That is (1 +meiψ) { (1− ξ) Φℸf(z) z + ξ(Φℸf(z)) ′ } −meiψ = ϱ⅁(q̃, ℓ;w) (12) and (1 +meiψ) { (1− ξ) Φℸg(w) w + ξ(Φℸg(w)) ′ } −meiψ = ϱ⅁(q̃, ℓ; v), (13) when x falls within the interval [−1, 1], thefunctiong(w), definedby(2), isprovided., m ≥ 0,−π < ψ ≤ π, and 0 < ℓ < π. The Meixner-Pollaczek polynomials ϱ⅁(q̃, ℓ; z) are provided by (6). Example 1. Consider ξ to be a positive integer. The function f ∈ Σ, which is represented by the equation (1), is considered to be a member of the class GΣ ( ℸ, 0,m, ψ, ϱ⅁(q̃, ℓ; z) ) if the requirements listed below are met: (1 +meiψ) { Φℸf(z) z } −meiψ ≺ ϱ⅁(q̃, ℓ; z) (14) O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7036 7 of 19 and (1 +meiψ) { Φℸg(w) w } −meiψ ≺ ϱ⅁(q̃, ℓ;w), (15) when x falls within the interval [−1, 1], thefunctiong(w), definedby(2), isprovided., m ≥ 0,−π < ψ ≤ π, and 0 < ℓ < π. The Meixner-Pollaczek polynomials ϱ⅁(q̃, ℓ; z) are provided by (6). Example 2. Consider ξ to be a positive integer. The function f ∈ Σ, which is represented by the equation (1), is considered to be a member of the class GΣ ( ℸ, 1,m, ψ, ϱ⅁(q̃, ℓ; z) ) if the requirements listed below are met: (1 +meiψ) { (Φℸf(z)) ′}−meiψ ≺ ϱ⅁(q̃, ℓ; z) (16) and (1 +meiψ) { (Φℸg(w)) ′}−meiψ ≺ ϱ⅁(q̃, ℓ;w), (17) when x falls within the interval [−1, 1], thefunctiong(w), definedby(2), isprovided., m ≥ 0,−π < ψ ≤ π, and 0 < ℓ < π. The Meixner-Pollaczek polynomials ϱ⅁(q̃, ℓ; z) are provided by (6). Example 3. Consider m to be a positive integer. The function f ∈ Σ, which is represented by the equation (1), is considered to be a member of the class GΣ ( ℸ, ξ, 0, ψ, ϱ⅁(q̃, ℓ; z) ) if the requirements listed below are met and go back to [56]: (1− ξ) Φℸf(z) z + ξ(Φℸf(z)) ′ ≺ ϱ⅁(q̃, ℓ; z) (18) and (1− ξ) Φℸg(w) w + ξ(Φℸg(w)) ′ ≺ ϱ⅁(q̃, ℓ; v), (19) when x falls within the interval [−1, 1], thefunctiong(w), definedby(2), isprovided., m ≥ 0,−π < ψ ≤ π, and 0 < ℓ < π. The Meixner-Pollaczek polynomials ϱ⅁(q̃, ℓ; z) are provided by (6). Example 4. Consider m, ξ to be a positive integer. The function f ∈ Σ, which is repre- sented by the equation (1), is considered to be a member of the class GΣ ( ℸ, 0, 0, ψ, ϱ⅁(q̃, ℓ; z) ) if the requirements listed below are met and go back to [56]: Φℸf(z) z ≺ ϱ⅁(q̃, ℓ; z) (20) and Φℸg(w) w ≺ ϱ⅁(q̃, ℓ;w), (21) O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7036 8 of 19 when x falls within the interval [−1, 1], thefunctiong(w), definedby(2), isprovided., m ≥ 0,−π < ψ ≤ π, and 0 < ℓ < π. The Meixner-Pollaczek polynomials ϱ⅁(q̃, ℓ; z) are provided by (6). Example 5. Consider m, ξ to be a positive integer. The function f ∈ Σ, which is repre- sented by the equation (1), is considered to be a member of the class GΣ ( ℸ, 1, 0, ψ, ϱ⅁(q̃, ℓ; z) ) if the requirements listed below are met and go back to [56]: (Φℸf(z)) ′ ≺ ϱ⅁(q̃, ℓ; z) (22) and (Φℸg(w)) ′ ≺ ϱ⅁(q̃, ℓ;w), (23) when x falls within the interval [−1, 1], thefunctiong(w), definedby(2), isprovided., m ≥ 0,−π < ψ ≤ π, and 0 < ℓ < π. The Meixner-Pollaczek polynomials ϱ⅁(q̃, ℓ; z) are provided by (6). 4. Bounds of the class GΣ (ℸ, ξ,m, ψ) for equations To begin, let us present the estimates of the coefficients for the class GΣ ( ℸ, ξ,m, ψ, ϱ⅁(q̃, ℓ; z) ) using the definition given in Definition 12. Theorem 1. Function f ∈ Σ, indicated by (1), belongs to the class GΣ ( ℸ, ξ,m, ψ, ϱ⅁(q̃, ℓ; z) ) , if the conditions in the subsequent subordinations are fulfilled. That is |a2| ≤ eℸ 2−1 ℸ ∣∣q̃ + δ⅁ ∣∣√2 (q̃ + δ⅁)√√√√√√√ ∣∣∣∣∣∣∣∣ (5(1 +meiψ)(1 + 2ξ)eℸ 2−1 − 8(1 +meiψ)2(1 + ξ)2)q̃2 + (10(1 +meiψ)2δ⅁(1 + 2ξ)eℸ 2−1 − 8(1 +meiψ)2(δ⅁+ ⅁+ 1)(1 + ξ)2)q̃ + 8(1 +meiψ)2(1 + ξ)2 ( 2δ2⅁− δ⅁2 − δ⅁+ 2⅁ ) ∣∣∣∣∣∣∣∣ and |a3| ≤ ( eℸ 2−1 )2 (q̃ + δ⅁)2 4(1 +meiψ)2(1 + ξ)2ℸ2 + 2eℸ 2−1|q̃ + δ⅁| 5(1 +meiψ)(1 + 2ξ)ℸ2 . Proof. Consider f ∈ GΣ ( ℸ, ξ,m, ψ, ϱ⅁(q̃, ℓ; z) ) . Definition 12 states that there are analytic functions w and v where w(0) = v(0) = 0 and |w(z)| < 1. If |v(w)| < 1 for any z, w ∈ U, it can be expressed as follows: (1 +meiψ) { (1− ξ) Φℸf(z) z + ξ(Φℸf(z)) ′ } −meiψ = ϱ⅁(q̃, ℓ;w(z)) (24) O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7036 9 of 19 and (1 +meiψ) { (1− ξ) Φℸg(w) w + ξ(Φℸg(w)) ′ } −meiψ = ϱ⅁(q̃, ℓ; v(w)), (25) Using equalities (24 and (25), we can conclude that (1 +meiψ) { (1− ξ) Φℸf(z) z + ξ(Φℸf(z)) ′ } −meiψ = 1 + λ (⅁) 1 (q̃; δ)c1z + [ λ (⅁) 1 (q̃; δ)c2 + λ (⅁) 2 (q̃; δ)c21 ] z2 + · · · (26) and (1 +meiψ) { (1− ξ) Φℸg(w) w + ξ(Φℸg(w)) ′ } −meiψ = 1 + λ (⅁) 1 (q̃; δ)d1w + [ λ (⅁) 1 (q̃; δ)d2 + λ (⅁) 2 (q̃; δ)d21 ] )w2 + · · · . (27) It’s commonly understood that if |w(z)| = ∣∣c1z + c2z 2 + c3z 3 + · · · ∣∣ < 1, (z ∈ U) and |v(w)| = ∣∣d1w + d2w 2 + d3w 3 + · · · ∣∣ < 1, (w ∈ U), then |cj | ≤ 1 and |dj | ≤ 1 for all j ∈ N. (28) Comparing the coefficients in (26) with (27) yields 2(1 +meiψ)(1 + ξ)ℸ eℸ2−1 a2 = λ (⅁) 1 (q̃; δ)c1, (29) 5(1 +meiψ)(1 + 2ξ)ℸ2 2eℸ2−1 a3 = λ (⅁) 1 (q̃; δ)c2 + λ (⅁) 2 (q̃; δ)c21, (30) and −2(1 +meiψ)(1 + ξ)ℸ eℸ2−1 a2 = λ (⅁) 1 (q̃; δ)d1, (31) 5(1 +meiψ)(1 + 2ξ)ℸ2 2eℸ2−1 ( 2a22 − a3 ) = λ (⅁) 1 (q̃; δ)d2 + λ (⅁) 2 (q̃; δ)d21, (32) According to (29) and (31), c1 = −d1 (33) and 2 ( 2(1 +meiψ)(1 + ξ)ℸ eℸ2−1 )2 a22 = [ λ (⅁) 1 (q̃; δ) ]2 ( c21 + d21 ) c21 + d21 = 8(1 +meiψ)2(1 + ξ)2ℸ2( eℸ2−1 )2 [ λ (⅁) 1 (q̃; δ) ]2 a22 (34) O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7036 10 of 19 If we add (30) and (32), we get 5(1 +meiψ)(1 + 2ξ)ℸ2 eℸ2−1 a22 = λ (⅁) 1 (q̃; δ) (c2 + d2) + λ (⅁) 2 (q̃; δ) ( c21 + d21 ) . (35) Substituting the expression for c21 + d21 from (34) into the right-hand side of (35) and rearranging the resulting identity, we obtain an equality relating a22 to c2 + d2. Moreover, by (28) — which follows from Carathéodory’s lemma for analytic Schwarz functions — we have |cj | ≤ 1 and |dj | ≤ 1 for all j. Hence |c21 + d21| ≤ 2 and |c2 + d2| ≤ 2; applying these bounds and carrying out elementary algebraic simplifications yields Equation (36).5(1 + 2ξ)(1 +meiψ)− 8(1 +meiψ)2(1 + ξ)2λ (⅁) 2 (q̃; δ)( eℸ2−1 ) [ λ (⅁) 1 (q̃; δ) ]2  ℸ2 eℸ2−1 a22 = λ (⅁) 1 (q̃; δ) (c2 + d2) a22 = ( eℸ 2−1 )2 [ λ (⅁) 1 (q̃; δ) ]3 ℸ2 ( 5(1 +meiψ)(1 + 2ξ) ( eℸ2−1 ) [ λ (⅁) 1 (q̃; δ) ]2 − 8(1 +meiψ)2(1 + ξ)2λ (⅁) 2 (q̃; δ) ) (c2 + d2) (36) Furthermore, computations utilising (11), (28), and (36) reveal that |a2| ≤ eℸ 2−1 ℸ ∣∣q̃ + δ⅁ ∣∣√2 (q̃ + δ⅁)√√√√√√√ ∣∣∣∣∣∣∣∣ (5(1 +meiψ)(1 + 2ξ)eℸ 2−1 − 8(1 +meiψ)2(1 + ξ)2)q̃2 + (10(1 +meiψ)2δ⅁(1 + 2ξ)eℸ 2−1 − 8(1 +meiψ)2(δ⅁+ ⅁+ 1)(1 + ξ)2)q̃ + 8(1 +meiψ)2(1 + ξ)2 ( 2δ2⅁− δ⅁2 − δ⅁+ 2⅁ ) ∣∣∣∣∣∣∣∣ Furthermore, when we subtract (32) from (30), we get 5(1 +meiψ)(1 + 2ξ)ℸ2 eℸ2−1 ( a3 − a22 ) = λ (⅁) 1 (q̃; δ) (c2 − d2) + λ (⅁) 2 (q̃; δ) ( c21 − d21 ) . (37) Then, in view of (28) and (34), Eq. (37) becomes a3 = ( eℸ 2−1 )2 [ λ (⅁) 1 (q̃; δ) ]2 8(1 +meiψ)2(1 + ξ)2ℸ2 ( c21 + d21 ) + eℸ 2−1λ (⅁) 1 (q̃; δ) 5(1 +meiψ)(1 + 2ξ)ℸ2 (c2 − d2) Thus, using (11) and (28), we deduce that |a3| ≤ ( eℸ 2−1 )2 (q̃ + δ⅁)2 4(1 +meiψ)2(1 + ξ)2ℸ2 + 2eℸ 2−1|q̃ + δ⅁| 5(1 +meiψ)(1 + 2ξ)ℸ2 . This completes the proof of Theorem. O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7036 11 of 19 Theorem 2. Function f ∈ Σ, indicated by (1), belongs to the class GΣ ( ℸ, ξ,m, ψ, ϱ⅁(q̃, ℓ; z) ) , if the conditions in the subsequent subordinations are fulfilled. That is ∣∣∣a3 − Φa22 ∣∣∣ ≤  2eℸ 2−1|q̃+δ⅁| 5(1+meiψ)(1+2ξ)ℸ2 , |1− Φ| ≤ J (ξ, q̃,⅁, δ,ℸ), 2 |q̃ + δ⅁| |K(Φ)| , |1− Φ| ≥ J (ξ, q̃,⅁, δ,ℸ), where J (ξ, q̃,⅁, δ,ℸ) = ∣∣∣∣∣1− 8(1 +meiψ)2(1 + ξ)2 ( q̃2 + ( δ⅁+ ⅁+ 1 ) q̃ − 2δ2⅁+ δ⅁2 + δ⅁− 2⅁ ) 5(1 +meiψ)(1 + 2ξ)eℸ2−1 (q̃ + δ⅁)2 ∣∣∣∣∣ , and K(Φ) = eℸ 2−1 [ λ (⅁) 1 (q̃; δ) ]2 (1− Φ) ℸ2 ( 5(1 +meiψ)(1 + 2ξ) ( eℸ2−1 ) [ λ (⅁) 1 (q̃; δ) ]2 − 8(1 +meiψ)2(1 + ξ)2λ (⅁) 2 (q̃; δ) ) . Proof. From (36) and (37) a3 − Φa22 = eℸ 2−1λ (⅁) 1 (q̃; δ) 5(1 +meiψ)(1 + 2ξ)ℸ2 (c2 − d2) + (1− Φ) ( eℸ 2−1 )2 [ λ (⅁) 1 (q̃; δ) ]3 (c2 + d2) ℸ2 ( 5(1 +meiψ)(1 + 2ξ) ( eℸ2−1 ) [ λ (⅁) 1 (q̃; δ) ]2 − 8(1 +meiψ)2(1 + ξ)2λ (⅁) 2 (q̃; δ) ) = λ (⅁) 1 (q̃; δ) ([ K(Φ) + eℸ 2−1 5(1 +meiψ)(1 + 2ξ)ℸ2 ] c2 + [ K(Φ)− eℸ 2−1 5(1 +meiψ)(1 + 2ξ)ℸ2 ] d2 ) , where K(Φ) = ( eℸ 2−1 )2 [ λ (⅁) 1 (q̃; δ) ]2 (1− Φ) ℸ2 ( 5(1 +meiψ)(1 + 2ξ) ( eℸ2−1 ) [ λ (⅁) 1 (q̃; δ) ]2 − 8(1 +meiψ)2(1 + ξ)2λ (⅁) 2 (q̃; δ) ) , Then, in view of (11), we conclude that ∣∣∣a3 − Φa22 ∣∣∣ ≤  2eℸ 2−1 ∣∣∣λ(⅁)1 (q̃;δ) ∣∣∣ 5(1+meiψ)(1+2ξ)ℸ2 , |K(Φ)| ≤ eℸ 2−1 5(1+meiψ)(1+2ξ)ℸ2 , 2 ∣∣∣λ(⅁)1 (q̃; δ) ∣∣∣ |K(Φ)| , |K(Φ)| ≥ eℸ 2−1 5(1+meiψ)(1+2ξ)ℸ2 . Which completes the proof of Theorem 2. O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7036 12 of 19 5. Corollaries and Consequences The results that are obtained from the application of Theorems 1 and 2 are in close agreement with the examples 1, 2, 3, 4, and 5. Corollary 1. Consider ξ to be a positive integer. The function f ∈ Σ, which is represented by the equation (1), is considered to be a member of the class GΣ ( ℸ, 0,m, ψ, ϱ⅁(q̃, ℓ; z) ) if the requirements listed below: |a2| ≤ eℸ 2−1 ℸ ∣∣q̃ + δ⅁ ∣∣√2 ( q̃ + δ⅁ )√√√√√√√ ∣∣∣∣∣∣∣∣ (5(1 +meiψ)eℸ 2−1 − 8(1 +meiψ)2)q̃2 + ( 10(1 +meiψ)δ⅁eℸ 2−1 − 8(1 +meiψ)2(δ⅁+ ⅁+ 1) ) q̃ + 8(1 +meiψ)2 ( 2δ2⅁− δ⅁2 − δ⅁+ 2⅁ ) ∣∣∣∣∣∣∣∣ |a3| ≤ ( eℸ 2−1 )2 (q̃ + δ⅁)2 4(1 +meiψ)2ℸ2 + 2eℸ 2−1|q̃ + δ⅁| 5(1 +meiψ)ℸ2 . and ∣∣∣a3 − Φa22 ∣∣∣ ≤  2eℸ 2−1|q̃+δ⅁| 5(1+meiψ)ℸ2 , |1− Φ| ≤ J (0, q̃,⅁, δ,ℸ), 2 |q̃ + δ⅁| |K(Φ)| , |1− Φ| ≥ J (0, q̃,⅁, δ,ℸ), where J (0, q̃,⅁, δ,ℸ) = ∣∣∣∣∣1− 8(1 +meiψ)2 ( q̃2 + ( δ⅁+ ⅁+ 1 ) q̃ − 2δ2⅁+ δ⅁2 + δ⅁− 2⅁ ) 5(1 +meiψ)eℸ2−1 (q̃ + δ⅁)2 ∣∣∣∣∣ , and K(Φ) = eℸ 2−1 [ λ (⅁) 1 (q̃; δ) ]2 (1− Φ) ℸ2 ( 5(1 +meiψ) ( eℸ2−1 ) [ λ (⅁) 1 (q̃; δ) ]2 − 8(1 +meiψ)2λ (⅁) 2 (q̃; δ) ) . Corollary 2. Consider ξ to be a positive integer. The function f ∈ Σ, which is represented by the equation (1), is considered to be a member of the class GΣ ( ℸ, 1,m, ψ, ϱ⅁(q̃, ℓ; z) ) if the requirements listed below: |a2| ≤ eℸ 2−1 ℸ ∣∣q̃ + δ⅁ ∣∣√2 ( q̃ + δ⅁ )√√√√√√√ ∣∣∣∣∣∣∣∣ (15(1 +meiψ)eℸ 2−1 − 32(1 +meiψ)2)q̃2 + ( 30(1 +meiψ)δ⅁eℸ 2−1 − 32(1 +meiψ)2(δ⅁+ ⅁+ 1) ) q̃ + 32(1 +meiψ)2 ( 2δ2⅁− δ⅁2 − δ⅁+ 2⅁ ) ∣∣∣∣∣∣∣∣ O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7036 13 of 19 |a3| ≤ ( eℸ 2−1 )2 (q̃ + δ⅁)2 16(1 +meiψ)2ℸ2 + 2eℸ 2−1|q̃ + δ⅁| 15(1 +meiψ)ℸ2 . and ∣∣∣a3 − Φa22 ∣∣∣ ≤  2eℸ 2−1|q̃+δ⅁| 15(1+meiψ)ℸ2 , |1− Φ| ≤ J (1, q̃,⅁, δ,ℸ), 2 |q̃ + δ⅁| |K(Φ)| , |1− Φ| ≥ J (1, q̃,⅁, δ,ℸ), where J (1, q̃,⅁, δ,ℸ) = ∣∣∣∣∣1− 32(1 +meiψ)2 ( q̃2 + ( δ⅁+ ⅁+ 1 ) q̃ − 2δ2⅁+ δ⅁2 + δ⅁− 2⅁ ) 15(1 +meiψ)eℸ2−1 (q̃ + δ⅁)2 ∣∣∣∣∣ , and K(Φ) = eℸ 2−1 [ λ (⅁) 1 (q̃; δ) ]2 (1− Φ) ℸ2 ( 15(1 +meiψ) ( eℸ2−1 ) [ λ (⅁) 1 (q̃; δ) ]2 − 32(1 +meiψ)2λ (⅁) 2 (q̃; δ) ) . Corollary 3. Consider m to be a positive integer. The function f ∈ Σ, which is repre- sented by the equation (1), is considered to be a member of the class GΣ ( ℸ, ξ, 0, ψ, ϱ⅁(q̃, ℓ; z) ) if the requirements listed below: |a2| ≤ eℸ 2−1 ℸ ∣∣q̃ + δ⅁ ∣∣√2 (q̃ + δ⅁)√√√√√√√ ∣∣∣∣∣∣∣∣ (5(1 + 2ξ)eℸ 2−1 − 8(1 + ξ)2)q̃2 + (10δ⅁(1 + 2ξ)eℸ 2−1 − 8(δ⅁+ ⅁+ 1)(1 + ξ)2)q̃ + 8(1 + ξ)2 ( 2δ2⅁− δ⅁2 − δ⅁+ 2⅁ ) ∣∣∣∣∣∣∣∣ and |a3| ≤ ( eℸ 2−1 )2 (q̃ + δ⅁)2 4(1 + ξ)2ℸ2 + 2eℸ 2−1|q̃ + δ⅁| 5(1 + 2ξ)ℸ2 . and ∣∣∣a3 − Φa22 ∣∣∣ ≤  2eℸ 2−1|q̃+δ⅁| 5(1+2ξ)ℸ2 , |1− Φ| ≤ J (ξ, q̃,⅁, δ,ℸ), 2 |q̃ + δ⅁| |K(Φ)| , |1− Φ| ≥ J (ξ, q̃,⅁, δ,ℸ), where J (ξ, q̃,⅁, δ,ℸ) = ∣∣∣∣∣1− 8(1 + ξ)2 ( q̃2 + ( δ⅁+ ⅁+ 1 ) q̃ − 2δ2⅁+ δ⅁2 + δ⅁− 2⅁ ) 5(1 + 2ξ)eℸ2−1 (q̃ + δ⅁)2 ∣∣∣∣∣ , O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7036 14 of 19 and K(Φ) = eℸ 2−1 [ λ (⅁) 1 (q̃; δ) ]2 (1− Φ) ℸ2 ( 5(1 + 2ξ) ( eℸ2−1 ) [ λ (⅁) 1 (q̃; δ) ]2 − 8(1 + ξ)2λ (⅁) 2 (q̃; δ) ) . Corollary 4. Consider m, ξ to be a positive integer. The function f ∈ Σ, which is repre- sented by the equation (1), is considered to be a member of the class GΣ ( ℸ, 0, 0, ψ, ϱ⅁(q̃, ℓ; z) ) if the requirements listed below: |a2| ≤ eℸ 2−1 ℸ ∣∣q̃ + δ⅁ ∣∣√2 (q̃ + δ⅁)√√√√√ ∣∣∣(5eℸ2−1 − 8 ) q̃2 + ( 10δ⅁eℸ2−1 − 8(δ⅁+ ⅁+ 1) ) q̃ +8 ( 2δ2⅁− δ⅁2 − δ⅁+ 2⅁ )∣∣∣ . |a3| ≤ ( eℸ 2−1 )2 (q̃ + δ⅁)2 4ℸ2 + 2eℸ 2−1|q̃ + δ⅁| 5ℸ2 . and ∣∣∣a3 − Φa22 ∣∣∣ ≤  2eℸ 2−1|q̃+δ⅁| 5ℸ2 , |1− Φ| ≤ J (0, q̃,⅁, δ,ℸ), 2 |q̃ + δ⅁| |K(Φ)| , |1− Φ| ≥ J (0, q̃,⅁, δ,ℸ), where J (0, q̃,⅁, δ,ℸ) = ∣∣∣∣∣1− 8 ( q̃2 + ( δ⅁+ ⅁+ 1 ) q̃ − 2δ2⅁+ δ⅁2 + δ⅁− 2⅁ ) 5eℸ2−1 (q̃ + δ⅁)2 ∣∣∣∣∣ , and K(Φ) = eℸ 2−1 [ λ (⅁) 1 (q̃; δ) ]2 (1− Φ) ℸ2 ( 5 ( eℸ2−1 ) [ λ (⅁) 1 (q̃; δ) ]2 − 8λ (⅁) 2 (q̃; δ) ) . Corollary 5. Consider m, ξ to be a positive integer. The function f ∈ Σ, which is repre- sented by the equation (1), is considered to be a member of the class GΣ ( ℸ, 1, 0, ψ, ϱ⅁(q̃, ℓ; z) ) if the requirements listed below: |a2| ≤ eℸ 2−1 ℸ ∣∣q̃ + δ⅁ ∣∣√2 (q̃ + δ⅁)√√√√√ ∣∣∣(15eℸ2−1 − 32 ) q̃2 + ( 30δ⅁eℸ2−1 − 32(δ⅁+ ⅁+ 1) ) q̃ +32 ( 2δ2⅁− δ⅁2 − δ⅁+ 2⅁ )∣∣∣ . |a3| ≤ ( eℸ 2−1 )2 (q̃ + δ⅁)2 16ℸ2 + 2eℸ 2−1|q̃ + δ⅁| 15ℸ2 . O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7036 15 of 19 and ∣∣∣a3 − Φa22 ∣∣∣ ≤  2eℸ 2−1|q̃+δ⅁| 15ℸ2 , |1− Φ| ≤ J (1, q̃,⅁, δ,ℸ), 2 |q̃ + δ⅁| |K(Φ)| , |1− Φ| ≥ J (1, q̃,⅁, δ,ℸ), where J (1, q̃,⅁, δ,ℸ) = ∣∣∣∣∣1− 32 ( q̃2 + ( δ⅁+ ⅁+ 1 ) q̃ − 2δ2⅁+ δ⅁2 + δ⅁− 2⅁ ) 15eℸ2−1 (q̃ + δ⅁)2 ∣∣∣∣∣ , and K(Φ) = eℸ 2−1 [ λ (⅁) 1 (q̃; δ) ]2 (1− Φ) ℸ2 ( 15 ( eℸ2−1 ) [ λ (⅁) 1 (q̃; δ) ]2 − 32λ (⅁) 2 (q̃; δ) ) . Concluding Remark: We have presented and analyzed the problems that arise with the coefficients of a new subclass of bi-univalent functions! This particular subclass is referred to as GΣ ( ℸ, ξ,m, ψ, ϱ⅁(q̃, ℓ; z) ) due to the presence of the Bell polynomials and the Meixner-Pollaczek polynomials. When it comes to functions that belong to this newly introduced subclass, we possess estimations for the Fekete-Szegö functional issues as well as the Taylor-Maclaurin coefficients, which are denoted as a2 and a3, respectively. The purpose of this study is to investigate the connection that exists between Meixner- Pollaczek polynomials that belong to specific families and the Bell polynomials. The estimations on the bounds of |an| for n ≥ 4;n ∈ N for the classes that have been detailed throughout this article are an example of how this finding might motivate further research in other domains. References [1] W. Gautschi. Orthogonal Polynomials: Computation and Approximation. Oxford University Press, Oxford, 2004. [2] B. Doman. The Classical Orthogonal Polynomials. World Scientific, Singapore, 2015. [3] J. Meixner. Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeu- genden Funktion. Journal of the London Mathematical Society, 9:6–13, 1934. [4] R. Koekoek, P. A. Lesky, and R. F. Swarttouw. Hypergeometric orthogonal poly- nomials. In Hypergeometric Orthogonal Polynomials, pages 183–253. Springer, Berlin/Heidelberg, 2010. [5] S. S. Miller and P. T. Mocanu. Second order differential inequalities in the complex plane. Journal of Mathematical Analysis and Applications, 65:289–305, 1978. [6] S. S. Miller and P. T. Mocanu. Differential subordinations and univalent functions. Michigan Mathematical Journal, 28:157–172, 1981. [7] M. Fekete and G. Szegő. Eine Bemerkung über ungerade schlichte Funktionen. Jour- nal of the London Mathematical Society, 8:85–89, 1933. O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7036 16 of 19 [8] F. Castellares, S. L. Ferrari, and A. J. Lemonte. On the Bell distribution and its associated regression model for count data. Applied Mathematical Modelling, 56:172– 185, 2018. [9] E. T. Bell. Exponential polynomials. Annals of Mathematics, 35:258–277, 1934. [10] E. T. Bell. Exponential numbers. The American Mathematical Monthly, 41(7):411– 419, 1934. [11] R. Koekoek, P. A. Lesky, and R. F. Swarttouw. Hypergeometric Orthogonal Polyno- mials and Their q-Analogues. Springer, Berlin/Heidelberg, 2010. [12] F. W. J. Olver, D. W. Boisvert, and C. W. Clark. NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge, 2010. [13] A. S. Kelil and A. R. Appadu. On certain properties and applications of the perturbed Meixner–Pollaczek weight. Mathematics, 9:1064, 2021. [14] 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 Mathematics, 32:183–193, 2024. [15] O. Alnajar, K. A. Alshammari, A. Amourah, and M. Darus. Hankel determinant of analytical functions closely tied to Bell polynomials. European Journal of Pure and Applied Mathematics, 18(3):6108, 2025. [16] Omar Alnajar, Osama Ogilat, Ala Amourah, Maslina Darus, and Maryam Salem Alatawi. The Miller-Ross P)oisson distribution and its applications to certain classes of bi-univalent functions related to Horadam polynomials, journal = Heliyon, vol- ume = 10, pages = e28302, year = 2024, doi = 10.1016/j.heliyon.2024.e28302, url = https://doi.org/10.1016/j.heliyon.2024.e28302. [17] B. A. Frasin and M. K. Aouf. New subclasses of bi-univalent functions. Applied Mathematics Letters, 24:1569–1573, 2011. [18] 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. Afrika Matematika, 31:1001–1013, 2020. [19] O. Alnajar, K. Alshammari, and A. Amourah. The neutrosophic P)oisson distribution applied to Horadam polynomial-subordinate bi-univalent functions, journal = Euro- pean Journal of Pure and Applied Mathematics, volume = 18, year = 2025, pages = 5955, doi = 10.29020/nybg.ejpam.v18i3.5955. [20] A. Amourah, A. Alsoboh, O. Ogilat, G. M. Gharib, R. Saadeh, and M. Al Soudi. A generalization of Gegenbauer polynomials and bi-univalent functions. Axioms, 12:128, 2023. [21] Mohamed Illafe, Maisarah Haji Mohd, Feras Yousef, and Shamani Supramaniam. In- vestigating inclusion, neighborhood, and partial sums properties for a general subclass of analytic functions. Int. J. Neutrosophic Sci, 25:501–510, 2025. [22] A. A. Amourah and M. Illafe. A comprehensive subclass of analytic and bi-univalent functions associated with subordination. Palestine Journal of Mathematics, 9(1):187– 193, 2020. [23] F. Yousef, S. Alroud, and M. Illafe. New subclasses of analytic and bi-univalent functions endowed with coefficient estimate problems. Analysis and Mathematical O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7036 17 of 19 Physics, 11:69, 2021. [24] S. Bulut. Coefficient estimates for a class of analytic and bi-univalent functions. Novi Sad Journal of Mathematics, 43:59–65, 2013. [25] S. Bulut, N. Magesh, and C. Abirami. A comprehensive class of analytic bi-univalent functions by means of Chebyshev polynomials. Journal of Fractional Calculus and Applications, 8:32–39, 2017. [26] S. Bulut, N. Magesh, and V. K. Balaji. Initial bounds for analytic and bi-univalent functions by means of Chebyshev polynomials. Analysis, 11:83–89, 2017. [27] M. Al-Kaseasbeh, A. Alamoush, A. Amourah, A. Aljarah, and J. Jerash. Subclasses of spiralike functions involving convoluted differential operator. International Journal of Open Problems in Complex Analysis, 12:23–34, 2020. [28] O. Alnajar, A. Amourah, J. Salah, and M. Darus. Fekete–Szegő functional prob- lem for analytic and bi-univalent functions subordinate to Gegenbauer polynomials. Contemporary Mathematics, 751:5731–5742, 2024. [29] A. A. Amourah and F. Yousef. Some properties of a class of analytic functions involving a new generalized differential operator. Boletim da Sociedade Paranaense de Matemática, 38(6):33–42, 2020. [30] M. G. Khan, B. Ahmad, N. Khan, W. K. Mashwani, S. Arjika, B. Khan, and R. Chin- ram. Applications of Mittag-Leffler type P)oisson distribution to a subclass of analytic functions involving conic-type regions, journal = Journal of Function Spaces, volume = 2021, year = 2021, pages = 4343163, doi = 10.1155/2021/4343163. [31] T. Al-Hawary, A. Amourah, J. Salah, and F. Yousef. Two inclusive subfamilies of bi-univalent functions. Int. J. Neutro. Sci., 24:315–323, 2024. [32] F. Yousef, A. A. Amourah, and M. Darus. Differential sandwich theorems for p- valent functions associated with a certain generalized differential operator and integral operator. Italian Journal of Pure and Applied Mathematics, 36:543–556, 2016. [33] 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. [34] A. Malkawi, D. Mahmoud, A. M. Rabaiah, R. Al-Deiakeh, and W. Shatanawi. On fixed point theorems in MR-Metric spaces. Nonlinear Functional Analysis and Appli- cations, pages 1125–1136, 2024. [35] A. A. R. M. Malkawi. Convergence and fixed points of self-mappings in MR-Metric spaces: Theory and applications. European Journal of Pure and Applied Mathematics, 18(2):5952, 2025. [36] A. Alsoboh, A. Amourah, M. Darus, and C. A. Rudder. Studying the harmonic functions associated with quantum calculus. Mathematics, 11(10):2220, 2023. [37] A. Alsoboh, M. Çağlar, and M. Buyankara. Fekete-Szegö inequality for a subclass of bi-univalent functions linked to q-ultraspherical polynomials. Contemporary Mathe- matics, pages 2531–2545, 2024. [38] A. Alsoboh and G. I. Oros. A class of bi-univalent functions in a leaf-like domain defined through subordination via q-calculus. Mathematics, 12(10):1594, 2024. [39] A. Amourah, A. Alsoboh, D. Breaz, and S. M. El-Deeb. A bi-starlike class in a leaf- O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7036 18 of 19 like domain defined through subordination via q-calculus. Mathematics, 12(11):1735, 2024. [40] M. Ahmed, A. Alsoboh, A. Amourah, and J. Salah. On the fractional q-differintegral operator for subclasses of bi-univalent functions subordinate to q-ultraspherical poly- nomials. European Journal of Pure and Applied Mathematics, 18(3):6586, 2025. [41] T. Al-Hawary, A. Amourah, A. Alsoboh, O. Ogilat, I. Harny, and M. Darus. Appli- cations of q-ultraspherical polynomials to bi-univalent functions defined by q-Saigo’s fractional integral operators. AIMS Mathematics, 9(7):17063–17075, 2024. [42] A. Alsoboh and M. Darus. New subclass of analytic functions defined by q-differential operator with respect to k-symmetric points. Int. J. Math. Comput. Sci, 14:761–773, 2019. [43] T. Al-Hawary, A. Amourah, A. Alsoboh, A. M Freihat, O. Ogilat, I. Harny, and M. Darus. Subclasses of yamakawa-type bi-starlike functions subordinate to gegen- baur polynomials associated with quantum calculus. Results in Nonlinear Analysis, 7(4):75–83, 2024. [44] A. Alsoboh, A. S. Tayyah, A. Amourah, A. A. Al-Maqbali, K. Al Mashraf, and T. Sasa. Hankel determinant estimates for bi-Bazilevič-type functions involving q- Fibonacci numbers. European Journal of Pure and Applied Mathematics, 18(3):6698, 2025. [45] A. Alsoboh, A. Amourah, O. Alnajar, M. Ahmed, and T. M. Seoudy. Exploring q-Fibonacci numbers in geometric function theory: Univalence and shell-like starlike curves. Mathematics, 13(8):1294, 2025. [46] A. Alsoboh, A. Amourah, K. Al Mashrafi, and T. Sasa. Bi-starlike and bi-convex func- tion classes connected to shell-like curves and the q-analogue of Fibonacci numbers. International Journal of Analysis and Applications, 23:201, 2025. [47] O. Alnajar, O. Khabour, A. Amourah, and M. Darus. The relationship of Borel distri- bution and Horadam polynomials leads to analytical bi-univalent functions. European Journal of Pure and Applied Mathematics, 18:5929, 2025. [48] 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. Surveys in Mathematics and its Applications, 16:193–205, 2021. [49] M. Ahmed. Amenable quasi-lattice ordered groups and true representations. Boletim da Sociedade Paranaense de Matemática, 41:1–10, 2023. [50] M Ahmed. Universal covariant representations and positive elements. Azerbaijan Journal of Mathematics, 15(1):44–52, 2025. [51] M. Ahmed and F. Moh’d. The graded annihilating submodule graph. AKCE Inter- national Journal of Graphs and Combinatorics, pages 1–9, 2025. [52] Y. Al-Qudah, F. Al-Sharqi, M. Mishlish, and M. M. Rasheed. Hybrid integrated decision-making algorithm based on AO of possibility interval-valued neutrosophic soft settings. International Journal of Neutrosophic Science, 22(3):84–98, 2023. [53] Y. Al-Qudah, K. Alhazaymeh, N. Hassan, M. Almousa, and M. Alaroud. Transitive closure of vague soft set relations and its operators. International Journal of Fuzzy O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7036 19 of 19 Logic and Intelligent Systems, 22(1):59–68, 2022. [54] M. H. Darassi, O. Yasin, and M. Ahmed. A semi-analytical method to solve the FitzHugh–Nagumo equation. Journal of Interdisciplinary Mathematics, 28(4):1489– 1504, 2025. [55] H. Qoqazeh, Y. Al-Qudah, M. Almousa, and A. Jaradat. On D-Compact topological spaces. Journal of Applied Mathematics and Informatics, 39(5-6):883–894, 2021. [56] 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 Conference Proceedings, volume 3150, page 020004, 2024.