EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6698
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Hankel Determinant Estimates for Bi-Bazilevič-Type
Functions Involving q-Fibonacci Numbers

Abdullah Alsoboh1, Adel Salim Tayyah2, Ala Amourah3,4,
Abdullrahman A. Al-Maqbali1,∗, Khaled Al Mashrafi1, Tala Sasa5

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 Department of Computer Science, College of Computer Science and Information
Technology, University of Al-Qadisiyah, Diwaniyah, 58002, Iraq
3 Mathematics Education Program, Faculty of Education and Arts, Sohar University,
Sohar 311, Oman
4 Jadara University Research Center, Jadara University, Jordan
5 Department of Mathematics, Faculty of Science, Applied Science Private University,
Amman, Jordan

Abstract. This study focuses on a specific class of analytic and bi-univalent functions of Bazilevič-
type, formulated within a geometric context shaped by shell-like curves and influenced by the
q-analogue of Fibonacci numbers. By utilizing the subordination approach, we establish precise
bounds for the initial coefficients in the Taylor–Maclaurin expansion of these functions. Moreover,
the paper presents Fekete–Szegö-type inequalities and introduces novel bounds for the second
Hankel determinant, thereby contributing to a deeper analytical insight into the behavior of this
function class. These contributions not only broaden the scope of traditional coefficient problems
related to bi-univalent functions but also highlight the intricate connections among geometric
function theory, specialized function classes, and the principles of q-calculus. The outcomes pave
the way for future studies aimed at deriving bounds for higher-order coefficients and examining
determinant-related functionals under this theoretical model.

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

Key Words and Phrases: Analytic functions, Bi-univalent functions, Starlike class, Fekete–Szegö
functional, Fibonacci sequence, q-calculus, Shell-like curves

Let A denote the class of functions that are analytic in the open unit disk D, defined
by

D = {z = a+ ib ∈ C : a, b ∈ R, |z| < 1} ,

which represents the interior of the unit circle in the complex plane, centered at the
origin and excluding the boundary.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6698

Email addresses: abdullah.alsoboh@asu.edu.om (A. Alsoboh),
abdulrahman.almaqbali@asu.edu.om (A. A. Al-Maqbali)

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 (3) (2025), 6698 2 of 25

Each function f ∈ A is normalized such that

f(0) = 0 and f ′(0) = 1.

These normalization conditions remove translational and dilational ambiguities, ensuring
a standardized form that facilitates structural analysis and comparative study under
shared geometric constraints.

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 ∈ D, (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 D, satisfies
f(0) = 0, and its modulus remains strictly less than one within the disk, i.e., |f(z)| < 1
for all z ∈ D. 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 ∈ D.

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 D. We also introduce the class P,
defined as the family of functions in A whose real parts are strictly positive throughout
D. 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 ∈ D). (2)

where the coefficients satisfy the sharp bound

|pn| ≤ 2, for all n ≥ 1. (3)

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

1−z , i.e.,

φ(z) ≺ 1 + z

1 − z
, z ∈ D.



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6698 3 of 25

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 [2],
who defined the following class:

S∗(Ω) =

{
f ∈ A :

z f ′(z)

f(z)
≺ Ω(z), where Ω ∈ P and z ∈ D

}
.

In this formulation, Ω is assumed to be analytic in D 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 literature
for constructing refined categories of starlike mappings.

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

The class P forms the cornerstone for the development of numerous significant sub-
classes 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 ξ = f(f−1(ξ)), (r0(f) ≥ 0.25; |ξ| < r0(f); z ∈ D) . (4)

where

χ(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 D. A table illustrating certain
functions within the class Σ and their inverse functions is provided below.



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6698 4 of 25

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

In contemporary mathematical and physical research, the study of quantum calcu-
lus—commonly referred to as q-calculus—has emerged as a vibrant and impactful area
of inquiry. The foundational concepts of this theory were introduced in the late 19th
century by Jackson [7, 8], who formulated the q-difference operator along with its in-
tegral counterpart. These contributions laid the groundwork for a novel approach to
calculus, one that does not rely on traditional notions of limits. Expanding on Jackson’s
foundational work, Aral and Gupta [9] explored the q-extensions of classical mathemat-
ical tools and operators, particularly in the realm of geometric function theory.

The essence of q-calculus lies in its ability to generalize conventional calculus through
the use of q-differences, offering a robust analytical structure for examining complex
classes of analytic functions. It has proven especially effective in characterizing and ana-
lyzing subclasses such as starlike, convex, and bi-univalent functions. Within this frame-
work, the deformation parameter q, constrained to the interval 0 < q < 1, plays a pivotal
role. It ensures the convergence of q-series and the preservation of geometric and analytic
properties necessary for the coherent definition of these subclasses. The q-derivative op-
erator ðq, along with its associated constructs like q-numbers and q-factorials, provides
a natural extension of classical operators. This facilitates refined estimates of coefficient
bounds and enables the derivation of sharp inequalities within q-analytic function the-
ory. Consequently, q-calculus offers new perspectives and methodologies for advancing
both theoretical investigations and practical applications in complex analysis.

Definition 1. [10] The q-bracket ⌈κ⌋q is defined as follows:

⌈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∗

qγ−1 + qγ−2 + · · · + q + 1 =
γ−1∑
n=0

qn, 0 < q < 1, κ = γ ∈ N,

with the useful identity ⌈κ+ 1⌋q = ⌈κ⌋q + qκ.



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6698 5 of 25

Definition 2. [10] 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.

.

Remark 1. 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 ∈ D),

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

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

)
ξ2−⌈4⌋q

(
5a32 + a4 − 5a3a2

)
ξ3+· · · .

In a more recent advancement, Alsoboh et al. [11] 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 ∈ D
}
, (6)

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

Υ(z; q) =
1 + qϑ2qz

2

1 − ϑqz − qϑ2qz
2
, (7)

and

ϑq =
1 −

√
4q + 1

2q
(8)

represents the q-analog of the Fibonacci numbers. Additionally, Alsoboh et al. [11]
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

pnz
n,

the coefficients pn satisfy the following recurrence relation:

pn =



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.

(9)



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6698 6 of 25

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 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

The function Υ(z; q), defined in (7), maps the unit circle to a curve Ωq characterized
by the parametric representation

κ =

√
4q + 1

2(1 + 2q − 2q cosϕ)
, y =

sinϕ
2(1+cosϕ)(4q cosϕ− 1)

1 + 2q − 2q cosϕ
, ϕ ∈ [0, 2π) \ {π}. (11)

Notably, Υ(z; q) is not injective over the unit disk D, as demonstrated by

Υ(0; q) = Υ

(
− 1

2qϑq
; q

)
= 1.

This non-injectivity results in a shell-like structure for the image curve Ωq, which is
symmetric with respect to the real axis, as illustrated in Fig. 1.

We distinguish two qualitative behaviors of the curve Ωq depending on the value of
the parameter q:

• Case I: q ∈
(
0, 14
)

In this regime, the curve Ωq is smooth and resembles a conchoid without any self-
intersections. The image under Υ remains injective on the unit circle, and the
curve does not loop back on itself.



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6698 7 of 25

• Case II: q ∈
(
1
4 , 1
)

Here, the curve Ωq develops a self-intersecting loop. Specifically, the relation

Υ

(
e
±i arccos

(
1
4q

)
; q

)
=

√
4q + 1

4q + 1

implies that Ωq intersects itself on the real axis at

e2 =

√
4q + 1

4q + 1
.

Consequently, the curve crosses the real axis at two distinct points:

e1 =

√
4q + 1

2
, e2 =

√
4q + 1

4q + 1
.

This results in a closed loop in the geometry of Ωq.

Figure 1: The curve Ωq for various values of q.

(a) q = 1 (b) q = 0.84 (c) q = 0.5

(d) q = 0.3 (e) q = 0.2 (f) q = 0.1

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



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6698 8 of 25

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. 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
, (12)

and ϑ = 1−
√
5

2 denotes the classical Fibonacci constant.

The development of q-calculus has profoundly advanced the field of analytic function
theory by facilitating the construction and investigation of new subclasses characterized
by rich geometric structures and intricate algebraic behavior. This analytical frame-
work highlights the intrinsic adaptability of q-calculus in generalizing classical results
and revealing previously unexplored mathematical phenomena. Its influence extends
beyond mere theoretical curiosity, providing a unified platform for significant insights
and promising applications in diverse areas of mathematical analysis. As such, q-calculus
serves as a powerful bridge between traditional methodologies and contemporary inno-
vations, laying a solid groundwork for sustained research and future breakthroughs in
the discipline [12–30].

1. 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. A bi-univalent function f of the form (1) belongs to the class BΣ(µ; q) if
and only if

z1−µ ðq⟨f(z)⟩(
f(z)

)1−µ ≺ Υ(z; q) =
1 + qϑ2qz

2

1 − ϑq z − qϑ2qz
2
, (13)

and
ξ1−µ ðq⟨χ(ξ)⟩(
χ(ξ)

)1−µ ≺ Υ(ξ; q) =
1 + qϑ2qξ

2

1 − ϑq ξ − qϑ2qξ
2
, (14)

where µ ∈ R+ ∪ {0}, χ = f−1 given by (5), ϑq is given by (8) and z, ξ ∈ D.



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6698 9 of 25

By varying the parameters µ ∈ R+ ∪ {0} and q ∈ (0, 1), a broad spectrum of novel
subclasses of the bi-univalent function class

∑
can be systematically derived. These sub-

classes capture diverse geometric behaviors and provide a unified framework for further
analytical investigations.

Example 2. If µ = 0, we obtain the class SLΣ(Υ(z; q)), defined as consisting of functions
f ∈ Σ satisfying the conditions

zðq⟨f(z)⟩
f(z)

≺ Υ(z; q) =
1 + qϑ2qz

2

1 − ϑq z − qϑ2qz
2
,

and
ξðq⟨χ(ξ)⟩
χ(ξ)

≺ Υ(ξ; q) =
1 + qϑ2qξ

2

1 − ϑq ξ − qϑ2qξ
2
,

where ϑq is given by (8), χ = f−1 defined as in (5), ϑ = 1−
√
5

2 is the classical Fibonacci
constant, and z, ξ ∈ D. This class was studied by Alsoboh et al. [11].

Example 3. If q → 1− then BΣ(µ; q) is reduce to the subclass BΣ(µ) studied by Pulala
[31].

z1−µ f ′(z)(
f(z)

)1−µ ≺ Υ(z) =
1 + ϑ2z2

1 − ϑz − ϑ2z2
,

and
ξ1−µ χ′(ξ)(
χ(ξ)

)1−µ ≺ Υ(ξ) =
1 + ϑ2ξ2

1 − ϑξ − ϑ2ξ2
,

As demonstrated in [32], the most precise agreement is attained in the case where

BΣ(µ; q) := J (1,µ)
Σ

(
Y (z; 1−)

)
,

Example 4. In the limiting case as q → 1−, and µ = 0 we recover the classical sub-
class SLMΣ, consisting of all functions f ∈ Σ that satisfy the following subordination
conditions:

z f ′(z)

f(z)
≺ Υ(z) =

1 + ϑ2z2

1 − ϑz − ϑ2z2
,

and
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 ∈ D. This class was initially investigated by Sokó l [5, 6],
and subsequently studied in greater depth by Özgür and Sokó l [33].



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6698 10 of 25

2. Coefficient bounds of BΣ(µ; q)

In this section, we aim to estimate the initial Taylor coefficients |a2| and |a3| for
functions belonging to the class BΣ(µ; q), as defined in Definition 3.

Consider the analytic function

p(z) = 1 + p1z + p2z
2 + p3z

3 + . . . ,

which satisfies the subordination condition p(z) ≺ Υ(z; q). By the principle of subordi-
nation, there exists a Schwarz function φ ∈ P such that

|φ(z)| < 1 for all z ∈ D, and p(z) = Υ(φ(z); q).

Define the function

ℏ(z) =
1 + φ(z)

1 − φ(z)
= 1 + ℓ1z + ℓ2z

2 + · · · ∈ P, (z ∈ D). (15)

Since φ(z) is analytic in D and subordinate to Υ(z; q), it admits the following Taylor
expansion:

φ(z) =
ℓ1z

2
+

(
ℓ2 −

ℓ21
2

)
z2

2
+

(
ℓ3 − ℓ1ℓ2 −

ℓ31
4

)
z3

2
+ · · · . (16)

Substituting this into the function Υ(φ(z); q), we obtain:

Υ(φ(z); q) = 1 + p1

[
ℓ1z

2
+

(
ℓ2 −

ℓ21
2

)
z2

2
+

(
ℓ3 − ℓ1ℓ2 −

ℓ31
4

)
z3

2
+ · · ·

]
+ p2

[
ℓ1z

2
+ · · ·

]2
+ p3

[
ℓ1z

2
+ · · ·

]3
+ · · ·

= 1 +
p1ℓ1

2
z +

1

2

[(
ℓ2 −

ℓ21
2

)
p1 +

ℓ21
2
p2

]
z2

+
1

2

[(
ℓ3 − ℓ1ℓ2 +

ℓ31
4

)
p1 + ℓ1

(
ℓ2 −

ℓ21
2

)
p2 +

ℓ31
4
p3

]
z3 + · · ·

= 1 +
ϑqℓ1

2
z +

1

2

[(
ℓ2 −

ℓ21
2

)
ϑq +

(1 + 2q)ϑ2qℓ
2
1

2

]
z2

+
1

2

[(
ℓ3 − ℓ1ℓ2 +

ℓ31
4

)
ϑq + ℓ1

(
ℓ2 −

ℓ21
2

)
(1 + 2q)ϑ2q +

(1 + 3q)ϑ3qℓ
3
1

4

]
z3 + · · ·

(17)

Similarly, there exists an analytic function ν, defined in D, such that |ν(ξ)| < 1 and
p(ξ) = Υ(ν(ξ); q). Accordingly, we define

κ(ξ) =
1 + ν(ξ)

1 − ν(ξ)
= 1 + τ1ξ + τ2ξ

2 + · · · ∈ P. (18)



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6698 11 of 25

The corresponding expansion for ν(ξ) is then given by:

ν(ξ) =
τ1ξ

2
+

(
τ2 −

τ21
2

)
ξ2

2
+

(
τ3 − τ1τ2 −

τ31
4

)
ξ3

2
+ · · · . (19)

Therefore, the expansion of Υ(ν(ξ); q) becomes:

Υ(ν(ξ); q) = 1 +
p1τ1

2
ξ +

1

2

[(
τ2 −

τ21
2

)
p1 +

τ21
2
p2

]
ξ2

+
1

2

[(
τ3 − τ1τ2 +

τ31
4

)
ϑq + τ1

(
τ2 −

(1 + 2q)ϑ2qτ
2
1

2

)
+

(1 + 3q)ϑ3qτ
3
1

4

]
ξ3 + · · · .

(20)

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 BΣ(µ; q). These estimates not only offer insights into the geometric be-
havior 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.

To proceed, we first introduce the following lemma, which plays a fundamental role
in the theoretical analysis that follows.

Lemma 1. [34] If the function r ∈ P is given by the series

r(z) = 1 + ℓ1z + ℓ2z
2 + ℓ3z

3 + . . . ,

then the following coefficient estimates hold:

2ℓ2 = ℓ21 + x(4 − ℓ21),

4ℓ3 = ℓ31 + 2ℓ1(4 − ℓ21)x− ℓ1(4 − ℓ21)x
2 + 2(4 − ℓ21)(1 − |x|2)z,

for some x, z ∈ C with max{|x|, |z|} ≤ 1.

Theorem 1. For µ ∈ R+ ∪ {0}, let f ∈ BΣ(µ; q). Then

∣∣a2∣∣ ≤ min

 |ϑq|
µ+ q

,

√
2ϑ2q

ψ(q, µ)

 (21)

∣∣a3∣∣ ≤ min

{
ϑ2q

(µ+ q)2
+

|ϑq|
µ+ q2 + q

,
2ϑ2q

ψ(q, µ)
+

|ϑq|
µ+ q2 + q

}
, (22)

where
ψ(q, µ) = 2(µ+ q)2 ((1 + 2q)ϑq − 1) + ϑq

(
2q2 + (2q + 1)µ+ µ2

)
. (23)



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6698 12 of 25

Proof. Let f ∈ BΣ(µ; q) and ξ = f−1. Considering (13) and (14) we have

z1−µ ðq⟨f(z)⟩(
f(z)

)1−µ = Υ(φ(z); q), (z ∈ D), (24)

and
ξ1−µ ðq⟨χ(ξ)⟩(
χ(ξ)

)1−µ = Υ(ν(ξ); q), (ξ ∈ D). (25)

Since

z1−µ ðq⟨f(z)⟩(
f(z)

)1−µ = 1 + (µ+ q) a2z +

[(
µ+ q⌈2⌋q

)
a3 +

1

2

( (
−2q⌈2⌋q − 3

)
µ+ µ2

)
a22

]
z2 · · ·

+
[(
µ+ q⌈3⌋q

)
a4 +

(
−
(
⌈3⌋q + ⌈2⌋q − 2

)
+ (⌈3⌋q + ⌈2⌋q − 3)µ+ µ2

)
a2a3

+
1

6

(
6q + (11 − 9⌈2⌋q)µ+ 3(⌈2⌋q − 2)µ2 + µ3

)
a32

]
z3 + · · · ,

(26)
and

ξ1−µ ðq⟨χ(ξ)⟩(
χ(ξ)

)1−µ = 1 − (µ+ q)a2ξ

+

[
1

2

(
−2(⌈2⌋q − 2⌈3⌋q + 1) + (2⌈2⌋q + 1)µ+ µ2

)
a22 − (µ+ ⌈3⌋q − 1)a3

]
ξ2

+

[
− (µ+ ⌈4⌋q − 1)a4 +

(
5⌈4⌋q − ⌈2⌋q − ⌈3⌋q − 3 + (⌈3⌋q + ⌈2⌋q + 2)µ+ µ2

)
a2a3

+
1

6

(
12 − 30⌈4⌋q + 6⌈2⌋q + 12⌈3⌋q + (−5 − 3⌈2⌋q − 12⌈3⌋q)µ

+(−3⌈2⌋q − 6)µ2 − µ3
)
a32

]
ξ3 + · · · . (27)

It follows from the equations (24), (17), and (26) that

(µ+ q)a2 =
ϑqℓ1

2
(28)

(µ+ q⌈2⌋q)a3 +
1

2

(
−2q + (2⌈2⌋q − 3)µ+ µ2

)
a22 =

1

2

[(
ℓ2 −

ℓ21
2

)
ϑq +

(1 + 2qϑ2q)ℓ
2
1

2

]
(29)

(µ+ q⌈3⌋q)a4 +
(
−(⌈3⌋q + ⌈2⌋q − 2) + (⌈3⌋q + ⌈2⌋q − 3)µ+ µ2

)
a2a3

+
1

6

(
6(⌈2⌋q − 1) + (11 − 9⌈2⌋q)µ+ 3(⌈2⌋q − 2)µ2 + µ3

)
a32

=
ϑq
2

(
ℓ3 − ℓ1ℓ2 +

ℓ31
4

)
+

(1 + 2q)ϑ2q
2

ℓ1

(
ℓ2 −

ℓ21
2

)
+

(1 + 3q)ϑ3q
8

ℓ31.

(30)



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6698 13 of 25

Similarly, from the equations (25), (20), and (27) we obtain:

−(µ+ q)a2 =
ϑq
2
τ1, (31)

1

2

(
−2(⌈2⌋q − 2⌈3⌋q + 1) + (2⌈2⌋q + 1)µ+ µ2

)
a22 − (µ+ q⌈2⌋q)a3

=
ϑq
2

(
τ2 −

τ21
2

)
+

(1 + 2q)ϑ2q
4

τ21 ,

(32)

− (µ+ q⌈3⌋q)a4 +
(
5⌈4⌋q − ⌈2⌋q − ⌈3⌋q − 3 + (⌈3⌋q + ⌈2⌋q + 2)µ+ µ2

)
a2a3

+
1

6

(
12 − 30⌈4⌋q + 6⌈2⌋q + 12⌈3⌋q − (5 + 3⌈2⌋q + 12⌈3⌋q)µ− (3⌈2⌋q + 6)µ2 − µ3

)
a32

=
ϑq
2

(
τ3 − τ1τ2 +

τ31
4

)
+

(1 + 2q)ϑ2q
2

τ1

(
τ2 −

τ21
2

)
+

(1 + 3q)ϑ3q
8

τ31 .

(33)
From (28) and (31) we have:

a2 =
ϑq

2(µ+ q)
ℓ1 = − ϑq

2(µ+ q)
τ1, (34)

and this indicates that
ℓ1 = −τ1. (35)

Using (3) to equation (34), we obtain that

a2 ≤
∣∣∣∣ ϑq
µ+ q

∣∣∣∣ =
|ϑq|
µ+ q

. (36)

The squaring and addition of (28) and (31) lead to

a22 =
ϑ2q(ℓ

2
1 + τ21 )

8(µ+ q)2
. (37)

In addition, the sum of (29) and (32) gives

a22
(
2(⌈3⌋q − ⌈2⌋q) + (2⌈2⌋q − 1)µ+ µ2

)
=
ϑq
2

(ℓ2 + τ2) +
(1 + 2q)ϑ2q − ϑq

2
· ℓ

2
1 + τ21

2
.

From (37), we get

a22
(
2(⌈3⌋q − ⌈2⌋q) + (2⌈2⌋q − 1)µ+ µ2

)
=
ϑq
2

(ℓ2 + τ2) +
(1 + 2q)ϑ2q − ϑq

2
· 4(µ+ q)2a22

ϑ2q
,

which means

a22 =
(ℓ2 + τ2)ϑ

2
q

−4(µ+ q)2 (−1 + (1 + 2q)ϑq) + 2ϑq

(
2(⌈3⌋q − ⌈2⌋q) + (2⌈2⌋q − 1)µ+ µ2

) .



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6698 14 of 25

Since ⌈2⌋q = q + 1 and ⌈3⌋q = q2 + q + 1, then

a22 =
(ℓ2 + τ2)ϑ

2
q

2
[
2(µ+ q)2 ((1 + 2q)ϑq − 1) + ϑq (2q2 + (2q + 1)µ+ µ2)

] . (38)

Thus, (3) implies that

|a2| ≤

√
2ϑ2q

2(µ+ q)2 ((1 + 2q)ϑq − 1) + ϑq (2q2 + (2q + 1)µ+ µ2)
.

and hence,

|a2| ≤

√
2ϑ2q

ψ(q, µ)
, (39)

is satisfied for all µ ≥ 0 (see Figure 2), where ψ(q, µ) is given by (23).

Figure 2: The plot of the function ψ(q, µ).

Subsequently, by subtracting (32) from (29), we obtain

a3 = a22 +
ϑq(ℓ2 − τ2)

4(µ+ q⌈2⌋q)
= a22 +

ϑq(ℓ2 − τ2)

4(µ+ q2 + q)
. (40)

Hence,

|a3| ≤ |a2|2 +
|ϑq|

µ+ q2 + q
. (41)

Substituting equations (36) and (39) into (41), respectively, we obtain

|a3| ≤
ϑ2q

(µ+ q)2
+

|ϑq|
µ+ q2 + q

,

and

|a3| ≤
2ϑ2q

ψ(q, µ)
+

|ϑq|
µ+ q2 + q

.



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6698 15 of 25

Hence, the desired result follows, completing the proof with elegance and clarity.

In the next result, we derive the sharp bound for the functional
∣∣a3 − ηa22

∣∣ for func-
tions f ∈ BΣ(µ, q), where η ∈ R.

Theorem 2. For η ∈ R+ ∪ {0}, let f ∈ BΣ(µ, q). Then

∣∣a3 − η a22
∣∣ ≤


|ϑq|

µ+ q2 + q
, if |1 − η| ≤ ψ(q, µ)

2(µ+ q2 + q)|ϑq|
,

2(1 − η)ϑ2q
ψ(q, µ)

, if |1 − η| ≥ ψ(q, µ)

2(µ+ q2 + q)|ϑq|
,

(42)

where ψ(q, µ) is defined in (23).

Proof. Assuming that f ∈ BΣ(µ, q), it follows from equations (38) and (40) that

a3 − η a22 =
(1 − η)ϑ2q(ℓ2 + τ2)

2 [−2(µ+ q)2(−1 + (1 + 2q)ϑq) + ϑq(2q2 + (2q + 1)µ+ µ2)]
+

ϑq(ℓ2 − τ2)

4(µ+ q2 + q)

=

(
Ξ(q, µ, η) +

ϑq
4(µ+ q2 + q)

)
ℓ2 +

(
Ξ(q, µ, η) − ϑq

4(µ+ q2 + q)

)
τ2,

(43)
where

Ξ(q, µ, η) =
(1 − η)ϑ2q

2 [−2(µ+ q)2(−1 + (1 + 2q)ϑq) + ϑq(2q2 + (2q + 1)µ+ µ2)]
. (44)

Accordingly, taking the modulus of (43), we obtain:

∣∣a3 − η a22
∣∣ ≤


|ϑq|

µ+ q2 + q
, if 0 ≤ Ξ(q, µ, η) ≤ ϑq

4(µ+ q2 + q)
,

4Ξ(q, µ, η), if Ξ(q, µ, η) ≥ ϑq
4(µ+ q2 + q)

.

After straightforward computations, we obtain the result in (42).

3. On coefficient bounds of the second Hankel determinant for the
class BΣ(µ, q)

This section is devoted to deriving a coefficient inequality for the second Hankel
determinant associated with the class BΣ(µ, q), as presented in the following theorem:



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6698 16 of 25

Theorem 3. Let f ∈ BΣ(µ, q) . Then

∣∣H2,2(f)
∣∣ ≤



Y(2−) if X1 ≥ 0 and X2 ≥ 0,

max

{
ϑ2q

(q + q2 + µ)2
, Y(2−)

}
if X1 > 0 and X2 < 0,

ϑ2q
(q + q2 + µ)2

if X1 ≤ 0 and X2 ≤ 0,

max

{
Y(2−), Y

(√
−12X2

X1

)}
if X1 < 0 and X2 > 0,

(45)

where

X1 :=
∣∣ϑ4q(6q3(1 + 6q) + 6q2(5 + 18q)µ+ 3qµ(−1 + (11 + 36q)µ)

+µ
(
−2 + µ(−3 + (11 + 36q)µ)

))∣∣ (q + q2 + µ)2

− 6(q + µ)2
[
q3(1 + q(3 + q))ϑ2q − q(1 + q)

(
1 + 2q(1 + q(1 + q)(3 + 4q))

)
ϑ3q

−
(
−3q2(1 + 2q)ϑ2q + ϑ3q + q

(
3 + q(15 + 18q + 4q2 + 8(1 + q)(3 + q)q)

)
ϑ3q

)
µ

−
(
q(−3 + (−3 + q)q)ϑ2q + ϑ3q + 4q(3 + 2q)(1 + 2q)ϑ3q

)
µ2

+
(
ϑ2q − 4(ϑ3q + 2qϑ3q)

)
µ3
]
,

X2 := −(q + µ)2
[
2q5(3 + 4q)ϑ3q + qϑ3q

(
1 + 3µ+ 12(1 + 2q)µ2

)
+ ϑ3qµ

(
1 + µ+ (4 + 8q)µ2

)
+ 2q3

(
ϑ2q(−2 + µ)µ+ ϑ3q(4 + 4q + 9µ+ 16qµ)

)
+ q2

(
−2ϑ2qµ

2 + ϑ3q
(
3 + 3(5 + 8q)µ+ 8(1 + 2q)µ2

))
+ 2q4

(
ϑ2q(−1 + µ) + 2ϑ3q(3 + µ+ 2q(2 + µ))

)]
,

Y(2−) =
ϑ2q

(q + q2 + µ)2
+

X1 + 6X2c
2

6(q + µ)4(q + q2 + µ)2(q + q2 + q3 + µ)
,

and

Y

(√
−12X2

X1

)
=

ϑ2q
(q + q2 + µ)2

− 3X 2
2

2X1(q + µ)4(q + q2 + µ)2(q + q2 + q3 + µ)
.

Proof. Let f ∈ BΣ(µ, q). By employing an argument analogous to that used in the
proof of Theorem 1, subtracting equation (33) from (30) and utilizing (34) and (40)
yields

a4 =
1

−1 + ⌈4⌋
q

+ µ

(
−1

8
ϑq

(
−4ℓ3 − ℓ31 + 4ℓ1τ2 + 4τ3 + 2ℓ31ϑq − 4ℓ1τ2ϑq + 4ℓ31qϑq − 8ℓ1τ2qϑq



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6698 17 of 25

− ℓ31ϑ
2
q − 3ℓ31qϑ

2
q − 4ℓ1ℓ2(−1 + ϑq + 2qϑq) − ℓ31

(
1 − 2(1 + 2q)ϑq + (1 + 3q)ϑ2q

))
− 1

48(−1 + ⌈2⌋q + µ)3(−1 + ⌈3⌋q + µ)
ℓ1ϑ

2
q(−1 + µ)

(
6ℓ2(−1 + ⌈2⌋

q
+ µ)2(−2 + ⌈2⌋

q
+ ⌈3⌋

q
+ µ)

− 6τ2(−1 + ⌈2⌋
q

+ µ)2(−2 + ⌈2⌋
q

+ ⌈3⌋
q

+ µ) + ℓ21ϑq(−1 + ⌈3⌋
q

+ µ)(−6 + 6⌈3⌋
q

+ µ+ 3⌈2⌋
q
µ

+ µ2)
))

.

Since ⌈2⌋q = q + 1, ⌈3⌋q = q2 + q + 1, and ⌈4⌋q = q3 + q2 + q + 1, then, after some
simplifications, we obtain

a4 =
ϑq(ℓ3 − τ3)

2(q3 + q2 + q + µ)
− ϑq(1 − ϑq − 2qϑq)ℓ1(ℓ2 + τ2)

2(q3 + q2 + q + µ)

+
ϑ2q(µ− 1)(q2 + 2q + µ)ℓ1(ℓ2 − τ2)

8(q + µ)(q2 + q + µ)(q3 + q2 + q + µ)

+

(
−
ϑq(−2 + 4ϑq + 8qϑq − 2ϑ2q − 6qϑ2q)

8(q3 + q2 + q + µ)
−
ϑ3q(−1 + µ)(6q2 + 6q + 3qµ+ 4µ+ µ2)

48(q + µ)3(q3 + q2 + q + µ)

)
ℓ31.

(46)
Therefore, by using (34), (40), and (46), we deduce that

a2a4 − a23 =

ϑ4q(1 − µ)
(
6q2 + 3q(2 + µ) + µ(4 + µ)

)
+12(q + µ)3ϑ2q

[
1 − 2(1 + 2q)ϑq + (1 + 3q)ϑ2q

]
− 6ϑ4q(q + q2 + q3 + µ)

96(q + µ)4(q + q2 + q3 + µ)
ℓ41

−
ϑ2q(ℓ2 − τ2)

2

16(q + q2 + µ)2
−

ϑ3q(1 + 2q + 2q2 + 2q3 + µ)ℓ21(ℓ2 − τ2)

16(q + µ)2(q + q2 + µ)(q + q2 + q3 + µ)

−
(ϑ2q − ϑ3q − 2qϑ3q)ℓ

2
1(ℓ2 + τ2)

4(q + µ)(q + q2 + q3 + µ)
+

ϑ2qℓ1(ℓ3 − τ3)

4(q + µ)(q + q2 + q3 + µ)
.

(47)
From Lemma 1, it can be concluded that

2ℓ2 = ℓ21 + (4 − ℓ21)x, and 2τ2 = τ21 + (4 − τ21 )y.

Therefore, in view of (35), we obtain

ℓ2 − τ2 =
4 − ℓ21

2
(x− y), (48)

and

ℓ2 + τ2 = ℓ21 +
4 − ℓ21

2
(x+ y). (49)

Moreover, we have

4ℓ3 = ℓ31 + 2(4 − ℓ21)ℓ1x− ℓ1(4 − ℓ21)x
2 + 2(4 − ℓ21)(1 − |x|2)z,



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6698 18 of 25

and
4τ3 = τ31 + 2(4 − τ21 )τ1y − τ1(4 − τ21 )y2 + 2(4 − τ21 )(1 − |y|2)w.

for some x, y, z and w, with max{|x|, |y|, |z|, |w|} ≤ 1, and ℓ1, τ1 ∈ [0, 2]. Thus, we have

ℓ3−τ3 =
ℓ31
2

+
ℓ1(4 − ℓ21)

2
(x+y)− ℓ1(4 − ℓ21)

4
(x2+y2)+

4 − ℓ21
2

[
(1 − |x|2)z − (1 − |y|2)w

]
.

(50)
In addition, the substitution of (48)-(50) into (47) yields
a2a4 − a23 =

ℓ41ϑ
4
q

(
6q3(1 + 6q) + 6q2(5 + 18q)µ+ 3qµ(−1 + (11 + 36q)µ) + µ(−2 + µ(−3 + (11 + 36q)µ))

)
96(q + µ)4(q + q2 + q3 + µ)

+

(
−ϑ3q(1 + 2q + 2q2 + 2q3 + µ)(x− y)

32(q + µ)2(q + q2 + µ)(q + q2 + q3 + µ)
−

(−ϑ3q − 2qϑ3q)(x+ y)

8(q + µ)(q + q2 + q3 + µ)

)
ℓ21(4 − ℓ21)

−
ϑ2qℓ

2
1(4 − ℓ21)

16(q + µ)(q + q2 + q3 + µ)
(x2 + y2) −

ϑ2q(4 − ℓ21)
2(x− y)2

64(q + q2 + µ)2

+
ϑ2q(4 − ℓ21)ℓ1

8(q + µ)(q + q2 + q3 + µ)

[
(1 − |x|2)z − (1 − |y|2)w

]
.

Hence, |a2a4 − a23| ≤∣∣∣∣∣ϑ4q
(
6q3(1 + 6q) + 6q2(5 + 18q)µ+ 3qµ(−1 + (11 + 36q)µ) + µ(−2 + µ(−3 + (11 + 36q)µ))

)
96(q + µ)4(q + q2 + q3 + µ)

∣∣∣∣∣ ℓ41
+

(
−4(1 + 2q)(q + µ)(q + q2 + µ)ϑ3q − ϑ3q[1 + 2q(1 + q + q2) + µ]

)
32(q + µ)2(q + q2 + µ)(q + q2 + q3 + µ)

ℓ21(4 − ℓ21)(|x| + |y|)

+
ϑ2qℓ

2
1(4 − ℓ21)

16(q + µ)(q + q2 + q3 + µ)
(|x|2 + |y|2) +

ϑ2q(4 − ℓ21)2

64(q + q2 + µ)2
(|x| + |y|)2

+
ϑ2q(4 − ℓ21)ℓ1

8(q + µ)(q + q2 + q3 + µ)

(
2 − (|x|2 + |y|2)

)
=

∣∣∣∣∣ϑ4q
(
6q3(1 + 6q) + 6q2(5 + 18q)µ+ 3qµ(−1 + (11 + 36q)µ) + µ(−2 + µ(−3 + (11 + 36q)µ))

)
96(q + µ)4(q + q2 + q3 + µ)

∣∣∣∣∣ ℓ41
+

2ϑ2q(4 − ℓ21)ℓ1

8(q + µ)(q + q2 + q3 + µ)

+

(
−4(1 + 2q)(q + µ)(q + q2 + µ)ϑ3q − ϑ3q[1 + 2q(1 + q + q2) + µ]

)
32(q + µ)2(q + q2 + µ)(q + q2 + q3 + µ)

ℓ21(4 − ℓ21)(|x| + |y|)

+
ϑ2qℓ1(ℓ1 − 2)(4 − ℓ21)

16(q + µ)(q + q2 + q3 + µ)
(|x|2 + |y|2) +

ϑ2q(4 − ℓ21)2

64(q + q2 + µ)2
(|x| + |y|)2.

Letting |x| = ϵ1, |y| = ϵ2, and ℓ1 = c, the following result can be derived straightfor-
wardly:

|a2a4 − a23| ≤ Λ1 + Λ2(ϵ1 + ϵ2) + Λ3(ϵ
2
1 + ϵ22) + Λ4(ϵ1 + ϵ2)

2 =: F(ϵ1, ϵ2),



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6698 19 of 25

where

Λ1(c) =

∣∣∣∣∣ϑ4q
(
6q3(1 + 6q) + 6q2(5 + 18q)µ+ 3qµ(−1 + (11 + 36q)µ) + µ(−2 + µ(−3 + (11 + 36q)µ))

)
96(q + µ)4(q + q2 + q3 + µ)

∣∣∣∣∣ c4
+

2ϑ2q(4 − c2)c

8(q + µ)(q + q2 + q3 + µ)
≥ 0,

Λ2(c) =

(
−4(1 + 2q)(q + µ)(q + q2 + µ)ϑ3q − ϑ3q[1 + 2q(1 + q + q2) + µ]

32(q + µ)2(q + q2 + µ)(q + q2 + q3 + µ)

)
c2(4 − c2) ≥ 0,

Λ3(c) =
ϑ2qc(c− 2)(4 − c2)

16(q + µ)(q + q2 + q3 + µ)
≤ 0,

Λ4(c) =
ϑ2q(4 − c2)2

64(q + q2 + µ)2
≥ 0.

We aim to determine the maximum value of the function F(ϵ1, ϵ2) over the closed
square

Λ := {(ϵ1, ϵ2) : ϵ1, ϵ2 ∈ [0, 1]}.

Given that Λ3 < 0 and Λ3 + 2Λ4 > 0 for all c ∈ (0, 1), it follows that

Fϵ1ϵ1Fϵ2ϵ2 − (Fϵ1ϵ2)2 < 0

throughout the square Λ. This inequality implies that F cannot attain a local maximum
in the interior of the square Λ. Accordingly, we proceed to examine the boundary of
Λ in search of the maximum value. When setting ϵ1 = 0 and ϵ2 ∈ [0, 1] (similarly for
ϵ2 = 0 and ϵ1 ∈ [0, 1]), the function reduces to

F(0, ϵ2) = G(ϵ2) = Λ1 + Λ2ϵ2 + (Λ3 + Λ4)ϵ
2
2.

Case (i): When Λ3 + Λ4 ≥ 0. In this situation, for any fixed c ∈ [0, 2), the derivative
G′(ϵ2) = 2(Λ3 + Λ4)ϵ2 + Λ2 remains strictly positive for 0 < ϵ2 < 1. This implies that
G(ϵ2) is monotonically increasing on (0, 1). As a consequence, the function G achieves
its maximum value at ϵ2 = 1, and thus we have

maxG(ϵ2) = G(1) = Λ1 + Λ2 + Λ3 + Λ4.

Case (ii): When Λ3 + Λ4 < 0. Given that Λ2 + 2(Λ3 + Λ4) ≥ 0 for ϵ2 ∈ (0, 1) and any
fixed c ∈ [0, 2), it follows from the inequality

Λ2 + 2(Λ3 + Λ4) < 2(Λ3 + Λ4)ϵ2 + Λ2 < Λ2

that G′(ϵ2) > 0. Therefore, the function G(ϵ2) is increasing, and it attains its maximum
at ϵ2 = 1. Additionally, when c = 2, the expression for F(ϵ1, ϵ2) simplifies to

F(ϵ1, ϵ2)
∣∣
c=2

=∣∣∣∣∣ϑ4q
(
6p3(1 + 6q) + 6p2(5 + 18q)µ+ 3pµ(−1 + (11 + 36q)µ) + µ (−2 + µ(−3 + (11 + 36q)µ))

)
6(p+ µ)4(p+ p2 + p3 + µ)

∣∣∣∣∣ .
(51)



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6698 20 of 25

Taking into account the analysis in both subcases (i) and (ii), we deduce that for
ϵ2 ∈ [0, 1) and any c ∈ [0, 2], the maximum of G(ϵ2) is again given by

maxG(ϵ2) = G(1) = Λ1 + Λ2 + Λ3 + Λ4.

Now, considering the boundary condition ϵ1 = 1 and 0 ≤ ϵ2 ≤ 1 (similarly, ϵ2 = 1 and
0 ≤ ϵ1 ≤ 1), the function takes the form

F(1, ϵ2) = H(ϵ2) = (Λ3 + Λ4)ϵ
2
2 + (Λ2 + 2Λ4)ϵ2 + Λ1 + Λ2 + Λ3 + Λ4.

Following the same logic applied earlier for the cases involving Λ3 + Λ4, we find that
H(ϵ2) reaches its maximum at ϵ2 = 1, leading to

maxH(ϵ2) = H(1) = Λ1 + 2Λ2 + 2Λ3 + 4Λ4.

Since it holds that G(1) = H(1) for all c ∈ [0, 2], the maximum value of the function
F(ϵ1, ϵ2) over the boundary of the square Λ is attained at the point (1, 1). Consequently,
the maximum of F within the closed square Λ is achieved precisely at this corner.
Moreover, we define the function Y : [0, 2] → R by

Y(c) = maxF(ϵ1, ϵ2) = F(1, 1) = Λ1 + 2Λ2 + 2Λ3 + 4Λ4, (52)

which expresses the maximum value of F in terms of c. By substituting the explicit
expressions of Λ1,Λ2,Λ3, and Λ4 into the function Y as defined in equation (52), we
arrive at a more detailed representation of Y(c) in terms of the parameters involved:

Y(c) =
ϑ2q

(p+ p2 + µ)2
+

X1c
4 + 24X2c

2

96(p+ µ)4(p+ p2 + µ)2(p+ p2 + p3 + µ)
,

where

X1 :=
∣∣ϑ4q(6q3(1 + 6q) + 6q2(5 + 18q)µ+ 3qµ(−1 + (11 + 36q)µ)

+µ
(
−2 + µ(−3 + (11 + 36q)µ)

))∣∣ (q + q2 + µ)2

− 6(q + µ)2
[
q3(1 + q(3 + q))ϑ2q − q(1 + q)

(
1 + 2q(1 + q(1 + q)(3 + 4q))

)
ϑ3q

−
(
−3q2(1 + 2q)ϑ2q + ϑ3q + q

(
3 + q(15 + 18q + 4q2 + 8(1 + q)(3 + q)q)

)
ϑ3q

)
µ

−
(
q(−3 + (−3 + q)q)ϑ2q + ϑ3q + 4q(3 + 2q)(1 + 2q)ϑ3q

)
µ2

+
(
ϑ2q − 4(ϑ3q + 2qϑ3q)

)
µ3
]
,

and

X2 := −(q + µ)2
[
2q5(3 + 4q)ϑ3q + qϑ3q

(
1 + 3µ+ 12(1 + 2q)µ2

)
+ ϑ3qµ

(
1 + µ+ (4 + 8q)µ2

)
+ 2q3

(
ϑ2q(−2 + µ)µ+ ϑ3q(4 + 4q + 9µ+ 16qµ)

)



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6698 21 of 25

+ q2
(
−2ϑ2qµ

2 + ϑ3q
(
3 + 3(5 + 8q)µ+ 8(1 + 2q)µ2

))
+ 2q4

(
ϑ2q(−1 + µ) + 2ϑ3q(3 + µ+ 2q(2 + µ))

)]
,

Let us suppose that the function Y(c) attains its maximum at an interior point in
the interval c ∈ [0, 2]. Through straightforward computations, we obtain the following
expression:

Y ′(c) =
(X1c

2 + 12q)c

24(q + µ)4(q + q2 + µ)2(q + q2 + q3 + µ)
.

In the subsequent analysis, we investigate the sign of Y ′(c) by considering various com-
binations of the signs of X1 and q, as detailed below:

(i) Let X1 ≥ 0 and X2 ≥ 0, then Y ′(c) ≥ 0, so Y(c) is an increasing function. There-
fore,

max
{
Y(c) : c ∈ (0, 2)

}
= Y(2−)

=
ϑ2q

(q + q2 + µ)2
+

X1 + 6X2c
2

6(q + µ)4(q + q2 + µ)2(q + q2 + q3 + µ)
,

(53)

that is,

max
{

max
{
F (ϵ1, ϵ2) : ϵ1, ϵ2 ∈ [0, 1]

}
: c ∈ (0, 2)

}
= Y(2−).

(ii) Let X1 > 0 and X2 < 0, then c0 =
√

−12X2
X1

is a critical point of the function Y(c).

We assume that c0 ∈ (0, 2). Since Y ′′(c) > 0, c0 is a local minimum point of the
function Y(c). That is, the function Y(c) cannot have a local maximum.

(iii) Let X1 ≤ 0 and X2 ≤ 0, then Y ′(c) ≤ 0, so Y(c) is a decreasing function on the
interval (0, 2). Therefore,

max{Y(c) : c ∈ (0, 2)} = Y(0+) = 4Λ4 =
ϑ2q

(q + q2 + µ)2
. (54)

(iv) Let X1 < 0 and X2 > 0, then c0 is a critical point of the function Y(c). We assume
that c0 ∈ (0, 2). Since Y ′′(c) < 0, c0 is a local maximum point of the function Y(c),
and the maximum value occurs at c = c0. Therefore,

max{Y(c) : c ∈ (0, 2)} = Y(c0), (55)

where

Y(c0) =
ϑ2q

(q + q2 + µ)2
− 3X 2

2

2X1(q + µ)4(q + q2 + µ)2(q + q2 + q3 + µ)



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6698 22 of 25

Thus, from equations (51) to (55), the proof is complete.

If µ = 0, we obtain the following results for the class SLΣ(Υ(z; q)) defined in Exam-
ple (2)

Corollary 1. Let f given by (1) be in the class SLΣ(Υ(z); q). Then∣∣a2∣∣ ≤ ∣∣ϑq∣∣
q
√

1 − 2qϑq
. (56)

∣∣a3∣∣ ≤ ∣∣ϑq∣∣(q − (1 + q + 2q2)ϑq
)

q2(1 + q)
(
1 − 2qϑq

) . (57)

∣∣a3 − η a22
∣∣ ≤


|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 |

(58)

If q 7→ 1− and µ = 0, we obtain the following results for the class SLΣ(Υ(z)) defined
in Example (3)

Corollary 2. [35] Let f given by (1) be in the class SLΣ(Υ(z)). Then∣∣a2∣∣ ≤ ∣∣ϑ∣∣
√

1 − 2ϑ
,

∣∣a3∣∣ ≤ ∣∣ϑ∣∣(1 − 4ϑ
)

2
(
1 − 2ϑ

) .
and ∣∣a3 − η a22

∣∣ ≤


|ϑ|
2 ,

∣∣1 − α
∣∣ ≤ 1−2ϑ

2|ϑ|

(1−α)ϑ2

1−2ϑ ,
∣∣1 − α

∣∣ ≥ 1−2ϑ
2|ϑ|

4. Conclusion

In this study, we introduced and analyzed a subclass of bi-univalent functions associ-
ated with shell-like curves, formulated via the q-analogue of Fibonacci numbers, within
the framework of the Bazilevič-type class. By employing the subordination principle, we
obtained sharp bounds for the initial coefficients of functions in this class. Additionally,
we derived Fekete–Szegö-type inequalities and estimates for the second Hankel determi-
nant, thereby contributing to the deeper understanding of bi-univalent function theory
and its interplay with special function spaces.

These findings not only enrich the structural theory of bi-univalent functions but
also highlight the influence of the underlying q-calculus in shaping their analytic and
geometric behavior.

Future investigations may focus on extending the current results to higher-order
coefficients, refining extremal characterizations of these subclasses, and studying their
geometric features in more depth. Further directions include exploring sharp bounds as-
sociated with the Zalcman conjecture and analyzing the third-order Hankel determinant
within the same analytic framework.



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6698 23 of 25

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