Science and Technology Indonesia e-ISSN:2580-4391 p-ISSN:2580-4405 Vol.
No.
July 2022 Research Paper Subclasses of Analytic Functions with Negative Coefficients Involving q-Derivative Operator Andy Liew Pik Hern1 .
Aini Janteng1 *.
Rashidah Omar2 1 Faculty of Science and Natural Resources.
Universiti Malaysia Sabah.
Kota Kinabalu, 88400.
Malaysia 2 Faculty of Computer and Mathematical Sciences.
Universiti Teknologi Mara Cawangan Sabah.
Kota Kinabalu, 88997.
Malaysia *Corresponding author: aini-jg@ums.
Abstract Let A denote the class of functions f which are analytic in the open unit disk U .
The subclass of A consisting of univalent functions is denoted by M .
In this paper, we also consider a subclass of M which is denoted by V , consisting of functions with negative In addition, this paper also studies the q-derivative operator.
By combining the ideas, this paper introduced three subclasses of A with negative coefficients involving q-derivative.
Furthermore, the coefficient estimates, growth results and extreme points were obtained for all of these classes.
Keywords Analytic.
Univalent, q-Derivative Operator Received: 15 March 2022.
Accepted: 23 June 2022 https://doi.
org/10.
26554/sti.
INTRODUCTION
We denote A as the class of functions which has a Maclaurin series expansion of the form = yu OcA ayua yu yua .
yua=2 The function f is analytic in the open unit disk U = .
u OO EC:
u |<.
While we use M to represent the subclass of A and it is consisting of univalent functions.
In recent times, there are quite a number of researchers have studied different subclasses of A which associated with q-derivative .
ee Breaz and CotyrlE, 2021.
Ibrahim, 2020.
Jabeen et al.
, 2022.
Janteng et al.
, 2020.
Khan et al.
, 2022.
Karahuseyin et al.
, 2017.
Murugusundaramoorthy et al.
, 2015.
Najafzadeh, 2021.
Oshah and Darus, 2015.
Rashid and Juma, 2022.
Shilpa, 2.
From (Jackson, 1909.
Aral et al.
, 2.
, we have the qderivative of a function f OO A which given by .
with 0 < q < 1 as f .
Oe f .
Dq ( f .
) = , q O 1, yu O 0, .
Oe .
yu Dq ( f .
) = f 0 .
From .
, we can get Dq ( f .
) = 1
OcA .
q ayua yu yuaOe1 , yua=2 yua where .
q = 1Oeq 1Oeq .
As q Ie 1, .
q Ie yua.
For a function j .
= 2yu yua .
Dq ( j .
) = Dq .
yu yua ) = 2 1 Oe qyua .
u yuaOe1 ) = 2.
q yu yuaOe1 1Oeq lim Dq ( j .
) = lim 2.
q yu yuaOe1 = 2yua yu yuaOe1 = j 0 .
qIe1 qIe1 where j 0 is the ordinary derivative.
Furthermore, we denote V as a class with negative coefficients and a subclass of M, consisting of the following functions f .
= yu Oe OcA ayua yu yua yua=2 where ayua Ou 0.
For f OO V , there are some significant researchers for example in (Halim et al.
, 2.
, the authors studied the class MSOV .
uC , y.
consisting of starlike functions with respect to .
symmetric points.
Besides, there are various studies for example in (Al-Abbadi and Darus, 2010.
Al Shaqsi and Darus.
Science and Technology Indonesia, 7 .
327-332 Hern et.
Atshan and Ghawi, 2012.
Bucur and Breaz, 2020.
Choo and Janteng, 2013.
Halim et al.
, 2006.
Janteng and Halim.
Najafzadeh and Salleh, 2022.
Oluwayemi et al.
, 2022.
Porwal et al.
, 2.
In this paper, by considering functions f OO V and q-derivaO V .
uC , y.
M O V .
uC , tive operator, we introduce the classes MS,q C ,q O y.
and MSC ,qV .
uC , y.
The coefficient estimates, growth results, and extreme points are obtained for these classes.
First, we give the definitions for the 3 classes.
We note that as q Ie 1, we obtain the classes which were introduced by (Halim et al.
, 2.
yuDq f .
Oe ( f .
Oe f (Oey.
) Oe yu yuC yuDq f .
( f .
Oe f (Oey.
) = Oeyu Oe OcA q Oe .
Oe (Oe.
yua ) ayua yu yua Oe yu .
yuC)yuOe yua=2 O OcA OcA .
q yuC 1 Oe (Oe.
yua ayua z yua O .
q Oe .
Oe (Oe.
yua ) ayua r yua yua=2 yua=2 r Oe yu.
yuC)r O OcA yu .
q yuC 1 Oe (Oe.
yua ayua r yua yua=2 "O OcA O V .
uC , y.
if and only if it Definition 1.
A function f OO MS,q .
q Oe .
Oe (Oe.
yua ) ayua 1 Oe yu.
yuC) yua=2 yuDq f .
yuC yuDq f .
Oe1 < yu f .
Oe f (Oey.
Oe f (Oey.
O OcA yu .
q yuC 1 Oe (Oe.
yua ayua r yua=2 " O OcA (.
yuC y.
q yu 1 Oe (Oe.
yua yua=2 for 0 O yuC < 1, 0 < yu < 1, 0 O 2.
Oey.
1 yuC yu < 1 and yu OO U .
yua Oe 1 Oe (Oe.
)ayua Oe .
yuC) Oe .
r Definition 2.
A function f OO MCO ,qV .
uC , y.
if and only if it yuDq f .
Oe1 < yu yuC yuDq f .
By considering inequality .
, we get ysO yua yua yua=2 .
yuyuC) .
q yu.
Oe (Oe.
) Oe .
Oe (Oe.
) ayua Oe .
yuC) Oe .
O 0, and by applying this inequality, we obtain for 0 O yuC < 1, 0 < yu < 1, 0 O 2.
Oey.
1 yuC yu < 1 and yu OO U .
O V .
uC , y.
if and only if it Definition 3.
A function f OO MSC yuDq f .
Oe f (Oey.
Oe1 < yu yuC yuDq f .
Oe f (Oey.
yuDq f .
Oe ( f .
Oe f (Oey.
) Oe yu yuC yuDq f .
( f .
Oe f (Oey.
) " O OcA .
yuC y.
q yu.
Oe (Oe.
yua ) Oe .
Oe (Oe.
yua ) ayua yua=2 Oe .
yuC) Oe .
r O 0 for 0 O yuC < 1, 0 < yu < 1, 0 O 2.
Oey.
1 yuC yu < 1 and yu OO U .
Thus.
RESULTS yuDq f .
Oef (Oey.
Oe 1 yuC yuDq f .
Oe f (Oey.
1 Now, we give the properties for the 3 classes.
First, we proceed O V .
uC , y.
with the coefficient estimates for f OO MS,q O V .
uC , y.
if and Theorem 1.
Let f OO V .
A function f OO MS,q only if O OcA yuyuC) .
yua=2 yu.
yuC) Oe 1 Oe (Oe.
yua ) Oe .
Oe (Oe.
yua ) yu.
yuC) Oe 1 ayua O 1 for 0 O yuC < 1, 0 < yu < 1 and 0 O 2.
Oey.
1 yuC yu < 1.
<yu O V .
uC , y.
Conversely, let and hence f OO MS,q yuDq f .
Oe1 f .
Oef (Oey.
yuC yuDq f .
Oef (Oey.
ysO yua yuaOe1 yua=2 .
ua ] q Oe .
Oe (Oe.
) ayua yu ysO yua .
yuC) Oe yua=2 .
ua ] q yuC 1 Oe (Oe.
ayua z yuaOe1 Oe1 Oe < yu.
Since we know that the function f is analytic, continuous and non constant in U , then we apply the maximum modulus principle, so we can get Initially, we may prove the AoifAo part first.
We apply the method in (Clunie and Keogh, 1.
So, we write A 2022 The Authors.
Page 328 of 332 Science and Technology Indonesia, 7 .
327-332 Hern et.
ysO ysO Oe .
Oe (Oe.
yua ) .
yuC)Oe1 we obtain that ayua O .
yuyuC) .
ua ] q yuyu.
Oe(Oe.
yua )Oe.
Oe(Oe.
yua ) .
The proof is completed.
yuaOe1 yua=2 .
q Oe .
Oe (Oe.
) ayua yu yuC) Oe yua=2 .
q yuC 1 Oe (Oe.
yua ayua yu yuaOe1 Oe1 Oe ayua yu yuaOe1 Oe yua=2 .
q yuC 1 Oe (Oe.
yua ayua yu yuaOe1 yuaOe1 yua=2 .
q Oe .
Oe (Oe.
) .
yua |.
u | O yuC) Oe yua=2 .
q yuC 1 Oe (Oe.
yua |.
u | yuaOe1 yuaOe1 yua=2 .
q Oe .
Oe (Oe.
) ayua r = f .
ysO .
yuC) Oe yua=2 .
q yuC 1 Oe (Oe.
yua ayua r yuaOe1 yua=2 O V .
uC , y.
and 0 < r < 1, we obtain Since f OO MS,q Next, by applying similar way of methods, we may get the coefficient properties for the functions which belongs to O V .
uC , y.
The results are shown in TheMCO ,qV .
uC , y.
and MSC orem 2 and Theorem 3.
Theorem 2.
Let f OO V .
A function f OO MCO ,qV .
uC , y.
if and only if O OcA .
yuyuC) .
q yua=2 2.
u Oe .
ayua O 1 yu.
yuC) Oe 1 yu.
yuC) Oe 1 for 0 O yuC < 1, 0 < yu < 1 and 0 O 2.
Oey.
1 yuC yu < 1.
ysO .
q Oe .
Oe (Oe.
yua ) ayua r yuaOe1 < yu.
yua a r yuaOe1 .
yuC) Oe O yua yua=2 .
q yuC 1 Oe (Oe.
yua=2 .
Corollary 2.
If f OO MCO ,qV .
uC , y.
then ayua O Then, we let r Ie 1 in .
, we gain OcA O V .
uC , y.
if and Theorem 3.
Let f OO V .
A function f OO MSC only if .
q Oe .
Oe (Oe.
yua ) ayua O yu .
yuC) yua=2 Oe OcA q yuC 1 Oe (Oe.
OcA .
yuyuC) .
q ayua yua=2 yu .
Oe(Oe.
yua )Oe.
Oe(Oe.
yua ) yu .
yuC)Oe1 yu .
yuC)Oe1 ayua O 1 as and hence yua=2 This completes the proof of the theorem.
O V .
uC , y.
then Corollary 1.
If f OO MS,q for 0 O yuC < 1, 0 < yu < 1 and 0 O 2.
Oey.
1 yuC yu < 1.
O V .
uC , y.
then Corollary 3.
If f OO MS,q ayua O ayua O yu.
yuC) Oe 1 , yua Ou 2.
yuyuC) .
q yu .
Oe (Oe.
yua ) Oe .
Oe (Oe.
yua ) O V .
uC , y.
then Proof.
From Theorem 1, if f OO MS,q O OcA yuyuC) .
q yu.
Oe (Oe.
yua ) Oe .
Oe (Oe.
yua ) yu.
yuC) Oe 1 yua=2 .
yuyuC) .
Oe (Oe.
yua ) Oe .
Oe (Oe.
yua ) yu.
yuC) Oe 1
O 1,
A 2022 The Authors.
yuC) Oe 1 After that, we may get the growth property for functions in the O V .
uC , y.
in the next part.
class MS,q ayua O 1 yuC) Oe 1 2 yuC) Oe 1 2 r O | f .
| O r yuyuC) .
yuyuC) First, it is obvious that ayua OcA .
yuyuC) .
q yu.
Oe (Oe.
yua ) Oe .
Oe (Oe.
yua ) yua=2 yu.
yuC) Oe 1 , yua Ou 2.
yuyuC) .
q yu .
Oe (Oe.
yua ) Oe .
Oe (Oe.
yua ) Theorem 4.
Given that a function f be defined by .
and O V .
uC , y.
Then for belongs to the class MS,q .
u : 0 < .
u | = r < .
, for 0 O yuC < 1, 0 < yu < 1 and 0 O 2.
Oey.
1 yuC yu < 1.
Since yu .
Oe (Oe.
yua ) Oe .
Oe (Oe.
yua ) ayua O 1 yu.
yuC) Oe 1 yu.
yuC) Oe 1 yua=2 ysO .
yuyuC) .
ua ] q yu.
yuC) Oe 1 , yua Ou 2.
yuyuC) .
u Oe .
OcA .
yuyuC) OcA .
yuyuC) .
q ayua O yu.
yuC) Oe 1 yu.
yuC) Oe 1 yua=2 yua=2 yu .
Oe (Oe.
yua ) Oe .
Oe (Oe.
yua ) ayua yu.
yuC) Oe 1 O V .
uC , y.
, we use the inequality in Theorem 1 and as f OO MS,q and it gives Page 329 of 332 Science and Technology Indonesia, 7 .
327-332 Hern et.
OcA yua=2 yu.
yuC) Oe 1 yuyuC) .
OcA = yuI yua fyua .
yua=1 =yuOe From .
u | = r .
< .
, we can gain OcA yuIyua yua=2 | f .
| O r | f .
| Ou r Oe OcA ayua r yua O r OcA yua=2 yua=2 O OcA OcA ayua r yua Ou r Oe Next since ayua r 2 OcA ayua r 2 .
yuC) Oe 1 yuyuC) .
q yu.
Oe (Oe.
yua ) Oe .
Oe (Oe.
yua ) .
yuyuC) .
q yu.
Oe (Oe.
yua ) Oe .
Oe (Oe.
yua ) yu.
yuC) Oe 1 OcA yuI yua = 1 Oe yuI 1 O 1.
yuIyua yua=2 yua=2 yua=2 Lastly, by considering the inequalities .
, we may gain the result of Theorem 4.
In the next part, we shall gain the growth results for funcO V .
uC , y.
by using tions that belongs to MCO ,qV .
uC , y.
and MSC a similar method.
The results are shown in Theorem 5 and Theorem 6.
yua=2 O V .
uC , y.
Conversely.
Therefore by Theorem 1, f OO MS,q O V .
uC , y.
Since suppose f OO MS,q ayua O Theorem 5.
Given that a function f be defined by .
and belongs to the class MCO ,qV .
uC , y.
Then for .
: 0 < .
u |= r < .
, yu.
yuC) Oe 1 , yua Ou 2, .
yuyuC) .
q yu .
Oe (Oe.
yua ) Oe .
Oe (Oe.
yua ) we may set yuIyua = yu.
yuC) Oe 1 r 2 O | f .
| .
q Oe 1 yu .
q yuC 2 yu.
yuC) Oe 1 r2.
Or .
q Oe 1 yu .
q yuC 2 rOe .
yuyuC) .
q yu .
Oe (Oe.
yua ) Oe .
Oe (Oe.
yua ) ayua , yua Ou 2 yu.
yuC) Oe 1 Theorem 6.
Given that a function f be defined by .
and O V .
uC , y.
Then for belongs to the class MSC .
: 0 < .
u | = r < .
, rOe yu.
yuC) Oe 1 yuyua .
yuyuC) .
q yu.
Oe (Oe.
yua ) Oe .
Oe (Oe.
yua ) yu.
yuC) Oe 1 2 yu.
yuC) Oe 1 2 r O | f .
| O r .
yuyuC) .
yuyuC) yuI1 = 1 Oe Then O OcA yuI yua fyua .
yua=1 Theorem 7.
Let f1 .
= yu and fyua .
= yuI 1 f1 .
yuC) Oe 1 yuyua , yua .
yuyuC) .
q yu.
Oe (Oe.
yua ) Oe .
Oe (Oe.
yua ) Ou 2.
OcA Then yuI yua fyua .
yua=1 O f OO MS,q V .
uC , y.
where yuI yua Ou 0 if and only if O OcA OcA yuI yua fyua .
yua=2 =yuOe O OcA yua=2 yuIyuayu OcA yuIyuayu Oe yua=2 O OcA ayua yu yua yua=2 = f .
yuI yua = 1.
yua=1 We adopt the technique by (Silverman, 1.
, we assume that A 2022 The Authors.
yuIyua.
yua=2 Finally, we consider extreme points for these 3 classes.
=yu Oe OcA Hence, we complete the proof.
By using a similar method, we obtain the extreme points for the other 2 classes.
Page 330 of 332 Science and Technology Indonesia, 7 .
327-332 Hern et.
Theorem 8.
Let f1 .
= yu and fyua .
=yu Oe yu.
yuC) Oe 1 yu yua , yua Ou 2.
yuyuC) .
u Oe .
f OO MCO ,qV .
uC , y.
= O OcA yuI yua fyua .
Then if and only if where yuI yua Ou 0and yua=1 O OcA yuI yua = 1.
yua=1 Theorem 9.
Let f1 .
= yu and fyua .
=yu Oe yu.
yuC) Oe 1 yuyua , yua .
yuyuC) .
q yu.
Oe (Oe.
yua ) Oe .
Oe (Oe.
yua ) Ou 2.
= O OcA Then yuI yua fyua .
O f OO MSC ,qV .
uC , y.
where yuI yua Ou 0 yua=1 if and only if O OcA yuI yua = 1.
yua=1 CONCLUSIONS In this paper, we introduced 3 new subclasses of A with negative coefficients involving q-derivative and obtained their results for the coefficient estimates, growth results and extreme points.
ACKNOWLEDGMENT
We give our gratitude to the financial support (SBK0485-2.
and all the reference papers.
REFERENCES