EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6586
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On the Fractional q-Differintegral Operator for
Subclasses of Bi-univalent functions Subordinate to

q-Ultraspherical Polynomials

Mamoon Ahmed1, Abdullah Alsoboh2,∗, Ala Amourah3,4, Jamal Salah2

1 Department of Basic Sciences, Princess Sumaya University for Technology, Amman,
Jordan.
2 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
3 Mathematics Education Program, Faculty of Education and Arts, Sohar University,
Sohar 311, Oman
4 Jadara University Research Center, Jadara University, Jordan

Abstract. In this paper, we introduce a novel class of bi-univalent functions using the fractional
q-differintegral operator and q-Ultraspherical polynomials. We examine the Taylor-Maclaurin
coefficients |a2| and |a3| for functions in this new class. We also establish Fekete-Szegö functional
inequalities relevant to this subclass. By varying the parameters in our main results, we derive
several new findings that contribute to the theoretical development of the field.

2020 Mathematics Subject Classifications: 30A36, 30C45, 81P68, 11B37

Key Words and Phrases: Subordination; Ultraspherical Polynomial; q-Calculus; Analytic
functions; Univalent functions; bi-univalent functions; Orthogonal polynomials; Carathéodory’s
functions; Fekete-Szegö problem.

Orthogonal polynomials (OP) are a fundamental concept in mathematics, playing a
crucial role in various fields due to their unique properties. These polynomials, renowned
for their mutual orthogonality, have become indispensable tools for solving a wide range
of mathematical problems. Their applications are particularly prominent in the context
of ordinary differential equations that satisfy specific modelling criteria. The significance
of orthogonal polynomials extends beyond differential equations, finding utility in ap-
proximation theory and other areas of study. The discovery of orthogonal polynomials
can be traced back to 1784 when Legendre [1] made a groundbreaking contribution to
mathematics by identifying and exploring these polynomials. Since then, the importance
and application of orthogonal polynomials have continuously expanded, solidifying their
place as indispensable instruments in mathematical research and problem-solving [2–4].

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6586

Email addresses: abdullah.alsoboh@asu.edu.om (A. Alsoboh)

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



M. Ahmed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6586 2 of 15

Orthogonal polynomials, represented as Πd and Πt with degrees d and t respectively,
establish their orthogonality through the integral equation:∫ β

α
Πd(y)Πt(y)ξ(y)dy = 0, for d ̸= t,

where ξ(y) is a well-defined weight function within the interval (α, β).

Among the various types of orthogonal polynomials, Ultraspherical polynomials (UP)
hold significant importance. As elucidated in [5–8], a notable symbolic correlation known
as TR has been found between the generating function of orthogonal polynomials (OP)
and the integral representation of real functions. This correlation, which becomes appar-
ent through classical algebraic methods, has resulted in the identification of numerous
significant inequalities within the domain of orthogonal polynomials.

Fractional calculus operators are widely used in various disciplines within the ap-
plied sciences, especially in the study of geometric functions [3, 9–11]. A significant
development in traditional fractional calculus is the fractional q-calculus, which finds
applications in different domains. These applications are built upon the fundamental
principles established in ordinary fractional calculus. For a more comprehensive under-
standing of this subject, readers are advised to consult resources, including [12–17].

Fractional derivatives and integral operators have extensive applications in various
scientific and engineering fields. They are used to represent anomalous diffusion and
transport phenomena, characterize stress-strain relationships in viscoelastic materials,
and design controllers and filters in control theory. In signal processing, they enhance
edge detection and feature extraction, while in electrical engineering, they provide pre-
cise descriptions of non-integer order dynamics in circuits. Integral operators are crucial
in solving differential equations, potential theory, image reconstruction, probability and
statistics, and financial mathematics. These tools offer advanced models and solutions
that effectively capture the complexities of real-world behaviors and systems.

Examining the implications of this discovery on the theory of q-Ultraspherical poly-
nomials (q-UP) within this context is of significant significance. In this investigation, our
efforts have been directed toward this direction, leveraging the aforementioned findings
to formulate novel nonlinear connection equations for q-UP and drawing comparisons
with their classical counterparts.

This research explores the characteristics of a specific class by establishing correla-
tions between selected bi-univalent functions and q-Ultraspherical polynomials (q-UP).
The next section presents important mathematical notations and definitions, providing
a comprehensive framework for further analysis. In this study, we thoroughly examine
the properties and behaviors of a particular category. By connecting carefully chosen
bi-univalent functions and q-UP, our goal is to clarify their interactions and implications.



M. Ahmed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6586 3 of 15

The following section explains essential mathematical concepts and terminologies that
are necessary for understanding the subsequent discussions.

1. Preliminaries

A class of polynomials, known as q-analog of the (UP), was discovered by Askey and
Ismail in 1983 [6]. These polynomials are defined as follows:

B(κ)(ε, z; q) =
∞∑
n=0

C (κ)
n (ε; q)zn, (ε ∈ [−1, 1], z ∈ U), (1)

where U = {z ∈ C : |z| < 1} is the the open unit disk in the complex plane C.

In 2006, Chakrabarti et al. made a significant discovery regarding a set of poly-
nomials [18]. These polynomials can be regarded as the q-analog of the Ultraspherical
polynomials (UP). The first few terms of these polynomials are

C
(κ)
0 (ε; q) = 1, C

(κ)
1 (ε; q) = 2⟨κ; q⟩ε,

C
(κ)
2 (ε; q) = 2

(
⟨κ; q2⟩+ ⟨κ; q⟩2

)
ε2 − ⟨κ; q2⟩ ,

(2)

the q-bracket, denoted as ⟨κ; q⟩, is defined explicitly for 0 < q < 1 (as shown in [19]), by

⟨κ; q⟩ =



1−qκ

1−q , if κ ∈ C\{0}

qn−1 + qn−2 + · · ·+ q + 1 =
n−1∑
ȷ=0

qȷ , if κ = n ∈ N

1 , if q → 0+, κ ∈ C\{0}
κ , if q → 1−, κ ∈ C\{0}

. (3)

Let f(z) be an analytic (regular, holomorphic or monogenic) function defined in the
open unit disk U. If f(z) can be expressed as:

f(z) = z +
∞∑
n=2

anz
n, (4)

then we say that f(z) belongs to a specific class A. The subclass S of A contains the
analytic functions that are univalent and satisfy the normalization conditions:

f(0) = 0 and f ′(0) = 1. (5)

A regular function η that satisfying

|η(z)| < 1 and η(0) = 0, (z ∈ U),

is commonly referred to as a Schwartz function. If we have two functions, f1 and f2,
defined on A, we state that f1 is subordinated to f2, denoted as f1 ≺ f2, if there exists a



M. Ahmed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6586 4 of 15

Schwartz function η such that f1(z) = f2(η(z)) for all z ∈ U (for more details, see [20]).
The class P is connected to Carathéodory’s functions functions, as defined by Miller [21].
These functions satisfy the following criteria:

φ(0) = 1 and Re{φ(z)} > 0, (∀z ∈ U) .

A Taylor series expansion can provide an exact representation for any polynomial func-
tion φ(z) ∈ P. This expansion is given by:

φ(z) = 1 +
∞∑
n=1

φnz
n, (z ∈ U). (6)

where
|φn| ≤ 2, for all n ≥ 1. (7)

This is in accordance with Carathéodory’s Lemma (see [21]). Essentially, φ ∈ P if
and only if

φ(z) ≺ (1 + z)(1− z)−1, (z ∈ U).

Within the subfamily S, each function f has an inverse function denoted as f−1. This
inverse function is characterized by the equations

z = f−1(f(z)) and ξ = f(f−1(ξ)),

where

r0(f) ≥
1

4
, |ξ| < r0(f) , and (z ∈ U).

Additionally, the expression for the inverse function f−1(ξ) can be represented by the
equation

ℏ(ξ) = f−1(ξ) = ξ
(
1− a2ξ + ξ2(−a3 + 2a22)− ξ3(a4 + 5a23 − 5a3a2) + · · ·

)
. (8)

A function f(z) belongs to the subclass S and is called bi-univalent if its inverse function
f−1(ξ) also belongs to S. The subclass Σ of S contains all bi-univalent functions within
U. The table below provides examples of certain functions in the class Σ and their
respective inverse functions.

Table 1: Lists some of the functions in class Σ along with their inverses.

The function The corresponding inverse

f1(z) =
z

1−z f−1
1 (ξ) = ξ

1+ξ

f2(z) = − log (1− z) f−1
2 (ξ) = e2ξ−1

e2ξ+1

f3(z) =
1

2
log
(
1+z
1−z

)
f−1
3 (ξ) = e−ξ

(
eξ − 1

)



M. Ahmed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6586 5 of 15

We begin by revisiting the foundational concepts of q-difference calculus, a field that
underpins the analytic structure and behavior of q-difference equations and their multi-
faceted applications. The mathematical framework of q-calculus has garnered sustained
interest due to its extensive utility across disciplines such as physics, quantum mechan-
ics, and, more prominently, geometric function theory. Central to this framework is the
q-differential operator ðq, which serves as a critical tool in the study and classification of
various subclasses of univalent and multivalent functions. A seminal contribution in this
area was made by Ismail et al. in 1990 [22], who introduced a q-extension of starlike func-
tions defined on the open unit disk, thereby initiating a new line of inquiry within the
theory of geometric function spaces. More recently, Srivastava [23] provided a thorough
exposition on the theoretical underpinnings and geometric ramifications of q-analogues
of fractional derivative operators. To support continued exploration of q-calculus within
geometric function theory, a substantial body of literature is available, beginning with
foundational works [24, 25] and progressing through more recent advancements such
as [26–34].

Definition 1. [19] The q-difference operator, also known as the q-derivative, is defined
for a function f when 0 < q < 1 as follows:

ðqf(z) =



f(z)−f(qz)
z−qz , if z ̸= 0,

f ′(0), if z = 0,

f ′(z), if q → 1−, z ̸= 0.

.

Remark 1. For f ∈ A of the form (4), it can easily be seen that:

ðqf(z) = ðq

{
z +

∞∑
n=2

anz
n

}
= 1 +

∞∑
n=2

⟨n; q⟩anηn−1, (z ∈ U),

and for f−1 of the form (8), we have

ðq
(
f−1(ξ)

)
= 1− ⟨2; q⟩ a2 ξ + ⟨3; q⟩ (2a32 − a3) ξ

2 + · · · , (ξ ∈ U).

Ravikumar [35] introduced the fractional q-differintegral operator denoted by ψδ
qf(z)

for a function f(z) of the form (4). The operator is defined as follows:

ψδ
qf(z) = z +

∞∑
n=2

Γq(2− δ)Γq(n+ 1)

Γq(n+ 1− δ)
anz

n, (δ ≤ 2, z ∈ U) (9)

where Γq represents the q-gamma function defined by

Γq(n) =
(q; q)∞(1− q)1−n

(qn; q)∞
, q ∈ (0, 1), (10)



M. Ahmed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6586 6 of 15

and (κ; q)n denotes the q-shifted factorial defined for n ∈ C (see [36]). The recurrence
relation for the q-gamma function is given by

Γq(n+ 1) = ⟨n; q⟩Γq(n) (11)

where ⟨n; q⟩ represents the bracket of the form (3).

2. Definition and Examples

In this section, we will introduce novel classifications of bi-univalent functions that
exhibit a hierarchical relationship with the q-Ultraspherical polynomials (q-UP). These
newly introduced subclasses provide a more sophisticated insight into the connection
between bi-univalent functions and q-UP, enhancing our comprehension of their interre-
lationship and offering valuable insights into their properties and behaviors.

Definition 2. Let 𭟋 ∈ C \ {0} and 0 ≤ λ ≤ 1. A bi-univalent function f , defined
in equation (4), belongs to the class BΣ(𭟋, λ, δ,B(κ)(ε, z; q)) if it satisfies the following
conditions:

1 +
1

𭟋

(
ðq
(
ψδ
qf(z)

)
+ λzð2q

(
ψδ
qf(z)

)
− 1
)
≺ B(κ)(ε, z; q), (12)

and

1 +
1

𭟋

(
ðq
(
ψδ
qℏ(ξ)

)
+ λξð2q

(
ψδ
qℏ(ξ)

)
− 1
)
≺ B(κ)(ε, ξ; q). (13)

For ε ∈ (12 , 1] and δ ≤ 2, the function ℏ(ξ) is defined as the inverse of f(ξ), which is
given by equation (8). The generating function of the q-analog of the (UP), denoted as
B(κ)(ε, z; q), is defined by equation (1).

Example 1. Let λ = 1, 𭟋 ∈ C\{0}. A bi-univalent function f , defined in equation (4),
belongs to the class BΣ(𭟋, 1, δ,B(κ)(ε, z; q)) if it satisfies the following conditions:

1 +
1

𭟋

(
ðq
(
ψδ
qf(z)

)
+ zð2q

(
ψδ
qf(z)

)
− 1
)
≺ B(κ)(ε, z; q),

and

1 +
1

𭟋

(
ðq
(
ψδ
qℏ(ξ)

)
+ ξð2q

(
ψδ
qℏ(ξ)

)
− 1
)
≺ B(κ)(ε, ξ; q).

For ε ∈ (12 , 1] and δ ≤ 2, the function ℏ(ξ) is defined as the inverse of f(ξ), which is
given by equation (8). The generating function of the q-analog of the (UP), denoted as
B(κ)(ε, z; q), is defined by equation (1).

Example 2. Let λ = 0, 𭟋 ∈ C\{0}. A bi-univalent function f , defined in equation (4),
belongs to the class BΣ(𭟋, 0, δ,B(κ)(ε, z; q)) if it satisfies the following conditions:

1 +
1

𭟋

(
ðq
(
ψδ
qf(z)

)
− 1
)
≺ B(κ)(ε, z; q),



M. Ahmed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6586 7 of 15

and

1 +
1

𭟋

(
ðq
(
ψδ
qℏ(ξ)

)
− 1
)
≺ B(κ)(ε, ξ; q).

For ε ∈ (12 , 1] and δ ≤ 2, the function ℏ(ξ) is defined as the inverse of f(ξ), which is
given by equation (8). The generating function of the q-analog of the (UP), denoted as
B(κ)(ε, z; q), is defined by equation (1).

Recently, Amourah et al. [37] and Alsoboh et al. [38–40] have introduced new cate-
gories of bi-univalent functions that possess analyticity using q-Ultraspherical polynomi-
als (q-UP). These contributions primarily focused on clarifying Fekete-Szegö inequalities
and constraints related to the coefficients |a2| and |a3| for functions in these newly estab-
lished subclasses. The study of bi-univalent functions associated with q-UP has attracted
significant attention from scholars, as evident in previous works such as [35, 41–44].

The primary objective of our research is to examine the characteristics exhibited by
bi-univalent functions associated with q-Ultraspherical polynomials (q-UP). To achieve
this aim, we consider the following definitions as fundamental aspects of our exploratory
analysis.

3. Initial Bounds and the Fekete and Szegö Functional

In this section, we provide estimates for the coefficients pertaining to the class de-
fined in Definition 2, denoted by BΣ(𭟋, λ, δ,B(κ)(ε, z; q)). In 1933, Fekete and Szegö
[45] derived a rigorous upper limit for the functional ηa22 − a3, where η belongs to the
interval (0, 1). This bound, known as the traditional Fekete-Szegö inequality, remains a
challenging task to determine precisely for any compact family of functions f within the
set A with arbitrary complex values of η. Building upon the findings of Zaprawa [46], we
delve into the subsequent Szegö inequality concerning functions belonging to the class
BΣ(𭟋, λ, δ,B(κ)(ε, z; q)).

Theorem 1. Suppose f ∈ Σ as defined by equation (4) belongs to the class. Then, the
inequalities

|a2| ≤
2ε⟨κ; q⟩⟨2− δ; q⟩

∣∣𭟋∣∣√ 2 ε
Γq(3)

⟨3− δ; q⟩ ⟨κ; q⟩
√
Q1

,

and

|a3| ≤
4𭟋2 ⟨2− δ; q⟩2 ⟨κ; q⟩2ε2

⟨2; q⟩2 (Γq(3))
2 (1 + λ)2

+
2⟨κ; q⟩ ⟨3− δ; q⟩⟨2− δ; q⟩ ε

∣∣𭟋∣∣
⟨3; q⟩Γq(4)

∣∣1 + λ⟨2; q⟩
∣∣ .

apply, where

Q1 =

(
4⟨3; q⟩2⟨2− δ; q⟩ (1 + λ⟨2; q⟩)𭟋 ⟨κ; q⟩2−

2⟨2; q⟩3(1 + λ)2 ⟨3− δ; q⟩
(
⟨κ; q2⟩+ ⟨κ; q⟩2

)
ε2 + ⟨2; q⟩3(1 + λ)2 ⟨3− δ; q⟩ ⟨κ; q2⟩

)
.



M. Ahmed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6586 8 of 15

Proof. Let f ∈ BΣ(𭟋, λ, δ,B(κ)(ε, z; q)). Then, from Definition 2, for some regular
functions ψ and ϑ, |ψ(z)| < 1 and |ϑ(ξ)| < 1 for all z, ξ ∈ U, where ψ(0) = 0 = ϑ(0).
We have

1 +
1

𭟋

(
ðq
(
ψδ
qf(z)

)
+ λzð2q

(
ψδ
qf(z)

)
− 1
)
= B(κ)(ε, ψ(z); q) , (14)

and

1 +
1

𭟋

(
ðq
(
ψδ
qℏ(ξ)

)
+ λξð2q

(
ψδ
qℏ(ξ)

)
− 1
)
= B(κ)(ε, ϑ(ξ); q) . (15)

From the equalities (14) and (15)

1 +
1

𭟋

(
ðq
(
ψδ
qf(z)

)
+ λzð2q

(
ψδ
qf(z)

)
− 1
)
=

(
1 + C

(κ)
1 (ε; q)c1z+[

C
(κ)
1 (ε; q)c2 + C

(κ)
2 (ε; q)c21

]
z2 + · · ·

)
,

(16)
and

1 +
1

𭟋

(
ðq
(
ψδ
qℏ(ξ)

)
+ λξð2q

(
ψδ
qℏ(ξ)

)
− 1
)
=

(
1 + C

(κ)
1 (ε; q)d1ξ+[

C
(κ)
1 (ε; q)d2 + C

(κ)
2 (ε; q)d21

]
ξ2 + · · ·

)
.

(17)
It is a widely accepted fact that if

|ψ(z)| =
∣∣c1z + c2z

2 + c3z
3 + · · ·

∣∣ < 1, (z ∈ U)

and
|ϑ(ξ)| =

∣∣d1ξ + d2ξ
2 + d3ξ

3 + · · ·
∣∣ < 1, (ξ ∈ U),

then
|cj | ≤ 1 and |dj | ≤ 1 for all j ∈ N. (18)

In view of (4), (8), from (16) and (17), we obtain

1 +
1 + λ

𭟋

(
⟨2; q⟩Γq(3)Γq(2− δ)

Γq(3− δ)

)
a2z +

1 + ⟨2; q⟩λ
𭟋

(
⟨3; q⟩Γq(4)Γq(2− δ)

Γq(4− δ)

)
a3z

2 + · · ·

= 1 + C
(κ)
1 (ε; q)c1z +

[
C

(κ)
1 (ε; q)c2 + C

(κ)
2 (ε; q)c21

]
z2 + · · · ,

and

1− 1 + λ

𭟋

(
⟨2; q⟩Γq(3)Γq(2− δ)

Γq(3− δ)

)
a2ξ +

1 + λ⟨2; q⟩
𭟋

(
⟨3; q⟩Γq(4)Γq(2− δ)

Γq(4− δ)

)
(2a22 − a3)ξ

2

+ · · · = 1 + C
(κ)
1 (ε; q)d1ξ +

[
C

(κ)
1 (ε; q)d2 + C

(κ)
2 (ε; q)d21

]
ξ2 + · · · .

Upon carefully examining the relevant coefficients presented in (16) and (17), the fol-
lowing findings emerge:

1 + λ

𭟋

(
⟨2; q⟩Γq(3)Γq(2− δ)

Γq(3− δ)

)
a2 = C

(κ)
1 (ε; q)c1, (19)



M. Ahmed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6586 9 of 15

1 + λ⟨2; q⟩
𭟋

(
⟨3; q⟩Γq(4)Γq(2− δ)

Γq(4− δ)

)
a3 = C

(κ)
1 (ε; q)c2 + C

(κ)
2 (ε; q)c21, (20)

and

−1 + λ

𭟋

(
⟨2; q⟩Γq(3)Γq(2− δ)

Γq(3− δ)

)
a2 = C

(κ)
1 (ε; q)d1, (21)

1 + λ⟨2; q⟩
𭟋

(
⟨3; q⟩Γq(4)Γq(2− δ)

Γq(4− δ)

)
(2a22 − a3) = C

(κ)
1 (ε; q)d2 + C

(κ)
2 (ε; q)d21. (22)

It is clear from equations (19) and (21) that

c1 = −d1 (23)

and
2(1 + λ)2

𭟋2

(
⟨2; q⟩Γq(3)Γq(2− δ)

Γq(3− δ)

)2

a22 =
[
C

(κ)
1 (ε; q)

]2 (
c21 + d21

)
a22 =

𭟋2 ⟨2− δ; q⟩2
[
C

(κ)
1 (ε; q)

]2
2⟨2; q⟩2 (Γq(3))

2 (1 + λ)2

(
c21 + d21

) . (24)

By adding equation (20) to equation (22), we obtain

1 + λ⟨2; q⟩
𭟋

(
2⟨3; q⟩Γq(4)Γq(2− δ)

Γq(4− δ)

)
a22 = C

(κ)
1 (ε; q)(c2 + d2) +C

(κ)
2 (ε; q)(c21 + d21). (25)

By substituting the value of (c21 + d21) from (24) and performing some calculations, we
obtain:

a22 =
[2− δ]2q⟨3− δ; q⟩𭟋2

[
C

(κ)
1 (ε; q)

]3
(c2 + d2)

2Γq(3)

(
⟨3; q⟩2⟨2− δ; q⟩ (1 + λ⟨2; q⟩)𭟋

[
C

(κ)
1 (ε; q)

]2
− ⟨2; q⟩3(1 + λ)2 ⟨3− δ; q⟩C(κ)

2 (ε; q)

) .

By applying for the coefficients c2 and d2 and utilizing equations (11) and (2), we can
obtain.

|a2| ≤
2ε⟨κ; q⟩⟨2− δ; q⟩

∣∣𭟋∣∣√ 2 ε
Γq(3)

⟨3− δ; q⟩ ⟨κ; q⟩√√√√√
(
4⟨3; q⟩2⟨2− δ; q⟩ (1 + λ⟨2; q⟩)𭟋 ⟨κ; q⟩2 − 2⟨2; q⟩3(1 + λ)2 ⟨3− δ; q⟩×(

⟨κ; q2⟩+ ⟨κ; q⟩2
)
ε2 + ⟨2; q⟩3(1 + λ)2 ⟨3− δ; q⟩ ⟨κ; q2⟩

)
.

To construct the second assertion, we can subtract (22) from (20). This will yield

2
(
1 + λ⟨2; q⟩

)
𭟋

(
⟨3; q⟩Γq(4)Γq(2− δ)

Γq(4− δ)

)
(a3 − a22) = C

(κ)
1 (ε; q) (c2 − d2) + C

(κ)
2 (ε; q)

(
c21 − d21

)
.

(26)



M. Ahmed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6586 10 of 15

Then, considering (23) and (24), Equation (26) can be rewritten as

a3 =
𭟋2 ⟨2− δ; q⟩2

[
C

(κ)
1 (ε; q)

]2
2⟨2; q⟩2 (Γq(3))

2 (1 + λ)2

(
c21 + d21

)
+

𭟋⟨3− δ; q⟩⟨2− δ; q⟩C(κ)
1 (ε; q)

2⟨3; q⟩Γq(4)
(
1 + λ⟨2; q⟩

) (c2 − d2) .

Using (11) and (2), we can say that

|a3| ≤
4𭟋2 ⟨2− δ; q⟩2 ⟨κ; q⟩2ε2

⟨2; q⟩2 (Γq(3))
2 (1 + λ)2

+
2⟨κ; q⟩ ⟨3− δ; q⟩⟨2− δ; q⟩ ε

∣∣𭟋∣∣
⟨3; q⟩Γq(4)

∣∣1 + λ⟨2; q⟩
∣∣ .

The proof of the theorem is considered comprehensive and conclusive.

Theorem 2. Let f ∈ Σ given by (4) belongs to the class BΣ(𭟋, λ, δ,B(κ)(ε, z; q)) and
η ∈ R. Then, we have

∣∣∣a3 − ηa22

∣∣∣ ≤


2
∣∣𭟋⟨κ;q⟩

∣∣ε
⟨3;q⟩Γq(4)

(
1+λ⟨2;q⟩

) , |1− η| ≤

∣∣∣∣∣∣1−
(1+λ)2 ⟨2;q⟩3 ⟨3−δ;q⟩

(
2(⟨κ;q2⟩+⟨κ;q⟩2)ε2−⟨κ;q2⟩

)
8⟨κ;q⟩2⟨3;q⟩2

(
1+λ⟨2;q⟩

)
⟨2−δ;q⟩𭟋 ε

∣∣∣∣∣∣ ,

4 |⟨κ; q⟩| ε
∣∣∣H(η)

∣∣∣, |1− η| ≥

∣∣∣∣∣∣1−
⟨2;q⟩3(1+λ)2 ⟨3−δ;q⟩

(
2(⟨κ;q2⟩+⟨κ;q⟩2)ε2−⟨κ;q2⟩

)
8⟨κ;q⟩2⟨3;q⟩2

(
1+λ⟨2;q⟩

)
⟨2−δ;q⟩𭟋 ε

∣∣∣∣∣∣ .
where

H(η) =
(1− η) [2− δ]2q⟨3− δ; q⟩𭟋2

[
C

(κ)
1 (ε; q)

]2
2Γq(3)

(
⟨2− δ; q⟩ (1 + λ⟨2; q⟩)𭟋

[
C

(κ)
1 (ε; q)

]2
⟨3; q⟩2 − ⟨2; q⟩3(1 + λ)2 ⟨3− δ; q⟩C(κ)

2 (ε; q)

) .
Proof. If f belongs to the set BΣ(𭟋, λ, δ,B(κ)(ε, z; q)), as defined in equation (4),

then we can derive from equations (25) and (26) the following result:

a3 − ηa22 =

𭟋⟨3− δ; q⟩⟨2− δ; q⟩C(κ)
1 (ε; q)

2⟨3; q⟩Γq(4)
(
1 + λ⟨2; q⟩

) (c2 − d2)

+
(1− η) [2− δ]2q⟨3− δ; q⟩𭟋2

[
C

(κ)
1 (ε; q)

]3
(c2 + d2)

2Γq(3)

(
⟨3; q⟩2⟨2− δ; q⟩ (1 + λ⟨2; q⟩)𭟋

[
C

(κ)
1 (ε; q)

]2
− ⟨2; q⟩3(1 + λ)2 ⟨3− δ; q⟩C(κ)

2 (ε; q)

)
= C

(κ)
1 (ε; q)

[(
H(η) +

𭟋⟨3− δ; q⟩⟨2− δ; q⟩
2⟨3; q⟩Γq(4)

(
1 + λ⟨2; q⟩

)) c2 +(H(η)− 𭟋⟨3− δ; q⟩⟨2− δ; q⟩
2⟨3; q⟩Γq(4)

(
1 + λ⟨2; q⟩

)) d2] ,
where

H(η) =
(1− η) [2− δ]2q⟨3− δ; q⟩𭟋2

[
C

(κ)
1 (ε; q)

]2
2Γq(3)

(
⟨3; q⟩2⟨2− δ; q⟩ (1 + λ⟨2; q⟩)𭟋

[
C

(κ)
1 (ε; q)

]2
− ⟨2; q⟩3(1 + λ)2 ⟨3− δ; q⟩C(κ)

2 (ε; q)

)



M. Ahmed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6586 11 of 15

Then, it can be inferred that

∣∣∣a3 − ηa22

∣∣∣ ≤


|𭟋|
∣∣∣C(κ)

1 (ε;q)
∣∣∣

⟨3;q⟩Γq(4)
(
1+λ⟨2;q⟩

) , |H(η)| ≤ 𭟋⟨3−δ;q⟩⟨2−δ;q⟩
2⟨3;q⟩Γq(4)

(
1+λ⟨2;q⟩

) ,
2
∣∣∣C(κ)

1 (ε; q)
∣∣∣ ∣∣∣H(η)

∣∣∣, |H(η)| ≥ 𭟋⟨3−δ;q⟩⟨2−δ;q⟩
2⟨3;q⟩Γq(4)

(
1+λ⟨2;q⟩

) .
The proof of Theorem 2 is now complete.

4. Corollaries

The subsequent corollaries, which approximately correlate with Examples 1 and 2,
are deduced from Theorems 1 and 2.

Corollary 1. If f ∈ Σ given by (4) belongs to the class BΣ(𭟋, 1, δ,B(κ)(ε, z; q)). Then

|a2| ≤
⟨2− δ; q⟩

∣∣𭟋⟨κ; q⟩
∣∣ε√2 ⟨3− δ; q⟩ ⟨κ; q⟩ε√√√√√ Γq(3)

((
⟨3; q⟩2⟨2− δ; q⟩ (1 + ⟨2; q⟩)𭟋 ⟨κ; q⟩2 − 2⟨2; q⟩3 ⟨3− δ; q⟩

(
⟨κ; q2⟩+ ⟨κ; q⟩2

)
ε2

+⟨2; q⟩3 ⟨3− δ; q⟩ ⟨κ; q2⟩
))

,

|a3| ≤
𭟋2 ⟨2− δ; q⟩2 ⟨κ; q⟩2ε2

⟨2; q⟩2 (Γq(3))
2 +

2
∣∣𭟋[κ]q

∣∣ ⟨3− δ; q⟩⟨2− δ; q⟩ ε
⟨3; q⟩Γq(4)

∣∣1 + ⟨2; q⟩
∣∣ ,

and

∣∣∣a3 − ηa22

∣∣∣ ≤


2
∣∣𭟋⟨κ;q⟩

∣∣ε
⟨3;q⟩Γq(4)(1+⟨2;q⟩) , |1− η| ≤

∣∣∣∣∣∣1−
4⟨2;q⟩3 ⟨3−δ;q⟩

(
2(⟨κ;q2⟩+⟨κ;q⟩2)ε2−⟨κ;q2⟩

)
8⟨κ;q⟩2⟨3;q⟩2(1+⟨2;q⟩) ⟨2−δ;q⟩𭟋 ε

∣∣∣∣∣∣ ,

4 |⟨κ; q⟩| ε
∣∣∣H(η)

∣∣∣, |1− η| ≥

∣∣∣∣∣∣1−
4⟨2;q⟩3 ⟨3−δ;q⟩

(
2(⟨κ;q2⟩+⟨κ;q⟩2)ε2−⟨κ;q2⟩

)
8⟨κ;q⟩2⟨3;q⟩2(1+⟨2;q⟩) ⟨2−δ;q⟩𭟋 ε

∣∣∣∣∣∣ .
where

H(η) =
(1− η) [2− δ]2q⟨3− δ; q⟩𭟋2

[
C

(κ)
1 (ε; q)

]2
2Γq(3)

(
⟨3; q⟩2⟨2− δ; q⟩ (1 + ⟨2; q⟩)𭟋

[
C

(κ)
1 (ε; q)

]2
− 4⟨2; q⟩3 ⟨3− δ; q⟩C(κ)

2 (ε; q)

) .
Corollary 2. Let f ∈ Σ given by (4) belongs to the class BΣ(𭟋, 0, δ,B(κ)(ε, z; q)). Then

|a2| ≤
2 ⟨2− δ; q⟩

∣∣𭟋⟨κ; q⟩
∣∣ε√2⟨3− δ; q⟩⟨κ; q⟩ε√

Γq(3)
(
4⟨3; q⟩2⟨2− δ; q⟩𭟋 ⟨κ; q⟩2 − 2⟨2; q⟩3 ⟨3− δ; q⟩

(
⟨κ; q2⟩+ ⟨κ; q⟩2

)
ε2

+⟨2; q⟩3 ⟨3− δ; q⟩ ⟨κ; q2⟩
) ,



M. Ahmed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6586 12 of 15

|a3| ≤
4𭟋2⟨κ; q⟩2ε2(

⟨2; q⟩
+

2
∣∣𭟋[κ]q

∣∣ε
⟨3; q⟩

,

and

∣∣∣a3 − ηa22

∣∣∣ ≤


2
∣∣𭟋⟨κ;q⟩

∣∣ε
⟨3;q⟩Γq(4)

, |1− η| ≤

∣∣∣∣∣∣1−
⟨2;q⟩3 ⟨3−δ;q⟩

(
2(⟨κ;q2⟩+⟨κ;q⟩2)ε2−⟨κ;q2⟩

)
8⟨κ;q⟩2⟨3;q⟩2 ⟨2−δ;q⟩𭟋 ε

∣∣∣∣∣∣ ,

4 |⟨κ; q⟩| ε
∣∣∣H(η)

∣∣∣, |1− η| ≥

∣∣∣∣∣∣1−
⟨2;q⟩3 ⟨3−δ;q⟩

(
2(⟨κ;q2⟩+⟨κ;q⟩2)ε2−⟨κ;q2⟩

)
8⟨κ;q⟩2⟨3;q⟩2 ⟨2−δ;q⟩𭟋 ε

∣∣∣∣∣∣ .
where

H(η) =
(1− η) [2− δ]2q⟨3− δ; q⟩𭟋2

[
C

(κ)
1 (ε; q)

]2
2Γq(3)

(
⟨3; q⟩2⟨2− δ; q⟩𭟋

[
C

(κ)
1 (ε; q)

]2
− ⟨2; q⟩3 ⟨3− δ; q⟩C(κ)

2 (ε; q)

) .
5. Conclusion

The present study investigates the coefficient of three novel subclasses of bi-univalent
functions defined in the open unit disk U:

BΣ(𭟋, λ, δ,B(κ)(ε, z; q),BΣ(𭟋, 0, δ,B(κ)(ε, z; q), and BΣ(𭟋, 1, δ,B(κ)(ε, z; q)

as defined in Definition 2. We obtained the Taylor-Maclaurin coefficients |a2|, |a3| for
all three subclasses and provided estimates for the Fekete-Szegö functional problems.
We also explored additional results arising from specific parameter configurations within
our primary framework. Future studies could examine the Hankel determinants of these
subclasses to gain deeper insights into their structure. We expect practical applications
of the q-differintegral operator in various scientific domains, including mathematics and
technology.

In future studies, one could investigate the maximum bounds of the Zalcman conjec-
ture and analyze Hankel determinants for the classes of bi-convex and bi-close-to-convex
functions. These directions present promising opportunities for novel discoveries and
deeper exploration in the field.

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