EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6689 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Class of Bi-Univalent Functions Associated with Shell-Like Geometries and the q-Fibonacci Analogue Abdullah Alsoboh1, Ala Amourah2,3,∗, Abdullrahman A. Al-Maqbali1,∗, Omar Alnajar4, Feras Awad5, Tala Sasa6 1 College of Applied and Health Sciences, A’Sharqiyah University, Post Box No. 42, Post Code No. 400, Ibra, Sultanate of Oman 2 Mathematics Education Program, Faculty of Education and Arts, Sohar University, Sohar 311, Oman 3 Jadara University Research Center, Jadara University, Jordan 4 Department of Mathematics, Faculty of Science and Technology, Irbid National University, P.O. Box: 2600, Irbid 21110, Jordan 5 Department of Mathematics, Faculty of Science, Philadelphia University, Amman 19392, Jordan 6 Department of Mathematics, Faculty of Science, Applied Science Private University, Amman, Jordan Abstract. Using the subordination principle, this study explores two subclasses of bi-univalent functions associated with shell-like curves via the q-analogue of Fibonacci numbers, namely the starlike and convex classes. We derive coefficient bounds for the initial terms of these function classes and establish the corresponding Fekete-Szegö inequalities. Our findings contribute to the advancement of biunivalent function theory and its interaction with special function spaces. 2020 Mathematics Subject Classifications: 30A36, 11B37, 30C45, 81P68 Key Words and Phrases: Analytic functions, bi-univalent functions, starlike class, Fekete-Szegö functional, Fibonacci sequence, q-calculus, shell-like curves 1. Introduction and Definitions We begin by considering the collection A of functions that are complex analytic within the open unit disk O. This domain is defined as O = {z = a + i b ∈ C where a, b ∈ R, and |z| < 1} , ∗Corresponding author. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6689 Email addresses: abdullah.alsoboh@asu.edu.om (A. Alsoboh), AAmourah@su.edu.om (A. Amourah), abdulrahman.almaqbali@asu.edu.om (A. A. Al-Maqbali), o.alnjar@inu.edu.jo (O. Alnajar), fawad@phialadelphia.edu.jo (F. Awad), t sasa@asu.edu.jo (T. Sasa) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6689 2 of 19 which geometrically corresponds to the interior of the unit circle in the complex plane, centered at the origin and excluding its boundary. All functions f ∈ A are subject to a standard normalization, namely: f(0) = 0 and f ′(0) = 1. These initial conditions eliminate translational and scaling ambiguities, ensuring that each function is uniquely defined at the origin with a prescribed rate of change. This allows for coherent structural analysis and comparison of such functions under common geometric constraints. Each member f ∈ A possesses a Maclaurin series representation about the origin, which can be written as: f(z) = z + ∞∑ n=2 δn z n, for z ∈ O, (1) where the coefficients δn determine the nonlinear components of f . The leading term z arises from the derivative condition f ′(0) = 1, and subsequent terms capture the analytic structure beyond linearity. A function f is called a Schwarz function if it is analytic throughout O, satisfies f(0) = 0, and its modulus remains strictly less than one within the disk, i.e. |f(z)| < 1 for all z ∈ O. These functions are of central importance in geometric function theory, particularly in the context of conformal and univalent mappings. Furthermore, for any two functions f1, f2 ∈ A , the function f1 is said to be subordinate to f2, denoted f1 ≺ f2, if there exists a Schwarz function η such that f1(z) = f2(η(z)) for all z ∈ O. This relation implies that f1 is functionally dependent on f2 through composition with η, preserving analyticity while embedding geometric structure. The notion of subordination is a key analytical tool for examining inclusion relations, growth estimates, and mapping behavior in complex analysis. In addition, let us consider the subclass S, S ⊂ A , which comprises all functions that are univalent (i.e., one-to-one) within the unit disk O. We also introduce the class P, defined as the family of functions in A whose real parts are strictly positive throughout O. A typical function φ ∈ P admits the following expansion of the power series: p(z) = 1 + ∞∑ n=1 pnz n = 1 + p1z + p2z 2 + p3z 3 + . . . , (z ∈ O). (2) where the coefficients satisfy the sharp bound, |pn| ≤ 2, for all n ≥ 1. (3) A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6689 3 of 19 in accordance with the classical lemma Carathéodory (see [1] for further details). Further- more, a function φ ∈ P if and only if it is subordinate to the Móbius transformation 1+z 1−z , i.e., φ(z) ≺ 1 + z 1 − z , z ∈ O. The class of starlike functions, denoted S∗, can be characterized in various ways using subordination techniques. Ma and Minda [2], who defined the following class, proposed a notable generalization. S∗(Ω) = { f ∈ A : z f ′(z) f(z) ≺ Ω(z), where Ω ∈ P and z ∈ O } . In this formulation, Ω is assumed to be analytic in O and have a positive real part through- out the disk. Table 1 provides a variety of subclasses of S∗, arising from specific choices of the function Ω, reflecting the diversity of approaches adopted in the literature to construct refined categories of starlike mappings. The class P forms the cornerstone for the devel- Table 1: Enumerates various starlike function classes characterized via the principle of subordination. The subclasses of starlike functions Ref. Author/s 1 S∗ ( 1+z 1−z ) = { f ∈ A : zf ′(z) f(z) ≺ 1+z 1−z } [3] Janowski 2 S∗(ϑ) = { f ∈ A : zf ′(z) f(z) ≺ 1+(1−2ϑ)z 1−z } , where 0 ≤ ϑ < 1 [4] Robertson 3 SL(ϑ) = { f ∈ A : zf ′(z) f(z) ≺ 1+ϑ2z2 1−ϑz−ϑ2z2 } , where ϑ = 1− √ 5 2 [5] Sokól 4 SK(ϑ) = { f ∈ A : zf ′(z) f(z) ≺ 3 3+(ϑ−3)z−ϑ2z2 } , where ϑ ∈ (−3, 1] [6] Sokól opment of numerous significant subclasses of analytic functions, making it a key target of study in complex analysis. For any function f in the subclass S ⊂ A , there exists an inverse function, denoted f−1, which is defined as z = f−1(f(z)) and ξ = f(f−1(ξ)), (r0(f) ≥ 0.25; |ξ| < r0(f); z ∈ O) . (4) where χ(ξ) = f−1(ξ) = ξ − δ2ξ 2 + ( 2δ22 − δ3 ) ξ3 − ( 5δ32 + δ4 − 5δ3δ2 ) ξ4 + · · · . (5) The function f ∈ S is said to be bi-univalent if its inverse function f−1 ∈ S. The subclass of S denoted by ∑ contains all bi-univalent functions in O. The table below illustrates certain functions within the class ∑ and their inverse functions. A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6689 4 of 19 Table 2: Representative examples of bi-univalent functions along with their corresponding inverse functions. f f−1 f1(z) = z 1 + z f−1 1 (z) = z 1 − z f2 = − log(1 − z) f−1 1 (z) = e2z − 1 e2z + 1 f3 = 1 2 log ( 1 + z 1 − z ) f−1 1 (z) = ez − 1 ez Quantum calculus, also known as q-calculus, extends beyond the conventional frame- work of ordinary calculus by incorporating the parameter q ∈ (0, 1), thus generalizing clas- sical analytical techniques. This field has garnered significant interest because of its deep connections with physics, quantum mechanics, and Geometric Function Theory (GFT). A foundational resource for understanding the q-difference calculus and its diverse applica- tions is the work of Gasper and Rahman [7], which provides a comprehensive exposition on the subject. Central to the study of analytic functions within this framework is the q-difference operator ∂q, which plays a crucial role in function theory. Notable advance- ments in this area include the work of Seoudy and Aouf [8], who extended the q-calculus to functions within the unit disk, further enriching GFT. For further exploration, numerous classical and contemporary studies provide valuable insights, including [9–29]. Polynomials play a significant role in Geometric Function Theory (GFT) as both an- alytic test functions and approximation tools. In GFT, polynomial mappings are used to study geometric behaviors such as starlikeness, convexity, and univalence through simpler, finite-degree cases. Many univalent and bi-univalent functions can be represented or ap- proximated by polynomial expansions, making it possible to estimate coefficient bounds and distortion theorems more effectively [30–46]. Furthermore, orthogonal polynomials, such as Chebyshev or Legendre polynomials, are employed in the construction of subclasses of analytic functions with the desired geometric properties. Thus, polynomials bridge the gap between abstract complex analysis and computational modeling, allowing deeper ex- ploration of geometric mappings and their analytic behavior [47–51]. Some applications in operator theory can be found in [52–54]. A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6689 5 of 19 Definition 1. [38] The q-bracket ⌈κ⌋q is defined as follows: ⌈κ⌋q =  1−qλ 1−q , 0 < q < 1, λ ∈ C∗ = C \ {0} 1, q 7→ 0+, λ ∈ C∗ λ, q 7→ 1−, λ ∈ C∗ qγ−1 + qγ−2 + · · · + q + 1 = γ−1∑ n=0 qn, 0 < q < 1, λ = γ ∈ N, with the useful identity ⌈κ + 1⌋q = ⌈κ⌋q + qκ. Definition 2. [38] The q−derivative, also known as the q−difference operator, of a func- tion f is defined by ðq⟨f(z)⟩ =  (f(z) − f(q z))(z − q z)−1, if 0 < q < 1, z ̸= 0, f ′(0), if z = 0, f ′(z), if q 7→ 1−, z ̸= 0. . Remark 1. For f ∈ A of the form (1), it is straightforward to verify that ðq⟨f(z)⟩ = ðq 〈 z + ∞∑ n=2 δn z n 〉 = 1 + ∞∑ n=2 ⌈n⌋qδn zn−1, (z ∈ O), and for the inverse function χ = f−1 of the form (4), we have ðq⟨χ(ξ)⟩ = ðq⟨f−1(ξ)⟩ = 1−⌈2⌋qδ2ξ+⌈3⌋q ( 2δ22 − δ3 ) ξ2−⌈4⌋q ( 5δ32 + δ4 − 5δ3δ2 ) ξ3 + · · · . In a more recent advance, Alsoboh et al. [55] introduced a notable class of functions known as q starlike functions, denoted by SLq, which were defined using the q-Jackson difference operators. The formal definition of this class is given by SLq = { f ∈ A : z ðq⟨f(z)⟩ f(z) ≺ Υ(z; q), z ∈ O } , (6) where the function Υ(z; q) is expressed explicitly as Υ(z; q) = 1 + qϑ2 qz 2 1 − ϑqz − qϑ2 qz 2 , (7) and ϑq = 1 − √ 4q + 1 2q (8) A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6689 6 of 19 represents the q-analog of the Fibonacci numbers. Also, Alsoboh et al. [55] established a significant connection between these q-Fibonacci numbers, denoted as ϑq, and the related Fibonacci polynomials φn(q). Specifically, they demonstrated that if Υ(z; q) = 1 + ∞∑ n=1 p̂nz n, the coefficients p̂n satisfy the following recurrence relation: p̂n =  ϑq, for n = 1, (2q + 1)ϑ2 q , for n = 2, (3q + 1)ϑ3 q , for n = 3, (φn+1(q) + qφn−1(q))ϑn q , for n ≥ 4. (9) Here, the q-Fibonacci polynomials φs(q) are defined as φs(q) = (1 − qϑq) s − (ϑq) s √ 4q + 1 , s ∈ N. (10) This research presents a comprehensive framework for examining the relationship between the q-modified Fibonacci numbers and their corresponding polynomial representations. The initial terms of the q-Fibonacci sequence, which constitutes a natural generaliza- tion of the classical Fibonacci numbers and converges to them as q → 1−, are enumerated in Table 3. Table 3: Comparison of the classical Fibonacci numbers with their corresponding q-analogue terms from the q-Fibonacci sequence. The classical Fibonacci numbers The q-analogue of Fibonacci numbers φ0 = 0 φ0(q) = 0 φ1 = 1 φ1(q) = 1 φ2 = 1 φ2(q) = 1 φ3 = 2 φ3(q) = 1 + q φ4 = 3 φ4(q) = 1 + 2q It should be noted that the function Υ(z; q) is not injective in the domain O. Specifi- cally, there exist distinct points in O at which Υ(z; q) attains the same value. For example, Υ(0; q) = 1 and Υ ( − 1 2qϑq ; q ) = 1. A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6689 7 of 19 In the following example, we explore the behavior of the q-starlike functions as the parameter q approaches 1 below. This transition leads to the classical case of starlike functions, often referred to as class SL. By taking the limit as q → 1−, we observe how the q-starlike functions generalize to the traditional starlike functions, and the associated function Υ(z) simplifies to a form that connects directly with the classical Fibonacci numbers. This example illustrates the connection between the q-starlike functions and their classical counterparts. Example 1. To illustrate the asymptotic behavior of the q-starlike functions as q → 1−, we examine the limiting case of the class SLq. In the limit, this class converges to the classical starlike function class associated with the Fibonacci generating function, namely SL = lim q→1− SLq = { f ∈ A : z f ′(z) f(z) ≺ Υ(z) } , where the function Υ(z) is given by Υ(z; 1) = Υ(z) = 1 + ϑ2z2 1 − ϑz − ϑ2z2 , (11) and ϑ = 1− √ 5 2 denotes the classical Fibonacci constant. In addition to introducing the class of q-starlike functions, Alsoboh et al. [56] further extended the framework by defining a novel class of analytic functions termed the q-convex class, denoted by KSLq. This class is characterized by a subordination condition analogous to that of the q-starlike class, but involves the application of a second-order q-difference operator, thereby capturing a more nuanced geometric structure. Specifically, a function f is said to belong to the class KSLq if and only if the following subordination condition is satisfied: 1 + z ð2q⟨f(z)⟩ ðq⟨f(z)⟩ ≺ Υ(z; q), (z ∈ O), (12) where the function Υ(z; q) is defined by the rational expression in (7), and the parameter ϑq is specified in (8). 2. Definition and example Motivated by q-Fibonacci numbers, this section will now look at a novel subclass of bi-univalent functions related to shell-like curves. Definition 3. For β ∈ [0, 1]. A bi-univalent function f of the form (1) belongs to the class SLM∑(β; q) if and only if (1 − β) zðq⟨f(z)⟩ f(z) + β ðq (z ðq⟨f(z)⟩) ðq⟨f(z)⟩ ≺ Υ(z; q) = 1 + qϑ2 qz 2 1 − ϑq z − qϑ2 qz 2 , (13) A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6689 8 of 19 and (1 − β) ξðq⟨χ(ξ)⟩ χ(ξ) + β ðq (ξðq⟨χ(ξ)⟩) ðq⟨χ(ξ)⟩ ≺ Υ(ξ; q) = 1 + qϑ2 qξ 2 1 − ϑq ξ − qϑ2 qξ 2 , (14) where χ = f−1 given by (5), ϑq given by (8) and z, ξ ∈ O. By varying the parameters β ∈ [0, 1] and q ∈ (0, 1), a broad spectrum of novel sub- classes of the bi-univalent function class ∑ can be systematically derived. These subclasses capture diverse geometric behaviors and provide a unified framework for further analytical investigations. Example 2. If β = 0, we obtain the class SL∑(Υ(z; q)) consisting of functions f ∈ ∑ satisfying the conditions zðq⟨f(z)⟩ f(z) ≺ Υ(z; q) = 1 + qϑ2 qz 2 1 − ϑq z − qϑ2 qz 2 , and ξðq⟨χ(ξ)⟩ χ(ξ) ≺ Υ(ξ; q) = 1 + qϑ2 qξ 2 1 − ϑq ξ − qϑ2 qξ 2 , where ϑq is given by (8). Example 3. Letting β = 1, we arrive at the subclass KL∑(Υ(z; q)), which comprises all functions f ∈ ∑ satisfying the subordination conditions 1 + z ð2q⟨f(z)⟩ ðq⟨f(z)⟩ ≺ Υ(z; q) = 1 + qϑ2 qz 2 1 − ϑqz − qϑ2 qz 2 , and 1 + ξ ð2q⟨χ(ξ)⟩ ðq⟨χ(ξ)⟩ ≺ Υ(ξ; q) = 1 + qϑ2 qξ 2 1 − ϑqξ − qϑ2 qξ 2 , (15) where ϑq is the q-analogue of the Fibonacci number as defined in (8). Example 4. In the limiting case as q → 1−, we recover the classical subclass SLM∑(β), consisting of all functions f ∈ ∑ that satisfy the following subordination conditions: (1 − β) z f ′(z) f(z) + β z f ′′(z) f ′(z) ≺ Υ(z) = 1 + ϑ2z2 1 − ϑz − ϑ2z2 , and (1 − β) ξ χ′(ξ) χ(ξ) + β ξ χ′′(ξ) χ′(ξ) ≺ Υ(ξ) = 1 + ϑ2ξ2 1 − ϑξ − ϑ2ξ2 , where χ = f−1 is the inverse function defined as in (5), ϑ = 1− √ 5 2 is the classical Fibonacci constant, and z, ξ ∈ O. A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6689 9 of 19 Example 5. If q → 1− and β = 0, we obtain the class SL∑(Υ(z)) consisting of functions f ∈ ∑ satisfying the conditions z f ′(z) f(z) ≺ Υ(z) = 1 + ϑ2z2 1 − z − ϑ2z2 , and ξ χ′(ξ) χ(ξ) ≺ Υ(ξ) = 1 + qϑ2ξ2 1 − ϑξ − ϑ2ξ2 , where χ = f−1 is the inverse function defined as in (5), ϑ = 1− √ 5 2 is the classical Fibonacci constant, and z, ξ ∈ O. Example 6. If q 7→ 1− and β = 1, we obtain the class class KL∑(Υ(z)) consisting of functions f ∈ ∑ satisfying the conditions 1 + zf ′′ (z) f ′(z) ≺ Υ(z) = 1 + ϑ2z2 1 − z − ϑ2z2 , and 1 + ξχ ′′ (ξ) χ′(ξ) ≺ Υ(ξ) = 1 + qϑ2ξ2 1 − ϑ ξ − ϑ2ξ2 , where χ = f−1 is the inverse function defined as in (5), ϑ = 1− √ 5 2 is the classical Fibonacci constant, and z, ξ ∈ O. 3. Main Results In this section, we first obtain the estimate of the initial Taylor coefficients |δ2| and |δ2| for functions in the class SLM∑(β; q) according to Definition 3. Firstly, let us p(z) = 1 + p1z + p2z 2 + p3z 3 + . . . , and p(z) ≺ Υ(z; q). Then there exists φ ∈ P such that |φ(z)| < 1 in O and p(z) = Υ(φ(z); q). We have ℏ(z) = (1 + φ(z))(1 − φ(z))−1 = 1 + ℓ1z + ℓ2z 2 + · · · ∈ P (z ∈ O). (16) Consequently, the function φ(z), being analytic in O and subordinate to Υ(z; q), admits the following Taylor expansion: φ(z) = ℓ1z 2 + ( ℓ2 − ℓ21 2 ) z2 2 + ( ℓ3 − ℓ1ℓ2 − ℓ31 4 ) z3 2 + · · · , (17) A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6689 10 of 19 and Υ(φ(z); q) = 1 + p̂1 [ ℓ1z 2 + ( ℓ2 − ℓ21 2 ) z2 2 + ( ℓ3 − ℓ1ℓ2 − ℓ31 4 ) z3 2 + · · · ] + p̂2 [ ℓ1z 2 + ( ℓ2 − ℓ21 2 ) z2 2 + ( ℓ3 − ℓ1ℓ2 − ℓ31 4 ) z3 2 + · · · ]2 + p̂3 [ ℓ1z 2 + ( ℓ2 − ℓ21 2 ) z2 2 + ( ℓ3 − ℓ1ℓ2 − ℓ31 4 ) z3 2 + · · · ]3 + · · · = 1 + p̂1ℓ1 2 z + 1 2 [( ℓ2 − ℓ21 2 ) p̂1 + ℓ21 2 p̂2 ] z2 + 1 2 [( ℓ3 − ℓ1ℓ2 + ℓ31 4 ) p̂1 + ℓ1 ( ℓ2 − ℓ21 2 ) p̂2 + ℓ31 4 p̂3 ] z3 + · · · . (18) Similarly, there exists an analytic function ν defined in O, satisfying |ν(ξ)| < 1, such that p(ξ) = Υ(ν(ξ); q). This allows us to represent the corresponding function κ(ξ) = (1 + ν(ξ))(1 − ν(ξ))−1 = 1 + τ1ξ + τ2ξ 2 + · · · ∈ P. (19) As a result, the Taylor expansion of ν(ξ) takes the form: ν(ξ) = τ1ξ 2 + ( τ2 − τ21 2 ) ξ2 2 + ( τ3 − τ1τ2 − τ31 4 ) ξ3 2 + · · · , (20) and, accordingly, the composition Υ(ν(ξ); q) expands as: Υ(ν(ξ); q) = 1 + p̂1τ1 2 ξ + 1 2 [( τ2 − τ21 2 ) p̂1 + τ21 2 p̂2 ] ξ2 + 1 2 [( τ3 − τ1τ2 + τ31 4 ) p̂1 + τ1 ( τ2 − τ21 2 ) p̂2 + τ31 4 p̂3 ] ξ3 + · · · . (21) Having established the necessary groundwork and auxiliary results, we are now in a position to derive bounds for the initial coefficients of the functions belonging to the newly introduced class SLM∑(β; q). These estimates not only offer insights into the geometric behavior of such bi-univalent functions but also highlight the influence of the deformation parameter q and the parameter β on the coefficient structure. The following theorem presents sharp bounds for the second and third coefficients |δ2| and |δ3|, respectively. Theorem 1. For β ∈ [0, 1], let f ∈ SLM∑(β; q). Then∣∣δ2∣∣ ≤ |ϑq|√∣∣∣ϑq(K −X) + ( 1 − (2q + 1)ϑq ) C ∣∣∣ . (22) ∣∣δ3∣∣ ≤ |ϑq| {∣∣(K −X ) ϑq + ( 1 − (2q + 1)ϑq ) C ∣∣ + |ϑq|K } K ∣∣(K −X ) ϑq + ( 1 − (2q + 1)ϑq ) C ∣∣ , (23) A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6689 11 of 19 where K = q ⌈2⌋q ( 1 + q ⌈2⌋q β ) , (24) X = q [ 1 + β ( ⌈2⌋2 q − 1 )] , (25) C = q2 ( 1 + q β )2 . (26) Proof. Let f ∈ SL∑(Υ(z)) and ξ = f−1. Taking into account (13) and (14), we have (1 − β) zðq⟨f(z)⟩ f(z) + β ðq (z ðq⟨f(z)⟩) ðq⟨f(z)⟩ = Υ(φ(z); q), (z ∈ O), (27) and (1 − β) ξðq⟨χ(ξ)⟩ χ(ξ) + β ðq (ξðq⟨χ(ξ)⟩) ðq⟨χ(ξ)⟩ = Υ(ν(ξ); q), (ξ ∈ O). (28) Since (1 − β) zðq⟨f(z)⟩ f(z) + β ðq (z ðq⟨f(z)⟩) ðq⟨f(z)⟩ = 1 + p̂1ℓ1 2 z + 1 2 [( ℓ2 − ℓ21 2 ) p̂1 + ℓ21 2 p̂2 ] z2 + · · · . (29) and (1 − β) ξðq⟨χ(ξ)⟩ χ(ξ) + β ðq (ξðq⟨χ(ξ)⟩) ðq⟨χ(ξ)⟩ = 1 + p̂1τ1 2 ξ + 1 2 [( τ2 − τ21 2 ) p̂1 + τ21 2 p̂2 ] ξ2 + · · · . (30) Compared with (27) and (29), along (18), yields q(1 + qβ)δ2z+q⌈2⌋q(1 + q⌈2⌋qβ)δ3 − q ( 1 + β(⌈2⌋2 q − 1) ) δ22z 2 + · · · = p̂1ℓ1 2 z + 1 2 [( ℓ2 − ℓ21 2 ) p̂1 + ℓ21 2 p̂2 ] z2 + · · · . (31) Besied that By comparing (28) and (30), along (21), yields −q(1 + qβ)δ2z+ ( 2q⌈2⌋q(1 + q⌈2⌋qβ) − q ( 1 + β(⌈2⌋2 q − 1) ) δ22 − q⌈2⌋q(1 + q⌈2⌋qβ)δ3 ) z2 + · · · = p̂1τ1 2 ξ + 1 2 [( τ2 − τ21 2 ) p̂1 + τ21 2 p̂2 ] ξ2 + · · · . (32) Equating the pertinent coefficient in (31) and (32), using (24) and (25), we obtain q(1 + qβ)δ2 = p̂1ℓ1 2 (33) A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6689 12 of 19 −q(1 + qβ)δ2 = p̂1τ1 2 (34) Kδ3 −Xδ22 = 1 2 [( ℓ2 − ℓ21 2 ) p̂1 + ℓ21 2 p̂2 ] (35) (2K −X)δ22 −Kδ3 = 1 2 [( τ2 − τ21 2 ) p̂1 + τ21 2 p̂2 ] (36) From (33) and (34), we have ℓ1 = −τ1 ⇐⇒ ℓ21 = τ21 , (37) and δ22 = ϑ2 q 8q2(1 + qβ)2 ( ℓ21 + τ21 ) ⇐⇒ ℓ21 + τ21 = 8q2(1 + qβ)2 ϑ2 q δ22 . (38) Now, by summing (35) and (36), we obtain 2 ( K −X ) δ22 = (ℓ2 + τ2)ϑq 2 + [ (2q + 1)ϑ2 q 4 − ϑq 4 ] ( ℓ21 + τ21 ) . (39) Putting (38) in (39), we obtain δ22 = (ℓ2 + τ2)ϑ 2 q 4 (( K −X ) ϑq + ( 1 − (2q + 1)ϑq ) C ) , (40) where K,X,C is given by (24), (25) and (26), respectively. Using (3) for (40), we have∣∣δ2∣∣ ≤ |ϑq|√∣∣∣ϑq(K −X) + ( 1 − (2q + 1)ϑq ) C ∣∣∣ . (41) Now, so as to find the bound on |δ3|, let us subtract from (35) and (36) along (38), we obtain δ3 = δ22 + ϑq 4K ( ℓ2 − τ2 ) . (42) Therefore, we get ∣∣δ3∣∣ ≤ ∣∣δ2∣∣2 + |ϑq| K . (43) Then, in view of (41), we obtain ∣∣δ3∣∣ ≤ |ϑq| {∣∣(K −X ) ϑq + ( 1 − (2q + 1)ϑq ) C ∣∣ + |ϑq|K } K ∣∣(K −X ) ϑq + ( 1 − (2q + 1)ϑq ) C ∣∣ , (44) where K,X,C are given by (24), (25) and (26), respectively. This proves (49). A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6689 13 of 19 Theorem 2. For α ∈ C∗ and β ∈ [0, 1], let f ∈ SLM∑(β; q). Then ∣∣δ3 − αδ22 ∣∣ ≤  |ϑq | K , ∣∣1 − α ∣∣ ≤ ∣∣ϑq(K−X)+(1−(2q+1)ϑq)C ∣∣ |ϑq |K |1−α||ϑq |2∣∣(K−X)ϑq+(1−(2q+1)ϑq)C ∣∣ , ∣∣1 − α ∣∣ ≥ ∣∣ϑq(K−X)+(1−(2q+1)ϑq)C ∣∣ |ϑq |K (45) where K,X,C are given by (24), (25) and (26), respectively. Proof. Let f ∈ SLM∑(β; q), from (40) and (42) we have δ3 − αδ22 = (1 − α)ϑ2 q 4 (( K −X ) ϑq + ( 1 − (2q + 1)ϑq ) C )(ℓ2 + τ2) + ϑq 4K ( ℓ2 − τ2 ) = ( K (α) + ϑq 4K ) ℓ2 + ( K (α) − ϑq 4K ) τ2, (46) where K (α) = (1 − α)ϑ2 q 4 (( K −X ) ϑq + ( 1 − (2q + 1)ϑq ) C ) . (47) Then, by taking modulus of (46), we conclude that ∣∣δ3 − αδ22 ∣∣ ≤  |ϑq | K , 0 ≤ ∣∣K (α) ∣∣ ≤ |ϑq | 4K 4 ∣∣K (α) ∣∣, ∣∣K (α) ∣∣ ≥ |ϑq | 4K If β = 0, we obtain the following results for the class SL∑(Υ(z; q)) defined in Exam- ple (2) Corollary 1. Let f given by (1) be in the class SL∑(Υ(z); q). Then ∣∣δ2∣∣ ≤ ∣∣ϑq ∣∣ q √ 1 − 2qϑq . (48) ∣∣δ3∣∣ ≤ ∣∣ϑq ∣∣(q − (1 + q + 2q2)ϑq ) q2(1 + q) ( 1 − 2qϑq ) . (49) ∣∣δ3 − αδ22 ∣∣ ≤  |ϑq | q(1+q) , ∣∣1 − α ∣∣ ≤ q ( 1−2qϑq ) (1+q)|ϑq | |1−α|ϑ2 q q2 ( 1−2qϑq ) , ∣∣1 − α ∣∣ ≥ q ( 1−2qϑq ) (1+q)|ϑq | (50) If β = 1, we obtain the following results for the class KL∑(Υ(z; q)) defined in Exam- ple (3) A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6689 14 of 19 Corollary 2. Let f given by (1) be in the class KL∑(Υ(z); q). Then ∣∣δ2∣∣ ≤ ∣∣ϑq ∣∣√ ⌈2⌋q ( ⌈2⌋q − ( ⌈3⌋q + 2q ) ϑq ) ∣∣δ3∣∣ ≤ ∣∣ϑq ∣∣(⌈2⌋q − 2(⌈3⌋q + q)ϑq ) ⌈2⌋q⌈3⌋q ( ⌈2⌋q − ( ⌈3⌋q + 2q ) ϑq ) , and ∣∣δ3 − αδ22 ∣∣ ≤  |ϑq | ⌈2⌋q ⌈3⌋q , ∣∣1 − α ∣∣ ≤ ⌈2⌋q− ( ⌈3⌋q+2q ) ϑq ⌈3⌋q |ϑq | |1−α|ϑ2 q ⌈2⌋q ( ⌈2⌋q− ( ⌈3⌋q+2q ) ϑq ) , ∣∣1 − α ∣∣ ≥ ⌈2⌋q− ( ⌈3⌋q+2q ) ϑq ⌈3⌋q |ϑq | If q 7→ 1−, we obtain the following results for the class SLM∑(β) defined in Example (4) Corollary 3. For q 7→ 1−, let f ∈ SLM∑(β). Then ∣∣δ2∣∣ ≤ |ϑ|√∣∣∣ϑ(K −X) + ( 1 − 3ϑ ) C ∣∣∣ , ∣∣δ3∣∣ ≤ |ϑ| {∣∣(K −X ) ϑ + ( 1 − 3ϑ ) C ∣∣ + |ϑ|K } K ∣∣(K −X ) ϑ + ( 1 − 3ϑ ) C ∣∣ , and ∣∣δ3 − αδ22 ∣∣ ≤  |ϑ| K , ∣∣1 − α ∣∣ ≤ ∣∣ϑ(K−X)+(1−3ϑ)C ∣∣ |ϑ|K |1−α||ϑ|2∣∣(K−X)ϑ+(1−3ϑ)C ∣∣ , ∣∣1 − α ∣∣ ≥ ∣∣ϑ(K−X)+(1−3ϑ)C ∣∣ |ϑ|K where K,X,C are given by (24), (25) and (26), respectively. If q 7→ 1− and β = 0, we obtain the following results for the class SL∑(Υ(z)) defined in Example (5) Corollary 4. [57] Let f given by (1) be in class SL∑(Υ(z)). Then ∣∣δ2∣∣ ≤ ∣∣ϑ∣∣ √ 1 − 2ϑ , ∣∣δ3∣∣ ≤ ∣∣ϑ∣∣(1 − 4ϑ ) 2 ( 1 − 2ϑ ) . and ∣∣δ3 − αδ22 ∣∣ ≤  |ϑ| 2 , ∣∣1 − α ∣∣ ≤ 1−2ϑ 2|ϑ| (1−α)ϑ2 1−2ϑ , ∣∣1 − α ∣∣ ≥ 1−2ϑ 2|ϑ| A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6689 15 of 19 If q 7→ 1− and β = 1, we obtain the following results for the class KL∑(Υ(z)) defined in Example (6) Corollary 5. [57] Let f given by (1) be in the class KL∑(Υ(z)). Then ∣∣δ2∣∣ ≤ ∣∣ϑ∣∣ √ 4 − 10ϑ , ∣∣δ3∣∣ ≤ ∣∣ϑ∣∣(1 − 4ϑ ) 3 ( 1 − 2ϑ ) . and ∣∣δ3 − αδ22 ∣∣ ≤  |ϑ| 6 , ∣∣1 − α ∣∣ ≤ 2−5ϑ 3|ϑ| |1−α|ϑ2 2 ( 2−5ϑ ) , ∣∣1 − α ∣∣ ≥ 2−5ϑ 3|ϑ| 4. Conclusion In this work, we investigated two subclasses of bi-univalent functions associated with shell-like curves through the q-analogue of Fibonacci numbers, namely the starlike and convex classes. Using the subordination principle, we establish coefficient bounds for the initial terms of these function classes and derived the corresponding Fekete-Szegö inequal- ities. These results enhance the theoretical framework of bi-univalent function theory and elucidate its deeper connections with special function spaces. Future research could extend these findings by exploring higher-order coefficient esti- mates, refining the structural characteristics of these subclasses, and examining their geo- metric properties. Moreover, investigating upper bounds related to the Zalcman conjecture and analyzing Hankel determinants of orders two and three within these subclasses could provide new insights and open further avenues in the study of analytic and bi-univalent function theory. References [1] P. L. Duren. Univalent functions. Grundlehren der Mathematischen Wissenschaften Series. Springer, New York, 1983. [2] W. Ma and D. Minda. A unified treatment of some special classes of univalent functions. In Proc. Conf. Comp. Anal. Tianjin China, pages 157–169, 1992. [3] W. Janowski. Extremal problems for a family of functions with positive real part and for some related families. Annales Polonici Mathematici, 23(28):159–177, 1970. [4] M. S. Robertson. Certain classes of starlike functions. Mich. Math. J., 32:135–140, 1985. [5] J. Sokó l. On starlike functions connected with fibonacci numbers. Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka, 23(157):111–116, 1999. [6] J. Sokó l. A certain class of starlike functions. Computers & Mathematics with Appli- cations, 62(2):611–619, 2011. A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6689 16 of 19 [7] G. Gasper and M. Rahman. Positivity of the poisson kernel for the continuous q-jacobi polynomials and some quadratic transformation formulas for basic hypergeometric series. SIAM Journal on Mathematical Analysis, 17(4):970–999, 1986. [8] T. M. Seoudy and M. K. Aouf. Coefficient estimates of new classes of q-starlike and q-convex functions of complex order. J. Math. Inequal., 10(1):135–145, 2016. [9] A. Alsoboh, A. Amourah, M. Darus, and C. A. Rudder. Investigating new subclasses of bi-univalent functions associated with q-pascal distribution series using the subor- dination principle. Symmetry, 15(5):1109, 2023. [10] 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. [11] A. A. Amourah. Faber polynomial coefficient estimates for a class of analytic bi- univalent functions. In AIP Conference Proceedings, volume 2096, page 020024. AIP Publishing, 2019. [12] 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. [13] 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 Matematica, 38(6):33–42, 2020. [14] A. A. Amourah, F. Yousef, T. Al-Hawary, and M. Darus. A certain fractional deriva- tive operator for p-valent functions and new class of analytic functions with negative coefficients. Far East Journal of Mathematical Sciences, 99(1):75–87, 2016. [15] A. A. Amourah, F. Yousef, T. Al-Hawary, and M. Darus. On a class of p-valent non- bazilevic functions of order µ + iβ. International Journal of Mathematical Analysis, 10(13-16):701–710, 2016. [16] M. Arif, O. Barkub, H. M. Srivastava, S. Abdullah, and S. A. Khan. Some janowski type harmonic q-starlike functions associated with symmetrical points. Mathematics, 8:629, 2020. [17] B. Khan, H. M. Srivastava, N. Khan, M. Darus, M. Tahir, and Q. Z. Ahmad. Coeffi- cient estimates for a subclass of analytic functions associated with a certain leaf-like domain. Mathematics, 8:1334, 2020. [18] M. Illafe, A. Hussen, M. H. Mohd, and F. Yousef. On a subclass of bi-univalent functions affiliated with Bell and Gegenbauer polynomials. Boletim da Sociedade Paranaense de Matematica, 43:1–10, 2025. [19] M. Illafe, F. Yousef, M. H. Mohamed, and S. Supramaniam. Fundamental properties of a class of analytic functions defined by a generalized multiplier transformation operator. International Journal of Mathematics and Computer Science, 19(4):1203– 1211, 2024. [20] M. Illafe, M. H. Mohd, F. Yousef, and S. Supramaniam. A subclass of bi-univalent functions defined by asymmetric q-derivative operator and Gegenbauer polynomials. European Journal of Pure and Applied Mathematics, 17(4):2467–2480, 2024. [21] M. Illafe, M. Haji Mohd, F. Yousef, and S. Supramaniam. Bounds for the second A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6689 17 of 19 Hankel determinant of a general subclass of bi-univalent functions. International Journal of Mathematics, Engineering, and Management Sciences, 9(5):1226–1239, 2024. [22] S. Mahmood, H. M. Srivastava, N. Khan, Q. Z. Ahmad, B. Khan, and I. Ali. Upper bound of the third hankel determinant for a subclass of q-starlike functions. Symme- try, 11:347, 2019. [23] S. Mahmood, Q. Z. Ahmad, H. M. Srivastava, N. Khan, B. Khan, and M. Tahir. A certain subclass of meromorphically q-starlike functions associated with the janowski functions. J. Inequal. Appl., page 88, 2019. [24] V. Masih, A. Ebadian, and S. Yalçin. Some properties associated to a certain class of starlike functions. Mathematica Slovaca, 69(6):1329–1340, 2019. [25] M. Shafiq, H. M. Srivastava, N. Khan, Q. Z. Ahmad, M. Darus, and S. Kiran. An upper bound of the third hankel determinant for a subclass of q-starlike functions associated with k-fibonacci numbers. Symmetry, 12:1043, 2020. [26] H. M. Srivastava, M. K. Aouf, and A. O. Mostafa. Some properties of analytic func- tions associated with fractional q-calculus operators. Miskolc Math. Notes, 20:1245– 1260, 2019. [27] H. M. Srivastava and S. M. El-Deeb. A certain class of analytic functions of complex order connected with a q-analogue of integral operators. Miskolc Math. Notes, 21:417– 433, 2020. [28] T. Al-Hawary, M. Illafe, and F. Yousef. Certain constraints for functions provided by Touchard polynomials. International Journal of Mathematics and Mathematical Sciences, 2025(1):2581058, 2025. [29] 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. [30] T. Al-Hawary, A. Amourah, A. Alsoboh, I. Harny, and M. Darus. Applications of q-ultraspherical polynomials to bi-univalent functions defined by q-saigo’s fractional integral operators. AIMS Mathematics, 9(7):17063–17075, 2024. [31] T. Al-Hawary, A. Amourah, A. Alsoboh, I. Harny, and M. Darus. Subclasses of yamakawa-type bi-starlike functions subordinate to gegenbauer polynomials associ- ated with quantum calculus. Results in Nonlinear Analysis, 7(4):75–83, 2024. [32] O. Al-Refai, A. Amourah, T. Al-Hawary, and B. A. Frasin. A new method for esti- mating general coefficients to classes of bi-univalent functions. Journal of Function Spaces, 2024:9889253, 2024. [33] 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, 10(7):e28302, 2024. [34] A. Alsoboh, A. Amourah, M. Darus, and C. A. Rudder. Studying the harmonic functions associated with quantum calculus. Mathematics, 11(10):2220, 2023. [35] A. Alsoboh, A. Amourah, F. M. Sakar, G. M. Gharib, and N. Zomot. Coefficient estimation utilizing the faber polynomial for a subfamily of bi-univalent functions. Axioms, 12(6):512, 2023. A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6689 18 of 19 [36] A. Alsoboh, A. Amourah, and J. Salah. Bi-univalent functions using bell distribution associated with meixner–pollaczek polynomials. International Journal of Mathemat- ics and Computer Science, 19(4):1077–1092, 2024. [37] A. Alsoboh and M. Darus. New subclass of analytic functions defined by q-differential operator with respect to k-symmetric points. International Journal of Mathematics and Computer Science, 14(4):761–773, 2019. [38] A. Alsoboh and M. Darus. On fekete–szegö problems for certain subclasses of analytic functions defined by differential operator involving q-ruscheweyh operator. Journal of Function Spaces, 2020:8459405, 2020. [39] 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. [40] 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 Singapore, 5(2):2366–2380, 2024. [41] A. Amourah. Coefficient estimates for a subclass of bi-univalent functions associ- ated with symmetric q-derivative operator by means of the gegenbauer polynomials. Kyungpook Mathematical Journal, 62(2):257–269, 2022. [42] A. Amourah, A. Alsoboh, D. Breaz, and S. M. El-Deeb. A bi-starlike class in a leaf- like domain defined through subordination via q-calculus. Mathematics, 12(11):1735, 2024. [43] 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. [44] A. Amourah, A. Alsoboh, J. Salah, and K. Al Kalbani. Bounds on initial coefficients for bi-univalent functions linked to q-analog of le roy-type mittag-leffler function. WSEAS Transactions on Mathematics, 23:714–722, 2024. [45] M. El-Ityan, A. Amourah, A. Alsoboh, M. B. Raba’a, and S. Hammad. Fekete–szegö inequalities for new subclasses of bi-univalent functions defined by s’al’agean q- differential operator. European Journal of Pure and Applied Mathematics, 18(2):6115, 2025. [46] M. El-Ityan, A. Amourah, S. Hammad, R. Buti, and A. Alsoboh. New sub- class of bi-univalent functions involving the wright function associated with the jung–kim–srivastav operator. Gulf Journal of Mathematics, 19(2):451–462, 2025. [47] S. Al-Ahmad, M. Mamat, N. Anakira, and R. Alahmad. Modified differential transfor- mation method for solving classes of non-linear differential equations. TWMS Journal of Applied and Engineering Mathematics, 2022. [48] N. Anakira, A. Almalki, M. J. Mohammed, S. Hamad, O. Oqilat, A. Amourah, and S. Arbia. Analytical approaches for computing exact solutions to system of volterra integro-differential equations. WSEAS Transactions on Mathematics, 23:400–407, 2024. [49] N. R. Anakira, A. K. Alomari, and I. Hashim. Application of optimal homotopy asymptotic method for solving linear delay differential equations. In AIP Conference Proceedings, volume 1571, pages 1013–1019. American Institute of Physics, November A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6689 19 of 19 2013. [50] R. W. Ibrahim, M. Z. Ahmad, and M. J. Mohammed. Generalized population dynamic operator with delay based on fractional calculus. Journal of Environmental Biology, 37(5):1139, 2016. [51] R. W. Ibrahim, M. Z. Ahmad, and M. J. Mohammed. Symmetric-periodic solutions for some types of generalized neutral equations. Mathematical Sciences, 10(4):219– 226, 2016. [52] Y. Al-Qudah. A robust framework for the decision-making based on single-valued neutrosophic fuzzy soft expert setting. International Journal of Neutrosophic Science, 23(2):195–210, 2024. [53] 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. [54] Y. Al-Qudah, M. Alaroud, H. Qoqazeh, S.E. Alhazmi, and S. Al-Omari. Approximate analytic–numeric fuzzy solutions of fuzzy fractional equations using a residual power series approach. Symmetry, 14(4):804, 2022. [55] 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, 13:1294, 2025. [56] 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, 23:201–201, 2025. [57] H. Subclasses of bi-univalent functions related to shell-like curves connected with fibonacci numbers. Unpublished manuscript.