EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6499
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Estimates for the Coefficients of Subclasses Defined by
the q-Babalola Convolution Operator of Bi-Univalent
Functions Subordinate to the q-Fibonacci Analogue

Ahmad Almalkawi 1,∗, Abdullah Alsoboh 2,∗, Ala Amourah3,4, Tala Sasa5

1 Modern College of Business and Science, Muscat, Sultanate of Oman
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
5 Applied Science Research Center, Applied Science Private University, Amman, Jordan

Abstract. In this work, we introduce and investigate a new subclass of bi-univalent functions
defined via the q-Babalola operator and the q-Fibonacci analogue. The q-Babalola operator
generalizes classical convolution-type operators in the context of q-calculus, enabling the analysis
of geometric properties of analytic functions under quantum calculus frameworks. Meanwhile,
the q-Fibonacci analogue extends the classical Fibonacci sequence into the realm of q-theory,
offering new structural insights and recursive behavior in analytic function theory. For functions
in this subclass, we derive sharp coefficient bounds for the initial Taylor coefficients |a2| and |a3|.
Furthermore, we address the Fekete-Szegö functional problem associated with this class. The
interplay between q-calculus and bi-univalent function theory revealed through our approach
yields several novel and significant results, enriching the geometric function theory literature
with new analytical tools and perspectives.

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

Key Words and Phrases: Analytic function, Bi-univalent function, Starlike class, Fibonacci
sequence, q-calculus, Shell-like curves, Fekete-Szegö functional, Babalola convolution operator

1. Introduction

We begin by considering the collection A of functions that are complex analytic
within the open unit disk U. This domain is defined as

U = {z = a + i b ∈ C where a, b ∈ R, and |z| < 1} ,

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6499

Email addresses: ahmad.abdelqader@mcbs.edu.om (A. Almalkawi),
abdullah.alsoboh@asu.edu.om (A. Alsoboh)

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



A. Almalkawi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6499 2 of 16

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 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 geo-
metric constraints, see [1].

Each member f ∈ A possesses a Maclaurin series representation about the origin,
which can be written as:

f(z) = z +
∞∑
n=2

an z
n, for z ∈ U, (1)

where the coefficients an 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 U, satisfies
f(0) = 0, and its modulus remains strictly less than one within the disk, i.e., |f(z)| < 1
for all z ∈ U. 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 subordi-
nate to f2, denoted f1 ≺ f2, if there exists a Schwarz function η such that

f1(z) = f2(η(z)) for all z ∈ U.

This relation implies that f1 is functionally dependent on f2 through composition with
η, preserving analyticity while embedding geometric structure, see [2]. 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 U. We also introduce the class P,
defined as the family of functions in A whose real parts are strictly positive throughout
U. A typical function φ ∈ P admits the following power series expansion:

p(z) = 1 +

∞∑
n=1

pnz
n = 1 + p1z + p2z

2 + p3z
3 + . . . , (z ∈ U). (2)

where the coefficients satisfy the sharp bound

|pn| ≤ 2, for all n ≥ 1. (3)



A. Almalkawi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6499 3 of 16

In accordance with the classical Caratheodory lemma (refer to [1] for further details).
Furthermore, a function φ ∈ P if and only if it is subordinate to the Mobius transfor-
mation 1+z

1−z , i.e.,

φ(z) ≺ 1 + z

1 − z
, z ∈ U.

The class of starlike functions, denoted S∗, can be characterized in various ways using
subordination techniques. A notable generalization was proposed by Ma and Minda [3],
who defined the following class:

S∗(Ω) =

{
f ∈ A :

z f ′(z)

f(z)
≺ Ω(z), where Ω ∈ P and z ∈ U

}
.

In this formulation, Ω is assumed to be analytic in U and possess a positive real part
throughout 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 liter-
ature for constructing refined categories of starlike mappings. The class P forms the

Table 1: Enumerates various starlike function classes characterized via the principle of subordination.

The subclasses of starlike functions Ref. Author

1 S∗
(
1+z
1−z

)
=
{
f ∈ A : zf

′(z)
f(z) ≺ 1+z

1−z

}
[4] Janowsk

2 S∗(ϑ) =
{
f ∈ A : zf

′(z)
f(z) ≺ 1+(1−2ϑ)z

1−z

}
, where 0 ≤ ϑ < 1 [5] Robertson

3 SL(ϑ) =
{
f ∈ A : zf

′(z)
f(z) ≺ 1+ϑ2z2

1−ϑz−ϑ2z2

}
, where ϑ = 1−

√
5

2 [6] Sokol

cornerstone for the development of numerous significant subclasses of analytic functions,
making it a key object 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 z = f(f−1(z)), (r0(f) ≥ 0.25; |z| < r0(f); z ∈ U) . (4)

where

ℏ(ξ) = f−1(ξ) = ξ − a2ξ
2 +

(
2a22 − a3

)
ξ3 −

(
5a32 + a4 − 5a3a2

)
ξ4 + · · · . (5)

function f ∈ S is said to be bi-univalent if its inverse function f−1 ∈ S. The subclass
of S denoted by Σ contains all bi-univalent functions in U. A table illustrating certain
functions within the class Σ and their inverse functions is provided below.



A. Almalkawi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6499 4 of 16

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 (ξ) = ξ
1−ξ

f2 = − log(1 − z) f−1
2 (ξ) = e2ξ−1

e2ξ+1

f3 = 1
2 log

(
1+z
1−z

)
f−1
3 (ξ) = eξ−1

eξ

Quantum calculus, or q-calculus, has become an influential tool in geometric func-
tion theory, offering an alternative to classical calculus that avoids the traditional limit
process. Initiated by Jackson [7, 8] through the introduction of the q-difference oper-
ator and its integral, this framework was later expanded by Aral and Gupta [9–11] to
include q-analogues of classical operators relevant to analytic function theory. Central
to q-calculus is the deformation parameter q ∈ (0, 1), which ensures convergence and
structural integrity of function classes, particularly those defined by geometric proper-
ties such as starlikeness and convexity. This approach has opened new directions for
both theoretical exploration and application in complex analysis.

Definition 1. [12] The q-bracket ⌈λ⌋q is defined as follows:

⌈λ⌋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 ⌈λ+ 1⌋q = ⌈λ⌋q + qλ. Resided that the q-factorial [λ]q! is defined
by

⌈λ⌋q ! =


λ∏
n=1

⌈n⌋q = ⌈λ⌋q .⌈λ− 1⌋q · · · ⌈3⌋q .⌈2⌋q .⌈1⌋q , (λ ∈ N)

1 , λ = 0,

Definition 2. [12] The q−derivative, also known as the q−difference operator, of a
function 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.

.



A. Almalkawi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6499 5 of 16

Remark: For f ∈ A of the form (1), it is straightforward to verify that

ðq⟨f(z)⟩ = ðq

〈
z +

∞∑
n=2

an z
n

〉
= 1 +

∞∑
n=2

⌈n⌋qan zn−1, (z ∈ U),

and for the inverse function ℏ = f−1 of the form (4), we have

ðq⟨ℏ(ξ)⟩ = 1 − ⌈2⌋qa2ξ + ⌈3⌋q
(
2a22 − a3

)
ξ2 − ⌈4⌋q

(
5a32 + a4 − 5a3a2

)
ξ3 + · · · .

More recently, Al-shbeil et al. [13] defined the q-Babalola convolution operator

Da,α,β
z : A → A by

Dχ
q f(z) = z +

∞∑
n=2

(λ]cq anz
n, (6)

where

(λ]cq =
1 + q(a) + q2(a) + · · · + qn−1(a)

1 + q(χ) + q2(χ) + · · · + qn−1(χ)
, (χ = a− t > −1).

qn−1(a) =
qn−1 (a+ n− 2)!

(n− 1)! (a− 1)!
and qn−1(χ) =

qn−1 (χ+ n− 2)!

(n− 1)! (χ− 1)!
.

In a more recent advancement, Alsoboh et al. [14] introduced a noteworthy 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 ∈ U
}
, (7)

where the function Υ(z; q) is expressed explicitly as

Υ(z; q) =
1 + qϑ2qz

2

1 − ϑqz − qϑ2qz
2
, (8)

and

ϑq =
1 −

√
4q + 1

2q
(9)

represents the q-analog of the Fibonacci numbers. Additionally, Alsoboh et al. [14]
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

p̃nz
n,

the coefficients p̃n satisfy the following recurrence relation:

p̃n =


ϑq, for n = 1,

(2q + 1)ϑ2q , for n = 2,

(3q + 1)ϑ3q , for n = 3,

(φn+1(q) + qφn−1(q))ϑ
n
q , for n ≥ 4.

(10)



A. Almalkawi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6499 6 of 16

Here, the q-Fibonacci polynomials φs(q) are defined as

φs(q) =
(1 − qϑq)

s − (ϑq)
s

√
4q + 1

, s ∈ N. (11)

This research presents a thorough 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 is noteworthy that the function Υ(z; q) is not injective over the domain U. Specif-
ically, there exist distinct points in U at which Υ(z; q) attains the same value. For
instance,

Υ(0; q) = 1 and Υ

(
− 1

2qϑq
; q

)
= 1.

In the following example, we explore the behavior of the q-starlike functions as the
parameter q approaches 1 from below. This transition leads to the classical case of
starlike functions, often referred to as the 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. [6] 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)

}
,



A. Almalkawi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6499 7 of 16

where the function Υ(z) is given by

Υ(z; 1) = Υ(z) =
1 + ϑ2z2

1 − ϑz − ϑ2z2
, (12)

and ϑ = 1−
√
5

2 denotes the classical Fibonacci constant.

In addition to introducing the class of q-starlike functions, Alsoboh et al. [14] 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 ∈ U), (13)

where the function Υ(z; q) is defined by the rational expression in (8), and the parameter
ϑq is specified in (9).

Fractional calculus (FC) operators have been extensively utilized in a wide array
of applied scientific disciplines, including but not limited to the study of geometric
function theory, as highlighted in [15]. One notable extension of classical fractional
calculus is fractional q-calculus, which integrates the discrete framework of q-analysis
with the principles of fractional differentiation and integration. This fusion has led to
significant advancements and applications across various mathematical and engineering
fields. Specifically, fractional q-calculus has been employed to address complex problems
such as optimal control systems, q-difference equations, q-integral equations, and more
traditional areas of fractional calculus. Comprehensive treatments of these applications
can be found in specialized literature, including the foundational text [16] and recent
scholarly contributions [17–19]. The emergence of q-calculus has profoundly enriched the
study of analytic function theory by facilitating the development and exploration of new
subclasses of functions with rich geometric, algebraic, and topological properties. This
analytical framework reveals the inherent flexibility of q-calculus, showcasing its potential
to generalize classical theories, uncover novel mathematical structures, and provide deep
theoretical insights. The implications of these developments are far-reaching, influencing
both abstract theory and practical applications. As such, q-calculus serves not only as
a bridge between classical analysis and modern mathematical techniques but also as a
fertile ground for further research and innovation in the field [20–34].

2. Definition and examples

This section begins by defining the subclass SLMΣ(β, χ, ψ; q), which is specifically
linked to the q-Babalola convolution operator and q-Fibonacci sequence.



A. Almalkawi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6499 8 of 16

Definition 3. For 0 < q < 1, β ∈ C \ {0} and ψ ≥ 0. A function f ∈ Σ given by (1) is
said to be in the class SLMΣ(β, χ, ψ; q) if the following subordinations are satisfied:

1 +
1

β

(
z1−ψðq⟨Dχ

q f(z)⟩
(Dχ

q f(z))1−ψ

)
≺ Υ(z; q) =

1 + qϑ2qz
2

1 − ϑq z − qϑ2qz
2
, (z ∈ U) (14)

and

1 +
1

β

(
ξ1−ψ∂q (Dχ

q ℏ(ξ))

(Dχ
q ℏ(ξ))1−ψ

)
≺ Υ(ξ; q) =

1 + qϑ2qξ
2

1 − ϑq ξ − qϑ2qξ
2
, (ξ ∈ U) (15)

where κ ∈
(
1
2 , 1
]
, the function ℏ(ξ) = f−1(ξ) is defined by (2) and ϑq is given by (9).

Example 2. Suppose that ψ = 0, 0 < q < 1, and β ∈ C \ {0}. Then, a function f
belonging to the class Σ, as given by equation (1), is considered to be in the category
SLMΣ(β, χ, 0; q) if it satisfies the following subordination conditions:

1 +
1

β

(
zðq⟨Dχ

q f(z)⟩
Dχ
q f(z)

)
≺ Υ(z; q) =

1 + qϑ2qz
2

1 − ϑq z − qϑ2qz
2
, (z ∈ U) (16)

and

1 +
1

β

(
ξ∂q (Dχ

q ℏ(ξ))

Dχ
q ℏ(ξ)

)
≺ Υ(ξ; q) =

1 + qϑ2qξ
2

1 − ϑq ξ − qϑ2qξ
2
, (ξ ∈ U) (17)

where κ ∈
(
1
2 , 1
]
, the function ℏ(ξ) = f−1(ξ) is defined by (2) and ϑq is given by (9).

Example 3. Suppose that ψ = 1, 0 < q < 1, and β ∈ C \ {0}. Then, a function f
belonging to the class

∑
, as given by equation (1), is considered to be in the category

SLMΣ(β, χ, 1; q) if it satisfies the following subordination conditions:

1 +
ðq⟨Dχ

q f(z)⟩
β

≺ Υ(z; q) =
1 + qϑ2qz

2

1 − ϑq z − qϑ2qz
2

(z ∈ U) (18)

and

1 +
∂q (Dχ

q ℏ(ξ))

β
≺ Υ(ξ; q) =

1 + qϑ2qξ
2

1 − ϑq ξ − qϑ2qξ
2
, (ξ ∈ U) (19)

where κ ∈
(
1
2 , 1
]
, the function ℏ(ξ) = f−1(ξ) is defined by (2) and ϑq is given by (9).



A. Almalkawi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6499 9 of 16

3. Coefficient bounds of the class SLMΣ(β, χ, ψ; q)

In this section, we first obtain the estimate of the initial Taylor coefficients |a2| and
|a2| for functions in the class SLMΣ(β, χ, ψ; q) as per Definition 3.

Firstly, let
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 U and p(z) = Υ(φ(z); q).

We have

ℏ(z) = (1 + φ(z))(1 − φ(z))−1 = 1 + ℓ1z + ℓ2z
2 + · · · ∈ P (z ∈ U). (20)

Consequently, the function φ(z), being analytic in U 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
+ · · · , (21)

and

Υ(φ(z); q) = 1 + p̃1

[
ℓ1z

2
+

(
ℓ2 −

ℓ21
2

)
z2

2
+

(
ℓ3 − ℓ1ℓ2 −

ℓ31
4

)
z3

2
+ · · ·

]
+ p̃2

[
ℓ1z

2
+

(
ℓ2 −

ℓ21
2

)
z2

2
+

(
ℓ3 − ℓ1ℓ2 −

ℓ31
4

)
z3

2
+ · · ·

]2
+ p̃3

[
ℓ1z

2
+

(
ℓ2 −

ℓ21
2

)
z2

2
+

(
ℓ3 − ℓ1ℓ2 −

ℓ31
4

)
z3

2
+ · · ·

]3
+ · · ·

= 1 +
p̃1ℓ1

2
z +

1

2

[(
ℓ2 −

ℓ21
2

)
p̃1 +

ℓ21
2
p̃2

]
z2

+
1

2

[(
ℓ3 − ℓ1ℓ2 +

ℓ31
4

)
p̃1 + ℓ1

(
ℓ2 −

ℓ21
2

)
p̃2 +

ℓ31
4
p̃3

]
z3 + · · · .

(22)

Similarly, there exists an analytic function ν defined on U, satisfying |ν(ξ)| < 1, such
that p(ξ) = Υ(ν(ξ); q). This allows us to represent the corresponding function

λ(ξ) = (1 + ν(ξ))(1 − ν(ξ))−1 = 1 + τ1ξ + τ2ξ
2 + · · · ∈ P. (23)

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
+ · · · , (24)



A. Almalkawi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6499 10 of 16

and accordingly, the composition Υ(ν(ξ); q) expands as:

Υ(ν(ξ); q) = 1 +
p̃1τ1

2
ξ +

1

2

[(
τ2 −

τ21
2

)
p̃1 +

τ21
2
p̃2

]
ξ2

+
1

2

[(
τ3 − τ1τ2 +

τ31
4

)
p̃1 + τ1

(
τ2 −

τ21
2

)
p̃2 +

τ31
4
p̃3

]
ξ3 + · · · .

(25)

Having established the necessary groundwork and auxiliary results, we are now in
a position to derive bounds for the initial coefficients of functions belonging to the
newly introduced class SLMΣ(β, χ, ψ; 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 |a2| and
|a3|, respectively.

Theorem 1. Let f be a function belonging to the class Σ, as defined by equation (1).
Suppose further that f lies within the subclass SLMΣ(β, χ, ψ; q). Then,

∣∣a2∣∣ ≤ √
2|ϑq|√∣∣∣ ((M + N − (2q + 1)U

)
ϑq + U

) ∣∣∣ .
and ∣∣a3∣∣ ≤ 2ϑ2q∣∣(M + N − (2q + 1)U

)
ϑq + U

∣∣ +
β
∣∣ϑq∣∣

(3]cq
(
q + q2 + ψ

) ,
where

M =
2(3]cq

(
q + q2 + ψ

)
β

, (26)

N =
(ψ − 1)

(
(2]cq

)2
(2q + ψ)

β
, (27)

U =
2
(
(2]cq (q + ψ)

)2
β2

. (28)

Proof. Let f ∈ SLMΣ(β, χ, ψ; q) and ξ = f−1. Considering (14) and (15) we have

1 +
1

β

(
z1−ψðq⟨Dχ

q f(z)⟩
(Dχ

q f(z))1−ψ

)
= Υ(φ(z); q), (z ∈ U), (29)

and

1 +
1

β

(
ξ1−ψðq⟨Dχ

q ℏ(ξ)⟩
(Dχ

q ℏ(ξ))1−ψ

)
= Υ(ν(ξ); q), (ξ ∈ U). (30)

Since

1 +
1

β

(
z1−ψðq⟨Dχ

q f(z)⟩
(Dχ

q f(z))1−ψ

)
= 1 +

p̃1ℓ1
2
z +

1

2

[(
ℓ2 −

ℓ21
2

)
p̃1 +

ℓ21
2
p̃2

]
z2 + · · · . (31)



A. Almalkawi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6499 11 of 16

and

1 +
1

β

(
ξ1−ψðq⟨Dχ

q ℏ(ξ)⟩
(Dχ

q ℏ(ξ))1−ψ

)
= 1 +

p̃1τ1
2
ξ +

1

2

[(
τ2 −

τ21
2

)
p̃1 +

τ21
2
p̃2

]
ξ2 + · · · . (32)

In view of (1), (2), from (31) and (32), we obtain

1 +
(2]cq(q + ψ)

β
a2z +

1

β

(
(3]cq

(
q + q2 + ψ

)
a3 +

(ψ − 1)
(
(2]cq

)2
(2q + ψ)

2
a22

)
z2 + · · ·

= 1 +
p̃1ℓ1

2
z +

1

2

[(
ℓ2 −

ℓ21
2

)
p̃1 +

ℓ21
2
p̃2

]
z2 + · · · ,

(33)

and

1 −
(2]cq(q + ψ)

β
a2ξ +

1

β

[(
2(3]cq

(
q + q2 + ψ

)
+

(ψ − 1)
(
(2]cq

)2
(2q + ψ)

2

)
a22

− (3]cq
(
q + q2 + ψ

)
a3

]
ξ2 + · · · = 1 +

p̃1τ1
2
ξ +

1

2

[(
τ2 −

τ21
2

)
p̃1 +

τ21
2
p̃2

]
ξ2 + · · · .

(34)

Therefore, by comparing the coefficients in (33) and (34) that correspond to each
other, we can obtain:

(2]cq(q + ψ)

β
a2 =

p̃1ℓ1
2

(35)

−
(2]cq(q + ψ)

β
a2 =

p̃1τ1
2

(36)

(3]cq
(
q + q2 + ψ

)
a3 +

(ψ − 1)
(
(2]cq

)2
(2q + ψ)

2
a22 =

β

2

[(
ℓ2 −

ℓ21
2

)
p̃1 +

ℓ21
2
p̃2

]
(37)(

2(3]cq
(
q + q2 + ψ

)
+

(ψ − 1)
(
(2]cq

)2
(2q + ψ)

2

)
a22 − (3]cq

(
q + q2 + ψ

)
a3 =

β

2

[(
τ2 −

τ21
2

)
p̃1 +

τ21
2
p̃2

]
(38)

From (35) and (36), we have

ℓ1 = −τ1 ⇐⇒ ℓ21 = τ21 , (39)

and

a22 =
ϑ2qβ

2

8
(
(2]cq (q + ψ)

)2 (ℓ21 + τ21
)

⇐⇒ ℓ21 + τ21 =
8
(
(2]cq (q + ψ)

)2
ϑ2qβ

2
a22. (40)



A. Almalkawi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6499 12 of 16

Now, by summing (37) and (38), we obtain

(M + N ) a22 =
(ℓ2 + τ2)ϑq

2
+

[
(2q + 1)ϑ2q

4
− ϑq

4

] (
ℓ21 + τ21

)
. (41)

By putting (40) in (41), we obtain

a22 =
(ℓ2 + τ2)ϑ

2
q

2
((

M + N − (2q + 1)U
)
ϑq + U

) (42)

where M ,N ,U are given by (26), (27) and (28), respectively. Using (3) for (42),
we have ∣∣a2∣∣ ≤ √

2|ϑq|√∣∣∣ ((M + N − (2q + 1)U
)
ϑq + U

) ∣∣∣ . (43)

Now, so as to find the bound on |a3|, let’s subtract from (37) and (38) along (40),
we obtain

a3 = a22 +
βϑq

4(3]cq
(
q + q2 + ψ

)(ℓ2 − τ2
)
. (44)

Hence, we get

∣∣a3∣∣ ≤ 2ϑ2q∣∣(M + N − (2q + 1)U
)
ϑq + U

∣∣ +
β
∣∣ϑq∣∣

(3]cq
(
q + q2 + ψ

) , (45)

where M ,N ,U are given by (26), (27) and (28), respectively. The proof of the theorem
is now complete.

Theorem 2. Suppose that f is a function in the class Σ, as given by equation (1), and
it belongs to the category SLMΣ(β, χ, ψ; q), and χ > −1. Then, we have

∣∣∣a3 − µa22

∣∣∣ ≤


βϑq

(3]cq

(
q+q2+ψ

) , 0 ≤ |1 − µ| ≤
β
∣∣∣(M+N −(2q+1)U

)
ϑq+U

∣∣∣
2(3]cq ϑq

(
q+q2+ψ

) ,

2|1−µ|ϑ2q∣∣∣(M+N −(2q+1)U
)
ϑq+U

∣∣∣ , |1 − µ| ≥
β
∣∣∣(M+N −(2q+1)U

)
ϑq+U

∣∣∣
2(3]cq ϑq

(
q+q2+ψ

) ,

where M ,N ,U are given by (26), (27) and (28), respectively.

Proof. Let f ∈ SLMΣ(β, χ, ψ; q), from (42) and (44), we have

a3 − µa22 =
(1 − µ) (ℓ2 + τ2)ϑ

2
q

2
((

M + N − (2q + 1)U
)
ϑq + U

) +
βϑq

4(3]cq
(
q + q2 + ψ

)(ℓ2 − τ2
)

=

(
K (µ) +

βϑq

4(3]cq
(
q + q2 + ψ

)) ℓ2 +

(
K (µ) − βϑq

4(3]cq
(
q + q2 + ψ

)) τ2, (46)



A. Almalkawi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6499 13 of 16

where

K (µ) =
(1 − µ)ϑ2q

2
((

M + N − (2q + 1)U
)
ϑq + U

) . (47)

Then, by taking modulus of (46), we conclude that

∣∣∣a3 − µa22

∣∣∣ ≤


βϑq

(3]cq

(
q+q2+ψ

) , 0 ≤
∣∣K (µ)

∣∣ ≤ βϑq

4(3]cq

(
q+q2+ψ

) ,
4
∣∣K (µ)

∣∣, ∣∣K (µ)
∣∣ ≥ βϑq

4(3]cq

(
q+q2+ψ

) ,
The proof of the theorem is now complete.

4. Corollaries

The subsequent corollaries, which closely correspond to instances resembling Exam-
ples 1, 2, and 3, are derived from the implications of Theorems 1 and 2.

Corollary 1. Suppose that f is a function in the class Σ, as given by equation (1), and
it belongs to the category SLMΣ(β, χ, 0; q). Then

∣∣a2∣∣ ≤ √
2|ϑq|√∣∣∣ ((M + N − (2q + 1)U

)
ϑq + U

) ∣∣∣ ,
∣∣a3∣∣ ≤ 2ϑ2q∣∣(M + N − (2q + 1)U

)
ϑq + U

∣∣ +
β
∣∣ϑq∣∣

(3]cq
(
q + q2

) ,
where and

∣∣∣a3 − µa22

∣∣∣ ≤


βϑq

(3]cq

(
q+q2

) , 0 ≤ |1 − µ| ≤
β
∣∣∣(M+N −(2q+1)U

)
ϑq+U

∣∣∣
2(3]cq ϑq

(
q+q2

) ,

2|1−µ|ϑ2q∣∣∣(M+N −(2q+1)U
)
ϑq+U

∣∣∣ , |1 − µ| ≥
β
∣∣∣(M+N −(2q+1)U

)
ϑq+U

∣∣∣
2(3]cq ϑq

(
q+q2

) ,

where

M =
2(3]cq

(
q + q2

)
β

, N =
−
(
(2]cq

)2
(2q)

β
, and U =

2
(
(q) (2]cq

)2
β2

.

Corollary 2. Suppose that f is a function in the class Σ, as given by equation (1), and
it belongs to the category SLMΣ(β, χ, 1; q). Then

∣∣a2∣∣ ≤ √
2|ϑq|√∣∣∣ ((M − (2q + 1)U

)
ϑq + U

) ∣∣∣ .



A. Almalkawi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6499 14 of 16

and ∣∣a3∣∣ ≤ 2ϑ2q∣∣(M − (2q + 1)U
)
ϑq + U

∣∣ +
β
∣∣ϑq∣∣

(3]cq
(
1 + q + q2

) ,
and

∣∣∣a3 − µa22

∣∣∣ ≤


βϑq

(3]cq

(
1+q+q2

) , 0 ≤ |1 − µ| ≤
β
∣∣∣(M−(2q+1)U

)
ϑq+U

∣∣∣
2(3]cq ϑq

(
1+q+q2

) ,

2|1−µ|ϑ2q∣∣∣(M−(2q+1)U
)
ϑq+U

∣∣∣ , |1 − µ| ≥
β
∣∣∣(M−(2q+1)U

)
ϑq+U

∣∣∣
2(3]cq ϑq

(
1+q+q2

) ,

where

M =
2(3]cq

(
1 + q + q2

)
β

, and U =
2
(
(2]cq (q + 1)

)2
β2

.

Conclusion: In this investigation, we have investigated the concerns regarding the
coefficients of three recently introduced categories of bi-univalent functions in the unit
disk U: SLMΣ(β, χ, ψ; q), SLMΣ(β, χ, 0; q) and SLMΣ(β, χ, 1; q), as described in Defini-
tions 3. We have calculated the estimated values for the Taylor-Maclaurin coefficients
|a2| and |a3| for each of these subclasses.
In future investigations, it would be of significant interest to explore sharp bounds related
to the Zalcman conjecture and to conduct a detailed analysis of Hankel determinants
within the classes of bi-convex and bi-close-to-convex functions. These avenues offer
promising prospects for uncovering novel results and fostering a deeper understanding
of the structural properties inherent in geometric function theory.

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