On a Class of Multivalent Functions With Negative Coefficients
Involving (r, q)−Calculus
MA’MOUN I. Y. ALHARAYZEH1*, MASLINA DARUS2, HABIS S. AL-ZBOON3
1Department of Scientific Basic Sciences, Faculty of Engineering Technology
Al-Balqa Applied University, Amman 11134
JORDAN
J
2Department of Mathematical Sciences, Faculty of Science and Technology
Universiti Kebangsaan Malaysia, Bangi, 43600 Selangor D. Ehsan,
MALAYSIA
M
3Department of Curriculum and Instruction, College of Education
Al-Hussein Bin Talal University,
JORDAN
Abstract: - In this research, we focused on presenting a novel subclass of multivalent analytic functions situated
in the open unit disk, characterized by the use of Jackson’s derivative operator. Our investigation systematically
establishes the requisite inclusion conditions in this class, offering detailed coefficient characterizations. The
exploration encompasses an array of significant properties intrinsic to this subclass, encompassing coefficient
estimates, growth and distortion theorems, identification of extreme points, and the determination of the radius
of starlikeness and convexity for functions falling within this specialized category. Expanding the preliminary
findings, this research extended the inquiry to delve deeper into the intriguing features and implications associated
with this new subclass of multivalent analytic functions. The research concentrated the light on the nuanced
intricacies of coefficient estimates, providing a comprehensive understanding of how these functions evolve within
the open unit disk through exploring the growth and distortion theorems, unraveling the underlying mathematical
principles governing the behavior of functions in this subclass as they extend beyond the unit disk. The findings of
this research contribute to the broader understanding of multivalent analytic functions, paving the way for further
exploration and applications in diverse mathematical contexts.
Key-Words: -Analytic function, Unit disk, New subclass, p−valent function, Quantum or (r, q)-Calculus,
(r, q)-Derivative operator.
Received: October 24, 2023. Revised: February 25, 2024. Accepted: March 13, 2024. Published: April 10, 2024.
1 Commencement and Definition
The category of all analytic functions exhibiting the
following structure
f(z) = z+
∞
X
n=2
anzn, anis complex number
withen the open unit disk, denoted as U=
{z∈C:|z|<1}where Csignifies the set of com-
plex numbers. This class of all analytic functions is
denoted as ˆ
A. Also, let ˆ
A(p)(p∈N={1,2,3, ...})
be the class consisting of all analytic functions f. This
class is represented by a series that articulates the un-
derlying structure of these functions.
f(z) = zp+
∞
X
n=p+1
anzn, anis complex number
which called p-valent in the open unit disk Uover the
complex numbers C. It is important to observe that
ˆ
A(1) is equivalent to ˆ
A. Additionally, the collection
of all univalent functions within the open disk Uis
symbolized as S(p)which is a subclass of ˆ
A(p). Fur-
thermore, let S∗
p(α)and Cp(α)represent the classes
of p- valent functions respectively starlike of order α
and convex of order α, for 0≤α < p . Notably,
S∗
p(0) is synonymous with S∗
pand Cp(0) corresponds
to Cp, both being the well-known classes of starlike
and convex p-valent functions in U, respectively.
Next, let us assume that T(p)(p∈N={1,2,3, ...})
denote the subclass of S(p)of analytic functions
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having the structure
f(z) = zp−
∞
X
n=p+1
anzn, an>0.(1)
As it is defined on the open unit disk U=
{z∈C:|z|<1}. A function f∈ T (p)is denoted
as a p- valent function with negative coefficients.
Moreover, it is evident that S∗
T,p(α)and CT,p(α)
for 0≤α < p , that are p- valent functions,
respectively starlike of order αand convex of order
αwhich is subclasses of T(p). Clearly, the class
T(1) = Tin [1], he derived and investigated the
subclasses of T(1) denoted by S∗
T,1(α) = S∗
T(α)
and CT,1(α) = CT(α),for 0≤α < 1that are
respectively starlike of order αand convex of order α.
In this section, we revisit established concepts
and fundamental results of (r, q)-calculus. Through-
out this paper, we denote constants as let r, q be
constants with 0< q < r ≤1. We provide
some definitions and theorems pertinent to (r, q)-
calculus, which will be employed in the subsequent
sections of these papers. [2], [3], [4], [5], [6], and, [7].
For 0< q < r ≤1the Jackson‘s (r,q)-derivative of
a function f∈ˆ
A(p)is, by definition, given as follow
Dr,qf(z) :=
f(rz)−f(qz)
(r−q)z, z = 0,
f′(0), z = 0.
(2)
From (2), we have
Dr,qf(z) = [p]r,q zp−1+
∞
X
n=p+1
[n]r,q anzn−1,
where [p]r,q =rp−qp
r−q,[n]r,q =rn−qn
r−qand
0< q < r ≤1.
Note that for r= 1, the Jackson (r, q)-derivative
reduces to the Jackson q-derivative operator of the
function f,Dqf(z)(refer to [8], [9], and, [10]). Note
also that D1,qf(z)→f′(z)when q→1−, where f′
is the classical derivative of the function f.
Clearly for a function g(z) = zn, we obtain
Dr,qg(z) = Dr,qzn=rn−qn
r−qzn−1= [n]r,q zn−1.
And
lim
q→1−D1,qg(z) = lim
q→1−
1−qn
1−qzn−1=nzn−1=g′(z),
where g′is the ordinary derivative.
The theory of q-calculus finds application in the
adaptation and resolution of various problems in
applied science like ordinary fractional calculus,
quantum physics, optimal control, hypergeometric
series , operator theory, q-difference and q-integral
equations, as well as geometric function theory of
complex analysis. The application of q-calculus
was started by, [11]. In, [12], have utilized the
fractional q-calculus operators in examinations of
specific classes of functions which are analytic in
U. For further details on q-calculus one can refer to
[13], [14], [15], and, [16], as well as the additional
references mentioned therein.
Alongside the advancement of the theory and
application of q-calculus, the theory of q-calculus
dependent on two parameters (r, q)-integers has also
been introduced and received more consideration
during the last few decades. In 1991, [17], showed
the (r, q)-calculus. Next, [18], investigated the
fundamental theorem of (r, q)-calculus and a few
(r, q)-Taylor formulas. As of late, [19], studied the
(r, q)-derivative and (r, q)-integral on finite intervals.
Besides, they concentrated on certain properties of
(r, q)-calculus and (r, q)-associated of some im-
portant integral inequalities. The (r, q)-derivative
have been considered and quickly created during this
period by many creators.
Classes defined by derivative operators often
arise in complex analysis or functional analysis and
are crucial in characterizing specific sets of functions
that satisfy certain properties related to their deriva-
tives. These classes help classify functions based on
their behavior under differentiation or specialized
derivative operators. Utilizing the above defined
(r, q)-calculus, certain subclasses in the class ˆ
A(p)
have as of now been explored in geometric function
theory. Some classes defined by (r, q)-calculus
operators like q-starlike and q-convex functions,
these classes are defined using q-calculus operators.
For instance, q-starlike functions are those for which
the q-derivative has positive real part in the unit disk
under the q-calculus framework. Similarly, q-convex
functions satisfy certain conditions related to their
q-derivatives in the unit disk. Also (r, q)-starlike and
(r, q)-convex functions, these classes are extensions
of the q-starlike and q-convex classes, incorporating
additional parameters, rand qin the calculus op-
erators. (r, q)-starlike and (r, q)-convex functions
exhibit specific properties related to their derivatives
under the (r, q)-calculus framework. [20], were the
first who utilized the q-derivative operator Dqto
concentrate on the q-calculus comparable of the class
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S∗of starlike functions in U.
Now, let M(A, B, C)be the subclass of ˆ
A(1)
consisting of functions f∈ˆ
A(1) which satisfy the
inequality
f′(z)−1
Af′(z) + (1 −B)
< C,
where 0≤A≤1,0≤B < 1and 0< C ≤1for all
z∈ U. This class of functions was studied by, [21].
In, [21], defined the class MT(A, B, C)by
MT(A, B, C) = M(A, B, C)∩ T .
Further we present some general subclass of analytic
and multivalent functions related to (r, q)-derivative
operator as follows.
Definition 1.1 For 0≤α≤1,0≤β < 1,
0≤γ < 1, k ≥0,0< q < r ≤1and
p∈N={1,2,3, ...}, we let Υ(α, β, γ, k, r, q, p)
consist of functions f∈ T (P)satisfying the
condition
Re (Dr,qf(z))′−1
α(Dr,qf(z))′+ (1 −γ)
> k
(Dr,qf(z))′−1
α(Dr,qf(z))′+ (1 −γ)−1
+β. (3)
Certainly, the study outlined here appears to be
focused on a specific class of mathematical functions
denoted as f∈Υ(α, β, γ, k, r, q, p). The initial
finding, the coefficient estimate or determine the
coefficients of functions within the specified class,
for functions f∈Υ(α, β, γ, k, r, q, p). These
coefficients likely hold crucial information about the
behavior and properties of these functions. Growth
and distortion theorem are included, by including
growth and distortion theorems, the study aims
to investigate how these functions behave under
certain transformations or conditions. Delving into
their growth and distortion characteristics offers
valuable insights into their behavior and potential
practical uses. Furthermore, we obtain the extreme
points, the study seeks to identify extreme points
of these functions. These extreme points often hold
significance in understanding the behavior and nature
of the functions. After that, the radius of starlike-
ness and convexity, for the function in the class
Υ(α, β, γ, k, r, q, p)are determined. Understanding
the boundaries within these functions possess such
geometric properties is crucial in their characteriza-
tion.
As a matter of first importance, let us take at
the coefficient inequalities. And the technique which
studied in, [22], and also in [23].
2 Coefficient Inequalities
In this section we present a fundamental and suf-
ficient condition for the function fin the class
Υ(α, β, γ, k, r, q, p). Our main first result as follows:
Theorem 2.1 Let 0≤α≤1,0≤β < 1,
0≤γ < 1, k ≥0,0< q < r ≤1and
p∈N={1,2,3, ...}. A function fgiven by (1) is in
the class Υ(α, β, γ, k, r, q, p)if and only if
∞
X
n=p+1
µnan≤µp+ 1,(4)
where
µn=(k+ 1) (1 −α) + α(1 −β)
(k+β) (1 −γ) + k+ 1 (n−1)[n]r,q .(5)
Proof. We have f∈Υ(α, β, γ, k, r, q, p)if and only
if the condition (3) is satisfied.
Let
w=(Dr,qf(z))′−1
α(Dr,qf(z))′+ 1 −γ,
upon the fact that,
Re (w)≥k|w−1|+βif and only if
(k+ 1 ) |w−1| ≤ 1−β.
Now
(k+ 1) |w−1|
= (k+ 1) ( ∞
X
n=p+1
(α−1) [n]r,q (n−1) anzn−2!
−(α−1) [p]r,q (p−1)zp−2−(2 −γ)o
/nα[p]r,q (p−1)zp−2
−α ∞
X
n=p+1
[n]r,q (n−1) anzn−2!−(γ−1))
≤1−β. (6)
The above inequality reduces to
(k+ 1)
∞
X
n=p+1
(α−1) [n]r,q (n−1) anzn−2
−(α−1) [p]r,q (p−1)zp−2− |2−γ|
/α[p]r,q (p−1)zp−2
−α
∞
X
n=p+1
[n]r,q (n−1) anzn−2
− |γ−1|!
≤1−β. (7)
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After that,
(k+ 1) ∞
X
n=p+1
(1 −α) [n]r,q (n−1) an
−(1 −α) [p]r,q (p−1) −(2 −γ)
/α[p]r,q (p−1)
−α
∞
X
n=p+1
[n]r,q (n−1) an−(1 −γ)!
≤1−β. where |z|<1.
then, we have
∞
X
n=p+1
((k+ 1) (1 −α) + α(1 −β)) (n−1) [n]r,q an
≤((k+ 1) (1 −α) + α(1 −β)) (p−1) [p]r,q
−(1 −γ) (1 −β)+(k+ 1) (2 −γ).
Thus
∞
X
n=p+1
((k+ 1) (1 −α) + α(1 −β)) (n−1) [n]r,q an
≤((k+ 1) (1 −α) + α(1 −β)) (p−1) [p]r,q
+ (1 −γ) (k+β)+(k+ 1) ,
divide by (1 −γ) (k+β)+(k+ 1) for both side,
which yield to (4).
Suppose that (4) holds and we have to show (6)
holds. Here the inequality (4) is equivalent to (7). So
it suffices to show that,
( ∞
X
n=p+1
(α−1) [n]r,q (n−1) anzn−2!
−(α−1) [p]r,q (p−1)zp−2−(2 −γ)o
/nα[p]r,q (p−1)zp−2
−α ∞
X
n=p+1
[n]r,q (n−1) anzn−2!−(γ−1))
≤( ∞
X
n=p+1
(1 −α) [n]r,q (n−1) an!
−(1 −α) [p]r,q (p−1) −(2 −γ)o
/nα[p]r,q (p−1)
−α ∞
X
n=p+1
[n]r,q (n−1) an!−(1 −γ)).(8)
Since,
α[p]r,q (p−1)zp−2− α
∞
X
n=p+1
[n]r,q (n−1) anzn−2!
−(γ−1)|,
≥α[p]r,q (p−1)zp−2−
α
∞
X
n=p+1
[n]r,q (n−1) anzn−2
− |γ−1|,
we have
α[p]r,q (p−1)zp−2− α
∞
X
n=p+1
[n]r,q (n−1) anzn−2!
−(γ−1)|,
≥α[p]r,q (p−1) −α
∞
X
n=p+1
[n]r,q (n−1) an−(1 −γ),
where |z|<1, and hence, we obtain (8).
Theorem 2.2 Let 0≤α≤1,0≤β < 1,
0≤γ < 1, k ≥0,0< q < r ≤1and
p∈N={1,2,3, ...}. If the function fgiven by (1)
be in the class Υ(α, β, γ, k, r, q, p)then
an≤µp+ 1
µn
, n =p+ 1, p + 2, p + 3, ..., (9)
where µnis given by (5).
Equality holds for the functions fgiven by,
f(z) = zP−(µp+ 1) zn
µn
.(10)
Proof. Since f∈Υ(α, β, γ, k, r, q, p)Theorem 2.1
holds.
Now
∞
X
n=p+1
µnan≤µp+ 1,
we have,
an≤µp+ 1
µn
.
Clearly the function given by (10) satisfies (9) and
therefore fgiven by (10) is in Υ(α, β, γ, k, r, q, p)for
this function, the result is clearly sharp.
3 Growth and Distortion Theorems
for the Subclass Υ(α, β, γ, k, r, q, p)
The growth and distortion theorems represent foun-
dational principles within complex analysis, focusing
on comprehending how analytic functions behave,
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grow, and affect geometric properties particularly in
the context of univalent functions. These theorems
play a crucial role in understanding the mappings and
transformations of complex-valued functions. Both
the growth and distortion theorems are especially
important when dealing with univalent functions,
which are functions that are injective or one-to-one
within a certain domain. These theorems provide
insights into the behavior of such functions and
are crucial in various areas of mathematics and its
applications.
The growth and distortion theorem will be con-
sidered and the covering property for function in the
class Υ(α, β, γ, k, r, q, p)is given by the following
theorems.
Theorem 3.1 Let 0≤α≤1,0≤β < 1,
0≤γ < 1, k ≥0,0< q < r ≤1and
p∈N={1,2,3, ...}. If the function fgiven
by (1) be in the class Υ(α, β, γ, k, r, q, p)then for
0<|z|=l < 1,we have
lp−µp+ 1
µp+1
lp+1 ⩽|f(z)|⩽lp+µp+ 1
µp+1
lp+1.(11)
Equality holds for the function,
f(z) = zp−µp+ 1
µp+1
zp+1,(z=±l, ±il),
where µpand µp+1 can be found by (5).
Proof. We only prove the right hand side in-
equality in (11), since the other inequality can be
justified using similar arguments.
Since f∈Υ(α, β, γ, k, r, q, p)by Theorem 2.1
we have,
∞
X
n=p+1
µnan≤µp+ 1.
Now
µp+1
∞
X
n=p+1
an=
∞
X
n=p+1
µp+1an,
≤
∞
X
n=p+1
µnan
≤µp+ 1.
And therefore
∞
X
n=p+1
an⩽µp+ 1
µp+1
,(12)
since
f(z) = zp−
∞
X
n=p+1
anzn,
we have,
|f(z)|=
zp−
∞
X
n=p+1
anzn
,
≤ |z|p+|z|p+1
∞
X
n=p+1
an|z|n−(p+1),
≤lp+lp+1
∞
X
n=p+1
an.
By aid of inequality (12), yields the right hand side
inequality of (11).
Theorem 3.2 If the function fgiven by (1) is
in the class Υ(α, β, γ, k, r, q, p)for 0<|z|=l < 1
then, we have
p lp−1−(p+ 1)(µp+ 1)
µp+1
lp⩽
f′(z)
⩽p lp−1+(p+ 1)(µp+ 1)
µp+1
lp.(13)
Equality holds for the function fgiven by
f(z) = zp−µp+ 1
µp+1
zp+1,(z=±l, ±il),
where µpand µp+1 can be found by (5).
Proof. Since f∈Υ(α, β, γ, k, r, q, p)by The-
orem 2.1 we have
∞
X
n=p+1
µnan≤µp+ 1.
Now,
µp+1
∞
X
n=p+1
n an≤(p+ 1)
∞
X
n=p+1
µnan
≤(p+ 1)(µp+ 1).
Hence
∞
X
n=p+1
nan⩽(p+ 1)(µp+ 1)
µp+1
,(14)
since
f′(z) = pzp−1−
∞
X
n=p+1
n anzn−1.
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Then, we have
p|z|p−1− |z|p
∞
X
n=p+1
nan|z|n−1−p
⩽
f′(z)
⩽p|z|p−1+|z|p
∞
X
n=p+1
nan|z|n−1−p,
where |z|<1. By using the inequality (14), we get
Theorem 3.2 and this completes the proof.
Theorem 3.3 If the function fgiven by (1) is
in the class Υ(α, β, γ, k, r, q, p)then fis starlike of
order δ, where
δ= 1 −(µp+ 1)p
−(µp+ 1) + µp+1
.
The result is sharp with
f(z) = zp−µp+ 1
µp+1
zp+1,
where µpand µp+1 can be found by (5).
Proof. It is suffices to show that (4) implies
∞
X
n=p+1
an(n−δ)≤1−δ. (15)
That is,
n−δ
1−δ≤µn
µp+ 1 , n ≥p+ 1.(16)
The above inequality is equivalent to
δ⩽1−(n−1)(µp+ 1)
−(µp+ 1) + µn
=ψ(n),
where n≥p+ 1.
And ψ(n)≥ψ(p+ 1), (16) holds true for any 0≤
α≤1,0≤β < 1,0≤γ < 1, k ≥0,0< q <
r≤1and p∈N={1,2,3, ...}. This completes the
proof of Theorem 3.3.
4 Extreme Points of the Class
Υ(α, β, γ, k, r, q, p)
In spite of its elegance, the significance of extreme
point theory within complex function theory is rela-
tively limited. Several papers authored by Brickman,
Hallenbeck, Mac Gregor, and Wilken have specif-
ically determined extreme points within traditional
families of analytic functions. A comprehensive
overview of their findings can be found in [24].
Notably, the availability of extreme points within
the set Υ(α, β, γ, k, r, q, p), comprising functions f
that are analytic on the unit disc, possess a positive
real part, and are normalized by f(0) = 1, holds
fundamental importance. The theorem detailing the
extreme points of the class Υ(α, β, γ, k, r, q, p)is as
follows.
Theorem 4.1 Let fp(z) = zp,
and
fn(z) = zp−µp+ 1
µn
zn, n =p+ 1, p + 2, p + 3, ...,
where µnis given by (5).
Then f∈Υ(α, β, γ, k, r, q, p)if and only if it
can be represented in the form
f(z) =
∞
X
n=p
ynfn(z)(17)
where yn≥0and
∞
P
n=p
yn= 1.
Proof. Suppose fcan be expressed as in (17). Our
goal is to show that f∈Υ(α, β, γ, k, r, q, p).
By (17) we have
f(z) =
∞
X
n=p
ynfn(z)
=ypfp(z) +
∞
X
n=p+1
ynfn(z),
=ypzp+
∞
X
n=p+1
yn(zp−µp+ 1
µn
zn),
=ypzp+
∞
X
n=p+1
ynzp−
∞
X
n=p+1
yn
µp+ 1
µn
zn,
=zp−
∞
X
n=p+1
(µp+ 1)yn
µn
zn.
After that, f(z) = zp−
∞
P
n=p+1
anznwe see an=
(µp+1) yn
µn
, n ⩾p+ 1.
Now, we have
∞
P
n=p
yn=yp+
∞
P
n=p+1
yn= 1 then
∞
P
n=p+1
yn= 1 −yp≤1.
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Setting
∞
X
n=p+1
yn
µp+ 1
µn
×µn
µp+ 1
=
∞
X
n=p+1
yn= 1 −yp⩽1.
It follows from Theorem 2.1 that the function
f∈Υ(α, β, γ, k, r, q, p).
Conversely, it suffices to show that
an=µp+ 1
µn
yn.
Now we have f∈Υ(α, β, γ, k, r, q, p)then by pre-
vious Theorem 2.2.
an⩽µp+ 1
µn
, n ⩾p+ 1.
That is,
µnan
µp+ 1 ⩽1,
but yn≤1.
Setting,
yn=µnan
µp+ 1, n ⩾p+ 1.
Thus yields to the desired result and completes the
theorem.
Corollary 4.2 The extreme point of the class
Υ(α, β, γ, k, r, q, p)are the function
fp(z) = zp,
and
fn(z) = zp−µp+ 1
µn
zn, n =p+ 1, p + 2, p + 3, ...,
where µnis given by (5).
Finally, in this paper we consider the radius of
starlikeness and convexity.
5 Radius of Starlikeness and
Convexity
The radius of starlikeness and convexity for the
function in the class Υ(α, β, γ, k, r, q, p)will also be
considered.
Theorem 5.1 If the function fgiven by (1) is
in the class Υ(α, β, γ, k, r, q, p), then fis starlike of
order δ( 0 ≤δ < p), in the disk |z|< R where
R=inf µn
µp+ 1 ×p−δ
n−δ
1
n−p
,(18)
where n=p+1, p+2, p+3, ..., and µnis given by (5).
Proof. Here (18) implies
(µp+ 1) (n−δ)|z|n−P≤µn(p−δ).
It suffices to show that
zf′(z)
f(z)−p
≤p−δ,
for |z|< R, we have
zf′(z)
f(z)−p
≤
∞
P
n=p+1
(n−p)an|z|n−p
1−
∞
P
n=p+1
an|z|n−p
.(19)
By aid of (9), we have
zf′(z)
f(z)−p
⩽
∞
P
n=p+1
(µp+1)(n−p)|z|n−p
µn
1−
∞
P
n=p+1
(µp+1)|z|n−p
µn
.
The last expression is bounded above by p−δif.
∞
X
n=p+1
(µp+ 1) (n−p)|z|n−p
µn
≤
1−
∞
X
n=p+1
(µp+ 1) |z|n−p
µn
(p−δ),
and it follows that
|z|n−p⩽µn
µp+ 1 p−δ
n−δ, n ⩾p+ 1
which is equivalent to our condition (18) of the
theorem.
Theorem 5.2 If the function fgiven by (1) is
in the class Υ(α, β, γ, k, r, q, p), then fis convex of
order ε( 0 ≤ε < p ), in the disk |z|< w where
w=inf µn
µp+ 1 ×p(p−ε)
n(n−ε)
1
n−p
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where n=p+1, p+2, p+3, ..., and µnis given by (5).
Proof. By using the same technique in the proof of
Theorem 5.1, we can show that
zf′′(z)
f′(z)−(p−1)
≤p−ε, for |z| ≤ w,
with the aid of (9). Thus we have the assertion of
Theorem 5.2.
6 Conclusion
In this article, our primary focus lies in investigat-
ing a newfound subclass of multivalent analytic func-
tions within the open unit disc, delineated by the ap-
plication of Jackson’s derivative operator. Our explo-
ration begins with a thorough investigation by uncov-
ering the essential criteria for functions falling into
this category through the lens of Coefficients’ Char-
acterization. This approach flattens many of intrigu-
ing features, with notable highlights encompassing
coefficient estimates, theorems on growth and dis-
tortion, identification of extreme points, determina-
tion of the starlikeness radius, and exploration of con-
vexity within this unique subclass. It is clear from
the analysis in this article that the use of the Jackson
derivative operator is not limited only to this specific
subclass, but also opens up ways to derive broader
classes of multivalent analytical functions. During
the analysis, the importance of extending this ap-
proach to studying the description of parameters for
these broader classes was emphasized, thus enriching
our understanding of the diverse mathematical land-
scapes governed by the interaction of functions and
their distinctive properties. We are going to branch
out into uncharted territories, pushing the boundaries
of knowledge in the fascinating realm of multivalent
analytic functions.
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E-ISSN: 2224-2880
260
Volume 23, 2024
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Mamoun Harayzeh Al-Abbadi: Find the results,
Analysis, Methodology, Writing–original draft,
Supervision, Writing–review editing. Maslina
Darus: review and check the results, Data curation,
Investigation and Visualization. Habis S. Al-
zboon: Investigation, Writing–review and editing.
Acknowledgments
The authors would like to express deep-
est thanks to the reviewers for their
insightful comments on their paper.
No funding was received for conducting this study.
Conflicts of Interest
The authors have no conflicts of interest to
declare that are relevant to the content of this article.
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