Title


Science and Technology Indonesia
e-ISSN:2580-4391 p-ISSN:2580-4405
Vol. 7, No. 3, July 2022

Research Paper

Subclasses of Analytic Functions with Negative Coefficients Involving q-Derivative
Operator
Andy Liew Pik Hern1, Aini Janteng1*, Rashidah Omar2
1Faculty of Science and Natural Resources, Universiti Malaysia Sabah, Kota Kinabalu, 88400, Malaysia2Faculty of Computer and Mathematical Sciences, Universiti Teknologi Mara Cawangan Sabah, Kota Kinabalu, 88997, Malaysia
*Corresponding author: aini-jg@ums.edu.my

AbstractLet A denote the class of functions f which are analytic in the open unit diskU . The subclass of A consisting of univalent functionsis denoted by M. In this paper, we also consider a subclass of M which is denoted byV , consisting of functions with negativecoefficients. In addition, this paper also studies the q-derivative operator. By combining the ideas, this paper introduced threesubclasses of A with negative coefficients involving q-derivative. Furthermore, the coefficient estimates, growth results and extremepoints were obtained for all of these classes.
KeywordsAnalytic, Univalent, q-Derivative Operator

Received: 15 March 2022, Accepted: 23 June 2022
https://doi.org/10.26554/sti.2022.7.3.327-332

1. INTRODUCTION

We denote A as the class of functions which has a Maclaurin
series expansion of the form

f (𝛿) = 𝛿 +
∞∑︁
𝜏=2

a𝜏 𝛿
𝜏. (1)

The function f is analytic in the open unit diskU = {𝛿 ∈ ℂ:
|𝛿 |<1}.

While we use M to represent the subclass of A and it is con-
sisting of univalent functions. In recent times, there are quite a
number of researchers have studied dierent subclasses of A
whichassociatedwithq-derivative (seeBreazandCotîrlă,2021;
Ibrahim, 2020; Jabeen et al., 2022; Janteng et al., 2020; Khan
et al., 2022; Karahuseyin et al., 2017; Murugusundaramoor-
thy et al., 2015; Najafzadeh, 2021; Oshah and Darus, 2015;
Rashid and Juma, 2022; Shilpa, 2022).

From (Jackson, 1909; Aral et al., 2013), we have the q-
derivative of a function f ∈ A which given by (1) with 0 < q <
1 as

Dq( f (𝛿)) =
f (q𝛿) − f (𝛿)
(q − 1)𝛿

, q ≠ 1, 𝛿 ≠ 0, (2)

Dq( f (0)) = f ′(0). From (2), we can get

Dq( f (𝛿)) = 1 +
∞∑︁
𝜏=2

[𝜏]qa𝜏 𝛿𝜏−1,

where [𝜏]q =
1−q𝜏
1−q . As q → 1, [𝜏]q → 𝜏. For a function

j(𝛿) = 2𝛿𝜏,

Dq( j(𝛿)) = Dq(2𝛿𝜏) = 2
(
1 − q𝜏
1 − q

)
(𝛿𝜏−1) = 2[𝜏]q𝛿𝜏−1

lim
q→1

(
Dq( j(𝛿))

)
= lim
q→1

(
2[𝜏]q𝛿𝜏−1

)
= 2𝜏𝛿𝜏−1 = j′(𝛿)

where j′ is the ordinary derivative.

Furthermore, we denoteV as a class with negative coe-
cients and a subclass of M, consisting of the following functions

f (𝛿) = 𝛿 −
∞∑︁
𝜏=2

a𝜏 𝛿
𝜏 (3)

where a𝜏 ≥ 0.

For f ∈ V , there are some signicant researchers for ex-
ample in (Halim et al., 2005), the authors studied the class
M∗SV ([ , 𝜗) consistingofstarlikefunctionswithrespect to (w.r.t)
symmetric points. Besides, there are various studies for ex-
ample in (Al-Abbadi and Darus, 2010; Al Shaqsi and Darus,

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https://doi.org/10.26554/sti.2022.7.3.327-332


Hern et. al. Science and Technology Indonesia, 7 (2022) 327-332

2007; Atshan and Ghawi, 2012; Bucur and Breaz, 2020; Choo
and Janteng, 2013; Halim et al., 2006; Janteng and Halim,
2009; Najafzadeh and Salleh, 2022; Oluwayemi et al., 2022;
Porwal et al., 2022).

In this paper, by considering functions f ∈ V and q-deriva-
tive operator, we introduce the classes M∗S,qV ([ , 𝜗), M

∗
C,qV ([ ,

𝜗) andM∗SC,qV ([ , 𝜗). The coecient estimates, growth results,
and extreme points are obtained for these classes.

First, we give the denitions for the 3 classes. We note
that as q → 1, we obtain the classes which were introduced by
(Halim et al., 2005).

Denition 1. A function f ∈ M∗S,qV ([ , 𝜗) if and only if it
satises

���� 𝛿Dq f (𝛿)f (𝛿) − f (−𝛿) − 1
���� < 𝜗 ���� [𝛿Dq f (𝛿)f (𝛿) − f (−𝛿) + 1

����
for 0 ≤ [ < 1, 0 < 𝜗 < 1, 0 ≤ 2(1−𝜗)1+[𝜗 < 1 and 𝛿 ∈ U.

Denition 2. A function f ∈ M∗C,qV ([ , 𝜗) if and only if it
satises

����� 𝛿Dq f (𝛿)f (𝛿) + f (𝛿) − 1
����� < 𝜗

����� [𝛿Dq f (𝛿)f (𝛿) + f (𝛿) + 1
�����

for 0 ≤ [ < 1, 0 < 𝜗 < 1, 0 ≤ 2(1−𝜗)1+[𝜗 < 1 and 𝛿 ∈ U.

Denition 3. A function f ∈ M∗SC,qV ([ , 𝜗) if and only if it
satises

����� 𝛿Dq f (𝛿)f (𝛿) − f (−𝛿) − 1
����� < 𝜗

����� [𝛿Dq f (𝛿)f (𝛿) − f (−𝛿) + 1
�����

for 0 ≤ [ < 1, 0 < 𝜗 < 1, 0 ≤ 2(1−𝜗)1+[𝜗 < 1 and 𝛿 ∈ U.

2. RESULTS

Now, we give the properties for the 3 classes. First, we proceed
with the coecient estimates for f ∈ M∗S,qV ([ , 𝜗).

Theorem 1. Let f ∈ V . A function f ∈ M∗S,qV ([ , 𝜗) if and
only if

∞∑︁
𝜏=2

(
(1 + 𝜗[)[𝜏]q
𝜗(2 + [) − 1

+
𝜗 (1 − (−1)𝜏) − (1 − (−1)𝜏)

𝜗(2 + [) − 1

)
a𝜏 ≤ 1

(4)

for 0 ≤ [ < 1, 0 < 𝜗 < 1 and 0 ≤ 2(1−𝜗)1+[𝜗 < 1.

proof. Initially, we may prove the ’if’ part rst. We apply the
method in (Clunie and Keogh, 1960). So, we write

��𝛿Dq f (𝛿) − ( f (𝛿) − f (−𝛿))�� − 𝜗 ��[𝛿Dq f (𝛿) + ( f (𝛿) − f (−𝛿))��
=

�����−𝛿 − ∞∑︁
𝜏=2

(
[𝜏]q − (1 − (−1)𝜏)

)
a𝜏 𝛿

𝜏

����� − 𝜗
�����(2 + [)𝛿−

∞∑︁
𝜏=2

(
[𝜏]q[ + 1 − (−1)𝜏

)
a𝜏z

𝜏

����� ≤ ∞∑︁
𝜏=2

(
[𝜏]q − (1 − (−1)𝜏)

)
a𝜏r

𝜏

+ r − 𝜗(2 + [)r +
∞∑︁
𝜏=2

𝜗
(
[𝜏]q[ + 1 − (−1)𝜏

)
a𝜏r

𝜏

<

[ ∞∑︁
𝜏=2

(
[𝜏]q − (1 − (−1)𝜏)

)
a𝜏 + 1 − 𝜗(2 + [)

+
∞∑︁
𝜏=2

𝜗
(
[𝜏]q[ + 1 − (−1)𝜏

)
a𝜏

]
r

=

[ ∞∑︁
𝜏=2

((1 + [𝜗)[𝜏]q + 𝜗
(
1 − (−1)𝜏

)
−

(
1 − (−1)𝜏

)
)a𝜏 − (𝜗(2 + [) − 1)

]
r

By considering inequality (4), we get∑∞
𝜏=2

(
(1 + 𝜗[)[𝜏]q + 𝜗(1 − (−1)𝜏) − (1 − (−1)𝜏)

)
a𝜏 −

(𝜗(2 + [) − 1) ≤ 0,

and by applying this inequality, we obtain

��𝛿Dq f (𝛿) − ( f (𝛿) − f (−𝛿))�� − 𝜗 ��[𝛿Dq f (𝛿) + ( f (𝛿) − f (−𝛿))��
=

[
∞∑︁
𝜏=2

(
(1 + [𝜗)[𝜏]q + 𝜗(1 − (−1)𝜏) − (1 − (−1)𝜏)

)
a𝜏

− (𝜗(2 + [) − 1)
]
r ≤ 0

Thus,

������
𝛿Dq f (𝛿)

f (𝛿)−f (−𝛿) − 1
[𝛿Dq f (𝛿)
f (𝛿)−f (−𝛿) + 1

������ < 𝜗
and hence f ∈ M∗S,qV ([ , 𝜗). Conversely, let������

𝛿Dq f (𝛿)
f (𝛿)−f (−𝛿) − 1
[𝛿Dq f (𝛿)
f (𝛿)−f (−𝛿) + 1

������ =
����� −1 −

∑∞
𝜏=2

(
[𝜏]q − (1 − (−1)𝜏)

)
a𝜏 𝛿𝜏−1

(2 + [) −
∑∞

𝜏=2
(
[𝜏]q[ + 1 − (−1)𝜏

)
a𝜏z𝜏−1

����� < 𝜗.
Since we know that the function f is analytic, continuous and
non constant inU, then we apply the maximum modulus
principle, so we can get

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Hern et. al. Science and Technology Indonesia, 7 (2022) 327-332

����� −1 −
∑∞

𝜏=2

(
[𝜏]q − (1 − (−1)𝜏)

)
a𝜏 𝛿𝜏−1

(2 + [) −
∑∞

𝜏=2

(
[𝜏]q[ + 1 − (−1)𝜏

)
a𝜏 𝛿𝜏−1

�����
=

��1 + ∑∞
𝜏=2

(
[𝜏]q − (1 − (−1)𝜏)

)
a𝜏 𝛿𝜏−1

����(2 + 𝛼) − ∑∞
𝜏=2

(
[𝜏]q[ + 1 − (−1)𝜏

)
a𝜏 𝛿𝜏−1

��
≤

1 +
∑∞

𝜏=2

(
[𝜏]q − (1 − (−1)𝜏)

)
|a𝜏 ||𝛿 |𝜏−1

(2 + [) −
∑∞

𝜏=2

(
[𝜏]q[ + 1 − (−1)𝜏

)
|a𝜏 ||𝛿 |𝜏−1

≤
1 +

∑∞
𝜏=2

(
[𝜏]q − (1 − (−1)𝜏)

)
a𝜏r𝜏−1

(2 + [) −
∑∞

𝜏=2

(
[𝜏]q[ + 1 − (−1)𝜏

)
a𝜏r𝜏−1

= f (r).

Since f ∈ M∗S,qV ([ , 𝜗) and 0 < r < 1, we obtain

1 +
∑∞

𝜏=2

(
[𝜏]q − (1 − (−1)𝜏)

)
a𝜏r𝜏−1

(2 + [) −
∑∞

𝜏=2

(
[𝜏]q[ + 1 − (−1)𝜏

)
a𝜏r𝜏−1

< 𝜗. (5)

Then, we let r → 1 in (5), we gain

1 +
∞∑︁
𝜏=2

(
[𝜏]q − (1 − (−1)𝜏)

)
a𝜏 ≤ 𝜗

(
(2 + [)

−
∞∑︁
𝜏=2

(
[𝜏]q[ + 1 − (−1)𝜏

)
a𝜏

)
and hence

∑∞
𝜏=2

(
(1+𝜗[) [𝜏]q
𝜗(2+[)−1 +

𝜗(1−(−1)𝜏)−(1−(−1)𝜏)
𝜗(2+[)−1

)
a𝜏 ≤ 1 as

required. This completes the proof of the theorem.

Corollary 1. If f ∈ M∗S,qV ([ , 𝜗) then

a𝜏 ≤
𝜗(2 + [) − 1

(1 + 𝜗[)[𝜏]q + 𝜗 (1 − (−1)𝜏) − (1 − (−1)𝜏)
, 𝜏 ≥ 2.

Proof. From Theorem 1, if f ∈ M∗S,qV ([ , 𝜗) then

∞∑︁
𝜏=2

(
(1 + 𝜗[)[𝜏]q + 𝜗(1 − (−1)𝜏) − (1 − (−1)𝜏)

𝜗(2 + [) − 1

)
a𝜏 ≤ 1

for 0 ≤ [ < 1, 0 < 𝜗 < 1 and 0 ≤ 2(1−𝜗)1+[𝜗 < 1.
Since

(
(1 + 𝜗[)[𝜏]q + 𝜗(1 − (−1)𝜏) − (1 − (−1)𝜏)

𝜗(2 + [) − 1

)
a𝜏

≤
∞∑︁
𝜏=2

(
(1 + 𝜗[)[𝜏]q + 𝜗(1 − (−1)𝜏) − (1 − (−1)𝜏)

𝜗(2 + [) − 1

)
a𝜏

≤ 1,

we obtain that a𝜏 ≤
𝜗(2+[)−1

(1+𝜗[) [𝜏]q+𝜗(1−(−1)𝜏)−(1−(−1)𝜏) . The proof
is completed.

Next, by applying similar way of methods, we may get
the coecient properties for the functions which belongs to
M∗C,qV ([ , 𝜗) andM

∗
SC,qV ([ , 𝜗). TheresultsareshowninThe-

orem 2 and Theorem 3.

Theorem 2. Let f ∈ V . A function f ∈ M∗C,qV ([ , 𝜗) if and
only if

∞∑︁
𝜏=2

(
(1 + 𝜗[)[𝜏]q
𝜗(2 + [) − 1

+
2(𝜗 − 1)

𝜗(2 + [) − 1

)
a𝜏 ≤ 1

for 0 ≤ [ < 1, 0 < 𝜗 < 1 and 0 ≤ 2(1−𝜗)1+[𝜗 < 1.

Corollary 2. If f ∈ M∗C,qV ([ , 𝜗) then

a𝜏 ≤
𝜗(2 + [) − 1

(1 + 𝜗[)[𝜏]q + 2(𝜗 − 1)
, 𝜏 ≥ 2.

Theorem 3. Let f ∈ V . A function f ∈ M∗SC,qV ([ , 𝜗) if and
only if

∞∑︁
𝜏=2

(
(1 + 𝜗[)[𝜏]q
𝜗(2 + [) − 1

+
𝜗 (1 − (−1)𝜏) − (1 − (−1)𝜏)

𝜗(2 + [) − 1

)
a𝜏 ≤ 1

for 0 ≤ [ < 1, 0 < 𝜗 < 1 and 0 ≤ 2(1−𝜗)1+[𝜗 < 1.

Corollary 3. If f ∈ M∗S,qV ([ , 𝜗) then

a𝜏 ≤
𝜗(2 + [) − 1

(1 + 𝜗[)[𝜏]q + 𝜗 (1 − (−1)𝜏) − (1 − (−1)𝜏)
, 𝜏 ≥ 2.

After that, we may get the growth property for functions in the
class M∗S,qV ([ , 𝜗) in the next part.

Theorem 4. Given that a function f be dened by (4) and
belongs to the class M∗S,qV ([ , 𝜗). Then for
{𝛿 : 0 < |𝛿 | = r < 1},

r −
𝜗(2 + [) − 1
[2]q(1 + 𝜗[)

r2 ≤ | f (𝛿)| ≤ r +
𝜗(2 + [) − 1
[2]q(1 + 𝜗[)

r2.

proof. First, it is obvious that

[2]q(1 + 𝜗[)
𝜗(2 + [) − 1

∞∑︁
𝜏=2

a𝜏 ≤
∞∑︁
𝜏=2

(
(1 + 𝜗[)[𝜏]q
𝜗(2 + [) − 1

+
𝜗 (1 − (−1)𝜏) − (1 − (−1)𝜏)

𝜗(2 + [) − 1

)
a𝜏

and as f ∈ M∗S,qV ([ , 𝜗), we use the inequality in Theorem 1
and it gives

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Hern et. al. Science and Technology Indonesia, 7 (2022) 327-332

∞∑︁
𝜏=2

a𝜏 ≤
𝜗(2 + [) − 1
[2]q(1 + 𝜗[)

. (6)

From (4) with |𝛿 | = r (r < 1), we can gain

| f (𝛿)| ≤ r +
∞∑︁
𝜏=2

a𝜏r
𝜏 ≤ r +

∞∑︁
𝜏=2

a𝜏r
2

and

| f (𝛿)| ≥ r −
∞∑︁
𝜏=2

a𝜏r
𝜏 ≥ r −

∞∑︁
𝜏=2

a𝜏r
2.

Lastly, by considering the inequalities (6), we may gain the
result of Theorem 4.

In the next part, we shall gain the growth results for func-
tions that belongs to M∗C,qV ([ , 𝜗) and M

∗
SC,qV ([ , 𝜗) by using

a similar method. The results are shown in Theorem 5 and
Theorem 6.

Theorem 5. Given that a function f be dened by (4) and
belongs to the class M∗C,qV ([ , 𝜗). Then for {z : 0 < |𝛿 |=
r < 1},

r −
𝜗(2 + [) − 1(

[2]q − 1
)
+ 𝜗

(
[2]q[ + 2

) r2 ≤ | f (𝛿)|
≤ r +

𝜗(2 + [) − 1(
[2]q − 1

)
+ 𝜗

(
[2]q[ + 2

) r2.
Theorem 6. Given that a function f be dened by (4) and
belongs to the class M∗SC,qV ([ , 𝜗). Then for
{z : 0 < |𝛿 | = r < 1},

r −
𝜗(2 + [) − 1
[2]q(1 + 𝜗[)

r2 ≤ | f (𝛿)| ≤ r +
𝜗(2 + [) − 1
[2]q(1 + 𝜗[)

r2.

Finally, we consider extreme points for these 3 classes.

Theorem 7. Let f1(𝛿) = 𝛿 and f𝜏 (𝛿)

=𝛿 −
𝜗(2 + [) − 1

(1 + 𝜗[)[𝜏]q + 𝜗(1 − (−1)𝜏) − (1 − (−1)𝜏)
𝛿
𝜏 , 𝜏

≥ 2. Then f ∈ M∗S,qV ([ , 𝜗) if and only if f (𝛿)

=

∞∑︁
𝜏=1

_𝜏 f𝜏 (𝛿) where _𝜏 ≥ 0 and
∞∑︁
𝜏=1

_𝜏 = 1.

proof. We adopt the technique by (Silverman, 1975), we
assume that

f (𝛿) =
∞∑︁
𝜏=1

_𝜏 f𝜏 (𝛿)

= 𝛿 −
∞∑︁
𝜏=2

_𝜏

(
𝜗(2 + [) − 1

(1 + 𝜗[)[𝜏]q + 𝜗(1 − (−1)𝜏) − (1 − (−1)𝜏)

)
𝛿
𝜏.

Next since

∞∑︁
𝜏=2

_𝜏

(
𝜗(2 + [) − 1

(1 + 𝜗[)[𝜏]q + 𝜗(1 − (−1)𝜏) − (1 − (−1)𝜏)

)
(
(1 + 𝜗[)[𝜏]q + 𝜗(1 − (−1)𝜏) − (1 − (−1)𝜏)

𝜗(2 + [) − 1

)
=

∞∑︁
𝜏=2

_𝜏 = 1 − _1 ≤ 1.

Therefore by Theorem 1, f ∈ M∗S,qV ([ , 𝜗). Conversely,
suppose f ∈ M∗S,qV ([ , 𝜗). Since

a𝜏 ≤
𝜗(2 + [) − 1

(1 + 𝜗[)[𝜏]q + 𝜗 (1 − (−1)𝜏) − (1 − (−1)𝜏)
, 𝜏 ≥ 2,

we may set

_𝜏 =

{
(1 + 𝜗[)[𝜏]q + 𝜗 (1 − (−1)𝜏) − (1 − (−1)𝜏)

𝜗(2 + [) − 1

}
a𝜏 , 𝜏 ≥ 2

and

_1 = 1 −
∞∑︁
𝜏=2

_𝜏.

Then

∞∑︁
𝜏=1

_𝜏 f𝜏 (𝛿)

= _1 f1(𝛿) +
∞∑︁
𝜏=2

_𝜏 f𝜏 (𝛿)

= 𝛿 −
∞∑︁
𝜏=2

_𝜏 𝛿 +
∞∑︁
𝜏=2

_𝜏 𝛿 −
∞∑︁
𝜏=2

a𝜏 𝛿
𝜏

= f (𝛿).

Hence, we complete the proof.

By using a similar method, we obtain the extreme points
for the other 2 classes.

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Hern et. al. Science and Technology Indonesia, 7 (2022) 327-332

Theorem 8. Let f1(𝛿) = 𝛿 and

f𝜏 (𝛿) =𝛿 −
𝜗(2 + [) − 1

(1 + 𝜗[)[𝜏]q + 2(𝜗 − 1)
𝛿
𝜏 , 𝜏 ≥ 2. Then

f ∈ M∗C,qV ([ , 𝜗) if and only if

f (𝛿) =
∞∑︁
𝜏=1

_𝜏 f𝜏 (𝛿) where _𝜏 ≥ 0and
∞∑︁
𝜏=1

_𝜏 = 1.

Theorem 9. Let f1(𝛿) = 𝛿 and

f𝜏 (𝛿) =𝛿 −
𝜗(2 + [) − 1

(1 + 𝜗[)[𝜏]q + 𝜗(1 − (−1)𝜏) − (1 − (−1)𝜏)
𝛿
𝜏 , 𝜏

≥ 2. Then f ∈ M∗SC,qV ([ , 𝜗) if and only if

f (𝛿) =
∞∑︁
𝜏=1

_𝜏 f𝜏 (𝛿) where _𝜏 ≥ 0 and
∞∑︁
𝜏=1

_𝜏 = 1.

3. CONCLUSIONS

In thispaper, we introduced3newsubclassesof Awithnegative
coecients involving q-derivative and obtained their results for
the coecient estimates, growth results and extreme points.

4. ACKNOWLEDGMENT

Wegiveourgratitudetothenancial support (SBK0485-2021)
and all the reference papers.

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	INTRODUCTION
	RESULTS
	CONCLUSIONS
	ACKNOWLEDGMENT