EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 4, 2024, 2467-2480
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

A Subclass of Bi-univalent Functions Defined by a
Symmetric q-Derivative Operator and Gegenbauer

Polynomials

Mohamed Illafe1,∗, Maisarah Haji Mohd2, Feras Yousef3,4, Shamani Supramaniam2

1 School of Engineering, Math, & Technology, Navajo Technical University, Crownpoint,
NM 87313, USA
2 School of Mathematical Sciences, Universiti Sains Malaysia, Penang 11800, Malaysia
3 Department of Mathematics, The University of Jordan, Amman 11942, Jordan
4 Jadara University Research Center, Jadara University, Irbid 21110, Jordan

Abstract. This paper introduces a novel subclass of bi-univalent analytic functions by utilizing a
symmetric q-derivative operator in conjunction with Gegenbauer polynomials. Within this newly
defined subclass, we derive bounds for the first two Maclaurin coefficients and address the Fekete-
Szegő problem. By varying the parameters in our results between 0 and 1, we obtain a range of
new insights and rediscover some previously established results. This approach not only broadens
the scope of bi-univalent function theory but also deepens the understanding of coefficient bounds
and extremal problems within this context.

2020 Mathematics Subject Classifications: 30C45

Key Words and Phrases: Bi-univalent analytic functions, Gegenbauer (or Ultraspherical) poly-
nomials, Fekete-Szegö functional

1. Definitions and Preliminaries

Let A denote the class of all analytic functions f defined in the open unit disk U =
{z ∈ C : |z| < 1} and normalized by the conditions f(0) = 0 and f ′(0) = 1. Thus each
f ∈ A has a Taylor-Maclaurin series expansion of the form:

f(z) = z +
∞∑
n=2

an z
n, (z ∈ U). (1)

Let S denote the class of all functions f ∈ A which are univalent in U. In addition,
Subordination, denoted as f ≺ g, between functions f and g in S occurs when there exists

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5408

Email addresses: millafe@navajotech.edu (M. Illafe), maisarah hjmohd@usm.my (M. Haji Mohd),
fyousef@ju.edu.jo (F. Yousef), shamani@usm.my (S. Supramaniam)

https://www.ejpam.com 2467 Copyright: © 2024 The Author(s). (CC BY-NC 4.0)



M. Illafe et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2467-2480 2468

an analytic function w(z) such that w(0) = 0, |w(z)| < 1 for z ∈ U, and f(z) = g(w(z)).
The inverse function of the function f ∈ S is given by:

f−1(w) = w − a2w
2 +

(
2a22 − a3

)
w3 −

(
5a32 − 5a2a3 + a4

)
w4 + · · · . (2)

A function f is said to be bi-univalent in U if both f(z) and f−1(z) are univalent in
U. Let Σ denote the class of bi-univalent functions in U given by (1).

Lewin [25], Brannan and Clunie [9], and Netanyahu [28] are known to be the first
researchers who have studied the class Σ. Since then, the class Σ has attracted several
researchers, see [2, 3, 13, 16, 20, 27, 30, 33, 34].

Orthogonal polynomials have been widely studied since their discovery by Legendre in
1784 [24]. They have been used as a mathematical approach to solve ordinary differen-
tial equations associated with model problems under certain conditions. The advantages
of orthogonal polynomials in modern mathematics and their application in physics and
engineering cannot be ignored. Orthogonal polynomials play a key role in approximation
theory, differential integral equations, and mathematical statistics. Additionally, these
polynomials have been instrumental in various applications, such as scattering theory, sig-
nal analysis [1, 5, 8, 10, 12, 14, 15, 17, 18, 31].

Let Cα
n (x) be the Gegenbauer polynomial of degree n defined using the following re-

currence relation

Cα
n (x) =

1

n

[
2x(n+ α− 1)Cα

n−1(x)− (n+ 2α− 2)Cα
n−1(x)

]
,

with
Cα
0 (x) = 1,

Cα
1 (x) = 2αx,

Cα
2 (x) = 2α(1 + α)x2 − α.

(3)

The Gegenbauer polynomials generate Legendre polynomials and Chebyshev polyno-
mials when taking α equaling 1/2 and 1, respectively.

Amourah et al. [6] were the first to investigate the polynomials generated by Hα(x, z),
defining them as follows:

Hα(x, z) =
1

(1− 2xz + z2)α
, (−1 ≤ x ≤ 1, and z ∈ U).

Also, since Hα is an analytic function in U, it can be expressed as follows:

Hα(x, z) =
∞∑
n=0

Cα
n (x)z

n. (4)

The theory of q-calculus operators has many applications in science and engineering.
Notably, several researchers have made significant contributions to the study of q-calculus,
see [4, 7, 23, 26, 29].



M. Illafe et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2467-2480 2469

Definition 1. ([22]) Let f ∈ A, the Jackson’s q-derivative is defined by

Dqf(z) =

{
f(z)−f(qz)

(1−q)z for z ̸= 0,

f ′(0) for z = 0
(5)

where 0 < q < 1. From (5), we can write

Dqf(z) = 1 +

∞∑
n=2

[n]qanz
n−1 (6)

where [n]q denotes the basic number and given by

[n]q =
1− qn

1− q
, n ∈ N = {1, 2, . . .}.

Definition 2. For a function f given by (1), the symmetric q-derivative is defined as:(
D̃qf

)
(z) =

{
f(qz)−f(q−1z)

(q−q−1)z
z ̸= 0

f ′(0) z = 0
. (7)

Equation (7) implies D̃qz
n = [̃n]qz

n−1, and D̃qf of a function f given by (1) is defined
as (

D̃qf
)
(z) = 1 +

∞∑
n=2

[̃n]qanz
n−1

where the symbol [̃n]q is defined as

[̃n]q =
qn − q−n

q − q−1
.

Using equations (2) and (7), we obtain(
D̃qg

)
(w) =

g(qw)− g
(
q−1w

)
(q − q−1)w

= 1− [̃2]qa2w + [̃3]q
(
2a22 − a3

)
w2 − [̃4]q

(
5a32–5a2a3 + a4

)
w3 + · · ·

(8)

In recent times, numerous researchers have been investigating the concept of bi-univalent
functions linked to Gegenbour polynomials. Some notable studies in this area include ref-
erences [19] and [21]. In the present work, we propose the following novel subclasses.

2. The class Bα
Σ(t, γ, ν, ϵ)

Definition 3. ([32]) For γ ≥ 1, ν, ϵ ≥ 0, 0 ≤ α ≤ 1, ζ = 2γ+ν
2γ+1 and t ∈ (1/2, 1], a

function f ∈ Σ given by (1) is in Mα
Σ(γ, ν, ϵ) if for all z, w ∈ D, it satisfies the following

subordination:

Re

(
(1− γ)

(
f(z)

z

)ν

+ γf ′(z)

(
f(z)

z

)ν−1

+ ζϵzf ′′(z)

)
> α (9)



M. Illafe et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2467-2480 2470

and

Re

(
(1− γ)

(
g(w)

w

)ν

+ γg′(w)

(
g(w)

w

)ν−1

+ ζϵzg′′(w)

)
> α (10)

where f ∈ Σ defined by (1), and g = f−1 given by (2).

Definition 4. Let α > 0, γ ≥ 1, ν ≥ 0, ϵ ≥ 0, ζ = 2γ+ν
2γ+1 , t ∈ (1/2, 1], and f ∈ Σ that is

given by (4) is in B̃q
Σ(t, γ, ν, ϵ) if for all z, w ∈ D, it satisfies the following subordination

(1− γ)

(
f(z)

z

)ν

+ γD̃q (f(z))

(
f(z)

z

)ν−1

+ ζϵzD̃q

(
D̃q (f(z))

)
≺ Hα(t, z) (11)

and

(1− γ)

(
g(w)

w

)ν

+ γD̃q (g(w))

(
g(w)

w

)ν−1

+ ζϵzD̃q

(
D̃q (g(w))

)
≺ Hα(t, w), (12)

where g = f−1(w) is given by (2). given by (4).

Definition 5. The function f ∈ 1B̃q
Σ(t, γ, ν) := B̃q

Σ(t, γ, ν, 0) iff it satisfies the following
subordination

(1− γ)

(
f(z)

z

)ν

+ γD̃q (f(z))

(
f(z)

z

)ν−1

≺ Hα(t, z)

and

(1− γ)

(
g(w)

w

)ν

+ γD̃q (g(w))

(
g(w)

w

)ν−1

≺ Hα(t, w).

Definition 6. The function f ∈ 2B̃q
Σ(t, γ, ϵ) := B̃q

Σ(t, γ, 1, ϵ) iff it satisfies the following
subordination

(1− γ)

(
f(z)

z

)
+ γD̃q (f(z)) + ζϵzD̃q

(
D̃q (f(z))

)
≺ Hα(t, z)

and

(1− γ)

(
g(w)

w

)
+ γD̃q (g(w)) + ζϵzD̃q

(
D̃q (g(w))

)
≺ Hα(t, w).

Definition 7. The function f ∈ 3B̃q
Σ(t, γ) := B̃q

Σ(t, γ, 1, 0) iff it satisfies the following
subordination:

(1− γ)

(
f(z)

z

)
+ γD̃q (f(z)) ≺ Hα(t, z)

and

(1− γ)

(
g(w)

w

)
+ γD̃q (g(w)) ≺ Hα(t, w).



M. Illafe et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2467-2480 2471

Definition 8. The function f ∈ 4B̃q
Σ(t) := B̃q

Σ(t, 1, 1, 0) iff it satisfies the following subor-
dination:

D̃q (f(z)) ≺ Hα(t, z)

and (
D̃qg(w)

)
≺ Hα(t, w).

Let P = {p : U → C | p(z) = 1 +
∞∑
n=1

pn z
n, is analytic function, and Re(p) > 0}. The

following lemma will be used when proofing our main results.

Lemma 1. ([11]) If p ∈ P, then

|pn| ≤ 2, n ∈ N. (13)

Throughout the rest of the paper, we assume that 0 < q < 1, x ∈
(
1
2 , 1
]
and α is a

nonzero real constant.

3. Main Results

Theorem 1. Let f ∈ B̃q
Σ(t, γ, ν, ϵ). Then

|a2| ≤
2αx

√
x√√√√√√√

∣∣∣∣x2 [α(2[̃2]qγ (ν − 1) + 2[̃3]qγ + ν (ν − 2γ + 1) + 2[̃2]q [̃3]qζϵ
)
− 2 (1 + α)Υ

]
+(1 + 2x)Υ

∣∣∣∣
and

|a3| ≤
2
[
[̃3]qγ − [̃2]qζϵ

]
x2α2

Υ
− 4αx[

γ
(
[̃3]q + ν − 1

)
+ (1− γ) ν + [̃2]q [̃3]qζϵ

] ,
where

Υ :=
(
ν − γ + [̃2]q(γ + ζϵ)

)2
.

Proof. Let f ∈ B̃q
Σ(t, γ, ν, ϵ). By Definition 4, there exist u, v such that u(0) = v(0) = 0

and |u(z)| < 1, |v(w)| < 1 where z, w ∈ U, then

(1− γ)

(
f(z)

z

)ν

+ γD̃q (f(z))

(
f(z)

z

)ν−1

+ ζϵzD̃q

(
D̃q (f(z))

)
= Hα(x, u(z)) (14)



M. Illafe et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2467-2480 2472

and

(1− γ)

(
g(w)

w

)ν

+ γD̃q (g(w))

(
g(w)

w

)ν−1

+ ζϵzD̃q

(
D̃q (g(w))

)
= Hα(x, v(w)) (15)

Now, let p, q ∈ P given by

p(z) =
1 + u(z)

1− u(z)
= 1 + c1z + c2z

2 + · · ·

and

q(w) =
1 + v(w)

1− v(w)
= 1 + d1w + d2w

2 + · · · .

Hence, we can write

u(z) =
p(z)− 1

p(z) + 1
=

1

2
c1z +

1

2

(
c2 −

1

2
c21

)
z2 + · · · (16)

and

v(w) =
q(w)− 1

q(w) + 1
=

1

2
d1w +

1

2

(
d2 −

1

2
d21

)
w2 + · · · . (17)

Now, using equations (14), (15), (16) and (17), we can write

Hα(x, u(z)) = 1 +
1

2
Cα
1 (x)c1z +

[
1

4
Cα
2 (x)c

2
1 +

1

2
Cα
1 (x)

(
c2 −

1

2
c21

)]
z2 + · · · , (18)

and

Hα(x, v(w)) = 1 +
1

2
Cα
1 (x)d1w +

[
1

4
Cα
2 (x)d

2
1 +

1

2
Cα
1 (x)

(
d2 −

1

2
d21

)]
w2 + · · · . (19)

Also, from equations (18) and (19), we get(
ν − γ + [̃2]q(γ + ζϵ)

)
a2 =

1

2
Cα
1 (x)c1, (20)

(ν − 1)

[
γ [̃2]q +

γ (ν − 2)

2
+

(1− γ) ν

2

]
a22+

[
γ
(
[̃3]q + ν − 1

)
+ (1− γ) ν + [̃2]q [̃3]qζϵ

]
a3

=
1

2
Cα
1 (x)

(
c2 −

c21
2

)
+

1

4
Cα
2 (x) c21,

(21)

−
(
ν − γ + [̃2]q(γ + ζϵ)

)
a2 =

1

2
Cα
1 (x)d1, (22)

and

(ν − 1)

[
[̃2]qγ + 2[̃3]qγ +

(ν + 2) γ

2
+

ν (ν + 3) (1− γ)

2 (ν − 1)
+

2[̃2]q [̃3]qζϵ

(ν − 1)

]
a22

−

[
γ
(
[̃3]q + ν − 1

)
+ (1− γ) ν + [̃2]q [̃3]qζϵ

]
a3 =

1

2
Cα
1 (x)

(
d2 −

d21
2

)
+

1

4
Cα
2 (x) d21.

(23)



M. Illafe et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2467-2480 2473

Equations (20) and (22) implies
c1 = −d1 (24)

and

8
(
ν − γ + [̃2]q(γ + ζϵ)

)2
a22 = (Cα

1 (x))
2 (c21 + d21

)
. (25)

Adding (21) and (23), we deduce[
2[̃2]qγ (ν − 1) + 2[̃3]qγ + ν (ν − 2γ + 1) + 2[̃2]q [̃3]qζϵ

]
a22 =

1

2
Cα
1 (x) (c2 + d2)+

1

4
(Cα

2 (x)− Cα
1 (x))

(
c21 + d21

)
.

(26)

Plugging
(
c21 + d21

)
obtained from (25) into (26) yield

[
2[̃2]qγ (ν − 1) + 2[̃3]qγ + ν (ν − 2γ + 1) + 2[̃2]q [̃3]qζϵ−

2Υ [Cα
2 (x)− Cα

1 (x)]

[Cα
1 (x)]

2

]
a22

=
1

2
Cα
1 (x) (c2 + d2) ,

(27)

where

Υ :=
(
ν − γ + [̃2]q(γ + ζϵ)

)2
.

Furthermore, from (13), (19) and (27), it follows that

|a2| ≤
2αx

√
x√√√√√√√

∣∣∣∣x2 [α(2[̃2]qγ (ν − 1) + 2[̃3]qγ + ν (ν − 2γ + 1) + 2[̃2]q [̃3]qζϵ
)
− 2 (1 + α)Υ

]
+(1 + 2x)Υ

∣∣∣∣
Subtracting (21) from (23), we have

2
[
γ
(
[̃3]q + ν − 1

)
+ (1− γ) ν + [̃2]q [̃3]qζϵ

]
(a3 − a22) =

1

2
Cα
1 (x) (c2 − d2)+

1

4
(Cα

2 (x)− Cα
1 (x))

(
c21 − d21

)
.

(28)

Utilizing equations (3) and (25), we can write (28) as

a3 = a22 +
Cα
1 (x)

4
[
γ [̃3]q − (γ − ν) + [̃2]q [̃3]qζϵ

] (c2 − d2) . (29)

Now, using equations (3) and (13), we can write



M. Illafe et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2467-2480 2474

a3 =
2α2x3 (c2 + d2)

x2
[
4γ [̃2]q(ν − 1) + 4γ [̃3]q + 2ν(ν − 2γ + 1) + 4[̃2]q [̃3]qζϵ− 2(1 + α)Υ

]
+ (1 + 2x)Υ

+
αx (c2 − d2)

2
[
γ [̃3]q − (γ − ν) + [̃2]q [̃3]qζϵ

] .
(30)

This concludes the proof of Theorem 1.

4. The Fekete-Szegö Inequality |a3 − φa22|

Theorem 2. If f ∈ B̃q
Σ(t, γ, ν, ϵ), then

∣∣a3 − φa22
∣∣ ≤


αx

γ [̃3]q−(γ−ν)+[̃2]q [̃3]qζϵ
if 0 ≤ |h(φ)| ≤ 1

2
[
γ [̃3]q−(γ−ν)+[̃2]q [̃3]qζϵ

] ,
2αx |h(φ)| if |h(φ)| ≥ 1

2
[
γ [̃3]q−(γ−ν)+[̃2]q [̃3]qζϵ

] ,
where

h(φ) =
2(1− φ)αx2

αx2
[
4γ [̃2]q(ν − 1) + 4γ [̃3]q + 2ν(ν − 2γ + 1) + 4[̃2]q [̃3]qζϵ− 2(1 + (1/α))Υ

]
+ (1 + 2x)Υ

,

and

Υ :=
(
ν − γ + [̃2]q(γ + ζϵ)

)2
.

Proof: Consider f in Bα
Σ(x, τ, γ, ν, ϵ), then by (29) we obtain

a3 − φa22 = a22 +
Cα
1 (x)

4
[
γ [̃3]q − (γ − ν) + [̃2]q [̃3]qζϵ

] (c2 − d2)− φa22

= (1− φ)a22 +
Cα
1 (x)

4
[
γ [̃3]q − (γ − ν) + [̃2]q [̃3]qζϵ

] (c2 − d2) .



M. Illafe et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2467-2480 2475

Then, in view of (3), and the value of a2 in equation (27), we have

a3 − φa22 =
2(1− φ)α2x3(c2 + d2)

αx2
[
4γ [̃2]q(ν − 1) + 4γ [̃3]q + 2ν(ν − 2γ + 1) + 4[̃2]q [̃3]qζϵ− 2(1 + (1/α))Υ

]
+(1 + 2x)Υ

+
αx

2
[
γ [̃3]q − (γ − ν) + [̃2]q [̃3]qζϵ

] (c2 − d2)

= αx

([
h(φ) +

1

2
[
γ [̃3]q − (γ − ν) + [̃2]q [̃3]qζϵ

]]c2
+

[
h(φ)− 1

2
[
γ [̃3]q − (γ − ν) + [̃2]q [̃3]qζϵ

]]d2),
where

Υ :=
(
ν − γ + [̃2]q(γ + ζϵ)

)2
,

and

h(φ) =
2(1− φ)αx2

αx2
[
4γ [̃2]q(ν − 1) + 4γ [̃3]q + 2ν(ν − 2γ + 1) + 4[̃2]q [̃3]qζϵ− 2(1 + (1/α))Υ

]
+ (1 + 2x)Υ

.

This completes the proof of Theorem 2.

5. Consequences and Corollaries

Corollary 1. If f ∈ 1B̃q
Σ(t, γ, ν), then

|a2| ≤
2αx

√
x√√√√√√√

∣∣∣∣x2[α(2[̃2]qγ (ν − 1) + 2[̃3]qγ + ν (ν − 2γ + 1)
)
− 2 (1 + α)

(
ν − γ + [̃2]qγ

)2 ]
+(1 + 2x)

(
ν − γ + [̃2]qγ

)2 ∣∣∣∣
,

|a3| ≤
2[̃3]qγx

2α2(
ν − γ + [̃2]qγ

)2 − 4xα[
γ
(
[̃3]q + ν − 1

)
+ (1− γ) ν

]
and ∣∣a3 − φa22

∣∣ ≤


αx

γ [̃3]q−(γ−ν)
if 0 ≤ |h(φ)| ≤ 1

2
[
γ [̃3]q−(γ−ν)

] ,
2αx |h(φ)| if |h(φ)| ≥ 1

2
[
γ [̃3]q−(γ−ν)

] ,
where



M. Illafe et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2467-2480 2476

h(φ) =
2(1− φ)αx2

αx2
[
4γ [̃2]q(ν − 1) + 4γ [̃3]q + 2ν(ν − 2γ + 1)− 2(1 + (1/α))

(
ν − γ + [̃2]qγ

)2]
+(1 + 2x)

(
ν − γ + [̃2]qγ

)2
.

Next, making ν = 1, yields.

Corollary 2. If f ∈ 2B̃q
Σ(t, γ, ϵ), then

|a2| ≤
2αx

√
x√√√√√√√

∣∣∣∣x2[α(2[̃3]qγ + 2 (1− γ) + 2[̃2]q [̃3]qζϵ
)
− 2 (1 + α)

(
1− γ + [̃2]q(γ + ζϵ)

)2 ]
+(1 + 2x)

(
1− γ + [̃2]q(γ + ζϵ)

)2 ∣∣∣∣
,

|a3| ≤
2
[
[̃3]qγ − [̃2]qζϵ

]
x2α2(

1− γ + [̃2]q(γ + ζϵ)
)2 − 4αx[

[̃3]qγ + 1− γ + [̃2]q [̃3]qζϵ
]

and

∣∣a3 − φa22
∣∣ ≤


αx

γ [̃3]q−(γ−1)+[̃2]q [̃3]qζϵ
if 0 ≤ |h(φ)| ≤ 1

2
[
γ [̃3]q−(γ−1)+[̃2]q [̃3]qζϵ

] ,
2αx |h(φ)| if |h(φ)| ≥ 1

2
[
γ [̃3]q−(γ−1)+[̃2]q [̃3]qζϵ

] ,
where

h(φ) =
2(1− φ)αx2

αx2
[
4γ [̃3]q + 4(1− γ) + 4[̃2]q [̃3]qζϵ− 2(1 + (1/α))

(
1− γ + [̃2]q(γ + ζϵ)

)2]
+(1 + 2x)

(
1− γ + [̃2]q(γ + ζϵ)

)2
.

Setting ν = 1 and ϵ = 0, we obtain the following corollary.

Corollary 3. If f ∈ 3B̃q
Σ(t, γ), then

|a2| ≤
2αx

√
x√∣∣∣∣x2 [α(2[̃3]qγ + 2 (1− γ)

)
− 2 (1 + α)

(
1− γ + [̃2]qγ

)2]
+ (1 + 2x)

(
1− γ + [̃2]qγ

)2∣∣∣∣
,



M. Illafe et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2467-2480 2477

|a3| ≤
2[̃3]qγx

2α2(
1− γ + [̃2]qγ

)2 − 4αx[
[̃3]qγ + 1− γ

]
and ∣∣a3 − φa22

∣∣ ≤


αx

γ [̃3]q−(γ−1)
if 0 ≤ |h(φ)| ≤ 1

2
[
γ [̃3]q−(γ−1)

] ,
2αx |h(φ)| if |h(φ)| ≥ 1

2
[
γ [̃3]q−(γ−1)

] ,
where

h(φ) =
2(1− φ)αx2

αx2
[
4γ [̃3]q + 4(1− γ)− 2(1 + (1/α))

(
1− γ + [̃2]qγ

)2]
+ (1 + 2x)

(
1− γ + [̃2]qγ

)2 .
Next, letting γ = ν = 1 and ϵ = 0, yields.

Corollary 4. If f ∈ 4B̃q
Σ(t), then

|a2| ≤
2αx

√
x√∣∣∣∣x2 [2α[̃3]q − 2 (1 + α)

(
[̃2]q

)2]
+ (1 + 2x)

(
[̃2]q

)2∣∣∣∣
,

|a3| ≤
2[̃3]qx

2α2(
[̃2]q

)2 − 4αx

[̃3]q

and ∣∣a3 − φa22
∣∣ ≤


αx

[̃3]q
if 0 ≤ |h(φ)| ≤ 1

2[̃3]q
,

2αx |h(φ)| if |h(φ)| ≥ 1

2[̃3]q
,

where

h(φ) =
2(1− φ)αx2

αx2
[
4[̃3]q − 2(1 + (1/α))

(
[̃2]q

)2]
+ (1 + 2x)

(
[̃2]q

)2 .
6. Conclusion

In our current investigation, a novel subclass B̃q
Σ(t, γ, ν, ϵ) of normalized bi-univalent

analytic functions has been delineated. This subclass integrates Gegenbauer polynomials
and a symmetric q-derivative operator series. Initially, we have furnished an estimate for
the primary Taylor-Maclaurin coefficients, |a2| and |a3|. Subsequently, we have success-
fully addressed the Fekete-Szegö inequality problem.
Furthermore, through substituting some values for the parameters ϵ, ν, and γ,we derived
analogous outcomes for the following subclasses: 1B̃q

Σ(t, γ, ν) := B̃q
Σ(t, γ, ν, 0),

2B̃q
Σ(t, γ, ϵ) :=

B̃q
Σ(t, γ, 1, ϵ),

3B̃q
Σ(t, γ) := B̃q

Σ(t, γ, 1, 0), and
4B̃q

Σ(t) := B̃q
Σ(t, 1, 1, 0).



REFERENCES 2478

Acknowledgements

This study was conducted during the third author’s sabbatical leave from The Uni-
versity of Jordan in the USA. The authors would like to thank the editor and anonymous
reviewers for their valuable comments, which have helped to improve the quality of this
work.

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