EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 7065
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

A Study on Bi-Univalent Functions of Complex Order
Arising from the q-Fibonacci Analogue

Abdullah Alsoboh1,∗, Ala Amourah2,3, Khaled Almashrafi1,∗, Tala Sasa4

1 Department of Basic and Applied Sciences, College of Applied and Health Sciences,
A’Sharqiyah, Post Box No. 42, Post Code No. 400, Ibra, Sultanate of Oman
2 Mathematics Education Program, Faculty of Education and Arts, Sohar University,
Sohar 311, Oman
3 Jadara University Research Center, Jadara University, Jordan
4 Department of Mathematics, Faculty of Science, Applied Science Private University,
Amman, Jordan

Abstract. This paper introduces new subclasses of bi-univalent functions of complex order
linked with shell-like domains, formulated via the subordination principle and the q analog of
Fibonacci numbers. Motivated by recent advances in q-calculus and its applications in geometric
function theory, we construct and examine two distinct families of analytic bi-univalent functions.
For these subclasses, coefficient estimates are derived for the initial Taylor–Maclaurin coefficients,
together with sharp bounds for the Fekete–Szegö functional expressed in terms of the parameters
involved. The results of this study extend and unify earlier results in the theory of bi-univalent
functions, while providing new insights into the interaction between the theory of bi-univalent
functions, the q-Fibonacci framework, and shell-like geometries. Furthermore, the subclasses
established here may serve as a foundation for future studies on analytic function spaces, special
functions, and their operator-theoretic connections.
2020 Mathematics Subject Classifications: 30C45, 11B37, 81P68
Key Words and Phrases: Analytic mappings, classes of bi-univalent functions, Fekete–Szegö
operator, Fibonacci numbers, q-analysis, shell-shaped domains

1. Introduction and Preliminaries

Let A denote the class of analytic functions in the open unit disk

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

∗Corresponding author.
∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7065

Email addresses: abdullah.alsoboh@asu.edu.om (A. Alsoboh),
khaled.almashrafi@asu.edu.om (K. Almashrafi)

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), 7065 2 of 16

the interior of the unit circle centered at the origin. Every member f ∈ A is assumed
to satisfy the standard normalization

f(0) = 0, f ′(0) = 1,

so that translation and dilation effects at the origin are eliminated.
Each function in A can be represented by a Maclaurin series of the form

f(z) = z +

∞∑
s=2

δs z
s, z ∈ O, (1)

where the coefficients δs encode its higher-order analytic structure. In particular, nor-
malization ensures that the leading term is exactly z.

A function f is called a Schwarz function if it is analytic in O, vanishes at the origin
and satisfies |f(z)| < 1 throughout O. Such functions are central to geometric function
theory because of their role in univalent and conformal mapping problems.

For two analytic functions f1, f2 ∈ A , we write f1 ≺ f2 whenever there exists a
Schwarz function η with

f1(z) = f2(η(z)), z ∈ O.

This concept of subordination provides a powerful framework for analyzing inclusion,
growth, and distortion properties.

Let S ⊂ A denote the class of univalent functions in O. If f ∈ S, then f admits an
analytic inverse f−1 defined in a disk of radius at least 1/4, with series expansion

f−1(ξ) = ξ − δ2ξ
2 + (2δ22 − δ3)ξ

3 − (5δ32 + δ4 − 5δ2δ3)ξ
4 + · · · . (2)

A function is termed bi-univalent if both f and f−1 are univalent in O. The set of such
functions is denoted by Σ.

Another class of interest is P, which consists of analytic functions in O with a positive
real part. Each ψ ∈ P admits the series

ψ(z) = 1 +

∞∑
s=1

psz
s, z ∈ O, (3)

where the sharp estimate
|ps| ≤ 2, s ≥ 1, (4)

follows from Carathéodory’s lemma [1]. Moreover, it is well known that ψ ∈ P if and
only if

ψ(z) ≺ 1 + z

1− z
, z ∈ O.

The class of starlike functions, denoted S∗, admits elegant formulations in terms of
subordination. Ma and Minda [2] introduced the generalized family

S∗(Φ) =

{
f ∈ A :

zf ′(z)

f(z)
≺ Φ(z), Φ ∈ P

}
,



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7065 3 of 16

where Φ is analytic in O and satisfies <e (Φ(z)) > 0.
Prominent subclasses of S∗ emerge from specific selections of Φ(z). For example,

Janowski [3, 4] employed Φ(z) = 1+z
1−z . Robertson [5] examined the choice Φ(z) =

1+(1−2ϑ)z
1−z , where 0 ≤ ϑ < 1. Sokól [6] studied Φ(z) = 1+ϑ2z2

1−ϑz−ϑ2z2
with ϑ = 1−

√
5

2 ,
and later introduced the case Φ(z) = 3

3+(ϑ−3)z−ϑ2z2
for ϑ ∈ (−3, 1] [7]. These examples

underscore the diversity of subclasses obtained from suitable choices of the function Φ.
Quantum calculus (or q-calculus) generalizes classical calculus by introducing a pa-

rameter q ∈ (0, 1), providing a natural deformation with deep links to physics, quantum
mechanics, and geometric function theory. A fundamental tool is the q-difference op-
erator ðq. Seminal references include the monograph by Gasper and Rahman [8] and
analytic function applications studied by Seoudy and Aouf [9]. Further developments
appear in [10–31]. Thus, polynomials bridge the gap between abstract complex analysis
and computational modeling, allowing deeper exploration of geometric mappings and
their analytic behavior [32–36].

Definition 1 ([17]). The q-bracket is defined by

dκcq =



1− qλ

1− q
, 0 < q < 1, λ ∈ C∗,

1, q → 0+,

λ, q → 1−,

γ−1∑
s=0

qs, 0 < q < 1, λ = γ ∈ N.

Definition 2 ([17]). The q-derivative (or q-difference operator) of f is

ðq〈f(z)〉 =


f(z)− f(qz)

z − qz
, 0 < q < 1, z 6= 0,

f ′(0), z = 0,

f ′(z), q → 1−.

Remark 1. If f has the form (1), then

ðq〈f(z)〉 = 1 +

∞∑
s=2

dscq δs zs−1,

while for its inverse f−1 given by (2),

ðq〈f−1(ξ)〉 = 1− d2cqδ2ξ + d3cq(2δ22 − δ3)ξ
2 − d4cq(5δ32 + δ4 − 5δ2δ3)ξ

3 + · · · .

Alsoboh et al. [37] introduced the q-starlike class

SLq =

{
f ∈ A :

zðq〈f(z)〉
f(z)

≺ Υ(z; q)

}
, (5)



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7065 4 of 16

where
Υ(z; q) =

1 + qϑ2qz
2

1− ϑqz − qϑ2qz
2
, ϑq =

1−
√
4q + 1

2q
, (6)

with ϑq representing the q analog of the Fibonacci numbers.
Moreover, they established connections between ϑq and the q-Fibonacci polynomials

φs(q), with the recurrence

p̂s =


ϑq, s = 1,

(2q + 1)ϑ2q , s = 2,

(3q + 1)ϑ3q , s = 3,

(φs+1(q) + qφs−1(q))ϑ
s
q, s ≥ 4.

(7)

The initial terms of the q-Fibonacci sequence are listed in Table 1, reducing to the
classical Fibonacci numbers as q → 1−.

Table 1: Classical Fibonacci numbers and their q-analogues.

Classical Fibonacci q-Fibonacci

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

In the limit q → 1−, the class SLq recovers the classical starlike family associated
with the Fibonacci generating function:

SL =
{
f ∈ A : zf ′(z)

f(z) ≺ Υ(z)
}
, Υ(z) =

1 + ϑ2z2

1− ϑz − ϑ2z2
,

where ϑ = 1−
√
5

2 .
Furthermore, Alsoboh et al. [17] defined the q-convex class KSLq via

1 +
zð2q〈f(z)〉
ðq〈f(z)〉

≺ Υ(z; q), z ∈ O. (8)

This generalization involves a higher-order q-difference operator, capturing refined geo-
metric structures.



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7065 5 of 16

2. Definition and Examples

Motivated by the theory of q-Fibonacci numbers, we introduce a new subclass of
bi-univalent functions associated with shell-like curves.

Definition 3. Let β ∈ [0, 1] and ρ ∈ C \ {0}. A bi-univalent function f of the form (1)
is said to belong to the class SLMΣ(β, ρ; q) if and only if

1 +
1

ρ

[
(1− β)

z ðq〈f(z)〉
f(z)

+ β
ðq(z ðq〈f(z)〉)

ðq〈f(z)〉
− 1

]
≺ Υ(z; q), z ∈ O, (9)

and simultaneously

1 +
1

ρ

[
(1− β)

ξ ðq〈χ(ξ)〉
χ(ξ)

+ β
ðq(ξ ðq〈χ(ξ)〉)

ðq〈χ(ξ)〉
− 1

]
≺ Υ(ξ; q), ξ ∈ O, (10)

where χ = f−1 is given by (2), and the functions Υ(z; q) and ϑq are defined in (6).

Changing the parameters ρ ∈ C∗, β ∈ [0, 1], and q ∈ (0, 1), a family of subclasses of
Σ is generated, exhibiting diverse geometric characteristics.

Example 1. If ρ = 1, the class reduces to SLMΣ(β; q), where each f ∈ Σ satisfies

(1− β)
z ðq〈f(z)〉
f(z)

+ β
ðq(z ðq〈f(z)〉)

ðq〈f(z)〉
≺ Υ(z; q),

and
(1− β)

ξ ðq〈χ(ξ)〉
χ(ξ)

+ β
ðq(ξ ðq〈χ(ξ)〉)

ðq〈χ(ξ)〉
≺ Υ(ξ; q),

where χ = f−1.

Example 2. If β = 0 and ρ = 1, we obtain SLΣ(Υ(z; q)), consisting of f ∈ Σ such that

z ðq〈f(z)〉
f(z)

≺ Υ(z; q),
ξ ðq〈χ(ξ)〉
χ(ξ)

≺ Υ(ξ; q).

Example 3. If β = 1 and ρ = 1, the class reduces to KLΣ(Υ(z; q)), consisting of f ∈ Σ
such that

1 +
z ð2q〈f(z)〉
ðq〈f(z)〉

≺ Υ(z; q), 1 +
ξ ð2q〈χ(ξ)〉
ðq〈χ(ξ)〉

≺ Υ(ξ; q).

Example 4. If q → 1− and ρ = 1, we recover the classical class SLMΣ(β), where f ∈ Σ
satisfies

(1− β)
zf ′(z)

f(z)
+ β

zf ′′(z)

f ′(z)
≺ Υ(z), (1− β)

ξχ′(ξ)

χ(ξ)
+ β

ξχ′′(ξ)

χ′(ξ)
≺ Υ(ξ),

with ϑ = 1−
√
5

2 .



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7065 6 of 16

Example 5. If q → 1−, β = 0, and ρ = 1, we obtain SLΣ(Υ(z)), where f ∈ Σ satisfies

zf ′(z)

f(z)
≺ Υ(z),

ξχ′(ξ)

χ(ξ)
≺ Υ(ξ).

Example 6. If q → 1−, β = 1, and ρ = 1, the class reduces to KLΣ(Υ(z)), consisting
of f ∈ Σ such that

1 +
zf ′′(z)

f ′(z)
≺ Υ(z), 1 +

ξχ′′(ξ)

χ′(ξ)
≺ Υ(ξ).

3. Main Results

In this section, we establish coefficient bounds for the initial Taylor coefficients |δ2|
and |δ3| of functions belonging to the class SLMΣ(β, ρ; q), as introduced in Definition 3.

Firstly, let us

P(z) = 1 + P1z + P2z
2 + P3z

3 + · · · , P(z) ≺ Υ(z; q).

Then there exists a Schwarz function ψ ∈ P, with |ψ(z)| < 1 for all z ∈ O, such that

P(z) = Υ(ψ(z); q).

In this setting, we may define the following.

ℏ(z) =
1 + ψ(z)

1− ψ(z)
= 1 + ℓ1z + ℓ2z

2 + · · · , z ∈ O, (11)

which belongs to the class P.
Consequently, since ψ(z) is analytic in O and subordinate to Υ(z; q), it admits the

Taylor expansion

ψ(z) =
ℓ1
2
z +

1

2

(
ℓ2 −

ℓ21
2

)
z2 +

1

2

(
ℓ3 − ℓ1ℓ2 −

ℓ31
4

)
z3 + · · · , (12)

and

Υ(ψ(z); q) = 1 + P̂1

[
ℓ1z

2
+

(
ℓ2 −

ℓ21
2

)
z2

2
+

(
ℓ3 − ℓ1ℓ2 −

ℓ31
4

)
z3

2
+ · · ·

]
+ P̂2

[
ℓ1z

2
+

(
ℓ2 −

ℓ21
2

)
z2

2
+

(
ℓ3 − ℓ1ℓ2 −

ℓ31
4

)
z3

2
+ · · ·

]2
+ P̂3

[
ℓ1z

2
+

(
ℓ2 −

ℓ21
2

)
z2

2
+

(
ℓ3 − ℓ1ℓ2 −

ℓ31
4

)
z3

2
+ · · ·

]3
+ · · ·

= 1 +
P̂1ℓ1
2

z +
1

2

[(
ℓ2 −

ℓ21
2

)
P̂1 +

ℓ21
2

P̂2

]
z2

+
1

2

[(
ℓ3 − ℓ1ℓ2 +

ℓ31
4

)
P̂1 + ℓ1

(
ℓ2 −

ℓ21
2

)
P̂2 +

ℓ31
4

P̂3

]
z3 + · · · .

(13)



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7065 7 of 16

Similarly, one can find an analytic function ν in O, with |ν(ξ)| < 1, such that P(ξ) =
Υ(ν(ξ); q). Accordingly, we may express the associated function

κ(ξ) = (1 + ν(ξ))(1− ν(ξ))−1 = 1 + τ1ξ + τ2ξ
2 + · · · ∈ P. (14)

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
+ · · · , (15)

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

Υ(ν(ξ); q) = 1 +
P̂1τ1
2

ξ +
1

2

[(
τ2 −

τ21
2

)
P̂1 +

τ21
2

P̂2

]
ξ2

+
1

2

[(
τ3 − τ1τ2 +

τ31
4

)
P̂1 + τ1

(
τ2 −

τ21
2

)
P̂2 +

τ31
4

P̂3

]
ξ3 + · · · .

(16)

Having established the necessary groundwork and auxiliary results, we are now pre-
pared to derive coefficient bounds for functions belonging to the newly introduced class
SLMΣ(β, ρ; q). These estimates provide valuable insights into the geometric behavior of
such bi-univalent functions and, moreover, emphasize the role of the deformation pa-
rameter q and the weighting parameter β in shaping the coefficient structure. The next
theorem provides sharp bounds for the initial coefficients |α2| and |α3|.

Theorem 1. For ρ ∈ C∗ and β ∈ [0, 1], let f ∈ SLM∑(β, ρ; q). Then

∣∣α2

∣∣ ≤ |ρ||ϑq|√∣∣∣ρϑq(K −X) +
(
1− (2q + 1)ϑq

)
C
∣∣∣ . (17)

∣∣α3

∣∣ ≤ |ρ||ϑq|
{∣∣(K −X

)
ρϑq +

(
1− (2q + 1)ϑq

)
C
∣∣+ |ρ||ϑq|K

}
K

∣∣(K −X
)
ρϑq +

(
1− (2q + 1)ϑq

)
C
∣∣ , (18)

where

K = qd2cq
(
1 + qd2cqβ

)
, (19)

X = q
[
1 + β

(
d2c2

q
− 1

) ]
, (20)

C = q2(1 + qβ)2. (21)

Proof. Let f ∈ SL∑(Υ(z)) and ξ = f−1. Taking into account (9) and (10), we have

1 +
1

ρ

(
(1− β)

zðq〈f(z)〉
f(z)

+ β
ðq (z ðq〈f(z)〉)

ðq〈f(z)〉
− 1

)
= Υ(ψ(z); q), (z ∈ O), (22)

and

1 +
1

ρ

(
(1− β)

ξðq〈χ(ξ)〉
χ(ξ)

+ β
ðq (ξðq〈χ(ξ)〉)

ðq〈χ(ξ)〉
− 1

)
= Υ(ν(ξ); q), (ξ ∈ O). (23)



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7065 8 of 16

Since
1

ρ

(
(1− β)

zðq〈f(z)〉
f(z)

+ β
ðq (z ðq〈f(z)〉)

ðq〈f(z)〉
− 1

)
= 1 +

q(1 + qβ)

ρ
α2z

+

qd2cq(1 + qd2cqβ)α3 − q
(
1 + β(d2c2

q
− 1)

)
α2
2

ρ

 z2 +O
(
z3
)

= 1 +
q(1 + qβ)

ρ
α2z +

(
Kα3 −Xα2

2

ρ

)
z2 +O

(
z3
)
,

(24)

and

1

ρ

(
(1− β)

ξðq〈χ(ξ)〉
χ(ξ)

+ β
ðq (ξðq〈χ(ξ)〉)

ðq〈χ(ξ)〉
− 1

)
= 1− q(1 + qβ)

ρ
α2ξ

+

2qd2cq(1 + qd2cqβ)− q
(
1 + β(d2c2

q
− 1)

)
ρ

α2
2 −

qd2cq(1 + qd2cqβ)
ρ

α3

 ξ2 +O
(
ξ3
)

= 1− q(1 + qβ)

ρ
α2 ξ +

(
2K −X

ρ
α2
2 −

K

ρ
α3

)
ξ2 +O

(
ξ3
)
.

(25)

Compared with (22) and (24), along (13), yields

q(1 + qβ)

ρ
α2z +

(
Kα3 −Xα2

2

ρ

)
z2 +O

(
z3
)
=

P̂1ℓ1
2

z +
1

2

[(
ℓ2 −

ℓ21
2

)
P̂1 +

ℓ21
2

P̂2

]
z2 +O

(
z3
)
.

(26)

Besied that by comparing (23) and (25), along (16), yields

−q(1 + qβ)

ρ
α2 ξ+

(
2K −X

ρ
α2
2 −

K

ρ
α3

)
ξ2 +O

(
ξ3
)

+ · · · = P̂1τ1
2

ξ +
1

2

[(
τ2 −

τ21
2

)
P̂1 +

τ21
2

P̂2

]
ξ2 + · · · .

(27)

Equating the pertinent coefficient in (26) and (27), using (19) and (20), we obtain

q(1 + qβ)

ρ
α2 =

P̂1ℓ1
2

(28)

−q(1 + qβ)

ρ
α2 =

P̂1τ1
2

(29)

Kα3 −Xα2
2

ρ
=

1

2

[(
ℓ2 −

ℓ21
2

)
P̂1 +

ℓ21
2

P̂2

]
(30)

2K −X

ρ
α2
2 −

K

ρ
α3 =

1

2

[(
τ2 −

τ21
2

)
P̂1 +

τ21
2

P̂2

]
(31)



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7065 9 of 16

From (28) and (29), we have

ℓ1 = −τ1 ⇐⇒ ℓ21 = τ21 , (32)

and
α2
2 =

ρ2ϑ2q
8q2(1 + qβ)2

(
ℓ21 + τ21

)
⇐⇒ ℓ21 + τ21 =

8q2(1 + qβ)2

ρ2ϑ2q
α2
2. (33)

Now, by summing (30) and (31), we obtain

2(K −X)

ρ
α2
2 =

1

2

[
(ℓ2 + τ2)P̂1 −

ℓ21 + τ21
2

P̂1 +
ℓ21 + τ21

2
P̂2

]

=
1

2

[
(ℓ2 + τ2)P̂1 +

ℓ21 + τ21
2

(P̂2 − P̂1)

]
.

(34)

Since from the expansion of Υ(z; q) we have

P̂1 = ϑq, P̂2 = (2q + 1)ϑ2q ,

we can express
P̂2 − P̂1 = ϑq

(
(2q + 1)ϑq − 1

)
.

Substituting these into (34) yields

2(K −X)

ρ
α2
2 =

ϑq
2
(ℓ2 + τ2) +

[
(2q + 1)ϑ2q

4
− ϑq

4

]
(ℓ21 + τ21 ). (35)

Now, substituting (33) into (35), we obtain

2(K −X)

ρ
α2
2 =

ϑq
2
(ℓ2 + τ2) +

[
(2q + 1)ϑ2q

4
− ϑq

4

](
8q2(1 + qβ)2

ρ2ϑ2q
α2
2

)

=
ϑq
2
(ℓ2 + τ2) +

2q2(1 + qβ)2

ρ2

(
(2q + 1)− 1

ϑq

)
α2
2.

(36)

Rearranging terms, we get[
2(K −X)

ρ
− 2q2(1 + qβ)2

ρ2

(
(2q + 1)− 1

ϑq

)]
α2
2 =

ϑq
2
(ℓ2 + τ2). (37)

Finally, defining C = q2(1 + qβ)2, we can rewrite (37) as

α2
2 =

(ℓ2 + τ2) ρ
2ϑ2q

4
[
(K −X)ρϑq +

(
1− (2q + 1)ϑq

)
C
] . (38)



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7065 10 of 16

Since for functions ψ, ν ∈ P, we have |ℓ2|, |τ2| ≤ 2, it follows that |ℓ2+τ2| ≤ 4. Therefore,
we obtain the bound

|α2| ≤
|ρ| |ϑq|√∣∣ (K −X)ρϑq +

(
1− (2q + 1)ϑq

)
C
∣∣ .

Now, to find the bound of |α3|, subtract (31) from (30). On the left-hand side we get

Kα3 −Xα2
2

ρ
− (2K −X)α2

2 −Kα3

ρ
=

2K

ρ

(
α3 − α2

2

)
.

On the right-hand side,

1

2

[(
ℓ2 −

ℓ21
2

)
P̂1 +

ℓ21
2

P̂2

]
−1

2

[(
τ2 −

τ21
2

)
P̂1 +

τ21
2

P̂2

]
=

1

2

[
(ℓ2 − τ2)P̂1 +

ℓ21 − τ21
2

(P̂2 − P̂1)

]
.

Using (28)–(29) we have ℓ1 = −τ1, and hence ℓ21 = τ21 and the last term vanishes.
Therefore,

2K

ρ

(
α3 − α2

2

)
=

1

2
(ℓ2 − τ2) P̂1.

Recalling P̂1 = ϑq, we obtain the identity

α3 = α2
2 +

ρϑq
4K

(ℓ2 − τ2). (39)

Taking absolute values and using the Carathéodory bounds |ℓ2|, |τ2| ≤ 2 (that is,
|ℓ2 − τ2| ≤ 4), we get

|α3| ≤ |α2|2 +
|ρ| |ϑq|
K

. (40)

Finally, substituting the estimate for |α2| from (39) (or (38)) yields

|α3| ≤
|ρ| |ϑq|

{ ∣∣ ρϑq(K −X) +
(
1− (2q + 1)ϑq

)
C
∣∣+ |ρ| |ϑq|K

}
K

∣∣ ρϑq(K −X) +
(
1− (2q + 1)ϑq

)
C
∣∣ .

Theorem 2. For ρ ∈ C∗ and β ∈ [0, 1], let f ∈ SLM∑(β, ρ; q). Then

∣∣α3 − µα2
2

∣∣ ≤


|ρ||ϑq |
K ,

∣∣1− µ
∣∣ ≤ ∣∣ρϑq(K−X)+(1−(2q+1)ϑq)C

∣∣
|ρ||ϑq |K

|1−µ||ρ|2|ϑq |2∣∣(K−X)ρϑq+(1−(2q+1)ϑq)C
∣∣ , ∣∣1− µ

∣∣ ≥ ∣∣ρϑq(K−X)+(1−(2q+1)ϑq)C
∣∣

|ρ||ϑq |K

(41)
where K,X,C are given by (19), (20) and (21), respectively.



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7065 11 of 16

Proof. Let f ∈ SLM∑(β, ρ; q), from (36) and (39) we have

α3 − µα2
2 =

(1− µ)ρ2ϑ2q

4
((
K −X

)
ρϑq +

(
1− (2q + 1)ϑq

)
C
)(ℓ2 + τ2) +

ρϑq
4K

(
ℓ2 − τ2

)
=

(
K (µ) +

ρϑq
4K

)
ℓ2 +

(
K (µ)− ρϑq

4K

)
τ2,

(42)

where
K (µ) =

(1− µ)ρ2ϑ2q

4
((
K −X

)
ρϑq +

(
1− (2q + 1)ϑq

)
C
) . (43)

Then, taking the modulus of (42), we conclude that

∣∣α3 − µα2
2

∣∣ ≤


|ρ||ϑq |
K , 0 ≤

∣∣K (µ)
∣∣ ≤ |ρ||ϑq |

4K

4
∣∣K (µ)

∣∣, ∣∣K (µ)
∣∣ ≥ |ρ||ϑq |

4K

If ρ = 1, we obtain the following results for the class SLM∑(β; q) defined in Exam-
ple (1)

Corollary 1. For ρ ∈ C∗ and β ∈ [0, 1], let f ∈ SLM∑(β, ρ; q). Then

∣∣α2

∣∣ ≤ |ϑq|√∣∣∣ϑq(K −X) +
(
1− (2q + 1)ϑq

)
C
∣∣∣ ,

∣∣α3

∣∣ ≤ |ϑq|
{∣∣(K −X

)
ϑq +

(
1− (2q + 1)ϑq

)
C
∣∣+ |ϑq|K

}
K

∣∣(K −X
)
ϑq +

(
1− (2q + 1)ϑq

)
C
∣∣ ,

and

∣∣α3 − µα2
2

∣∣ ≤


|ϑq |
K ,

∣∣1− µ
∣∣ ≤ ∣∣ϑq(K−X)+(1−(2q+1)ϑq)C

∣∣
|ϑq |K

|1−µ||ϑq |2∣∣(K−X)ϑq+(1−(2q+1)ϑq)C
∣∣ , ∣∣1− µ

∣∣ ≥ ∣∣ϑq(K−X)+(1−(2q+1)ϑq)C
∣∣

|ϑq |K

where K,X,C is given by (19), (20), and (21), respectively.

If β = 0 and ρ = 1, we obtain the following results for the class SL∑(Υ(z; q)) defined
in Example (2)

Corollary 2. [19] Let f given by (1) be in the class SL∑(Υ(z); q). Then

∣∣α2

∣∣ ≤ ∣∣ϑq∣∣
q
√

1− 2qϑq
. (44)



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7065 12 of 16

∣∣α3

∣∣ ≤ ∣∣ϑq∣∣(q − (1 + q + 2q2)ϑq
)

q2(1 + q)
(
1− 2qϑq

) . (45)

∣∣α3 − µα2
2

∣∣ ≤


|ϑq |
q(1+q) ,

∣∣1− µ
∣∣ ≤ q

(
1−2qϑq

)
(1+q)|ϑq |

|1−µ|ϑ2
q

q2
(
1−2qϑq

) , ∣∣1− µ
∣∣ ≥ q

(
1−2qϑq

)
(1+q)|ϑq |

(46)

If β = 1 and ρ = 1, we obtain the following results for the class KL∑(Υ(z; q)) defined
in Example (3)

Corollary 3. Let f given by (1) be in the class KL∑(Υ(z); q). Then

∣∣α2

∣∣ ≤ ∣∣ϑq∣∣√
d2cq

(
d2cq −

(
d3cq + 2q

)
ϑq

)
∣∣α3

∣∣ ≤ ∣∣ϑq∣∣(d2cq − 2(d3cq + q)ϑq
)

d2cqd3cq
(
d2cq −

(
d3cq + 2q

)
ϑq

) ,
and

∣∣α3 −£α2
2

∣∣ ≤


|ϑq |
⌈2⌋q ⌈3⌋q

,
∣∣1−£

∣∣ ≤ ⌈2⌋q−
(
⌈3⌋q+2q

)
ϑq

⌈3⌋q |ϑq |

|1−£|ϑ2
q

⌈2⌋q

(
⌈2⌋q−

(
⌈3⌋q+2q

)
ϑq

) , ∣∣1−£
∣∣ ≥ ⌈2⌋q−

(
⌈3⌋q+2q

)
ϑq

⌈3⌋q |ϑq |

If q 7→ 1− and ρ = 1, we obtain the following results for the class SLM∑(β) defined
in Example (4)

Corollary 4. For q 7→ 1− and ρ = 1, let f ∈ SLM∑(β). Then

∣∣α2

∣∣ ≤ |ϑ|√∣∣∣ϑ(K −X) +
(
1− 3ϑ

)
C
∣∣∣ ,

∣∣α3

∣∣ ≤ |ϑ|
{∣∣(K −X

)
ϑ+

(
1− 3ϑ

)
C
∣∣+ |ϑ|K

}
K

∣∣(K −X
)
ϑ+

(
1− 3ϑ

)
C
∣∣ ,

and

∣∣α3 − µα2
2

∣∣ ≤


|ϑ|
K ,

∣∣1− µ
∣∣ ≤ ∣∣ϑ(K−X)+(1−3ϑ)C

∣∣
|ϑ|K

|1−µ||ϑ|2∣∣(K−X)ϑ+(1−3ϑ)C
∣∣ , ∣∣1− µ

∣∣ ≥ ∣∣ϑ(K−X)+(1−3ϑ)C
∣∣

|ϑ|K

where K,X,C are given by (19), (20) and (21), respectively.



A. Alsoboh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7065 13 of 16

If q 7→ 1−, β = 0 and ρ = 1, we obtain the following results for the class SL∑(Υ(z))
defined in Example (5)

Corollary 5. [38] Let f given by (1) be in class SL∑(Υ(z)). Then

∣∣α2

∣∣ ≤ ∣∣ϑ∣∣
√
1− 2ϑ

,
∣∣α3

∣∣ ≤ ∣∣ϑ∣∣(1− 4ϑ
)

2
(
1− 2ϑ

) .

and ∣∣α3 − µα2
2

∣∣ ≤


|ϑ|
2 ,

∣∣1− µ
∣∣ ≤ 1−2ϑ

2|ϑ|

(1−µ)ϑ2

1−2ϑ ,
∣∣1− µ

∣∣ ≥ 1−2ϑ
2|ϑ|

If q 7→ 1−, β = 1 and ρ = 1, we obtain the following results for the class KL∑(Υ(z))
defined in Example (6)

Corollary 6. [38] Let f given by (1) be in the class KL∑(Υ(z)). Then

∣∣α2

∣∣ ≤ ∣∣ϑ∣∣
√
4− 10ϑ

,
∣∣α3

∣∣ ≤
∣∣ϑ∣∣(1− 4ϑ

)
3
(
1− 2ϑ

) .
and ∣∣α3 −£α2

2

∣∣ ≤


|ϑ|
6 ,

∣∣1−£
∣∣ ≤ 2−5ϑ

3|ϑ|

|1−£|ϑ2

2
(
2−5ϑ

) , ∣∣1−£
∣∣ ≥ 2−5ϑ

3|ϑ|

4. Conclusion

In this paper, we study new subclasses of complex order bi-univalent functions that
are associated with shell-like curves through the subordination principle and the use
of the q-analogue of Fibonacci numbers. Motivated by recent developments in the q-
calculus and its fruitful applications in geometric function theory, we construct and
analyze two distinct families of analytic and bi-univalent functions. For these families,
we establish coefficient estimates for the initial Taylor–Maclaurin terms and we derive
sharp bounds for the Fekete–Szegö functional in terms of the relevant parameters. The
results obtained in this investigation not only extend and generalize several previous
contributions in the theory of bi-univalent functions, but also provide new insights into
the interplay between bi-univalent function theory, the q-Fibonacci numbers and the
shell-like geometries. Furthermore, the subclasses introduced here may serve as a useful
platform for future research in analytic function spaces, special functions, and their
associated operator-theoretic properties.

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