EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 7125 ISSN 1307-5543 – ejpam.com Published by New York Business Global Coefficient Problems for Bi-Univalent Functions via q–Rabotnov Kernels and q–Fibonacci Subordination Abdullah Alsoboh1, Ahmad Almalkawi2, Ala Amourah3, Khaled Al Mashrafi1,∗, Tala Sasa4 1 Department of Basic and Applied Sciences, College of Applied and Health Sciences, A’Sharqiyah University, Post Box No. 42, Post Code No. 400, Ibra, Sultanate of Oman 2 Modern College of Business and Science, Muscat, Sultanate of Oman 3 Mathematics Education Program, Faculty of Education and Arts, Sohar University, Sohar 311, Oman 4 Department of Mathematics, Faculty of Science, Applied Science Private University, Amman, Jordan Abstract. Motivated by the interplay between q–calculus and geometric function theory, this paper introduces and investigates a new subclass of bi-univalent functions associated with shell- like domains generated through the q–Rabotnov function and the q–analogue of Fibonacci num- bers. A central contribution of this work is the definition of a novel q–derivative operator, constructed via convolution with kernels involving the q–Rabotnov function. Employing the subordination principle, we derive sharp coefficient estimates for the initial Taylor–Maclaurin co- efficients |α2| and |α3|, and establish Fekete–Szegö-type inequalities for the proposed class. The results obtained here unify and extend several recent contributions in the theory of bi-univalent functions, while also highlighting the role of q–special functions in generating new analytic struc- tures. These findings enrich the structural understanding of bi-univalent functions and suggest future directions involving operator theory, convolution structures, and further applications of q–calculus in complex analysis. 2020 Mathematics Subject Classifications: 30A36, 30C45, 81P68, 11B37 Key Words and Phrases: Analytic functions, univalent functions, convolution, Fibonacci numbers, Fekete–Szegö, q-Rabotnov function, quantum calculus 1. Introduction Geometric function theory has long been recognized as a fertile area of complex anal- ysis, focusing on the structural, geometric, and analytic properties of functions that are analytic and univalent in the open unit disk U = {z ∈ C : |z| < 1}. A central theme in this field is the study of subclasses of analytic and bi-univalent functions, which often ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7125 Email addresses: khaled.almashrafi@asu.edu.om (K. Al Mashrafi) 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), 7125 2 of 18 arise through subordination, convolution operators, or fractional-calculus techniques. Classical problems such as estimating initial coefficients, growth and distortion theo- rems, and Fekete–Szegö type inequalities remain at the core of ongoing investigations, and their generalizations via q-calculus have opened new avenues for research. The advent of q-calculus, sometimes referred to as the calculus of finite differences, has significantly enriched analytic function theory by providing a powerful framework for developing q-analogues of well-known operators and function classes. Through this approach, several subclasses with deep geometric and algebraic structures have been introduced and analyzed. In particular, the q-calculus has established strong links with special functions, combinatorics, and orthogonal polynomials, thus extending the appli- cability of classical geometric function theory to discrete and fractional domains. This versatility underscores its role in the advancement of both theoretical and applied per- spectives (see, e.g., [1–21]). The q–gamma function Γq, regarded as the natural q–analogue of the Euler gamma function, is a cornerstone of the modern q–calculus and plays a fundamental role in the construction of analytical operators. It is defined recursively (see [22, 23]) by Γq(κ+ 1) = 1− qκ 1− q Γq(κ) = [κ]q Γq(κ), (1) where the q–integer [κ]q is given by [κ]q =  1− qκ 1− q , 0 < q < 1, κ ∈ C∗ = C \ {0}, 1, q 7→ 0+, κ ∈ C∗, κ, q 7→ 1−, κ ∈ C∗, γ−1∑ n=0 qn, 0 < q < 1, κ = γ ∈ N. This formulation shows that Γq not only preserves the essential structural features of the classical gamma function, but also incorporates the discrete deformation encoded by the parameter q. Consequently, it provides a unifying framework in which fractional-order operators and kernel-type generating functions can be generalized and studied within analytic function theory. Closely associated with Γq is the q–analogue of the Pochhammer symbol, or q–shifted factorial, defined by (see [23]) (κ; q)n = (1− κ)(1− κq) · · · (1− κqn−1), n = 1, 2, 3, . . . , 1, n = 0, which admits the representation; (κ; q)n = (1− q)n Γq(κ+ n) Γq(κ) , n > 0. A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7125 3 of 18 This identity highlights the intrinsic connection between the q–shifted factorial and the q–gamma function, a relationship that underpins many of the convolution and subor- dination operators employed in geometric function theory. In particular, it serves as a fundamental building block in the analytic modeling of q–extensions of bi-univalent function classes. In parallel, Rabotnov-type kernels, first introduced by Rabotnov [24] within the framework of linear viscoelasticity, have proven to be indispensable tools for modeling hereditary phenomena such as creep and relaxation. Expressed in terms of convolu- tion operators involving Mittag–Leffler-type functions, these kernels provide a rigorous representation of fractional-order operators in constitutive equations [25]. Their remark- able flexibility has made them standard in the mathematical modeling of stress–strain relations with memory effects in mechanics and engineering. Moreover, their intrinsic connection with fractional calculus places them as a cornerstone in the analysis of vis- coelastic materials and dynamical systems [26], and some applications can be found in [1, 2, 4, 27–41]. The analytic nature of Rabotnov-type kernels naturally invites their extension into geometric function theory, particularly when combined with the discrete framework of q-calculus. Such an interplay not only bridges fractional viscoelastic models with an- alytic operator theory but also enables the construction of novel subclasses of analytic and bi-univalent functions. These connections provide a robust mechanism for encoding hereditary behavior and nonlocal operators into analytic settings, thereby allowing clas- sical results—such as coefficient bounds, growth and distortion estimates, and Fekete– Szegö inequalities—to be extended in new directions. Motivated by these observations, Alsoboh et al. [42–44] recently employed subordi- nation techniques to define a new family of q-starlike functions associated with the q- analogue of Fibonacci numbers. Their construction revealed a fundamental connection between q-Fibonacci numbers κq and the associated q-Fibonacci polynomials, expressed through the mapping Ω(z; q) = 1 + qκ2 qz 2 1− κq z − qκ2 qz 2 , (2) introduced a new family of q–starlike functions. They also established a fundamen- tal connection between the q–analogue of Fibonacci numbers κq and their associated Fibonacci polynomials κq = 1− √ 4q + 1 2q . (3) In particular, they proved that if Ω(z; q) = 1 + ∞∑ n=1 p̂n z n, A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7125 4 of 18 then the coefficients p̂n satisfy the recurrence relation p̂n =  κq, n = 1, (2q + 1)κ2 q , n = 2, (3q + 1)κ3 q , n = 3,( δn+1(q) + q δn−1(q) ) κn q , s ≥ 4, . (4) In the present work, we introduce and investigate a novel subclass of bi-univalent functions generated by the q-Rabotnov function together with the q-analogue of Fi- bonacci numbers. Using the principle of subordination, we derive coefficient bounds for the initial Taylor–Maclaurin coefficients and establish sharp Fekete–Szegö-type inequal- ities for the proposed function class. Our results extend several recent frameworks in geometric function theory and build new bridges between fractional viscoelastic model- ing, q-calculus, and the analytic theory of bi-univalent functions. 2. Preliminaries Let A denote the family of all analytic functions defined on the open unit disk U, where U is the set of all complex numbers z = a + ib (with a, b ∈ R) satisfying |z| < 1. Geometrically, U represents the collection of all points in the complex plane that lie strictly inside the unit circle centered at the origin. The functions f ∈ A are normalized to satisfy the following initial conditions: f(0) = 0 and f ′(0) = 1. These normalization conditions ensure that the functions are uniquely determined and facilitate the study of their properties within the unit disk. For every function f ∈ A, the Taylor-Maclaurin series expansion can be expressed in the following form: f(z) = z + ∞∑ n=2 αn z n, (z ∈ U). (5) An analytic function f that satisfies |f(z)| < 1 and f(0) = 0 within the domain U is called a Schwartz function. When considering two functions f1 and f2 from A, f1 is referred to as subordinate to f2, denoted by f1 ≺ f2, if a Schwarz function g exists such that f1(z) = f2(g(z)) for all z ∈ U. Additionally, examine the class S, which includes all functions f ∈ A that are univalent (injective) on the unit disk U. Let P represent the collection of functions within A that possess positive real parts, defined as follows: p(z) = 1 + ∞∑ n=1 pnz n = 1 + p1z + p2z 2 + p3z 3 + . . . , (6) where |pn| ≤ 2, for all n ≥ 1. (7) A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7125 5 of 18 This is in accordance with the renowned Carathéodory’s Lemma (for more details, see [45]). Essentially, δ ∈ P if and only if ε(z) ≺ (1 + z)(1− z)−1 for z ∈ U. As the foundation upon which many important subclasses of analytic functions are built, the class P is crucial to the study of analytic functions. For any function f in the subfamily S of A, there exists an inverse function denoted f−1 and defined by z = f−1(f(z)) and ξ = f(f−1(ξ)), (r0(f) ≥ 0.25; |ξ| < r0(f); z ∈ U) . (8) where η(ξ) = f−1(ξ) = ξ − α2ξ 2 + ( 2α2 2 − α3 ) ξ3 − ( 5α3 2 + α4 − 5α3α2 ) ξ4 + · · · . (9) A function f ∈ S is said to be bi-univalent if both f and its inverse f−1 belong to the class S. The family of all such functions, denoted by Σ, forms a natural and significant subclass of S within the unit disk U. Several classical examples illustrate this concept. For example, f1(z) = z 1 + z with its inverse f−1 1 (z) = z 1− z is a typical element of Σ. Similarly, f2(z) = − log(1− z) admits the inverse f−1 2 (z) = e2z − 1 e2z + 1 , while f3(z) = 1 2 log ( 1 + z 1− z ) has the inverse f−1 3 (z) = ez − 1 ez . These examples emphasize the structural interplay between a bi-univalent function and its inverse, and they highlight the analytical richness of the class Σ in the context of geometric function theory. Definition 1. [46] Let β, δ,κ ∈ C with <(β) > 0, <(δ) > 0, <(κ) > 0, and |q| < 1. The generalized q–Mittag–Leffler function Eδ β,λ is defined by Eδ β,λ(z; q) = ∞∑ n=0 (qδ; q)n (q; q)n zn Γq(β n+ λ) , (10) where Γq denotes the q–gamma function given in (1). In the limiting case q → 1−, the function Eδ β,λ(z; q) reduces to the classical gen- eralized Mittag–Leffler function. This limiting behavior elegantly bridges the discrete q–framework with its continuous analog. Motivated by this connection, we now intro- duce the q–analogue of the Rabotnov function as follows. Definition 2. Let β ∈ C with <(β) > 0, λ > 0, and |q| < 1. The q–Rabotnov type function Φδ β,λ(z; q) is defined by Φδ β,λ(z; q) = zβ ∞∑ n=0 (qδ; q)n (q; q)n [λ]nq Γq((n+ 1)(1 + β)) zn(1+β). (11) It should be noted that in the limiting case q → 1−, the function Φδ β,λ(z; q) reduces to the classical Rabotnov function Φβ,λ(z) (see [24]), thus establishing a natural link A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7125 6 of 18 between the q framework and its classical analog. Since Φδ β,λ(z; q) is not normalized, we consider the following normalized form: Rδ β,λ(z; q) = z 1 1+β +1 Γq(1 + β)Φδ β,λ ( z 1 1+β ; q ) = z + ∞∑ n=2 (qδ; q)n−1 (q; q)n−1 [λ]n−1 q Γq(1 + β) Γq((1 + β)n) zn, z ∈ U. (12) Remark 1. The function Φδ β,λ(z; q) can be interpreted as a q–analogue of kernel-type generating functions that frequently arise in the investigation of analytic and bi-univalent function classes. It encapsulates the combined effect of the generalized q –Mittag - Leffler structure together with the parameter λ, and in the limiting case q → 1−, it reduces to its classical analog expressed in terms of the Euler gamma function. Such kernel functions serve as fundamental building blocks in the development of subclasses of analytic functions defined through subordination principles, convolution structures, and operators associated with fractional q–calculus. We now introduce a linear operator of Hadamard–convolution type associated with the q–Rabotnov kernel. Definition 3. For β, δ ∈ C with <(β) > 0 and λ > 0, the linear operator F δ β,λ : A → A is defined by F δ β,λ(f(z); q) = Rδ β,λ(z; q) ∗ f(z) = z + ∞∑ n=2 (qδ; q)n−1 (q; q)n−1 [λ]n−1 q Γq(1 + β) Γq(n(1 + β)) αnz n, = z + [δ]q [λ]q Γq(1 + β) Γq ( 2(1 + β) ) α2 z 2 + [δ]q [δ + 1]q [λ] 2 q Γq(1 + β) [2]q Γq ( 3(1 + β) ) α3 z 3 +O(z4) (13) where f is of the form (5), and ∗ denotes the Hadamard product (or coefficient-wise) of power series. Remark 2. The operator F δ β,λ generalizes the classical convolution operators by in- corporating q–Rabotnov kernels. Such operators play a crucial role in constructing and investigating subclasses of analytic and bi-univalent functions, particularly in deriving sharp coefficient bounds and Fekete–Szegö type inequalities. The advent of q-calculus has significantly advanced the study of analytic function the- ory by enabling the discovery of novel subclasses with intricate geometric and algebraic properties. These developments underscore the versatility of the q-calculus, demonstrat- ing its potential to enrich the classical function theory and uncover new mathematical phenomena. The relevance of these findings extends to both theoretical and applied settings, providing a solid foundation for future research and innovation in the field [1, 2, 4, 6, 8–10, 12–14, 37–41, 47, 48]. A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7125 7 of 18 3. Definition and example Motivated by q-Fibonacci numbers and the q–Rabotnov operator, this section will now look at a novel subclasses of bi-univalent functions related to shell-like curves. Definition 4. Let µ ≥ 0. A function f ∈ Σ, defined by (5), is said to belong to the class RΣµ q (β, δ, λ) if the following subordinations are satisfied: µ∂q ( F δ β,λ(f(z); q) ) + (1− µ) F δ β,λ(f(z); q) z ≺ Ω(z; q) := 1 + qκ2 qz 2 1− κqz − qκ2 qz 2 , (z ∈ U), (14) and µ∂q ( F δ β,λ(η(ξ); q) ) + (1− µ) F δ β,λ(η(ξ); q) ξ ≺ Ω(ξ; q) := 1 + qκ2 qξ 2 1− κqξ − qκ2 qξ 2 , (ξ ∈ U), (15) where η = f−1 denotes the inverse of f , ∂q represents the q–derivative, and κq is specified in (3). By prescribing suitable specializations of the parameters q and µ, a variety of familiar subclasses of the bi-univalent function class Σ. For clarity, we present below several representative examples, illustrating how the general class RΣµ q (β, δ, λ) reduces to well- known families under particular parameter choices. Example 1. If we take µ = 1 in Definition 4, then a function f ∈ Σ is said to belong to the class RΣ1 q (β, δ, λ) whenever the following subordinations hold: ∂q ( F δ β,λ(f(z); q) ) ≺ Ω(z; q) := 1 + qκ2 qz 2 1− κqz − qκ2 qz 2 , (z ∈ U), (16) and ∂q ( F δ β,λ(η(ξ); q) ) ≺ Ω(ξ; q) := 1 + qκ2 qξ 2 1− κqξ − qκ2 qξ 2 , (ξ ∈ U), (17) where η = f−1 denotes the inverse of f , ∂q is the q–derivative, and κq is given by (3). Example 2. If we take µ = 0 in Definition 4, then a function f ∈ Σ is said to belong to the class RΣ0 q (β, δ, λ) whenever the following subordinations hold: F δ β,λ(f(z); q) z ≺ Ω(z; q) := 1 + qκ2 qz 2 1− κqz − qκ2 qz 2 , (z ∈ U), (18) and F δ β,λ(η(ξ); q) ξ ≺ Ω(ξ; q) := 1 + qκ2 qξ 2 1− κqξ − qκ2 qξ 2 , (ξ ∈ U), (19) where η = f−1 denotes the inverse of f , and κq is given by (3). A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7125 8 of 18 Example 3. If we let q → 1− in Definition 4, then the class RΣµ q (β, δ, λ) reduces to its classical analogue RΣµ(β, δ, λ). In this case, a function f ∈ Σ belongs to the class if the subordinations µ f ′(z) + (1− µ) f(z) z ≺ Ω(z) := 1 + κ2z2 1− κz − κ2z2 , (z ∈ U), (20) and µ η′(ξ) + (1− µ) η(ξ) ξ ≺ Ω(ξ) := 1 + κ2ξ2 1− κξ − κ2ξ2 , (ξ ∈ U), (21) hold, where η = f−1 is the inverse of f , and κ = 1− √ 5 2 = limq→1− κq. Here the operator ∂q is replaced by the classical derivative. 4. Main Results In this section, we obtain the initial Taylor coefficients |α2| and |α3| for the bi- univalent starlike and convex subclass RΣµ q (β, δ, λ). Firstly, let p(z) = 1 + p1z + p2z 2 + p3z 3 + . . . , and p(z) ≺ Ω(z; q). Then there exist δ ∈ P such that |ε(z)| < 1 in U and p(z) = Ω(ε(z); q), we have ℏ(z) = (1 + ε(z))(1− ε(z))−1 = 1 + ϑ1z + ϑ2z 2 + · · · ∈ P (z ∈ U). (22) It follows that ε(z) = ϑ1z 2 + ( ϑ2 − ϑ2 1 2 ) z2 2 + ( ϑ3 − ϑ1ϑ2 − ϑ3 1 4 ) z3 2 + · · · , (23) and Ω(ε(z); q) = 1 + p̂1 [ ϑ1z 2 + ( ϑ2 − ϑ2 1 2 ) z2 2 + ( ϑ3 − ϑ1ϑ2 − ϑ3 1 4 ) z3 2 + · · · ] + p̂2 [ ϑ1z 2 + ( ϑ2 − ϑ2 1 2 ) z2 2 + ( ϑ3 − ϑ1ϑ2 − ϑ3 1 4 ) z3 2 + · · · ]2 + p̂3 [ ϑ1z 2 + ( ϑ2 − ϑ2 1 2 ) z2 2 + ( ϑ3 − ϑ1ϑ2 − ϑ3 1 4 ) z3 2 + · · · ]3 + · · · = 1 + p̂1ϑ1 2 z + 1 2 [( ϑ2 − ϑ2 1 2 ) p̂1 + ϑ2 1 2 p̂2 ] z2 + 1 2 [( ϑ3 − ϑ1ϑ2 + ϑ3 1 4 ) p̂1 + ϑ1 ( ϑ2 − ϑ2 1 2 ) p̂2 + ϑ3 1 4 p̂3 ] z3 + · · · . (24) Similarly, there exists an analytic function ν such that |ν(ξ)| < 1 in U and p(ξ) = Ω(ν(ξ); q). Therefore, the function κ(ξ) = (1 + ν(ξ))(1− ν(ξ))−1 = 1 + υ1ξ + υ2ξ 2 + · · · ∈ P. (25) A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7125 9 of 18 It follows that ν(ξ) = υ1ξ 2 + ( υ2 − υ21 2 ) ξ2 2 + ( υ3 − υ1υ2 − υ31 4 ) ξ3 2 + · · · , (26) and Ω(ν(ξ); q) = 1 + p̂1υ1 2 ξ + 1 2 [( υ2 − υ21 2 ) p̂1 + υ21 2 p̂2 ] ξ2 + 1 2 [( υ3 − υ1υ2 + υ31 4 ) p̂1 + υ1 ( υ2 − υ21 2 ) p̂2 + υ31 4 p̂3 ] ξ3 + · · · . (27) In the following theorem we determine the initial Taylor coefficients |α2| and |α2| for the class RΣµ q (β, δ, λ). Later we will reduce these bounds to other classes for special cases. Theorem 1. Let f given by (5) be in the class RΣµ q (β, δ, λ). Then |α2| ≤ min  |κq | [λ]q √√√√√√ [2]q Γq ( 3(1+β) ) Γ2 q ( 2(1+β) ) κq ( 1 + q µ [2]q ) [δ]q [δ + 1]q Γq(1 + β) Γ2 q ( 2(1 + β) ) − ( 1 + µ q )2 [δ]2q [2]q Γ 2 q(1 + β) Γq ( 3(1 + β) ) ( (2q + 1)κq − 1 )  , κ2 q Γ2 q ( 2(1+β) )( 1+µ q )2 [δ]2q [λ] 2 q Γ 2 q(1+β)  , and ∣∣α3 ∣∣ ≤ κ2 q Γ 2 q ( 2(1 + β) )( 1 + µ q )2 [δ]2q [λ] 2 q Γ 2 q(1 + β) + [2]q ∣∣κq ∣∣Γq(3(1 + β) )( 1 + q µ [2]q ) [δ]q [δ + 1]q [λ]2q Γq(1 + β) . Proof. Let f ∈ RΣµ q (β, δ, λ) and η = f−1. Considering (14) and (15) we have µ∂q ( F δ β,λ(f(z); q) ) + (1− µ) F δ β,λ(f(z); q) z = Ω(ε(z); q), (z ∈ U), (28) and µ∂q ( F δ β,λ(η(ξ); q) ) + (1− µ) F δ β,λ(η(ξ); q) ξ = Ω(ν(ξ); q), (ξ ∈ U). (29) Using (12), we have µ∂q ( F δ β,λ(f(z); q) ) + (1− µ) F δ β,λ(f(z); q) z = 1 + ( 1 + µ q ) [δ]q [λ]q Γq(1 + β) Γq ( 2(1 + β) ) α2 z + ( 1 + q µ [2]q ) [δ]q [δ + 1]q [λ] 2 q Γq(1 + β) [2]q Γq ( 3(1 + β) ) α3 z 2 + O(z3). (30) A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7125 10 of 18 and µ∂q ( F δ β,λ(η(ξ); q) ) + (1− µ) F δ β,λ(η(ξ); q) ξ = 1− ( 1 + µ q ) [δ]q [λ]q Γq(1 + β) Γq ( 2(1 + β) ) α2 ξ + ( 1 + q µ [2]q ) [δ]q [δ + 1]q [λ] 2 q Γq(1 + β) [2]q Γq ( 3(1 + β) ) ( 2α2 2 − α3 ) ξ2 + O(ξ3). (31) By comparing (28) and (30), along (24), yields( 1 + µ q ) [δ]q [λ]q Γq(1 + β) Γq ( 2(1 + β) ) α2 z + ( 1 + q µ [2]q ) [δ]q [δ + 1]q [λ] 2 q Γq(1 + β) [2]q Γq ( 3(1 + β) ) α3 z 2 + · · · = p̂1ϑ1 2 z + 1 2 [( ϑ2 − ϑ2 1 2 ) p̂1 + ϑ2 1 2 p̂2 ] z2 + · · · . (32) Besied that, by comparing (24) and (31), along (27), yields − ( 1 + µ q ) [δ]q [λ]q Γq(1 + β) Γq ( 2(1 + β) ) α2 ξ + ( 1 + q µ [2]q ) [δ]q [δ + 1]q [λ] 2 q Γq(1 + β) [2]q Γq ( 3(1 + β) ) ( 2α2 2 − α3 ) ξ2 + · · · = p̂1 υ1 2 ξ + 1 2 [( υ2 − υ2 1 2 ) p̂1 + υ2 1 2 p̂2 ] ξ2 + · · · . (33) Equating the pertinent coefficient in (32) and (33), we obtain( 1 + µ q ) [δ]q [λ]q Γq(1 + β) Γq ( 2(1 + β) ) α2 = p̂1ϑ1 2 (34) − ( 1 + µ q ) [δ]q [λ]q Γq(1 + β) Γq ( 2(1 + β) ) α2 = p̂1υ1 2 (35)( 1 + q µ [2]q ) [δ]q [δ + 1]q [λ] 2 q Γq(1 + β) [2]q Γq ( 3(1 + β) ) α3 = 1 2 [( ϑ2 − ϑ2 1 2 ) p̂1 + ϑ2 1 2 p̂2 ] (36)( 1 + q µ [2]q ) [δ]q [δ + 1]q [λ] 2 q Γq(1 + β) [2]q Γq ( 3(1 + β) ) ( 2α2 2 − α3 ) = 1 2 [( υ2 − υ21 2 ) p̂1 + υ21 2 p̂2 ] (37) From (34) and (35), we have ϑ1 = −υ1 ⇐⇒ ϑ2 1 = υ21, (38) and using (4), we have A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7125 11 of 18 α2 2 = κ2 q Γ 2 q ( 2(1 + β) ) 8 ( 1 + µ q )2 [δ]2q [λ] 2 q Γ 2 q(1 + β) ( ϑ2 1 + υ21 ) , (39) or equivalent to ( ϑ2 1 + υ21 ) = 8 ( 1 + µ q )2 [δ]2q [λ] 2 q Γ 2 q(1 + β) κ2 q Γ 2 q ( 2(1 + β) ) α2 2, (40) Now, by summing (36) and (37), we obtain 2 ( 1 + q µ [2]q ) [δ]q [δ + 1]q [λ] 2 q Γq(1 + β) [2]q Γq ( 3(1 + β) ) α2 2 = (ϑ2 + υ2)κq 2 + [ (2q + 1)κ2 q 4 − κq 4 ] ( ϑ2 1 + υ21 ) . (41) By putting (39) in (41), with doing some calculations, yields to α2 2 = (ϑ2 + υ2)κ2 q [2]q Γq ( 3(1 + β) ) Γ2 q ( 2(1 + β) ) 4 [λ]2q { κq ( 1 + q µ [2]q ) [δ]q [δ + 1]q Γq(1 + β) Γ2 q ( 2(1 + β) ) − ( 1 + µ q )2 [δ]2q [2]q Γ 2 q(1 + β) Γq ( 3(1 + β) ) ( (2q + 1)κq − 1 ) } . (42) Using (7) for (42), we have |α2| ≤ |κq| [λ]q √√√√√√√ [2]q Γq ( 3(1 + β) ) Γ2 q ( 2(1 + β) ){ κq ( 1 + q µ [2]q ) [δ]q [δ + 1]q Γq(1 + β) Γ2 q ( 2(1 + β) ) − ( 1 + µ q )2 [δ]2q [2]q Γ 2 q(1 + β) Γq ( 3(1 + β) ) ( (2q + 1)κq − 1 ) } . (43) Besided that, from (39)∣∣α2 ∣∣ ≤ κ2 q Γ 2 q ( 2(1 + β) )( 1 + µ q )2 [δ]2q [λ] 2 q Γ 2 q(1 + β) . Now, so as to find the bound on |α3|, let’s subtract from (36) and (37) along (39), we obtain α3 = α2 2 + [2]q κq Γq ( 3(1 + β) ) 4 ( 1 + q µ [2]q ) [δ]q [δ + 1]q [λ]2q Γq(1 + β) (ϑ2 − υ2) . (44) Hence, we get∣∣α3 ∣∣ = ∣∣α2 ∣∣2 + [2]q ∣∣κq ∣∣Γq(3(1 + β) )( 1 + q µ [2]q ) [δ]q [δ + 1]q [λ]2q Γq(1 + β) . (45) Then, in view of (39), we obtain∣∣α3 ∣∣ ≤ κ2 q Γ 2 q ( 2(1 + β) )( 1 + µ q )2 [δ]2q [λ] 2 q Γ 2 q(1 + β) + [2]q ∣∣κq ∣∣Γq(3(1 + β) )( 1 + q µ [2]q ) [δ]q [δ + 1]q [λ]2q Γq(1 + β) . (46) In the following theorem, we find the Fekete-Szegö functional for f ∈ RΣµ q (β, δ, λ). A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7125 12 of 18 Theorem 2. Let f given by (5) be in the class RΣµ q (β, δ, λ) and ρ ∈ R. Then we have ∣∣α3 − ρα2 2 ∣∣ ≤  κq [2]q Γq ( 3(1+β) ) [λ]2q Γq(1+β) [δ]q ( 1+q µ [2]q ) [δ+1]q , 0 ≤ ∣∣D(ρ) ∣∣ ≤ 1( 1+q µ [2]q ) [δ+1]q 4 ∣∣D(ρ) ∣∣, ∣∣D(ρ) ∣∣ ≥ 1( 1+q µ [2]q ) [δ+1]q . where D(ρ) = (1− ρ)κq Γ 2 q ( 2(1 + β) ){ κq ( 1 + q µ [2]q ) [δ + 1]q Γ 2 q ( 2(1 + β) ) − ( 1 + µ q )2 [δ]q [2]q Γq(1 + β) Γq ( 3(1 + β) ) ( (2q + 1)κq − 1 ) } (47) Proof. Let f ∈ RΣµ q (β, δ, λ), from (42) and (44) we have α3 − ρα2 2 = [2]q κq Γq ( 3(1 + β) ) 4 ( 1 + q µ [2]q ) [δ]q [δ + 1]q [λ]2q Γq(1 + β) (ϑ2 − υ2) + (1− ρ)(ϑ2 + υ2)κ2 q [2]q Γq ( 3(1 + β) ) Γ2 q ( 2(1 + β) ) 4 [λ]2q { κq ( 1 + q µ [2]q ) [δ]q [δ + 1]q Γq(1 + β) Γ2 q ( 2(1 + β) ) − ( 1 + µ q )2 [δ]2q [2]q Γ 2 q(1 + β) Γq ( 3(1 + β) ) ( (2q + 1)κq − 1 ) } = κq [2]q Γq ( 3(1 + β) ) 4 [λ]2q Γq(1 + β) [δ]q [( D(ρ) + 1( 1 + q µ [2]q ) [δ + 1]q ) ϑ2 + ( D(ρ)− 1( 1 + q µ [2]q ) [δ + 1]q ) υ2 ] (48) where D(ρ) is given by (47). Then, by taking modulus of (48), we conclude that ∣∣α3 − ρα2 2 ∣∣ ≤  κq [2]q Γq ( 3(1+β) ) [λ]2q Γq(1+β) [δ]q ( 1+q µ [2]q ) [δ+1]q , 0 ≤ ∣∣D(ρ) ∣∣ ≤ 1( 1+q µ [2]q ) [δ+1]q 4 ∣∣D(ρ) ∣∣, ∣∣D(ρ) ∣∣ ≥ 1( 1+q µ [2]q ) [δ+1]q . 5. Corollaries The general coefficient estimates established in Theorems 1 and 2 give rise to several noteworthy special cases under suitable choices of the parameters µ and q. In particular, when one considers the purely q–differential subclass (µ = 1), the ratio-type subclass (µ = 0), and the classical limiting case (q → 1−), the results simplify to the following corollaries. A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7125 13 of 18 Corollary 1 (µ = 1). Let f given by (5) belong to RΣ1 q (β, δ, λ). Then |α2| ≤ min  |κq| [λ]q √√√√√√ [2]q Γq ( 3(1 + β) ) Γ2 q ( 2(1 + β) ) κq ( 1 + q[2]q ) [δ]q [δ + 1]q Γq(1 + β) Γ2 q ( 2(1 + β) ) − (1 + q)2 [δ]2q [2]q Γ 2 q(1 + β) Γq ( 3(1 + β) ) ( (2q + 1)κq − 1 ) , κ2 q Γ 2 q ( 2(1 + β) ) (1 + q)2 [δ]2q [λ] 2 q Γ 2 q(1 + β)  , and |α3| ≤ κ2 q Γ 2 q ( 2(1 + β) ) (1 + q)2 [δ]2q [λ] 2 q Γ 2 q(1 + β) + [2]q |κq|Γq ( 3(1 + β) )( 1 + q[2]q ) [δ]q [δ + 1]q [λ]2q Γq(1 + β) . Moreover, for any ρ ∈ R, ∣∣α3−ρα2 2 ∣∣ ≤  κq [2]q Γq ( 3(1 + β) ) [λ]2q Γq(1 + β) [δ]q ( 1 + q[2]q ) [δ + 1]q , 0 ≤ ∣∣D1(ρ) ∣∣ ≤ 1( 1 + q[2]q ) [δ + 1]q , 4 ∣∣D1(ρ) ∣∣, ∣∣D1(ρ) ∣∣ ≥ 1( 1 + q[2]q ) [δ + 1]q , where D1(ρ) = (1− ρ)κq Γ 2 q ( 2(1 + β) ) κq ( 1 + q[2]q ) [δ + 1]q Γ 2 q ( 2(1 + β) ) − (1 + q)2 [δ]q [2]q Γq(1 + β) Γq ( 3(1 + β) ) ( (2q + 1)κq − 1 ) . Corollary 2 (µ = 0). Let f given by (5) belong to RΣ0 q (β, δ, λ). Then |α2| ≤ min  |κq| [λ]q √√√√√√ [2]q Γq ( 3(1 + β) ) Γ2 q ( 2(1 + β) ) κq [δ]q [δ + 1]q Γq(1 + β) Γ2 q ( 2(1 + β) ) − [δ]2q [2]q Γ 2 q(1 + β) Γq ( 3(1 + β) ) ( (2q + 1)κq − 1 ) , κ2 q Γ 2 q ( 2(1 + β) ) [δ]2q [λ] 2 q Γ 2 q(1 + β)  , and |α3| ≤ κ2 q Γ 2 q ( 2(1 + β) ) [δ]2q [λ] 2 q Γ 2 q(1 + β) + [2]q |κq|Γq ( 3(1 + β) ) [δ]q [δ + 1]q [λ]2q Γq(1 + β) . Moreover, for any ρ ∈ R, ∣∣α3 − ρα2 2 ∣∣ ≤  κq [2]q Γq ( 3(1 + β) ) [λ]2q Γq(1 + β) [δ]q [δ + 1]q , 0 ≤ ∣∣D0(ρ) ∣∣ ≤ 1 [δ + 1]q , 4 ∣∣D0(ρ) ∣∣, ∣∣D0(ρ) ∣∣ ≥ 1 [δ + 1]q , A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7125 14 of 18 where D0(ρ) = (1− ρ)κq Γ 2 q ( 2(1 + β) ) κq [δ + 1]q Γ 2 q ( 2(1 + β) ) − [δ]q [2]q Γq(1 + β) Γq ( 3(1 + β) ) ( (2q + 1)κq − 1 ) . Corollary 3 (Classical limit q → 1−). Let f given by (5) belong to RΣµ(β, δ, λ), the classical limit of RΣµ q (β, δ, λ) as q → 1−. With [n]q → n, Γq → Γ, [δ]q → δ, [δ + 1]q → δ + 1, [λ]q → λ, and κq → κ, we obtain |α2| ≤ min  |κ| λ √√√√√√ 2Γ ( 3(1 + β) ) Γ2 ( 2(1 + β) ) κ ( 1 + 2µ ) δ(δ + 1)Γ(1 + β) Γ2 ( 2(1 + β) ) − (1 + µ)2 δ2 2Γ2(1 + β) Γ ( 3(1 + β) ) ( 3κ − 1 ) , κ2 Γ2 ( 2(1 + β) ) (1 + µ)2 δ2 λ2 Γ2(1 + β)  , and |α3| ≤ κ2 Γ2 ( 2(1 + β) ) (1 + µ)2 δ2 λ2 Γ2(1 + β) + 2 |κ|Γ ( 3(1 + β) )( 1 + 2µ ) δ(δ + 1)λ2 Γ(1 + β) . Moreover, for any ρ ∈ R, ∣∣α3 − ρα2 2 ∣∣ ≤  κ 2Γ ( 3(1 + β) ) λ2 Γ(1 + β) δ ( 1 + 2µ ) (δ + 1) , 0 ≤ ∣∣Dcl(ρ) ∣∣ ≤ 1( 1 + 2µ ) (δ + 1) , 4 ∣∣Dcl(ρ) ∣∣, ∣∣Dcl(ρ) ∣∣ ≥ 1( 1 + 2µ ) (δ + 1) , where Dcl(ρ) = (1− ρ)κ Γ2 ( 2(1 + β) ) κ ( 1 + 2µ ) (δ + 1)Γ2 ( 2(1 + β) ) − (1 + µ)2 δ 2Γ(1 + β) Γ ( 3(1 + β) ) (3κ − 1) . 6. Conclusion In this work, we have introduced and studied the class RΣµ q (β, δ, λ), constructed through convolution operators involving the q–Rabotnov function and subordinated to the q– Fibonacci structure. A key feature of our investigation is the definition of a new q– derivative operator based on q–Rabotnov kernels, which provides a flexible framework for analyzing subclasses of bi-univalent functions. Within this setting, we have derived sharp coefficient estimates for the initial Taylor–Maclaurin coefficients and established corresponding Fekete–Szegö type inequalities. The general results obtained in Theorems 1 and 2 unify and extend several recent contributions to the theory of bi-univalent functions, while naturally reducing to im- portant special cases under suitable parameter choices. In particular, the framework recovers the purely q–differential subclass (µ = 1), the ratio-type subclass (µ = 0), and A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7125 15 of 18 the classical limit (q → 1−), thereby illustrating both the flexibility and the unifying character of the class RΣµ q (β, δ, λ) in geometric function theory. For future research, it would be of significant interest to develop analogous subclasses generated by other q–special functions or higher-order convolution operators, and to examine possible applications of the proposed q–derivative operator in operator theory, multivariable geometric mappings, and related analytic inequalities. References [1] A. Alsoboh and G. I. Oros. 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