EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 3, 2019, 846-856
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Inequalities for the Taylor coefficients of spiralike
functions involving q-differential operator

K. Amarender Reddy1, K. R. Karthikeyan1, G. Murugusundaramoorthy2,∗

1 Department of Mathematics and Statistics, College of Engineering,
National University of Science and Technology, Muscat, Sultanate of Oman
2 Department of Mathematics, School of Advanced Sciences,
Vellore Institute of Technology, Deemed to be University, Vellore, Tamilnadu, India

Abstract. Making use of q-analogue of the well-known differential operator, we provide a formal
extension of a bi-univalent spiralike and bi-univalent strongly spiralike functions. We obtain the in-
equalities for the Maclaurin-Taylor coefficients of the functions belonging to the defined subclasses.
Further we have provided some applications of our main results.

2010 Mathematics Subject Classifications: 30C45

Key Words and Phrases: Starlike Functions, Spiralike Functions, Bi-Univalent Functions, Co-
efficient Inequalities, q-Calculus Operator

1. Introduction of Quantum Calculus in dual with Univalent Functions

Quantum calculus popularly called as q-calculus is based on the idea of finite difference
rescaling. The difference of quantum differentials from the ordinary ones is that notion of
limit is removed in q-calculus, that is q-derivative is merely a ratio which is given by

Dqf(z) =
f(qz)− f(z)

(q − 1)z
.

Notice that as limit q → 1−, Dqf(z) = f ′(z). q-calculus has numerous applications
in variety of disciplines such as theory of special functions, operator theory, quantum-
mechanics, relativity etc. Notations and symbols play an very important role in the study
of q-calculus. Throughout this paper, we let

[n]q =

n∑
k=1

qk−1, [0]q = 0, (q ∈ C)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v12i3.3429

Email addresses: amarenderkommula@gmail.com (K. A. Reddy),
kr karthikeyan1979@yahoo.com (K. R. Karthikeyan),
gmsmoorthy@yahoo.com (G. Murugusundaramoorthy)

http://www.ejpam.com 846 c© 2019 EJPAM All rights reserved.



K. A. Reddy, K. R. Karthikeyan, G. Murugusundaramoorthy / Eur. J. Pure Appl. Math, 12 (3) (2019), 846-856 847

and the q-shifted factorial by

(a; q)n =

{
1, n = 0

(1− a)(1− aq) . . .
(
1− aqn−1

)
, n = 1, 2, . . . .

The q-hypergeometric series was developed by Heine as a generalization of the hyper-
geometric series

2F1[a, b; c|q, z] =

∞∑
n=0

(a; q)n(b; q)n
(q; q)n(c; q)n

zn. (1)

Generalizing the Heine’s series, we define rφs the basic hypergeometric series by

rφs (a1, a2, . . . , ar; b1, b2, . . . , bs; q, z)

=
∞∑
n=0

(a1; q)n(a2; q)n . . . (ar; q)n
(q; q)n(b1; q)n . . . (bs; q)n

[
(−1)nq(

n
2)
]1+s−r

zn
(2)

with
(
n
2

)
= n(n−1)

2 , where q 6= 0 when r > s + 1. In (1) and (2), it is assumed that the
parameters b1, b2, . . . , bs are such that the denominator factors in the terms of the series
are never zero.

Let A denote the class of all functions having a Taylor series expansion of the form

f(z) = z +
∞∑
n=2

knz
n, (z ∈ U). (3)

For complex parameters a1, . . . , ar and b1, . . . , bs (βj ∈ C\Z−0 ; Z−0 = 0,−1, −2, . . . ; j =
1, . . . , s), we define the generalized q-hypergeometric function rΨs(a1, . . . , aq; b1, . . . , bs; q, z)
by

rΨs(a1, a2, . . . , aq; b1, b2, . . . , bs; q, z) =

∞∑
n=0

(a1; q)n(a2; q)n . . . (ar; q)n
(q; q)n(b1; q)n . . . (bs; q)n

zn (4)

(r = s+ 1; r, s ∈ N0 = N ∪ {0}; z ∈ U),

where N denotes the set of positive integers. By using the ratio test, we should note
that, if |q| < 1, the series (4) converges absolutely for |z| < 1 and r = s + 1. For more
mathematical background of these functions, one may refer to [2].

Corresponding to a function Gr, s(ai, bj ; q, z) (ai, bj are real; i = 1, 2, . . . , r; j = 1, 2, . . . , s)
defined by

Gr, s(ai, bj ; q, z) := z qΨs(a1, a2, . . . , ar; b1, b2, . . . , bs; q, z). (5)

We now define the following operator Jmλ (a1, b1; q, z)f : U −→ U by

J 0
λ (a1, b1; q, z)f(z) = f(z) ∗ Gr, s(ai, bj ; q, z)

J 1
λ (a1, b1; q, z)f(z) = (1−λ)(f(z)∗Gr, s(ai, bj ; q, z))+λ zDq(f(z)∗Gr, s(ai, bj ; q, z)) (6)



K. A. Reddy, K. R. Karthikeyan, G. Murugusundaramoorthy / Eur. J. Pure Appl. Math, 12 (3) (2019), 846-856 848

Jmλ (a1, b1; q, z)f(z) = J 1
λ (Jm−1λ (a1, b1; q, z)f(z)). (7)

If f ∈ A, then from (6) and (7) we may easily deduce that

Jmλ (a1, b1; q, z)f = z +
∞∑
n=2

[1− λ+ [n]qλ]m Υnknz
n, (8)

(m ∈ N0 = N ∪ {0} and λ ≥ 0) ,

where

Υn =
(a1; q)n−1(a2; q)n−1 . . . (ar; q)n−1
(q; q)n−1(b1; q)n−1 . . . (bs; q)n−1

, (|q| < 1) .

Remark 1. We note that the linear operator (8) is q-analogue of the operator defined by
Selvaraj and Karthikeyan [5]. Here we list some special cases of the operator Jmλ (a1, b1; q, z)f .

1. For a choice of the parameter m = 0, the operator J 0
λ (α1, β1)f(z) reduces to the

q-analogue of Dziok- Srivastava operator [1].

2. For ai = qαi , bj = qβj , αi, βj ∈ C, βj 6= 0, , (i = 1, . . . , r, j = 1, . . . , s) and q → 1−,
we get the operator defined by Selvaraj and Karthikeyan [5].

3. For r = 2, s = 1; a1 = b1, a2 = q, and λ = 1, we get the q- analogue of the well
known Sălăgean operator (see [4]).

Also many (well known and new) integral and differential operators can be obtained by
specializing the parameters.

We let S∗, C and K to denote the well known classes of starlike,convex and close to
convex function respectively. We refer Goodman[3] which provides the study of vari-
ous subclasses of univalent functions in slow motion. Another very important class in the
study of various subclasses of univalent functions is the class of functions with positive real
part. We denote by P (ρ) the class of functions with p(0) = 1 which satisfies R{p(z)} > ρ.
It is well known that p(z) = 1+c1z+c2z

2+· · · ∈ P (ρ) implies | pn |≤ 2(1−ρ) for all n ≥ 1.

The coefficients for the inverse of a function f(z) of the form (3) is given by

g(w) = f−1(z) = w − k2w2 + (2k22 − k3)w3 − (5k32 − 5k2k3 + k4)w
4 + · · · , (9)

for details on the coefficients of the inverse of a function, we refer to chapter 5 in [3].

The area of a closed disc of radius r of a function f ∈ S, provided us with an inequality
which in turn was used to prove several central theorems in the field of univalent func-
tions. In the class S, the upper bound on a2 was very useful in establishing the growth,
distortion and radius problems of univalent functions. So finding the initial coefficients
of various subclasses of analytic functions has always been a very attractive topic in the
study of univalent function theory. The main purpose of this paper is to obtain the ini-
tial coefficients of the two classes of spiralike functions namely α − SP∗(β, a, b; q, z) and
α− SP(ρ, a, b; q, z).



K. A. Reddy, K. R. Karthikeyan, G. Murugusundaramoorthy / Eur. J. Pure Appl. Math, 12 (3) (2019), 846-856 849

Now we begin with the following definitions.

Definition 1. The function f(z), given by (3), is said to be a member of α−SP∗(β, a, b; q, z),
if each of the following conditions are satisfied.

∣∣∣∣arg

(
eiα

z [Dq(J
m
λ (a1, b1; q, z)f)]

(Jmλ (a1, b1; q, z)f)

)∣∣∣∣ < β
π

2
, (z ∈ U ; | α |≤ π/2, 0 ≤ β < 1)

and

∣∣∣∣arg

(
eiα

w [Dq(J
m
λ (a1, b1; q, w)f)]

(Jmλ (a1, b1; q, w)f)

)∣∣∣∣ < β
π

2
, (w ∈ U ; | α |≤ π/2, 0 ≤ β < 1) .

Definition 2. The function f(z) given by (3), is said to be a member of α−SP(ρ, a, b; q, z),
if each of the following conditions are satisfied.

R
(
eiα

z [Dq(J
m
λ (a1, b1; q, z)f)]

(Jmλ (a1, b1; q, z)f)

)
> ρcos(α), (z ∈ U ; | α |≤ π/2, 0 ≤ ρ < 1)

and

R
(
eiα

w [Dq(J
m
λ (a1, b1; q, w)f)]

(Jmλ (a1, b1; q, w)f)

)
> ρcos(α), (w ∈ U ; | α |≤ π/2, 0 ≤ ρ < 1) .

The classes of α−SP∗(β, a, b; q, z) and α−SP(ρ, a, b; q, z) were motivated by [6]. If we
let m = 0, r = 2, s = 1; a1 = b1, a2 = q and by taking limit q → 1− in α−SP∗(β, a, b; q, z)
and α − SP(ρ, a, b; q, z), we get the classes introduced by M. M. Soren and A. K. Misra
[6].

2. Main Results

Theorem 1. Let f(z) given by (3), be in the class α−SP(ρ, a, b; q, z), (| α |≤ π
2 , 0 ≤ ρ <

1). Then

| k2 |≤
√

2 cosα(1− ρ)√
q [1− λ+ (1 + q)λ]2m γ22 + [1− λ+ [3]qλ]m ([3]q − 1)γ3

,

| k3 |≤ (1− ρ) cosα

(
2

q [1− λ+ (1 + q)λ]2m γ22 + [1− λ+ [3]qλ]m ([3]q − 1)γ3

)



K. A. Reddy, K. R. Karthikeyan, G. Murugusundaramoorthy / Eur. J. Pure Appl. Math, 12 (3) (2019), 846-856 850

and

| k4 |≤
2(1− ρ) cosα

[1− λ+ [4]qλ]m ([4]q − 1)γ4
+

10
√

2[(1− ρ) cosα]
3
2

[[1− λ+ (1 + q)λ]2m (q)γ22 + [1− λ+ [3]qλ]m ([3]q − 1)γ3]
3
2

+

2
√

2[(1− ρ) cosα]
3
2

[[1− λ+ (1 + q)λ]2m (q)γ22 + [1− λ+ [3]qλ]m ([3]q − 1)γ3]
3
2[

2 [1− λ+ (1 + q)λ]m [1− λ+ [3]qλ]m γ2γ3(1 + q + [3]q)

[1− λ+ [4]qλ]m ([4]q − 1)γ4
+ 5

]
.

Proof.
Let f ∈ α− SP(ρ, a, b; q, z). Then the inequalities in Definition 2 can be equivalently

rewritten as, (
eiα

z [Dq(J
m
λ (a1, b1; q, z)f)]

(Jmλ (a1, b1; q, z)f)

)
= P1(z) cosα+ i sinα (10)

and (
eiα

w [Dq(J
m
λ (a1, b1; q, w)f)]

(Jmλ (a1, b1; q, w)f)

)
= Q1(w) cosα+ i sinα (11)

respectively, where R(P1(z)) > ρ and R(Q1(z)) > ρ,

P1(z) = 1 + c1z + c2z
2 + · · · (z ∈ U)

and
Q1(w) = 1 + l1w + l2w

2 + · · · (w ∈ U).

By comparing the coefficients in (10), we have

eiα [1− λ+ (1 + q)λ]m qγ2k2 = c1 cosα (12)

eiα
[
[1− λ+ (1 + q)λ]2m (q)γ22k

2
2 + [1− λ+ [3]qλ]m ([3]q − 1) γ3k3

]
= c2 cosα (13)

eiα
[

[1− λ+ [4]qλ]m ([4]q − 1)γ4k4 − (1 + q + [3]q) [1− λ+ (1 + q)λ]m [1− λ+ [3]qλ]m γ2γ3k2k3 +

(1 + q) [1− λ+ (1 + q)λ]2m γ22k
2
2

]
= c3 cosα.

(14)
Similarly by equating the coefficients in (11), we get

−eiα [1− λ+ (1 + q)λ]m (q) γ2k2 = l1 cosα (15)

eiα
[
[1− λ+ (1 + q)λ]2m (q)γ22k

2
2 + [1− λ+ [3]qλ]m ([3]q − 1) γ3(2k

2
2 − k3)

]
= l2 cosα.

(16)



K. A. Reddy, K. R. Karthikeyan, G. Murugusundaramoorthy / Eur. J. Pure Appl. Math, 12 (3) (2019), 846-856 851

eiα
[

[1− λ+ [4]qλ]m ([4]q − 1)γ4(5k2k3 − 5k32 − k4)−

(1 + q + [3]q) [1− λ+ (1 + q)λ]m [1− λ+ [3]qλ]m γ2γ3k2k3+

(1 + q) [1− λ+ (1 + q)λ]2m γ22k
2
2

]
= l3 cosα.

(17)
From (12) and (15), we get l1 = −c1. Before computing | a2 | and | a3 |, we will obtain

a refined estimate of | c1 |. For this purpose, we first add (13) and (16)

2k22 = (c2 + l2)
cosα

eiα[[1− λ+ (1 + q)λ]2m (q)γ22 + [1− λ+ [3]qλ]m (1− [3]q)γ3]
.

Using (12) in the above equation in conjunction with the known result that
|cn| ≤ 2(1− ρ) and |ln| ≤ 2(1− ρ), we have

| c21 | =

∣∣∣∣∣(c2 + l2)
eiα [1− λ+ (1 + q)λ]2m (q)2γ22

2 cosα[[1− λ+ (1 + q)λ]2m (q)γ22 + [1− λ+ [3]qλ]m (1− [3]q)γ3]

∣∣∣∣∣
≤ |c2|+ |l2|

2

1 [1− λ+ (1 + q)λ]2m (q)2γ22
cosα[[1− λ+ (1 + q)λ]2m (q)γ22 + [1− λ+ [3]qλ]m ([3]q − 1)γ3]

≤ 2(1− ρ) [1− λ+ (1 + q)λ]2m (q)2γ22
cosα[[1− λ+ (1 + q)λ]2m (q)γ22 + [1− λ+ [3]qλ]m ([3]q − 1)γ3]

| c1 |≤
√

2(1− ρ) [1− λ+ (1 + q)λ]m (q)γ2√
cosα[[1− λ+ (1 + q)λ]2m (q)γ22 + [1− λ+ [3]qλ]m ([3]q − 1)γ3]

(18)

and

| k2 | ≤
| c1 | cosα

[1− λ+ (1 + q)λ]m ([2]q − 1)γ2

=

√
2(1− ρ) cosα√

[[1− λ+ (1 + q)λ]2m (q)γ22 + [1− λ+ [3]qλ]m ([3]q − 1)γ3]

which proves the assertion.
We next find the upper bound on a3. For this we subtract (13) and (16) and using c1 = −l1,
we get

k3 =
(c2 − l2) cosα

2eiα [1− λ+ [3]qλ]m ([3]q − 1)γ3
+ k22. (19)

On simplification, we get



K. A. Reddy, K. R. Karthikeyan, G. Murugusundaramoorthy / Eur. J. Pure Appl. Math, 12 (3) (2019), 846-856 852

k3 =
(c2 − l2) cosα

2eiα [1− λ+ [3]qλ]m ([3]q − 1)γ3
+

c21 cos2 α

e2iα [1− λ+ (1 + q)λ]2m (q)2

=
(c2 − l2) cosα

2eiα [1− λ+ [3]qλ]m ([3]q − 1)γ3
+

(c2 + l2) cosα

2eiα[[1− λ+ (1 + q)λ]2m (q)γ22 + [1− λ+ [3]qλ]m ([3]q − 1)γ3]

=
cosα

2eiα

[
c2

[1− λ+ [3]qλ]m ([3]q − 1)γ3
+

c2

[1− λ+ (1 + q)λ]2m (q)γ2 + [1− λ+ [3]qλ]m ([3]q − 1)γ3

]
+

cosα

2eiα

[
l2

[1− λ+ (1 + q)λ]2m (q)γ2 + [1− λ+ [3]qλ]m ([3]q − 1)γ3
−

l2
[1− λ+ [3]qλ]m (1− [3]q)γ3

]
.

On taking the modulus, we have

| k3 |≤ (1− ρ) cosα

[
2

[1− λ+ (1 + q)λ]2m (q)γ22 + [1− λ+ [3]qλ]m ([3]q − 1)γ3

]
. (20)

Hence the upper bound of k3. Now we shall we move onto find the estimate on | k4 |.
By subtracting the equations (14) and (17), we get

2k4 =
e−iα cosα(c3 − l3)

[1− λ+ [4]qλ]m ([4]q − 1)γ4
+

5c1 cosαk3
eiα [1− λ+ (1 + q)λ]m (q)γ2

−

c31 cos3 α

e3iα [1− λ+ (1 + q)λ]3m (q)3γ32

[
2 [1− λ+ (1 + q)λ]m [1− λ+ [3]qλ]m γ2γ3((1 + q) + [3]q)

[1− λ+ [4]qλ]m ([4]q − 1)γ4
+ 5

]
.

| k4 |≤
2(1− ρ) cosα

[1− λ+ [4]qλ]m ([4]q − 1)γ4
+

10
√

2[(1− ρ) cosα]
3
2

[[1− λ+ (1 + q)λ]2m (q)γ22 + [1− λ+ [3]qλ]m ([3]q − 1)γ3]
3
2

+

2
√

2[(1− ρ) cosα]
3
2

[[1− λ+ (1 + q)λ]2m (q)γ22 + [1− λ+ [3]qλ]m ([3]q − 1)γ3]
3
2[

2 [1− λ+ (1 + q)λ]m [1− λ+ [3]qλ]m γ2γ3(1 + q + [3]q)

[1− λ+ [4]qλ]m ([4]q − 1)γ4
+ 5

]
.

This completes the proof of the Theorem 1.



K. A. Reddy, K. R. Karthikeyan, G. Murugusundaramoorthy / Eur. J. Pure Appl. Math, 12 (3) (2019), 846-856 853

Theorem 2. Let f(z), given by (3), be in the class α − SP∗(β, a, b; q, z) (| α |≤ π
2 , 0 ≤

β < 1). Then

| k2 |≤
β
√

2 cos(αβ )

q [1− λ+ (1 + q)λ]m | γ2 |
√

[1− λ+ [3]qλ]m ([3]q − 1)γ3 cos(αβ )δ

and

| k3 |≤
2β cos(αβ )

[1− λ+ [3]qλ]m ([3]q − 1)γ3

[
β

q2 [1− λ+ (1 + q)λ]2m γ22δ
− 1

]

where

δ =
β

q2 [1− λ+ (1 + q)λ]2m γ22
−

[
1 + β−1

2 (q + 2)
]

q [1− λ+ [3]qλ]m ([3]q − 1)γ3
.

Proof. From Definition 1, we have

Dq(J
m
λ (a1, b1; q, z)f) =

Jmλ (a1, b1; q, z)f

z
e−iαh(z) (21)

where h(z) is analytic in U and satisfies

h(0) = eiα and | arg h(z) |< βπ/2 (z ∈ U).

It can be checked that the function q(z) defined by

h(z)
1
β = cos

(
α

β

)
q(z) + i sin

(
α

β

)
(z ∈ U)

is a member of the class P. Suppose that

q(z) = 1 + c1z + c2z
2 + . . . . (z ∈ U).

By comparing coefficients in (21), we have

k2 =
βc1e

−i(α
β
)
cos(αβ )

q [1− λ+ (1 + q)λ]m γ2
(22)

and

k3 =
βc2e

−i(α
β
)
cos(αβ ) + β

q c
2
1e
−2i(α

β
)
cos2(αβ )

[
1 + β−1

2 (q + 2)
]

[1− λ+ [3]qλ]m ([3]q − 1)γ3
. (23)

Similarly, we take

Dq(J
m
λ (a1, b1; q, w)f) =

Jmλ (a1, b1; q, w)f

w
e−iαh(w), (24)



K. A. Reddy, K. R. Karthikeyan, G. Murugusundaramoorthy / Eur. J. Pure Appl. Math, 12 (3) (2019), 846-856 854

where h(w) is analytic in U and satisfies

H(0) = eiα and | arg h(w) |< βπ/2 (w ∈ U).

It can be checked that the function p(w) defined by:

h(w)
1
β = cos

(
α

β

)
p(w) + i sin

(
α

β

)
(w ∈ U)

is a member of the class P. If

p(w) = 1 + l1w + l2w
2 + . . . (w ∈ U),

then again by comparing the coefficients in (24), we have the following

−k2 =
βl1e

−i(α
β
)
cos(αβ )

q [1− λ+ (1 + q)λ]m γ2
(25)

and

2k22 − k3 =
βl2e

−i(α
β
)
cos(αβ ) + β

q l
2
1e
−2i(α

β
)
cos2(αβ )

[
1 + β−1

2 (q + 2)
]

[1− λ+ [3]qλ]m ([3]q − 1)γ3
. (26)

It is obvious from (22) and (25) that l1 = −c1. From (23) and (26), we get

c21 =
(c2 + l2)

2 [1− λ+ [3]qλ]m ([3]q − 1)γ3e
−i(α

β
)
cos(αβ )δ

(27)

where

δ =
β

q2 [1− λ+ (1 + q)λ]2m γ22
−

[
1 + β−1

2 (q + 2)
]

q [1− λ+ [3]qλ]m ([3]q − 1)γ3
.

By applying the familiar inequalities | c2 |≤ 2 and | l2 |≤ 2, we get

| c1 |≤
√

2√
[1− λ+ [3]qλ]m ([3]q − 1)γ3 cos(αβ )δ

(28)

and

| k2 | =
β | c1 | cos(αβ )

q [1− λ+ (1 + q)λ]m | γ2 |
(29)

≤
β
√

2 cos(αβ )

q [1− λ+ (1 + q)λ]m | γ2 |
√

[1− λ+ [3]qλ]m ([3]q − 1)γ3 cos(αβ )δ
. (30)

We next find a upper bound on | a3 |. For this we subtract (26) from (23) and get

2k3 = 2k22 −
βl2e

−i(α
β
)
cos(αβ ) + β

q l
2
1e
−2i(α

β
)
cos2(αβ )

[
1 + β−1

2 (q + 2)
]

[1− λ+ [3]qλ]m ([3]q − 1)γ3
+



REFERENCES 855

βc2e
−i(α

β
)
cos(αβ ) + β

q c
2
1e
−2i(α

β
)
cos2(αβ )

[
1 + β−1

2 (q + 2)
]

[1− λ+ [3]qλ]m ([3]q − 1)γ3
.

Now putting that c21 = l21 and k2 values in above equation, we obtain

2k3 =
βe
−i(α

β
)
cos(αβ )

[1− λ+ [3]qλ]m ([3]q − 1)γ3

[
β(c2 + l2)

q2 [1− λ+ (1 + q)λ]2m γ22δ
− (c2 − l2)

]
. (31)

By applying the familiar inequalities | c2 |≤ 2 and | l2 |≤ 2 we get

| k3 |≤
2β cos(αβ )

[1− λ+ [3]qλ]m ([3]q − 1)γ3

[
β

q2 [1− λ+ (1 + q)λ]2m γ22δ
− 1

]
. (32)

The proof of Theorem 2 is thus completed.

3. Concluding Remarks

Remark 2. For the choice of the parameters, m = 0, r = 2, s = 1; a1 = b1, a2 = q, and
by taking limit q → 1− in Theorem 1 and Theorem 2, we get the results obtained in [6].

Remark 3. For appropriate choice of the parameter in Theorem 1, we get the following
inequalities for a class of functions bi-starlike of order ρ (0 ≤ ρ < 1).

|k2| ≤
√

2(1− ρ) and |k3| ≤ 2(1− ρ).

Remark 4. Similarly for the appropriate choice of the parameter in Theorem 1, we get the
following inequalities for a class of functions which are bi-convex of order ρ (0 ≤ ρ < 1).

|k2| ≤
√

(1− ρ) and |k3| ≤ (1− ρ).

References

[1] M. Darus. A new look at q-hypergeometric functions. TWMS J. Appl. Eng. Math.,
4(1):16–19, 2014.

[2] G. Gasper and M. Rahman. Basic hypergeometric series, volume 35 of Encyclopedia
of Mathematics and its Applications. Cambridge University Press, Cambridge, 1990.
With a foreword by Richard Askey.

[3] A. W. Goodman. Univalent functions. Vol. II. Mariner Publishing Co., Inc., Tampa,
FL, 1983.

[4] K. R. Karthikeyan, M. Ibrahim, and S. Srinivasan. Fractional class of analytic functions
defined using q-differential operator. Aust. J. Math. Anal. Appl., 15(1):Art. 9, 15, 2018.



REFERENCES 856

[5] C. Selvaraj and K. R. Karthikeyan. Differential sandwich theorems for certain sub-
classes of analytic functions. Math. Commun., 13(2):311–319, 2008.

[6] M. Mohan Soren and Akshaya Kumar Mishra. Coefficient bounds for bi-spirallike
analytic functions. Kyungpook Math. J., 58(4):697–709, 2018.