Defining and Analyzing New Classes Associated with (λ,γ)-Symmetrical Functions and Quantum Calculus

Abstract

In this paper, we introduce new classes of functions defined within the open unit disk by integrating the concepts of (λ,γ)-symmetrical functions, generalized Janowski functions, and quantum calculus. We derive a structural formula and a representation theorem for the class Sqλ,γ(x,y,z). Utilizing convolution techniques and quantum calculus, we investigate convolution conditions supported by examples and corollary, establishing sufficient conditions. Additionally, we derive properties related to coefficient estimates, which further elucidate the characteristics of the defined function classes.

Full text

Citation: Louati, H.; Al-Rezami, A.Y.;
Darem, A.A.; Alsarari, F. Defining and
Analyzing New Classes Associated
with (λ,γ)-Symmetrical Functions
and Quantum Calculus. Mathematics
2024,12, 2603. https://doi.org/
10.3390/math12162603
Academic Editor: Michael M. Tung
Received: 5 June 2024
Revised: 17 August 2024
Accepted: 21 August 2024
Published: 22 August 2024
Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
mathematics
Article
Defining and Analyzing New Classes Associated
with (λ,γ)-Symmetrical Functions and Quantum Calculus
Hanen Louati 1,2, Afrah Y. Al-Rezami 3,4,*, Abdulbasit A. Darem 5and Fuad Alsarari 6
1Department of Mathematics, College of Science, Northeren Border University, Arar 73222, Saudi Arabia;
hanen.louati@nbu.edu.sa
2Laboratory of PDEs and Applications (LR03ES04), Faculty of Science of Tunis, University of Tunis El Manar,
Tunis 1068, Tunisia
3Mathematics Department, Prince Sattam Bin Abdulaziz University, Al-Kharj 16278, Saudi Arabia
4Department of Statistics and Information, Sana’a University, Sana’a 1247, Yemen
5
Department of Computer Science, College of Science, Northern Border University, Arar 73222, Saudi Arabia;
basit.darem@nbu.edu.sa
6
Department of Mathematics and Statistics, College of Sciences, Taibah University, Yanbu 46423, Saudi Arabia;
alsrary@gmail.com or fsarari@taibahu.edu.sa
*Correspondence: a.alrezamee@psau.edu.sa or razwanniaz11@gmail.com
Abstract: In this paper, we introduce new classes of functions defined within the open unit disk by
integrating the concepts of
(λ
,
γ)
-symmetrical functions, generalized Janowski functions, and quan-
tum calculus. We derive a structural formula and a representation theorem for the class
Sλ,γ
q(x
,
y
,
z)
.
Utilizing convolution techniques and quantum calculus, we investigate convolution conditions
supported by examples and corollary, establishing sufficient conditions. Additionally, we derive
properties related to coefficient estimates, which further elucidate the characteristics of the defined
function classes.
Keywords: convolution; Janowski functions; q-calculus; (λ,γ)-symmetric points
MSC: 81Q12; 30C45
1. Introduction
Geometric function theory (GFT) is a branch of complex analysis that studies holomor-
phic functions by exploring their geometric properties and behaviors. This field combines
techniques from complex analysis, topology, and differential geometry to investigate map-
pings in the complex plane and higher-dimensional complex spaces. This work focuses on
the space analytic functions.
b
A(Ψ)
denotes the set of analytic functions within the open
unit disk
Ψ={µ∈C:|µ|<1}
, and
b
A
represents a specific subset, characterized by a
class h∈b
A(Ψ), and expressed using the following form:
h(µ) = µ+
∞
∑
k=2
akµk. (1)
Let
S
denote the subclass of
b
A
consisting of all functions which are univalent in
Ψ
.
Let
h
and
g
be analytic in
Ψ
. We say that the function
h
is subordinate to
g
in
Ψ
, denoted
by
h(µ)≺g(µ)
, if there exists an analytic function
ω
in
Ψ
, such that
|ω(µ)|<
1 with
ω(
0
) =
0, and
h(µ) = g(ω(µ))
. If
g
is univalent in
Ψ
, then the subordination is equivalent
to
h(
0
) = g(
0
)
and
h(Ψ)⊂g(Ψ)
. Let
h
and
g
be analytic in
Ψ
. The convolution (or
Hadamard product) of hand g, denoted by (h∗g)(µ), which has the following definition.
If his given by (1) and g(µ) = ∑∞
n=0bnµn, then the following is true:
Mathematics 2024,12, 2603. https://doi.org/10.3390/math12162603 https://www.mdpi.com/journal/mathematics

Mathematics 2024,12, 2603 2 of 11
(h∗g)(µ) = µ+
∞
∑
k=2
akbkµk.
Let us utilize the concept of subordination to define the well-known Carathéodory
class. The Carathéodory class Pis defined as follows:
P=np∈b
A(Ψ)|p(0) = 1, Re p(µ)>0 for all µ∈Ψo.
Any function
p
in
P
has the representation
p(µ) = 1+ω(µ)
1−ω(µ)
, where
ω∈Ω
, and the
following is true:
Ω={ω∈b
A:ω(0) = 0, |ω(µ)|<1}. (2)
In reference [
1
], Janowski introduced a specific subclass of
P
, denoted as
P[e
x
,
y]
−
1
≤y<e
x≤
1. For this class, an analytic function
p
in
Ψ
with
p(
0
) =
1 belongs to
P[e
x
,
y]
,
if
p(µ) = 1+e
xω(µ)
1−yω(µ)
. The class
P[x
,
y]
of generalized Janowski functions was introduced
in [2] for x= (1−z)e
x+zy with 0 ≤z<1.
This work introduces a novel class of
q
-Janowski symmetrical functions defined
within the open unit disk
Ψ
. These functions are intrinsically linked to the
q
-derivative
operator. Before delving into the specifics of this new class, we begin by providing a concise
overview of the fundamental concepts of
(λ
,
γ)
-symmetrical functions and
q
-calculus. This
foundational background is essential for understanding the subsequent development and
analysis of our proposed class of functions.
We first recall the concept of
γ
-fold symmetric functions defined within a
γ
-fold
symmetric domain, where
γ
is any positive integer. A domain
Q
is considered
γ
-fold
symmetric if a rotation of
Q
around the origin by an angle of
2π
γ
maps
Q
onto itself. A
function his deemed γ-fold symmetric in Qif, for every µin Q, the following holds:
h(εµ)=εh(µ),(ε=e2πi
γ),µ∈ Q.
The family of all
γ
-fold symmetric functions is denoted by
e
Sb
. Notably, when
γ=
2, we obtain the class of odd univalent functions. Liczberski and Polubinski, in [
3
],
extended this notion by developing the theory of
(λ
,
γ)
-symmetrical functions, where
λ=
0, 1, 2,
. . .
,
γ−
1 and
γ=
2, 3,
. . .
. In a
γ
-fold symmetric domain
Q
, a function
h:Q → C
is termed
(λ
,
γ)
-symmetrical if, for every
µ∈ Q
,
h(εµ) = ελh(µ)
. It is impor-
tant to note that
(λ
,
γ)
-symmetrical functions generalize the concepts of even, odd, and
γ
-symmetrical functions. We observe that
Ψ
is a
γ
-fold symmetric domain with the
γ
of
any integer. We use the unique decomposition [
3
] of every mapping
h:Ψ7→ C
, as follows:
h(µ) =
γ−1
∑
λ=0
hλ,γ(µ), where hλ,γ(µ) = γ−1γ−1
∑
r=0
ε−rλh(εrµ),µ∈Ψ. (3)
Equivalently, (3) may be written as follows:
hλ,γ(µ) =
∞
∑
v=1
avαv
λµv,a1=1, (4)
where
αk
λ=1
γ
γ−1
∑
r=0
ε(k−λ)r=(1, k=lγ+λ;
0, k=lγ+λ;, (5)
(l∈N,γ=1, 2, . . . , λ=0, 1, 2, . . . , γ−1).
The family of all starlike functions, with respect to
(λ
,
γ)
-symmetric points, is denoted
by
e
S(λ,γ)
, which generalizes several well-known subclasses of starlike functions, such as

Mathematics 2024,12, 2603 3 of 11
e
S(0,2)
,
e
S(1,2)
, and
e
S(1,γ)
. These correspond to the classes of even, odd, and
γ
-symmetric
functions, respectively. The study of starlike functions is a significant area in the field of
geometric function theory, due to its applications in complex analysis and mathematical
modeling. Traditional starlike functions map the unit disk onto starlike domains with
respect to the origin. However, recent research has extended this concept to starlike
functions with respect to
(λ
,
γ)
-symmetric points. These studies have investigated various
properties of the class
e
S(λ,γ)
, including coefficient estimates, distortion theorems, and
subordination results; please see [4–6].
In [
7
], Jackson introduced and studied the concept of the
q
-derivative operator
Dqh(µ)
,
where qsatisfies the condition 0 <q<1, as follows:
Dqh(µ) = (h(µ)−h(qµ)
µ(1−q),µ=0,
h′(0),µ=0. (6)
Equivalently (6), may be written as follows:
Dqh(µ) = 1+
∞
∑
k=2
[k]qakµk−1µ=0,
where
[k]q=1−qk
1−q=1+q+q2+... +qk−1. (7)
Note that, as
q→
1
−
,
[k]q→k
. For the function
h(µ) = µk
, we note that the following
is true:
Dqh(µ) = Dq(µk) = 1−qk
1−qµk−1= [k]qµk−1.
Then, the following is also true:
lim
q→1−Dqh(µ) = lim
q→1−[k]qµk−1=kµk−1=h′(µ),
where h′(µ)is the ordinary derivative.
Assuming the definition of the q-difference operator, the following rules hold:
•Dq(ah(µ)±bg(µ)) = aDqh(µ)±bDqg(µ)
, where
a
and
b
are real (or complex) con-
stants.
•Dq(h(µ)g(µ)) = h(qµ)Dqg(µ) + g(µ)Dqh(µ) = h(µ)Dqg(µ) + g(qµ)Dqh(µ)
•Dqh(µ)
g(µ)=g(µ)Dqh(µ)−h(µ)Dqg(µ)
g(qµ)g(µ).
•µDqh(µ)∗g(µ) = h(µ)∗µDqg(µ).
In [
8
], Jackson introduced the
q
-integral of a function
h
as a right inverse, expressed as
follows: Zµ
0h(w)dqw=µ(1−q)
∞
∑
k=0
qkh(µqk),
provided the
q
-series converges. The connection between quantum calculus and geometric
function theory was first established by Ismail et al. [
9
]. This groundbreaking work opened
a new avenue for exploring the geometric properties of analytic functions using the powerful
tools of quantum calculus. In recent years, there has been a surge of interest in applying
quantum calculus to investigate various subclasses of analytic functions. For example, Naeem
et al. [
10
] delved into the properties of
q
-convex functions, while Srivastava et al. [
11
] explored
subclasses of
q
-starlike functions. Alsarari et al. [
12
] analyzed convolution conditions for
q
-Janowski symmetrical function classes. Ovindaraj and Sivasubramanian [
13
] identified
subclasses associated with
q
-conic domains. Khan et al. [
14
] utilized the symmetric
q
-derivative
operator to further expand the field. Srivastava’s [
15
] comprehensive survey-cum-expository
review paper has been instrumental in guiding researchers in this burgeoning area.

Mathematics 2024,12, 2603 4 of 11
By leveraging the powerful tools of generalized Janowski functions and
(λ
,
γ)
-
symmetrical functions, in conjunction with the concept of the
q
-calculus, we embark
on defining a novel set of classes.
Definition 1. The function
h
in
b
A
is said to belong to the class
Sλ,γ
q(x
,
y)
,
(−
1
≤y<x≤
1
)
, if
the following holds:
µDqh(µ)
hλ,γ(µ)≺1+xµ
1+yµ,µ∈Ψ,
where hλ,γ(µ)is given by (3).
The general framework we propose encompasses various existing classes. By selecting
particular values for
q
,
x
,
y
,
λ
and
γ
, we can retrieve specific classes as special cases. We
will list some of these recovered classes for illustration.
1. S1,γ
1(x,y):=Sγ(x,y), introduced and studied by Darus et al. [16].
2. S1,1
q(1−2κ,−1)=Sq(κ), the class motivated by Agrawal and Sahoo in [17].
3. S1,1
1(
1
−
2
κ
,
−
1
)
=
S(κ)
, the well-known class of starlike function of order
κ
by Robert-
son [18].
4. S1,γ
1(1, −1):=Sy, motivated by Sakaguchi [19].
5. S1,1
q(1, −1)=Sq, which was first introduced by Ismail et al. [9].
6. S1,1
1(x,y):=S[x,y], which reduces to the well-known class defined by Janowski [1].
7. S1,1
1(1, −1) = S∗, the class introduced by Nevanlinna [20].
We denote, using
Kλ,γ
q(x
,
y)
, the subclass of
b
A
, which consists of all functions
h
, such
that the following is true:
µDqh(µ)∈ Sλ,γ
q(x,y). (8)
The class
Sλ,γ
q(x
,
y)
consists of functions with specific properties. Here are some
examples of functions belonging to this class, using the following parameter values:
Example 1.
1.
In the basic case with
γ=
1, the definition simplifies, since there are no roots of unity involved.
Let
λ=
0,
x=−
0.25,
y=−
0.5. In this case, the condition becomes the following expression:
µDqh(µ)
h(µ)≺1−0.25µ
1−0.5µ.
Consider the following function:
h(µ) = µ
(1−µ)2.
We compute its q-derivative as follows:
Dqh(µ) = h(qµ)−h(µ)
(q−1)µ=
qµ
(1−qµ)2−µ
(1−µ)2
(q−1)µ.
Simplifying the above expression can verify the condition. For this specific example, if the
subordination holds, h(µ)belongs to S0,1
q(−0.25, −0.5).
2. In the case of symmetric points with γ=2, we use the primitive 2nd roots of unity ε=−1.
Let
λ=
1,
x=−
0.25, and
y=−
0.5. In this case, the condition becomes the following
expression:
µDqh(µ)
h1,2(µ)≺1+0.25µ
1−0.5µ.
Consider the following function:

Mathematics 2024,12, 2603 5 of 11
h(µ) = µ
1−µ.
We construct h1,2(µ)as follows:
h1,2(µ) = 1
2(h(µ)−h(−µ))=1
2µ
1−µ−−µ
1+µ=µ(1+µ) + µ(1−µ)
2(1−µ)(1+µ)=µ
(1−µ2).
Then, the following is true:
µDqh(µ) = µ·h(qµ)−h(µ)
(q−1)µ=µqµ
1−qµ−µ
1−µ
(q−1)µ.
Simplifying this and checking the subordination condition verifies that
h(µ)
satisfies the
condition for S1,2
q(0.5, −0.5).
3.
In the general case with
γ=
3, we use the primitive 3rd roots of unity
ε=e2πi/3
Let
λ=
1,
x=0.5, and y =0. In this case, the condition becomes the following expression:
µDqh(µ)
h1,3(µ)≺1+0.5µ.
Consider the following function:
h(µ) = µ
(1−µ)3. (9)
We construct h1,3(µ)as follows:
h1,3(µ) = 1
3h(µ) + ε−1h(εµ) + ε−2h(ε2µ).
Computing h(εµ)and h(ε2µ)and averaging them can verify the function’s symmetry properties.
The examples illustrate how to verify functions belonging to the class
Sλ,γ
q(x
,
y)
by
constructing symmetric points and checking the subordination condition. These specific
functions and parameters help demonstrate the membership in the defined class, ensuring
that the functions are starlike with respect to symmetric points.
In this work, we derive a structural formula and a representation theorem for the
class
Sλ,γ
q(x
,
y)
. Utilizing convolution techniques and quantum calculus, we investigate
convolution conditions, providing supporting examples and corollaries to establish suffi-
cient conditions. Additionally, we derive properties related to coefficient estimates, further
elucidating the characteristics of the defined function classes.
We need the following lemma to prove our main results.
Lemma 1 ([
21
] (Lemma 2.1)).Let
p(µ) =
1
+∑∞
k=1ckµk∈ P[x
,
y]
; then, for
k≥
1, the
following is true:
|ck| ≤ x−y.
2. Main Results
Theorem 1. The function h belongs to the class Sλ,γ
q(x,y)if and only if
h(µ) = Zµ
0p(ω)f(ω)dω, (10)
where f (ω) = expnRω
01
λu∑λ−1
v=0p(εvu)−λdquo.

Mathematics 2024,12, 2603 6 of 11
Proof. For the arbitrary function h∈ Sλ,γ
q(x,y,z), we have the following expression:
µDqh(µ)
hλ,γ(µ)=p(µ),p∈ P[x,y]. (11)
Replacing µwith εvµin (11), we obtain the following:
εv(1−λ)µDqh(εvµ)
hλ,γ(µ)=p(εvµ), (12)
From (11) and (12), we obtain the following:
Dqh(εvµ) = p(εvµ)εv(λ−1)Dqh(µ)
p(µ)(13)
Through the q-differentiation of (11), we obtain the following:
Dqhλ,γ(µ) = qµDq(Dqh(µ)) + Dqh(µ)
p(µ)−qµDqh(µ)Dqp(µ)
p(µ)p(qµ). (14)
From (5) and (13), we obtain the following:
Dqhλ,γ(µ) = 1
γ
Dqh(µ)
p(µ)
λ−1
∑
v=0
p(εvµ), (15)
From (14) and (15), we obtain the following:
Dq(Dqh(µ))
Dqh(µ)=Dqp(µ)
p(µ)+1
λµ λ−1
∑
v=0
p(εvµ)−λ!.
By repeatedly
q
-integrating the above equation, we obtain the required structural
formula, as follows:
h(µ) = Zµ
0p(ω)f(ω)dqω.
This proves the necessity. To prove the sufficiency of
(10)
, we suppose that
(10)
holds
with
p∈ P[x
,
y]
. The function
h
defined by
(10)
is obviously in
b
A
with
h(
0
) =
0 and
h(
0
) =
1. To confirm the validity of
(10)
, we proceed by verifying the given identity
through q−differentiation, as follows:
µf(µ) = Zεvµ
0"1
λ
λ−1
∑
v=0
ε−ηvp(ω)f(ω)#dqω, (16)
where fis given in (10). Furthermore, using (10), we obtain the following:
Dqh(µ) = p(µ)f(µ), (17)
which shows that Dqf=0 in Ψ.
From (10), since εis the root of unity, we conclude that the following holds:
hλ,γ(µ) = Zεvµ
0"1
λ
λ−1
∑
v=0
ε−λvp(ω)f(ω)#dqω. (18)
Using (16)–(18), we arrive at the following result:
hλ,γ(µ) = tDqh(µ)
p(µ).

Mathematics 2024,12, 2603 7 of 11
In this way, we have demonstrated the sufficiency of Equation (10).
Remark 1. For specific selections of
λ
,
γ
,
x
,
y
, and
q→
1, we obtain the structural formula, which
was previously derived for classes cited in [22,23].
Theorem 2. The function h ∈ S λ,γ
q(x,y)if and only if
1
µh∗µ(1+yeiθ)
(1−µ)(1−qµ)−1+e
xµeiθ
(1−αλµ)=0, (19)
where q ∈(0, 1),−1≤y<x≤1, and θ∈[0, 2π).
Proof. If we suppose that h∈ Sλ,γ
q(x,y), then the following holds true:
µDqh(µ)
hλ,γ(µ)=p(µ),p∈ P[x,y],
if and only if
µDqh(µ)
hλ,γ(µ)=1+xeiθ
1+yeiθ, (20)
for all µ∈Ψand 0 ≤θ<2π. The condition (20) can be written as follows:
1
µ[µDqh(µ)(1+yeiθ)−hλ,γ(µ)(1+xeiθ)] =0. (21)
Setting h(µ) = µ+∑∞
k=2akµk, we obtain the following:
Dqh(µ) = 1+
∞
∑
k=2
[k]qakµk−1,=h∗µ
(1−µ)(1−qµ). (22)
hλ,γ(µ) = h∗µ
(1−α1
λµ)=
∞
∑
k=1
αk
λakµk, (23)
where αk
λis given by (5).
We derive (19) from (22) and (23), leading to (21).
Example 2. Now, we apply Theorem 2to explain that
h(µ) = µ
(1−µ)3
, given by
(9)
, belongs to the
class ∈ Sλ,γ
q(x,y); we only need to prove that h(µ)∗G(µ)=0.
G(µ) = µ(1+yeiθ)
(1−µ)(1−qµ)−1+xµeiθ
(1−αλµ)is given by Equation (19).
First, we expand h(µ)and G(µ)into their respective power series, as follows:
h(µ) = µ
(1−µ)3=µ
∞
∑
k=0k+2
2µk=
∞
∑
k=1k+1
2µk.
To expand G(µ), we need to break it into simpler parts, as follows:
G(µ) = µ(1+yeiθ)
(1−µ)(1−qµ)−1+xµeiθ
(1−αλµ)
Using geometric series expansions for each term, we obtain the following:
µ(1+yeiθ)
(1−µ)(1−qµ)=µ(1+yeiθ) ∞
∑
n=0
µn! ∞
∑
m=0
(qµ)m!=µ(1+yeiθ)
∞
∑
k=0 k
∑
j=0
qk−j!µk,

Mathematics 2024,12, 2603 8 of 11
1+xµeiθ
(1−αλµ)=1+xµeiθ∞
∑
k=0
(αλµ)k
Combining these into a single series expansion for G(µ), we can write the following:
G(µ) =
∞
∑
n=0
bnµn,
where bnare the coefficients obtained from the above series expansions.
Since
(k+1
2)=
0for all
k≥
1, and the coefficients
bk
in
G(µ)
are derived from a combination
of geometric series expansions which are typically non-zero unless specifically designed to cancel
out, it is clear that not all bkare zero.
Thus, there exist some k, such that (k+1
2)bk=0.
Note that, from Theorem 2, we can easily derive the equivalent condition for a function
h∈ Sλ,γ
q(x,y), as stated in the following corollary.
Corollary 1. For q ∈(0, 1),−1≤y<x≤1, and ϕ∈[0, 2π),the following is true:
h∈ Sλ,γ
q(x,y)⇔(h∗T)(µ)
µ=0, , µ∈Ψ, (24)
where T(µ)has the following form:
T(µ) = µ+
∞
∑
k=2
tkµk,
tk=[k]q−αk
λ+ ([k]qy−αk
λxeiϕ)
(y−x)eiϕ. (25)
By applying Corollary 1, we derive the sufficient condition stated in the theorem.
Theorem 3. Let
h(µ) = µ+∑∞
k=2akµk
,be analytic in
Ψ
, for
−
1
≤y<x≤
1and 0
<q<
1; if
∞
∑
k=2([k]q−αk
λ+|[k]qy−αk
λx|
|(x−y)|)|av| ≤ 1, (26)
then h(µ)∈ Sλ,γ
q(x,y).
Proof.
To prove Theorem 3, it is sufficient to demonstrate that
(h∗T)(µ)
µ=
0, where
T
is
defined as in Equation
(25)
. Let us assume that
h(µ) = µ+∑∞
k=2akµv
. Then, we can
consider the convolution
(h∗T)(µ)
µ=1+
∞
∑
k=2
tkakµk−1,µ∈Ψ.
Through Corollary 1, we know that
h(µ)
belongs to
Sλ,γ
q(x
,
y)
if and only if
(h∗
T)(µ)/µ
is non-zero, where
T
is defined by
(25)
. Using
(25)
and
(26)
, we can derive the
following inequality:

(h∗T)(µ)
µ≥1−
∞
∑
k=2
[k]q−αk
λ+|[k]qy−αk
λx|
|(x−y)||ak||µ|k−1>0, for all µ∈Ψ.
This implies that (h∗T)(µ)/µis non-zero, and, hence, h(µ)∈ S λ,γ
q(x,y).

Mathematics 2024,12, 2603 9 of 11
Theorem 4. Let
h(µ)∈ Sλ,γ
q(x
,
y)
. For
−
1
≤y<x≤
1and
q∈(
0, 1
)
, then the following
holds:
|ak| ≤
k−1
∏
m=0
[(x−y)−1]αk
λ+ [m]q
[m+1]q−αm+1
λ
,k≥2, (27)
where αm
λis given by (3).
Proof. According to definition (1), we know that the following is true:
µDqh(µ)
hλ,γ(µ)=p(µ), where p(µ) = 1+
∞
∑
k=1
ckµk∈ P[x,y].
This gives the following expression:
µDqh(µ) = "∞
∑
k=1
ckµk#hλ,γ(µ).
after simplifying the expression below:
(1−α1
λ) +
∞
∑
k=2
([k]q−αk
λ)akµk="∞
∑
k=1
ckµk#"∞
∑
k=1
αk
λakµk#.
Applying the Cauchy product formula to the inequality above and equating coeffi-
cients of µk,k≥2, we obtain the following:
ak=1
[k]q−αk
λ
k−1
∑
m=1
cmαk−m
λak−m, (28)
Using Lemma 1, we obtain the following:
|ak| ≤ (x−y)
[k]q−αk
λ
k−1
∑
m=1
αm
λ|am|. (29)
The proof is completed by showing the following:
(x−y)
[k]q−αk
λ
k−1
∑
m=1
αm
λ|am| ≤
k−2
∏
m=0
[(x−y)−1]αk
λ+ [m]q
[m+1]q−αm+1
λ
. (30)
To prove this, we will employ the method of mathematical induction. It is clear that
(30) is true for k=2 and 3.
Let the hypothesis be true for k=m; in this case, we obtain the following:
(x−y)
[m]q−αm
λ
m−1
∑
r=1
αr
λ|ar| ≤
m−1
∏
r=0
[(x−y)−1]αr
λ+ [r]q
[r+1]q−αr+1
λ
.
Multiplying both sides by [(x−y)−1]αm
λ+[m]q
[m+1]q−αm+1
λ
, we then obtain the following:
m
∏
r=0
[(x−y)−1]αr
λ+ [r]q
[r+1]q−αr+1
λ
≥[(x−y)−1]αm
λ+ [m]q
[m+1]q−αm+1
λ
.(x−y)
[m]q−αm
λ
m−1
∑
r=1
αr
λ|ar|,
=(x−y)
[m+1]q−αm+1
λ
.1+(x−y)
[m]q−αm
λm−1
∑
r=1
αr
λ|ar|,

Mathematics 2024,12, 2603 10 of 11
≥(x−y)
[m+1]q−αm+1
λ
."m−1
∑
r=1
αr
λ|ar|+αm
λ|am|#,
=(x−y)
[m+1]q−αm+1
λ
."m
∑
r=1
αr
λ|ar|#.
That is, the following holds:
|am+1| ≤ (x−y)
[m+1]q−αm+1
λ
m
∑
r=1
αr
λ|ar| ≤
m
∏
r=0
[(x−y)−1]αr
λ+ [r]q
[r+1]q−αr+1
λ
.
This completes the proof, showing that the inequality in Equation
(30)
holds true for
the value of kequal to m+1.
Corollary 2. Let
h(µ)∈ Kλ,γ
q(x
,
y)
. For
−
1
≤y<x≤
1and
q∈(
0, 1
)
, the following holds
true:
|ak| ≤ 1
[k]q
k−1
∏
m=0
[(x−y)−1]αk
λ+ [m]q
[m+1]q−αm+1
λ
,k≥2, (31)
where αm
λis given by (3).
3. Conclusions
In this research paper, we have introduced a novel class of analytic functions,
Sλ,γ
q(x
,
y)
,
in the open unit disk
Ψ
. This class combines the concepts of
(λ
,
γ)
-symmetrical functions,
generalized Janowski functions, and
q
-calculus in a unique and innovative manner. By
deriving a structural formula and a representation theorem, we have established a solid
foundation for understanding the nature of these functions.
The powerful tools of convolution and quantum calculus have been employed to
explore the convolution conditions for functions within the class. This has led to a crucial
supporting result for determining a sufficient condition for membership in the class, which
we have illustrated through a relevant example and corollary. Furthermore, we have
investigated the properties of coefficient estimates, providing valuable insights into the
behavior and characteristics of these functions.
Our findings contribute to a deeper understanding of the analytical properties of the
class
Sλ,γ
q(x
,
y)
. This research opens up exciting avenues for further exploration. Future
studies could focus on investigating other geometric properties, such as distortion theorems,
the radius of starlikeness and convexity, and potential applications in the field of univalent
functions. Additionally, exploring the connections between this class and other function
classes within the framework of
q
-calculus could lead to a richer understanding of the
interplay between different mathematical concepts.
In conclusion, this paper has successfully introduced and analyzed a new class of
analytic functions, offering valuable insights and paving the way for further research in the
field of complex analysis and q-calculus.
Author Contributions: The researchers F.A., H.L., A.Y.A.-R. and A.A.D. formulated the concept
for the present investigation. They verified the data and proposed recommendations that signifi-
cantly augmented the existing article. After perusing the final draft, each author made individual
contributions. All authors have read and agreed to the published version of the manuscript.
Funding: The authors extend their appreciation to the Deanship of Scientific Research at Northern
Border University, Arar, Saudi Arabia, for funding this research work through the project number
“NBU-FPEJ-2024- 2920-01”. This study was also supported via funding from Prince Sattam bin
Abdulaziz University, project number (PSAU/2024/R/1445).

Mathematics 2024,12, 2603 11 of 11
Data Availability Statement: The original contributions presented in the study are included in the
article, further inquiries can be directed to the corresponding author.
Conflicts of Interest: The authors declare no conflicts of interest.
References
1. Janowski, W. Some extremal problems for certain families of analytic functions. Ann. Pol. Math. 1973,28, 297–326. [CrossRef]
2.
Polatoglu, Y.; Bolcal, M.; Sen, A.; Yavuz, E. A study on the generalization of Janowski functions in the unit disc. Acta Math. Acad.
Paedag. Nyhazi. 2006,28, 27–31.
3. Liczberski, P.; Połubi´nski, J. On (j,k)-symmetrical functions. Math. Bohemca 1995,120, 13–28. 10.21136/MB.1995.125897. [CrossRef]
4.
Al-Sarari, F.; Frasin, B.; AL-Hawary, T.; Latha, S. A few results on generalized Janowski type functions associated with
(j
,
k)
-
symmetrical functions. Acta Univ. Sapientiae Math. 2016,8, 195–205. [CrossRef]
5.
Gandhi, S.; Ravichandran, V. Starlike functions with respect to
(j
,
k)
-symmetric points of complex order. Bull. Korean Math. Soc.
2017,54, 807–822.
6.
Al-Sarari, F.; Latha, S. A few results on functions that are Janowski starlike related to
(j
,
k)
-symmetric points, Octag. Math. Mag.
2013,21, 556–563.
7. Jackson, F.H. On q-functions and a certain difference operator. Trans. R. Soc. Edinb. 1909,46, 253–281. [CrossRef]
8. Jackson, F.H. On q-definite integrals. Q. J. Pure Appl. Math. 1910,41, 193–203.
9.
Mourad, E.; Ismail, H.; Merkes, E.; Styer, D. A generalization of starlike functions. Complex Var. Theory Appl. 1990,14, 77–84.
[CrossRef]
10.
Naeem, M.; Hussain, S.; Khan, S.; Mahmood, T.; Darus, M.; Shareef, Z. Janowski type
q
-convex and qclose-to-convex functions
associated with q-conic domain. Mathematics 2020,8, 440. [CrossRef]
11.
Srivastava, M.; Tahir, M.; Khan, B.; Ahmad, Z.; Khan, N. Some general classes of q-starlike functions associated with the Janowski
functions. Symmetry 2019,11, 292. [CrossRef]
12.
Alsarari, F.; Alzahrani, S. Convolution Properties of q-Janowski-Type Functions Associated with
(x
,
y)
-Symmetrical Functions.
Mathematics 2022,14, 1406. [CrossRef]
13.
Govindaraj, M.; Sivasubramanian, S. On a class of analytic functions related to conic domains involving q-calculus. Anal. Math.
2017,43, 475–487. [CrossRef]
14.
Khan, B.; Liu, Z.G.; Shaba, T.G.; Araci, S.; Khan, N.; Khan, M.G. Applications of
q
-Derivative Operator to the Subclass of
Bi-Univalent Functions Involving q-Chebyshev Polynomials. J. Math. 2022,2022, 8162182. [CrossRef]
15.
Srivastava, M. Operators of Basic (or
q−
) Calculus and Fractional
q
-Calculus and Their Applications in Geometric Function
Theory of Complex Analysis. Iran. Sci. Technol. Trans. Sci. 2020,44, 327–344. [CrossRef]
16.
Al-Sarari, F.; Latha, S.; Darus, M.A. few results on Janowski functions associated with
k
-symmetric points. Korean J. Math. 2017,
25, 389–403. [CrossRef]
17. Agrawal, S.; Sahoo, S.K. A generalization of starlike functions of order alpha. Hokkaido Math. J. 2017,46, 15–27. [CrossRef]
18. Robertson, M.S. On the theory of univalent functions. Ann. Math. 1936,37, 374–408. [CrossRef]
19. Sakaguchi, K. On a certain univalent mapping. J. Math. Soc. Jpn. 1959,11, 72–75. [CrossRef]
20. Nevanlinna, R. Über die konforme Abbildung von Sterngebieten. Översikt Vetensk.-Soc. Förhandlingar/A 1920,63, 1–21. [CrossRef]
21. Noor, K.; Malik, S. On a Subclass of Quasi-Convex Univalent Functions. World Appl. Sci. J. 2011,12, 2202–2209.
22.
Stankiewicz, J. Some remarks on functions starlike with respect to symmetric points. Ann. Univ. Mariae-Curie Skledowke 1965,
19, 53–59. Available online: https://9lib.org/document/zlgvn9xr-remarks-functions-starlike-respect-symmetric-points.html
(accessed on 5 June 2024).
23.
Nezhmetdinov, I.R.; Ponnusamy, S. On the class of univalent functions starlike with respect to N-symmetric points. Hokkaido
Math. J. 2002,31, 61–77. [CrossRef]
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