EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 6115
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Fekete–Szegő Inequalities for New Subclasses of
Bi-Univalent Functions Defined by Sălăgean

q-Differential Operator

Mohammad Al-Ityan1, Ala Amourah2,∗, Abdullah Alsoboh3,∗, Sultan Alsaadi2,
Mohammad Bani Raba’a4, Suha Hammad5

1 Department of Mathematics, Faculty of Science, Al-Balqa Applied University, 19117, Salt,
Jordan
2 Mathematics Education Program, Faculty of Education and Arts, Sohar University,
Sohar 311, Sultanate of Oman
3 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
4 Department of Mathematics, Faculty of Science and Technology, Irbid National University,
Irbid, Jordan
5 Department of Mathematics, College of Education for Pure Sciences ?University of Tikrit,
Iraq

Abstract. In this paper, we introduce a new operator based on the Sălăgean q-differential ap-
proach to define a new class of analytic functions. Using this operator, we obtain estimates for the
first two coefficients in the Taylor series, |a2| and |a3|. A significant part of the study focuses on the

Fekete–Szegő inequalities for the function classes Mζ,m
σ,q,Σ(⋋, κ, α) and Mζ,m

σ,q,Σ(γ,⋋, κ). Through
our analysis, we derive several important results, including some special cases that we present in
this paper as Corollaries.

2020 Mathematics Subject Classifications: 30C45, 30C50, 33D15, 47B38

Key Words and Phrases: Analytic Functions, q-Sălăgean Operator, Starlike Functions, Taylor
Coefficients

1. Introduction

Let Λ denote the class of all analytic functions I defined in the open unit disk ⋓ =
{z ∈ C : |z| < 1} and normalized by the conditions I(0) = 0 and I′(0) = 1. Each I ∈ Λ

∗Corresponding author.
∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6115

Email addresses: Mohammad65655vv22@gmail.com (M. EL-Ityan), AAmourah@su.edu.om (A. Amourah),
abdullah.alsoboh@asu.edu.om (A. Alsoboh), alsaad99@hotmail.com (S. Alsaadi),
0779382684mohammad@gmail.com (M. Bani Raba’a), suhajumaa1987@tu.edu.iq (S. Hammad)

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



M. EL-Ityan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6115 2 of 16

has a Taylor series expansion of the form:

I(z) = z +
∞∑
n=2

anz
n, z ∈ ⋓. (1)

For every I ∈ S, there exists an inverse map I−1 satisfying the following conditions:

I−1(I(z)) = z, z ∈ ⋓,

I(I−1(ϖ)) = ϖ, |ϖ| < r0(I); r0(I) ≥ 1

4
.

The inverse function is given by the series:

I−1(ϖ) = ϖ − a2ϖ
2 + (2a22 − a3)ϖ

3 − (5a32 − 5a2a3 + a4)ϖ
4 + · · · . (2)

Definition 1. A single-valued complex function I is said to be univalent in a simply
connected domain D if it does not take the same value twice in D; that is, I(z1) ̸= I(z2)
whenever z1 ̸= z2, for all z1, z2 ∈ D.

Definition 2. A function I ∈ Λ is said to be bi-univalent in U if both I(z) and I−1(z)
are univalent in ⋓.

Let Σ denote the class of bi-univalent functions in ⋓ defined by (1). Examples of
functions in Σ include:

z

1 − z
, − log(1 − z),

1

2
log

(
1 + z

1 − z

)
, . . . .

It is worth noting that the familiar Koebe function is not a member of Σ because it
maps the unit disk ⋓ univalently onto the entire complex plane except for the part of the
negative real axis from −1

4 to −∞.
The class S∗(α) of starlike functions of order α in ⋓ has been extensively studied and

is a subset of S. By definition:

S∗(α) =

{
I ∈ S : Re

(
I′(z)

I(z)

)
> α, z ∈ ⋓, 0 ≤ α < 1

}
. (3)

Ezrohi [1] introduced the class H(α), defined as:

H(α) =
{
I ∈ S : Re{I′(z)} > α, z ∈ ⋓, 0 ≤ α < 1

}
. (4)

Similarly, the class K(α) was introduced by [2]:

K(α) =

{
F ∈ S : Re

(
1 +

zF ′′(z)

F ′(z)

)
> α, z ∈ ⋓, 0 ≤ α < 1

}
. (5)

A function I ∈ Λ belongs to the class S∗
Σ(α) of strongly bi-starlike functions of order α

(0 < α ≤ 1) if:

| arg

(
zI′(z)

I(z)

)
| < απ

2
, z ∈ ⋓,



M. EL-Ityan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6115 3 of 16

| arg

(
ϖG′(ϖ)

G(ϖ)

)
| < απ

2
, ϖ ∈ ⋓,

where G = I−1.
Here, we revisit the q-difference operator, a fundamental tool in q-calculus that plays

a key role in various fields such as hypergeometric series, quantum physics, and opera-
tor theory. The q-calculus framework, introduced by Jackson [3], has been extended to
fractional q-calculus operators, as utilized by Kanas and Răducanu [4]. For more details,
readers are referred to [3, 5–31]. Below, we outline key definitions and concepts, assuming
0 < q < 1.

The Jackson q-derivative of a function I ∈ Λ is defined as [3]:

DqI(z) =

{
I(z)−I(qz)
(1−q)z , z ̸= 0,

I′(0), z = 0,
(6)

with the second q-derivative given by:

D2
qI(z) = Dq(DqI(z)).

Using the above, DqI(z) can be expressed as:

DqI(z) = 1 +
∞∑
n=2

[n]qanz
n−1, (7)

where the q-basic number [n]q is defined as:

[n]q =
1 − qn

1 − q
.

As q → 1−, [n]q → n. For h(z) = zn, the q-derivative becomes:

Dqh(z) = [n]qz
n−1.

This result converges to the classical derivative h′(z) = nzn−1 as q → 1−.
Recently, Govindaraj and Sivasubramanian [32] introduced the Sălăgean q-differential

operator:
D0

qI(z) = I(z), D1
qI(z) = zDqI(z),

Dm
q I(z) = zDm

q

(
Dm−1

q I(z)
)
,

Dm
q I(z) = z +

∞∑
n=2

[n]mq anz
n, m ∈ N0, z ∈ ⋓. (8)

[33] Define the generalized operator:

D0I(z) = Dm
q I(z),



M. EL-Ityan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6115 4 of 16

D1,m
σ,q I(z) = (1 − σ)Dm

q I(z) + σz
(
Dm

q I(z)
)′
,

= z +

∞∑
n=2

[n]mq
[
1 + (n− 1)σ

]
anz

n, (9)

Dζ,m
σ,q I(z) = z +

∞∑
n=2

[n]mq
[
1 + (n− 1)σ

]ζ
anz

n, σ > 0, ζ ∈ N0. (10)

As q → 1−, the operator reduces to:

Dζ,m
σ I(z) = z +

∞∑
n=2

nm
[
1 + (n− 1)σ

]ζ
anz

n, σ > 0, m, ζ ∈ N0. (11)

Definition 3. A function I(z), as described in (1), belongs to the class Mζ,m
σ,q,Σ(⋋, κ, α)

if:

| arg

(
1 +

1

κ
[(1 −⋋)(

d

dq
Dζ,m
σ,q I(z)) + ⋋

Dζ,m
σ,q I(z)

z
− 1]

)
| < απ

2
,

and

| arg

(
1 +

1

κ
[(1 −⋋)(

d

dq
Dζ,m
σ,q G(ϖ)) + ⋋

Dζ,m
σ,q G(ϖ)

z
− 1]

)
| < απ

2
,

where 0 < α ≤ 1, ⋋ ≥ 0, κ ≥ 1, σ > 0, m, ζ ∈ N0 z,ϖ ∈ ⋓.

Definition 4. A function I(z), as described in (1), belongs to the class Mζ,m
σ,q,Σ(γ,⋋, κ)

if:

Re

(
1 +

1

κ
[(1 −⋋)(

d

dq
Dζ,m
σ,q I(z)) + ⋋

Dζ,m
σ,q I(z)

z
− 1]

)
> γ,

and

Re

(
1 +

1

κ
[(1 −⋋)(

d

dq
Dζ,m
σ,q G(ϖ)) + ⋋

Dζ,m
σ,q G(ϖ)

z
− 1]

)
> γ,

where 0 ≤ γ < 1, ⋋ ≥ 0, κ ≥ 1, σ > 0, m, ζ ∈ N0 z,ϖ ∈ ⋓.

To prove our theorem, we will make use of the following lemma:

Lemma 1 ([34]). If h belongs to the family H, where H represents all analytic functions
in ⋓ satisfying Re(h(z)) > 0 and h(z) = 1 + h1z + h2z

2 + · · ·, then |hi| ≤ 2 for each index
i.



M. EL-Ityan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6115 5 of 16

Coefficients Bounds for Classes Mζ,m
σ,q,Σ(⋋, κ, α) and Mζ,m

σ,q,Σ(γ,⋋, κ)

Theorem 1. Let I(z) given by (1) be in the class Mζ,m
σ,q,Σ(⋋, κ, α), with 0 < α ≤ 1, ⋋ ≥

0, κ ≥ 1, σ > 0, m, ζ ∈ N0 z,ϖ ∈ ⋓. Then

|a2| ≤
8ακ√

4ακ
[
1 + 2σ

]ζ
((1 −⋋)[3]m+1

q + ⋋[3]mq ) − (α− 1)
[
1 + σ

]2ζ
((1 −⋋)[2]m+1

q + ⋋[2]mq )2
,

and

|a3| ≤
2α

|
[
1+2σ

]ζ
κ ((1 −⋋)[3]m+1

q + ⋋[3]mq )|
+

8α2κ2

|
[
1 + σ

]2ζ
((1 −⋋)[2]m+1

q + ⋋[2]mq )2|
.

Proof. To establish the theorem, the definition (3) is utilized in its equivalent forms:

1 +
1

κ
[(1 −⋋)(

d

dq
Dζ,m
σ,q I(z)) + ⋋

Dζ,m
σ,q I(z)

z
− 1] = [v(z)]α, (12)

1 +
1

κ
[(1 −⋋)(

d

dq
Dζ,m
σ,q G(ϖ)) + ⋋

Dζ,m
σ,q G(ϖ)

z
− 1] = [c(ϖ)]α, (13)

where v(z) and c(w) belong to the class H and satisfy the conditions defined in (1) .
These functions can be expressed as:

v(z) = 1 + v1z + v2z
2 + v3z

3 + · · · , (14)

c(ϖ) = 1 + c1ϖ + c2ϖ
2 + c3ϖ

3 + · · · . (15)

By equating coefficients in the above equations, the following relations are obtained:[
1 + σ

]ζ
κ

((1 −⋋)[2]m+1
q + ⋋[2]mq )a2 = αv1, (16)

[
1 + 2σ

]ζ
κ

((1 −⋋)[3]m+1
q + ⋋[3]mq )a3 = αv2 +

α(α− 1)

2
v21, (17)

−
[
1 + σ

]ζ
κ

((1 −⋋)[2]m+1
q + ⋋[2]mq )a2 = αc1, (18)

and [
1 + 2σ

]ζ
κ

((1 −⋋)[3]m+1
q + ⋋[3]mq )(2a22 − a3) = αc2 +

α(α− 1)

2
c21. (19)

Using the equations (16),(18) , it follows that:

v1 = −c1, (20)



M. EL-Ityan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6115 6 of 16[
1 + σ

]2ζ
κ2

((1 −⋋)[2]m+1
q + ⋋[2]mq )2a22 = α2(v21 + c21). (21)

From equations (17) and (19) , it can be concluded that:

4ακ
[
1 + 2σ

]ζ
((1 −⋋)[3]m+1

q + ⋋[3]mq )a22 = 2α2κ2(v2 + c2)+

(α− 1)
[
1 + σ

]2ζ
((1 −⋋)[2]m+1

q + ⋋[2]mq )2a22. (22)

Consequently, we obtain:

a22 =
2α2κ2(v2 + c2)

4ακ
[
1 + 2σ

]ζ
((1 −⋋)[3]m+1

q + ⋋[3]mq ) − (α− 1)
[
1 + σ

]2ζ
((1 −⋋)[2]m+1

q + ⋋[2]mq )2
,

(23)
and the upper bound for |a2| is determined as:

|a2| ≤
8ακ√

4ακ
[
1 + 2σ

]ζ
((1 −⋋)[3]m+1

q + ⋋[3]mq ) − (α− 1)
[
1 + σ

]2ζ
((1 −⋋)[2]m+1

q + ⋋[2]mq )2
.

By using equations (17) , (19) and (21) we have:

a3 =
α(v2 − c2)

2

[
1+2σ

]ζ
κ ((1 −⋋)[3]m+1

q + ⋋[3]mq )

+
α2κ2(v21 + c21)[

1 + σ
]2ζ

((1 −⋋)[2]m+1
q + ⋋[2]mq )2

, (24)

and the following upper bound is obtained:

|a3| ≤
2α

|
[
1+2σ

]ζ
κ ((1 −⋋)[3]m+1

q + ⋋[3]mq )|
+

8α2κ2

|
[
1 + σ

]2ζ
((1 −⋋)[2]m+1

q + ⋋[2]mq )2|
.

This concludes the proof.

Theorem 2. Let I(z) given by (1) be in the class Mζ,m
σ,q,Σ(γ,⋋, κ), where 0 ≤ γ <

1, ⋋, δ ≥ 0, κ ≥ 1, σ > 0, m, ζ ∈ N0 z,ϖ ∈ ⋓. Then

|a2| ≤
√

2κ(1 − γ)

|
[
1 + 2σ

]ζ
((1 −⋋)[3]m+1

q + ⋋[3]mq )|

and

|a3| ≤
8κ2(1 − γ)2

|
[
1 + σ

]2ζ
((1 −⋋)[2]m+1

q + ⋋[2]mq )2|
+

2κ(1 − γ)

|
[
1 + 2σ

]ζ
((1 −⋋)[3]m+1

q + ⋋[3]mq )|



M. EL-Ityan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6115 7 of 16

Proof. It can be inferred from definition (4) that there exist v(z) and c(ϖ) ∈ H such
that

1 +
1

κ
[(1 −⋋)(

d

dq
Dζ,m
σ,q I(z)) + ⋋

Dζ,m
σ,q I(z)

z
− 1] = γ + (1 − γ)v(z), (25)

and

1 +
1

κ
[(1 −⋋)(

d

dq
Dζ,m
σ,q G(ϖ)) + ⋋

Dζ,m
σ,q G(ϖ)

z
− 1] = γ + (1 − γ)c(ϖ). (26)

Equating coefficients in (25) and (26) , we obtain:[
1 + σ

]ζ
κ

((1 −⋋)[2]m+1
q + ⋋[2]mq )a2 = (1 − γ)v1, (27)

[
1 + 2σ

]ζ
κ

((1 −⋋)[3]m+1
q + ⋋[3]mq )a3 = (1 − γ)v2, (28)

−
[
1 + σ

]ζ
κ

((1 −⋋)[2]m+1
q + ⋋[2]mq )a2 = (1 − γ)c1, (29)

and [
1 + 2σ

]ζ
κ

((1 −⋋)[3]m+1
q + ⋋[3]mq )(2a22 − a3) = (1 − γ)c2. (30)

Utilizing equations (27) and (29) , we deduce the following:

v1 = −c1, (31)

and [
1 + σ

]2ζ
κ2

((1 −⋋)[2]m+1
q + ⋋[2]mq )2a22 = (1 − γ)2(v21 + c21). (32)

From equations (28) and (30) , it can be concluded that:

2
[
1 + 2σ

]ζ
κ

((1 −⋋)[3]m+1
q + ⋋[3]mq )a22 = (1 − γ)(v2 + c2). (33)

Consequently, we obtain:

a2 =

√
κ(1 − γ)(v2 + c2)

2
[
1 + 2σ

]ζ
((1 −⋋)[3]m+1

q + ⋋[3]mq )
. (34)

This determines the upper bound for |a2|:

|a2| ≤
√

2κ(1 − γ)

|
[
1 + 2σ

]ζ
((1 −⋋)[3]m+1

q + ⋋[3]mq )|
(35)



M. EL-Ityan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6115 8 of 16

Next, for the purpose of establishing the constraint on |a3|, we subtract (28) and (30),
using (32), we get:

2
[
1 + 2σ

]ζ
κ

((1 −⋋)[3]m+1
q + ⋋[3]mq )(a3 − a22) = (1 − γ)(v2 − c2). (36)

Alternatively, it can be expressed as:

a3 = a22 +
κ(1 − γ)(v2 − c2)

2
[
1 + 2σ

]ζ
((1 −⋋)[3]m+1

q + ⋋[3]mq )
. (37)

By using equation (31) in (32), we have:

a3 =
κ2(1 − γ)2(v21 + c21)[

1 + σ
]2ζ

((1 −⋋)[2]m+1
q + ⋋[2]mq )2

+
κ(1 − γ)(v2 − c2)

2
[
1 + 2σ

]ζ
((1 −⋋)[3]m+1

q + ⋋[3]mq )
(38)

We can establish the following upper bound for |a3|:

|a3| ≤
8κ2(1 − γ)2

|
[
1 + σ

]2ζ
((1 −⋋)[2]m+1

q + ⋋[2]mq )2|
+

2κ(1 − γ)

|
[
1 + 2σ

]ζ
((1 −⋋)[3]m+1

q + ⋋[3]mq )|
(39)

This completes the proof.

2. Corollaries and Consequences

By substituting ⋋ = 1 in Theorem (1) and Theorem (2), we arrive at the following
corollaries, respectively:

Corollary 1. Let I(z) given by (1) be in the class Mζ,m
σ,q,Σ(1, κ, α), with 0 < α ≤ 1, κ ≥

1, σ > 0,⋋ = 1,m, ζ ∈ N0 z,ϖ ∈ ⋓. Then

|a2| ≤
8ακ√

4ακ
[
1 + 2σ

]ζ
([3]mq ) − (α− 1)

[
1 + σ

]2ζ
([2]mq )2

.

and

|a3| ≤
2α

|
[
1+2σ

]ζ
κ ([3]mq )|

+
8α2κ2

|
[
1 + σ

]2ζ
([2]mq )2|

.

Corollary 2. Let I(z) given by (1) be in the class Mζ,m
σ,q,Σ(γ, 1, κ), where 0 ≤ γ <

1, κ ≥ 1, σ > 0,⋋ = 1,m, ζ ∈ N0 z,ϖ ∈ ⋓. Then

|a2| ≤
√

2κ(1 − γ)

|
[
1 + 2σ

]ζ
([3]mq )|

and

|a3| ≤
8κ2(1 − γ)2

|
[
1 + σ

]2ζ
([2]mq )2|

+
2κ(1 − γ)

|
[
1 + 2σ

]ζ
([3]mq )|



M. EL-Ityan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6115 9 of 16

By substituting κ = 1 in Theorem (1) and Theorem (2), we arrive at the following
corollaries, respectively:

Corollary 3. Let I(z) given by (1) be in the class Mζ,m
σ,q,Σ(⋋, 1, α), with 0 < α ≤ 1, ⋋ ≥

0, κ = 1, σ > 0, m, ζ ∈ N0 z,ϖ ∈ ⋓. Then

|a2| ≤
8α√

4α
[
1 + 2σ

]ζ
((1 −⋋)[3]m+1

q + ⋋[3]mq ) − (α− 1)
[
1 + σ

]2ζ
((1 −⋋)[2]m+1

q + ⋋[2]mq )2
.

and

|a3| ≤
2α

|
[
1 + 2σ

]ζ
((1 −⋋)[3]m+1

q + ⋋[3]mq )|
+

8α2

|
[
1 + σ

]2ζ
((1 −⋋)[2]m+1

q + ⋋[2]mq )2|
.

Corollary 4. Let I(z) given by (1) be in the class Mζ,m
σ,q,Σ(γ,⋋, 1), where 0 ≤ γ <

1, ⋋ ≥ 0, κ = 1, σ > 0, m, ζ ∈ N0 z,ϖ ∈ ⋓. Then

|a2| ≤
√

2(1 − γ)

|
[
1 + 2σ

]ζ
((1 −⋋)[3]m+1

q + ⋋[3]mq )|

and

|a3| ≤
8(1 − γ)2

|
[
1 + σ

]2ζ
((1 −⋋)[2]m+1

q + ⋋[2]mq )2|
+

2(1 − γ)

|
[
1 + 2σ

]ζ
((1 −⋋)[3]m+1

q + ⋋[3]mq )|

By substituting α = 1 and γ = 0 respectively in The previous corollaries , we arrive:

Corollary 5. Let I(z) given by (1) be in the class Mζ,m
σ,q,Σ(⋋, 1, 1), with 0 < α ≤ 1, ⋋ ≥

0, κ = 1, σ > 0, m, ζ ∈ N0 z,ϖ ∈ ⋓. Then

|a2| ≤
8√

4
[
1 + 2σ

]ζ
((1 −⋋)[3]m+1

q + ⋋[3]mq )
.

and

|a3| ≤
2

|
[
1 + 2σ

]ζ
((1 −⋋)[3]m+1

q + ⋋[3]mq )|
+

8

|
[
1 + σ

]2ζ
((1 −⋋)[2]m+1

q + ⋋[2]mq )2|
.

Corollary 6. Let I(z) given by (1) be in the class Mζ,m
σ,q,Σ(0,⋋, 1), where 0 ≤ γ <

1, ⋋ ≥ 0, κ = 1, σ > 0, m, ζ ∈ N0 z,ϖ ∈ ⋓. Then

|a2| ≤
√

2

|
[
1 + 2σ

]ζ
((1 −⋋)[3]m+1

q + ⋋[3]mq )|

and

|a3| ≤
8

|
[
1 + σ

]2ζ
((1 −⋋)[2]m+1

q + ⋋[2]mq )2|
+

2

|
[
1 + 2σ

]ζ
((1 −⋋)[3]m+1

q + ⋋[3]mq )|



M. EL-Ityan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6115 10 of 16

3. Fekete–Szegő Inequalities for the Functions in the Classes
Mζ,m

σ,q,Σ(⋋, κ, α) and Mζ,m
σ,q,Σ(γ,⋋, κ)

In this section, the focus is on the Fekete–Szegő inequalities for the Functions in the
Classes Mζ,m

σ,q,Σ(⋋, κ, α) and Mζ,m
σ,q,Σ(γ,⋋, κ).

Theorem 3. Let I(z) given by (1) be in the class Mζ,m
σ,q,Σ(⋋, κ, α), with 0 < α ≤ 1, ⋋ ≥

0, κ ≥ 1, σ > 0, m, ζ ∈ N0 z,ϖ ∈ ⋓. Then

|a3 − Θa22| ≤



2α

|

[
1+2σ

]ζ
κ

(
(1−⋋)[3]m+1

q +⋋[3]mq

)
|

for |h(Θ)| ≤ 1

|

[
1+2σ

]ζ
κ

(
(1−⋋)[3]m+1

q +⋋[3]mq

)
|

2α|h(Θ)| for |h(Θ)| ≥ 1

|

[
1+2σ

]ζ
κ

(
(1−⋋)[3]m+1

q +⋋[3]mq

)
|

(40)

Proof. From equations (23) and (24), it is derived that:

a3 − Θa22 =
α(v2 − c2)

2

[
1+2σ

]ζ
κ ((1 −⋋)[3]m+1

q + ⋋[3]mq )

+ (1 − Θ)a22

Also,

a3 − Θa22 =
α(v2 − c2)

2

[
1+2σ

]ζ
κ

(
(1 −⋋)[3]m+1

q + ⋋[3]mq
)

+
2(1 − Θ)α2κ2(v2 + c2)

4ακ
[
1 + 2σ

]ζ (
(1 −⋋)[3]m+1

q + ⋋[3]mq
)
− (α− 1)

[
1 + σ

]2ζ (
(1 −⋋)[2]m+1

q + ⋋[2]mq
)2

Simplify to:

a3 − Θa22 = α

[(
h(Θ) +

1[
1+2σ

]ζ
κ

(
(1 −⋋)[3]m+1

q + ⋋[3]mq
)
)
v2

+

(
h(Θ) − 1[

1+2σ
]ζ

κ

(
(1 −⋋)[3]m+1

q + ⋋[3]mq
)
)
c2

] (41)

where

h(Θ) =
2α(1 − Θ)κ2

4ακ
[
1 + 2σ

]ζ (
(1 −⋋)[3]m+1

q + ⋋[3]mq
)
− (α− 1)

[
1 + σ

]2ζ (
(1 −⋋)[2]m+1

q + ⋋[2]mq
)2

(42)



M. EL-Ityan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6115 11 of 16

Theorem 4. Let I(z) given by (1) be in the class Mζ,m
σ,q,Σ(γ,⋋, κ), where 0 ≤ γ <

1, ⋋, δ ≥ 0, κ ≥ 1, σ > 0, m, ζ ∈ N0 z,ϖ ∈ ⋓. Then

|a3 − ϑa22| ≤


2κ(1−γ)

|2
[
1+2σ

]ζ
((1−⋋)[3]m+1

q +⋋[3]mq )|
for |h(ϑ)| ≤ 1

|2
[
1+2σ

]ζ
((1−⋋)[3]m+1

q +⋋[3]mq )|

2κ(1 − γ)|h(ϑ)| for |h(ϑ)| ≥ 1

|2
[
1+2σ

]ζ
((1−⋋)[3]m+1

q +⋋[3]mq )|
(43)

Proof. From equations (37) and (33), it is derived that:

a3 − ϑa22 =
κ(1 − γ)(v2 − c2)

2
[
1 + 2σ

]ζ
((1 −⋋)[3]m+1

q + ⋋[3]mq )
+ (1 − ϑ)a22

Also,

a3 − ϑa22 =
κ(1 − γ)(v2 − c2)

2
[
1 + 2σ

]ζ
((1 −⋋)[3]m+1

q + ⋋[3]mq )

+
κ(1 − ϑ)(1 − γ)(v2 + c2)

2
[
1 + 2σ

]ζ
((1 −⋋)[3]m+1

q + ⋋[3]mq )

Simplify to:

a3 − ϑa22 = κ(1 − γ)

[(
h(ϑ) +

1

2
[
1 + 2σ

]ζ
((1 −⋋)[3]m+1

q + ⋋[3]mq )

)
v2

+

(
h(ϑ) − 1

2
[
1 + 2σ

]ζ
((1 −⋋)[3]m+1

q + ⋋[3]mq )

)
c2

] (44)

where

h(ϑ) =
(1 − ϑ)

2
[
1 + 2σ

]ζ
((1 −⋋)[3]m+1

q + ⋋[3]mq )
(45)

4. Corollaries and Consequences

By substituting ⋋ = 1 in Theorem (1) and Theorem (2), we arrive at the following
corollaries, respectively:

Corollary 7. Let I(z) given by (1) be in the class Mζ,m
σ,q,Σ(1, κ, α), with 0 < α ≤ 1, ⋋ ≥

0, κ ≥ 1, σ > 0, m, ζ ∈ N0 z,ϖ ∈ ⋓. Then



M. EL-Ityan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6115 12 of 16

|a3−Θa22| ≤



2κα

|
[
1+2σ

]ζ(
[3]mq

)
|

for |h(Θ)| ≤ 1

|

[
1+2σ

]ζ
κ

(
[3]mq

)
|

2α| 2α(1−Θ)κ2

4ακ
[
1+2σ

]ζ
([3]mq )−(α−1)

[
1+σ
]2ζ

([2]mq )
2
| for |h(Θ)| ≥ 1

|

[
1+2σ

]ζ
κ

(
[3]mq

)
|

(46)

where

h(Θ) =
2α(1 − Θ)κ2

4ακ
[
1 + 2σ

]ζ (
[3]mq

)
− (α− 1)

[
1 + σ

]2ζ (
[2]mq

)2 (47)

Corollary 8. Let I(z) given by (1) be in the class Mζ,m
σ,q,Σ(γ, 1, κ), where 0 ≤ γ <

1, ⋋, δ ≥ 0, κ ≥ 1, σ > 0, m, ζ ∈ N0 z,ϖ ∈ ⋓. Then

|a3 − ϑa22| ≤


2κ(1−γ)

|2
[
1+2σ

]ζ
([3]mq )|

for | (1−ϑ)

2
[
1+2σ

]ζ
([3]mq )

| ≤ 1

|2
[
1+2σ

]ζ
([3]mq )|

2κ(1 − γ)| (1−ϑ)

2
[
1+2σ

]ζ
([3]mq )

| for | (1−ϑ)

2
[
1+2σ

]ζ
([3]mq )

| ≥ 1

|2
[
1+2σ

]ζ
([3]mq )|

(48)

By substituting κ = 1 in Theorem (1) and Theorem (2), we arrive at the following
corollaries, respectively:

Corollary 9. Let I(z) given by (1) be in the class Mζ,m
σ,q,Σ(⋋, 1, α), with 0 < α ≤ 1, ⋋ ≥

0, σ > 0, m, ζ ∈ N0 z,ϖ ∈ ⋓. Then

|a3 − Θa22| ≤


2α

|
[
1+2σ

]ζ(
(1−⋋)[3]m+1

q +⋋[3]mq

)
|

for |h(Θ)| ≤ 1

|
[
1+2σ

]ζ(
(1−⋋)[3]m+1

q +⋋[3]mq

)
|

2α|h(Θ)| for |h(Θ)| ≥ 1

|
[
1+2σ

]ζ(
(1−⋋)[3]m+1

q +⋋[3]mq

)
|

(49)
where

h(Θ) =
2α(1 − Θ)

4α
[
1 + 2σ

]ζ (
(1 −⋋)[3]m+1

q + ⋋[3]mq
)
− (α− 1)

[
1 + σ

]2ζ (
(1 −⋋)[2]m+1

q + ⋋[2]mq
)2

(50)

Corollary 10. Let I(z) given by (1) be in the class Mζ,m
σ,q,Σ(γ,⋋, 1), where 0 ≤ γ <

1, ⋋, δ ≥ 0, σ > 0, m, ζ ∈ N0 z,ϖ ∈ ⋓. Then



M. EL-Ityan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6115 13 of 16

|a3 − ϑa22| ≤


2(1−γ)

|2
[
1+2σ

]ζ
((1−⋋)[3]m+1

q +⋋[3]mq )|
for |h(ϑ)| ≤ 1

|2
[
1+2σ

]ζ
((1−⋋)[3]m+1

q +⋋[3]mq )|

2(1 − γ)|h(ϑ)| for |h(ϑ)| ≥ 1

|2
[
1+2σ

]ζ
((1−⋋)[3]m+1

q +⋋[3]mq )|
(51)

where

h(ϑ) =
(1 − ϑ)

2
[
1 + 2σ

]ζ
((1 −⋋)[3]m+1

q + ⋋[3]mq )
(52)

By substituting α = 1 and γ = 0 respectively in The previous corollaries , we arrive:

Corollary 11. Let I(z) given by (1) be in the class Mζ,m
σ,q,Σ(⋋, 1, 1), with 0 < α ≤

1, ⋋ ≥ 0, σ > 0, m, ζ ∈ N0 z,ϖ ∈ ⋓. Then

|a3 − Θa22| ≤


2

|
[
1+2σ

]ζ(
(1−⋋)[3]m+1

q +⋋[3]mq

)
|

for |h(Θ)| ≤ 1

|
[
1+2σ

]ζ(
(1−⋋)[3]m+1

q +⋋[3]mq

)
|

2|h(Θ)| for |h(Θ)| ≥ 1

|
[
1+2σ

]ζ(
(1−⋋)[3]m+1

q +⋋[3]mq

)
|

(53)
where

h(Θ) =
2(1 − Θ)

4
[
1 + 2σ

]ζ (
(1 −⋋)[3]m+1

q + ⋋[3]mq
) (54)

Corollary 12. Let I(z) given by (1) be in the class Mζ,m
σ,q,Σ(0,⋋, 1), where 0 ≤ γ <

1, ⋋, δ ≥ 0, σ > 0, m, ζ ∈ N0 z,ϖ ∈ ⋓. Then

|a3−ϑa22| ≤


1

|
[
1+2σ

]ζ
((1−⋋)[3]m+1

q +⋋[3]mq )|
for |h(ϑ)| ≤ 1

|2
[
1+2σ

]ζ
((1−⋋)[3]m+1

q +⋋[3]mq )|

2|h(ϑ)| for |h(ϑ)| ≥ 1

|2
[
1+2σ

]ζ
((1−⋋)[3]m+1

q +⋋[3]mq )|

(55)

where

h(ϑ) =
(1 − ϑ)

2
[
1 + 2σ

]ζ
((1 −⋋)[3]m+1

q + ⋋[3]mq )
(56)

5. Conclusions

In this paper, we introduced a new operator based on the Salagean q-differential ap-
proach to define a new class of analytic functions. We provided estimates for the Maclau-
rin coefficients |a2| and |a3|, and addressed the Fekete–Szegő problems. Additionally, by



M. EL-Ityan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6115 14 of 16

specializing the parameters Mζ,m
σ,q,Σ(⋋, κ, α) and Mζ,m

σ,q,Σ(γ,⋋, κ), we hope this study will
inspire other researchers to extend this family to harmonic functions and symmetric q-
calculus. Our approach can also be adapted to incorporate the symmetric q-sine and
q-cosine domains as alternatives to the current domain.

Acknowledgements

The authors express their gratitude to the editor and the anonymous reviewers for
their valuable comments and suggestions, which significantly enhanced the quality of this
work.

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