Title


Science and Technology Indonesia
e-ISSN:2580-4391 p-ISSN:2580-4405
Vol. 8, No. 3, July 2023

Research Paper

The Subclasses of Analytic Functions of Complex Order with Application of q-Derivative
Operators
Aini Janteng1*, Desmond Lee Ching Yiing2, Yong Enn Lun1*, Jaludin Janteng3, Lee See Keong4, Rashidah Omar5, Andy Liew Pik Hern1
1Faculty of Science and Natural Resources, Universiti Malaysia Sabah, Kota Kinabalu, Sabah, 88400, Malaysia2Faculty of Science, Universiti Teknologi Malaysia, Johor Bahru, Johor, 81310, Malaysia3Labuan Faculty of International Finance, Universiti Malaysia Sabah, Federal Territory of Labuan, 87000, Malaysia4School of Mathematical Sciences, Universiti Sains Malaysia, Penang, 11800, Malaysia5Faculty of Computer and Mathematical Sciences, Universiti Teknologi Mara Cawangan Sabah, Kota Kinabalu, Sabah, 88997, Malaysia
*Corresponding author: aini−jg@ums.edu.my; yongel@ums.edu.my

AbstractIn this article, we represent A as the of analytic functions in the open unit disk. Further, new subclasses of analytic functions ofcomplex order utilising q-derivative operator are generated. The subclasses are symbolised by Hq,b(𝜑) and Iq,b(𝜑). Additionally, wediscover that these function classes are implicated with the Fekete-Szegö inequalities.
KeywordsAnalytic, q-Derivative Operator, Fekete-Szegö Inequality

Received: 24 March 2023, Accepted: 10 June 2023
https://doi.org/10.26554/sti.2023.8.3.436-442

1. INTRODUCTION

In recent times, research on the field of quantum calculus is
actively being done by mathematicians. This is demonstrated
by the use of it in the study of complex analysis, particularly
geometry functions theories. Quantum calculus referred to
as calculus without limits, is a kind of standard infinitesimal
calculus that does not include the concept of limits.

The terms q-calculus and h-calculus are defined, where
q is quantum and h is Planck’s constant. q-calculus bridges
the gap between physics and mathematics. Its applications
are often seen in numerous branches of mathematics such as
discrete mathematics, complex numbers, fundamental hyper-
geometric polynomials, symmetric coefficients, as well as other
fields like quantum physics, general relativity and mechan-
ics. Jackson (1908) and Jackson (1910) started the imple-
mentation of q-calculus. Jackson pioneered the q-integral and
q-derivative systematically. Motivated by the research by Ald-
weby and Darus (2016) and Aldweby and Darus (2017) , this
study focuses on the q-derivative in defining several subclasses
of analytic functions.

In the current study, let A symbolise the class of analytic
functions in the open unit disk 𝕆 = {Z : Z ∈ ℂ, |Z | < 1}. The
analytic function is represented by function f and has a Maclau-
rin series expansion as below:

f (Z) = Z +
∞∑︁
𝜏=2

a𝜏 Z
𝜏 , a𝜏 ∈ ℂ, Z ∈ 𝕆 (1)

By Jeyaraman and Suresh (2014) , if f and g are represented
by the analytic functions in 𝕆, then the subordinate to g is f
denoted as f ≺ g in 𝕆 or f (Z) ≺ g(Z) for all Z ∈ 𝕆 if the
presence of the Schwarz function w(Z) is analytic in 𝕆 that has
the properties |w(Z)| < 1 and w(0) = 0 for all Z ∈ 𝕆 such that

f (Z) = g(w(Z)), Z ∈ 𝕆

In the case g is univalent, we can state that f ≺ g, if and
only if f (0) = g(0) and f (𝕆) ⊆ g(𝕆).

Now, we provide the basic definition of q-derivative in this
study. Let Dq f symbolise the q-derivative operator of f with
0 < q < 1. It is defined in Aral et al. (2013); Jackson (1908)
as below:

Dq f (Z) =
{
f (qZ )− f (Z )

(q−1)Z , Z ≠ 0 and q ≠ 1

f ′(0), Z = 0
(2)

In view of (1) and (2), it can be shown that

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https://doi.org/10.26554/sti.2023.8.3.436-442


Janteng et. al. Science and Technology Indonesia, 8 (2023) 436-442

Dq( f (Z)) = 1 +
∞∑︁
𝜏=2

[𝜏]qa𝜏 Z 𝜏−1

where the formulae of [𝜏]q is given as below:

[𝜏]q =
1 − q𝜏
1 − q

and noted that as q → 1−, [𝜏]q → 𝜏.

In the studies of previous researchers, several new subclasses
of analytic functions utilising q-derivative operator have been
proposed. Aldweby and Darus (2016) have introduced the
following subclasses:

S∗q (𝛾) =
{
f ∈ A : Re

(
ZDq( f (Z))
f (Z)

)
> 𝛾, Z ∈ 𝕆

}
and

Cq(𝛾) =
{
f ∈ A : Re

(
1 +

ZqDq(Dq( f (Z))
Dq( f (Z))

)
> 𝛾, Z ∈ 𝕆

}
where 0 ≤ 𝛾 < 1.

Further, if q → 1−, the class S∗q (𝛾) and Cq(𝛾) reduces to
the starlike functions of order 𝛾, S∗(𝛾), and convex functions
of order 𝛾, C(𝛾), respectively. Furthermore, there are several
new subclasses of A involving q-derivative that have been intro-
duced by other mathematicians (see Aldweby and Darus, 2017;
Alsoboh and Darus, 2019; Altinkaya and Yalçin, 2017; Bulut,
2017; Janteng and Halim, 2009b; Karahuseyin, 2017; Lashin
et al., 2021; Hern et al., 2022; Hern et al., 2020; Olatunji and
Dutta, 2018; Piejko and Sokół, 2020; Ramachandran et al.,
2017; Shamsan et al., 2021; Shilpa, 2022). In fact Seoudy
and Aouf (2016) have introduced how classes of q-convex and
q-starlike of complex order can be obtained using the princi-
ple of subordination and q-derivative. This study has inspired
other mathematicians to study subclasses of analytic functions
of complex order utilising q-derivative (see Ali and El Ashwah,
2021; Ibrahim et al., 2020; Purohit and Raina, 2014; Selvaraj
et al., 2017; Srivastava and El Deeb, 2020; Srivastava and
Zayed, 2019).

The research from Seoudy and Aouf (2016) and the re-
search from Janteng and Halim (2009a) ; Janteng and Halim
(2020) serves as our inspiration as we utilise the q-derivative
of f ∈ A and the subordination principle to propose new sub-
classes of analytic functions of complex order.

Definition 1.
Let P represents the class of analytic and univalent functions

𝜑(Z) in 𝕆. 𝜑(Z) is convex with the properties 𝜑(0) = 1 and
Re(𝜑(Z)) > 0 for all Z ∈ 𝕆. f ∈ A is considered as belonging
to class Hq,b(𝜑) if f fulfils the subordination criteria as below:

1+
1
b

[
ZDq( f (Z))
f (Z)

+
𝜖qZ2Dq

(
Dq( f (Z))

)
f (Z)

− 1
]
≺ 𝜑(Z) (3)

with 𝜖 ≥ 0, b ∈ ℂ\{0} , and 𝜑(Z) ∈ P.

Definition 2.
f ∈ A is considered as belonging to class Iq,b(𝜑) if f fulfils the

subordination criteria as below:

1+
1
b

[(
ZDq( f (Z))
f (Z)

)𝜖 (
1 +

ZqDq(Dq( f (Z)))
Dq( f (Z))

)1−𝜖
− 1

]
≺ 𝜑(Z)

with 𝜖 ≥ 0, b ∈ ℂ \ {0} ,and 𝜑(Z) ∈ P.
The demonstration of our key findings requires the use of

the preceding lemmas.

Lemma 1.
Ma and Minda (1992) If 𝜐 is a complex number and p(Z) =
1 + c1Z + c2Z2 + · · · is a function that has a positive real part in
𝕆, then

|c2 − 𝜐c21 | ≤ 2max {1, |2𝜐 − 1|} (4)

The Equality (4) holds for functions provided by

p(Z) =
1 + Z
1 − Z

and

p(Z) =
1 + Z2

1 − Z2

Lemma 2.
Ma and Minda (1992) If p(Z) = 1 + c1Z + c2Z2 + · · · is a

function that has a positive real part in 𝕆, then

|c2 − 𝜐c21 | ≤



−4𝜐 + 2 if 𝜐 ≤ 0

2 if 0 ≤ 𝜐 ≤ 1

4𝜐 − 2 if 𝜐 ≥ 1

When 𝜐 < 0 or 𝜐 > 1, the result is sharp if and only if p(Z) =
1 + Z
1 − Z

or a rotation of itself. If 0 < 𝜐 < 1, the result is sharp if

and only if p(Z) =
1 + Z2

1 − Z2
or a rotation of itself. If 𝜐 = 0, the

result is sharp if and only if

p(Z) =
(
1
2
+
1
2
𝛿

)
·
1 + Z
1 − Z

+
(
1
2
−
1
2
𝛿

)
·
1 − Z
1 + Z

(0 ≤ 𝛿 ≤ 1)

© 2023 The Authors. Page 437 of 442



Janteng et. al. Science and Technology Indonesia, 8 (2023) 436-442

or a rotation of itself. If 𝜐 = 1, the result is sharp if and only if
p(Z) is the reciprocal of a function of itself such that the result
is sharp when 𝜐 = 0.

It is a sharp upper bound, with the possible improvements
when 0 < 𝜐 < 1:

|c2 − 𝜐c21 | + 𝜐 |c1|
2 ≤ 2 (0 ≤ 𝜐 ≤ 1/2)

and

|c2 − 𝜐c21 | + (1 − 𝜐)|c1|
2 ≤ 2 (1/2 ≤ 𝜐 ≤ 1)

2. RESULTS AND DISCUSSION

Throughout this study, the Fekete-Szegö inequalities for classes
Hq,b(𝜑) and Iq,b(𝜑) provided 0 < q < 1, b ∈ ℂ \ {0} and
𝜑(Z) ∈ P and Z ∈ 𝕆 are obtained.

Theorem 1.
Let 𝜑(Z) = 1 + B1Z + B2Z2 + · · · with B1 ≠ 0. If f is provided
by Equation (1) in class Hq,b(𝜑) with Y ≥ 0, then

|a3 − `a22 | ≤
|B1b|

q[2]q(Y[3]q + 1)

max
{
1;

����B2B1 + B1bq(Y[2]q + 1)
(
1 −

[2]q(Y[3]q + 1)
Y[2]q + 1

`

)����}
Proof. Let f ∈ Hq,b(𝜑), then f fulfils subordination (3).

According to the subordination principle, the Schwarz function
w(Z) is present, and it is analytic in 𝕆, which has the properties
w(0) = 0 and |w(Z)| < 1 such that

1+
1
b

[
ZDq( f (Z))
f (Z)

+
YqZ2Dq(Dq( f (Z)))

f (Z)
− 1

]
= 𝜑(w(Z)) (5)

Let p(Z) be a function which is defined by

p(Z) =
1 + w(Z)
1 − w(Z)

= 1 + c1Z + c2Z2 + · · · (6)

Since w(Z) is a Schwarz function, it is obvious for Re(p(Z)) >
0 and p(0) = 1.

In view of Equation (6), we obtain

w(Z) =
p(Z) − 1
p(Z) + 1

=
1
2

[
c1Z +

(
c2 −

c21
2

)
Z
2 + . . .

]
(7)

From Equation (5) and (7), we obtain

𝜑(w(Z)) = 1+
1
2
B1c1Z +

[
1
2
B1

(
c2 −

c21
2

)
+
1
4
B2c

2
1

]
Z
2+. . . (8)

Now, by utilising Equation (1) and (2), we obtain

1 +
1
b

[
ZDq( f (Z))
f (Z)

+
𝜖qZ2Dq(Dq( f (Z)))

f (Z)
− 1

]
= 1 +

q(Y[2]q + 1)a2Z
b

+
q
(
[2]q(Y[3]q + 1)a3 − (Y[2]q + 1)a22

)
Z2

b
+ . . .

(9)

By utilising Equation (8) and (9), we compare the coefficients
of Z and Z2,

q(Y[2]q + 1)a2
b

=
1
2
B1c1

q([2]q(Y[3]q + 1)a3 − (Y[2]q + 1)a22)
b

=
1
2
B1

(
c2 −

c21
2

)
+
1
4
B2c

2
1

or equivalently,

a2 =
B1c1b

2q(𝜖 [2]q + 1)
(10)

a3 =
B1c2b

2q[2]q(Y[3]q + 1)
−

B1c21b

4q[2]q(Y[3]q + 1)

+
B2c21b

4q[2]q(𝜖 [3]q + 1)
+

B21c
2
1b
2

4q2[2]q(𝜖 [2]q + 1)(𝜖 [3]q + 1)

(11)

Next, by utilising Equation (10) and (11), we obtain

a3 − `a22 =
B1b

2q[2]q(Y[3]q + 1)[
c2 −

1
2

[
1 −

B2
B1

−
B1b

q(𝜖 [2]q + 1)

(
1 −

[2]q(𝜖 [3]q + 1)
𝜖 [2]q + 1

`

)]
c21

]
(12)

If we take

© 2023 The Authors. Page 438 of 442



Janteng et. al. Science and Technology Indonesia, 8 (2023) 436-442

𝜐 =
1
2

[
1 −

B2
B1

−
B1b

q(𝜖 [2]q + 1)

(
1 −

[2]q(𝜖 [3]q + 1)
𝜖 [2]q + 1

`

)]

then, from Equation (12),

|a3 − `a22 | =
|B1b|

2q[2]q(𝜖 [3]q + 1)

���c2 − 𝜐c21��� (13)
As we demonstrate, the Fekete-Szegö inequality is derived for
the class Hq,b(𝜑) by applying inequality (4) of Lemma 1 in
Equation (13). Theorem 1 has been successfully proved.

By setting b = 1, Theorem 1 has a corollary as below.

Corollary 1.
Let 𝜑(Z) = 1 + B1Z + B2Z2 + · · · with B1 ≠ 0. If f provided
by Equation (1) is in class Hq(𝜑) with Y ≥ 0, then

|a3 − `a22 | ⩽
|B1|

q[2]q(Y[3]q + 1)

max
{
1,

����B2B1 + B1q(Y[2]q + 1)
(
1 −

[2]q(Y[3]q + 1)
Y[2]q + 1

`

)����}

Theorem 2.
Let 𝜑(Z) = 1 + B1Z + B2Z2 + · · · with B1 > 0 and B2 ≥ 0. Let

𝜎1 =
(Y[2]q + 1)B21b + q(Y[2]q + 1)

2(B2 − B1)
[2]q(Y[3]q + 1)B21b

𝜎2 =
(𝜖 [2]q + 1)B21b + q(𝜖 [2]q + 1)

2(B2 + B1)
[2]q(𝜖 [3]q + 1)B21b

and

𝜎3 =
(𝜖 [2]q + 1)B21b + q(𝜖 [2]q + 1)

2B2
[2]q(𝜖 [3]q + 1)B21b

If f is provided by Equation (1) in class Hq,b(𝜑) with 𝜖 ≥ 0
and b > 0, then

|a3 − `a22 | ≤



B2b
q[2]q (𝜖 [3]q+1) +

B21b
2

q2[2]q (𝜖 [2]q+1) (𝜖 [3]q+1)(
1 − [2]q(𝜖 [3]q+1)

𝜖 [2]q+1 `
)
, ` ≤ 𝜎1

B1b
q[2]q (𝜖 [3]q+1) , 𝜎1 ≤ ` ≤ 𝜎2
− B2bq[2]q (𝜖 [3]q+1) −

B21b
2

q2[2]q (𝜖 [2]q+1) (𝜖 [3]q+1)(
1 − [2]q(𝜖 [3]q+1)

𝜖 [2]q+1 `
)
, ` ≥ 𝜎2

Further, if 𝜎1 ≤ ` ≤ 𝜎3, then

|a3 − `a22 | +
q(𝜖 [2]q + 1)2

[2]q(𝜖 [3]q + 1)B21b[
B1 − B2 −

B21b

q(𝜖 [2]q + 1)

(
1 −

`[2]q(𝜖 [3]q + 1)
𝜖 [2]q + 1

)]
|a2|2

≤
B1b

q[2]q(𝜖 [3]q + 1)

and if 𝜎3 ≤ ` ≤ 𝜎2, then

|a3 − `a22 | +
q(𝜖 [2]q + 1)2

[2]q(𝜖 [3]q + 1)B21b[
B1 + B2 +

B21b

q(𝜖 [2]q + 1)

(
1 −

`[2]q(𝜖 [3]q + 1)
𝜖 [2]q + 1

)]
|a2|2

≤
B1b

q[2]q(𝜖 [3]q + 1)

Proof. By utilising Lemma 2 in Equation (12), we can obtain
the results in Theorem 2.

By setting b = 1, Theorem 2 has a corollary as below.

Corollary 2.
Let 𝜑(Z) = 1 + B1Z + B2Z2 + · · · with B1 > 0 and B2 ≥ 0. Let

𝜎1 =
(Y[2]q + 1)B21 + q(Y[2]q + 1)

2(B2 − B1)
[2]q(Y[3]q + 1)B21

𝜎2 =
(𝜖 [2]q + 1)B21 + q(𝜖 [2]q + 1)

2(B2 + B1)
[2]q(𝜖 [3]q + 1)B21

and

𝜎3 =
(𝜖 [2]q + 1)B21 + q(𝜖 [2]q + 1)

2B2
[2]q(𝜖 [3]q + 1)B21

If f provided by Equation (1) is in class Hq(𝜑) with 𝜖 ≥ 0,
then

|a3 − `a22 | ≤



B2
q[2]q (𝜖 [3]q+1) +

B21
q2[2]q (𝜖 [2]q+1) (𝜖 [3]q+1)(

1 − [2]q(𝜖 [3]q+1)
𝜖 [2]q+1 `

)
, ` ≤ 𝜎1

B1
q[2]q (𝜖 [3]q+1) , 𝜎1 ≤ ` ≤ 𝜎2
− B2q[2]q (𝜖 [3]q+1) −

B21
q2[2]q (𝜖 [2]q+1) (𝜖 [3]q+1)(

1 − [2]q(𝜖 [3]q+1)
𝜖 [2]q+1 `

)
, ` ≥ 𝜎2

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Janteng et. al. Science and Technology Indonesia, 8 (2023) 436-442

Further, if 𝜎1 ≤ ` ≤ 𝜎3, then

|a3 − `a22 | +
q(𝜖 [2]q + 1)2

[2]q(𝜖 [3]q + 1)B21[
B1 − B2 −

B21
q(𝜖 [2]q + 1)

(
1 −

`[2]q(𝜖 [3]q + 1)
𝜖 [2]q + 1

)]
|a2|2

≤
B1

q[2]q(𝜖 [3]q + 1)

and if 𝜎3 ≤ ` ≤ 𝜎2, then

|a3 − `a22 | +
q(𝜖 [2]q + 1)2

[2]q(𝜖 [3]q + 1)B21[
B1 + B2 +

B21
q(𝜖 [2]q + 1)

(
1 −

`[2]q(𝜖 [3]q + 1)
𝜖 [2]q + 1

)]
|a2|2

≤
B1

q[2]q(𝜖 [3]q + 1)

In addition, as shown above, the Fekete-Szegö inequality is
discovered for the class Iq,b(𝜑). We get the required outcome
by carrying out the processes as in Theorem 1.

Theorem 3.
Let 𝜑(Z) = 1 + B1Z + B2Z2 + · · · with B1 ≠ 0. If f is provided
by Equation (1) in class Iq,b(𝜑) with 𝜖 ≥ 0, then

|a3 − `a22 | ≤
|B1b|
q𝛿3[2]q

max

{
1,

�����B2B1 + B1bq𝛿23 ( 𝜌 + `𝛿3[2]q)
�����
}

where

𝜌 = 𝜖 −
𝜖 (𝜖 − 1)

2
q([2]2q + 1) + (1 − 𝜖)[2]q(𝜖q − [2]q)

and

𝛿𝜏 = 𝜖 + (1 − 𝜖)[𝜏]q.

By setting b = 1, Theorem 3 has a corollary as below.

Corollary 3.
Let 𝜑(Z) = 1 + B1Z + B2Z2 + · · · with B1 ≠ 0. If f is provided
by Equation (1) in class Iq(𝜑) with 𝜖 ≥ 0, then

|a3 − `a22 | ≤
|B1|

q𝛿3[2]q
max

{
1,

�����B2B1 + B1q𝛿23 ( 𝜌 + `𝛿3[2]q)
�����
}

where

𝜌 = 𝜖 −
𝜖 (𝜖 − 1)

2
q([2]2q + 1) + (1 − 𝜖)[2]q(𝜖q − [2]q)

and

𝛿𝜏 = 𝜖 + (1 − 𝜖)[𝜏]q.

Theorem 4.
Let 𝜑(Z) = 1 + B1Z + B2Z2 + · · · with B1 > 0 and B2 ≥ 0. Let

𝜎4 =
𝜌B21b + q𝛿

2
2(B2 − B1)

𝛿3[2]qB21b
,

𝜎5 =
𝜌B21b + q𝛿

2
2(B2 + B1)

𝛿3[2]qB21b
,

and

𝜎6 =
𝜌B21b + q𝛿

2
2B2

𝛿3[2]qB21b
.

If f is provided by Equation (1) in class Iq,b(𝜑) with 𝜖 ≥ 0 and
b > 0, then

|a3− `a22 |


B2b

𝛿3q[2]q
+ B

2
1b

q2𝛿22𝛿3[2]q
( 𝜌 − `𝛿3[2]q), ` ≤ 𝜎4

B1b
𝛿3q[2]q

, 𝜎4 ≤ ` ≤ 𝜎5
− B2b

𝛿3q[2]q
− B

2
1b

q2𝛿22𝛿3[2]q
( 𝜌 − `𝛿3[2]q), ` ≥ 𝜎5

Further, if 𝜎4 ≤ ` ≤ 𝜎6, then

|a3 − `a22 | +
q𝛿22

𝛿3[2]qB21b

[
B1 − B2 −

B21b

q𝛿22
( 𝜌 − `𝛿3[2]q)

]
|a2|2 ≤

B1b
𝛿3q[2]q

and if 𝜎6 ≤ ` ≤ 𝜎5, then

|a3 − `a22 | +
q𝛿22

𝛿3[2]qB21b

[
B1 + B2 +

B21b

q𝛿22
( 𝜌 − `𝛿3[2]q)

]
|a2|2 ≤

B1b
𝛿3q[2]q

where

𝜌 = 𝜖 −
𝜖 (𝜖 − 1)

2
q([2]2q + 1) + (1 − 𝜖)[2]q(𝜖q − [2]q)

© 2023 The Authors. Page 440 of 442



Janteng et. al. Science and Technology Indonesia, 8 (2023) 436-442

and

𝛿𝜏 = 𝜖 + (1 − 𝜖)[𝜏]q.

By setting b = 1, Theorem 4 has a corollary as below.

Corollary 4.
Let 𝜑(Z) = 1 + B1Z + B2Z2 + · · · with B1 > 0 and B2 ≥ 0. Let

𝜎4 =
𝜌B21 + q𝛿

2
2(B2 − B1)

𝛿3[2]qB21
,

𝜎5 =
𝜌B21 + q𝛿

2
2(B2 + B1)

𝛿3[2]qB21
and

𝜎6 =
𝜌B21 + q𝛿

2
2B2

𝛿3[2]qB21

If f is provided by Equation (1) in class Iq(𝜑) with 𝜖 ≥ 0 then

|a3−`a22 | ≤


B2

𝛿3q[2]q
+ B

2
1

q2𝛿22𝛿3[2]q
( 𝜌 − `𝛿3[2]q), ` ≤ 𝜎4

B1
𝛿3q[2]q

, 𝜎4 ≤ ` ≤ 𝜎5
− B2

𝛿3q[2]q
− B

2
1

q2𝛿22𝛿3[2]q
( 𝜌 − `𝛿3[2]q), ` ≥ 𝜎5

Further, if 𝜎4 ≤ ` ≤ 𝜎6, then

|a3 − `a22 | +
q𝛿22

𝛿3[2]qB21

[
B1 − B2 −

B21
q𝛿22

( 𝜌 − `𝛿3[2]q)
]

|a2|2 ≤
B1

𝛿3q[2]q

and if 𝜎6 ≤ ` ≤ 𝜎5, then

|a3 − `a22 | +
q𝛿22

𝛿3[2]qB21

[
B1 + B2 +

B21
q𝛿22

( 𝜌 − `𝛿3[2]q)
]

|a2|2 ≤
B1

𝛿3q[2]q

where

𝜌 = 𝜖 −
𝜖 (𝜖 − 1)

2
q([2]2q + 1) + (1 − 𝜖)[2]q(𝜖q − [2]q)

and

𝛿𝜏 = 𝜖 + (1 − 𝜖)[𝜏]q.

3. CONCLUSION

This study demonstrates two discoveries of subclasses belong-
ing to the analytic functions of complex order by applying the
q-derivative operator, Hq,b(𝜑) and Iq,b(𝜑), for finding results
for the Fekete-Szegö inequalities.

4. ACKNOWLEDGMENT

We thank the research funding (SBK0485-2021) provided by
Universiti Malaysia Sabah, the reviewers’ time with constructive
feedback, and all the references.

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	INTRODUCTION
	RESULTS AND DISCUSSION
	CONCLUSION
	ACKNOWLEDGMENT