EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 10, No. 2, 2017, 348-362
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

New Subclasses of Analytic Function Associated with
q-Difference Operator

C. Ramachandran 1, T. Soupramanien2, B.A. Frasin3,∗

1 Department of Mathematics, University College of Engineering Villupuram, Anna Univer-
sity, Villupuram, Tamilnadu, India.
2 Department of Mathematics, IFET College of Engineering, Gangarampalayam, Villupu-
ram, Tamilnadu, India.
3 Department of Mathematics, Faculty of Science, Al al-Bayt University, Mafraq, Jordan.

Abstract. The aim of this paper is to establish the coefficient bounds for certain classes of ana-
lytic functions associated with q-difference operator. Certain applications of these results for the
functions defined through convolution are also obtained.

2010 Mathematics Subject Classifications: Primary 30C45; Secondary 30C50

Key Words and Phrases: Univalent function, Schwarz function, q-starlike function, q-convex
function, q-derivative operator, subordination, Fekete-Szego inequality.

1. Introduction

Recently, the area of q-analysis has attracted the serious attention of researchers. The
q-difference calculus or quantum calculus was initiated at the beginning of 19th century,
that was initiated by Jackson [6, 7]. He was the first to develop q-integral and q-derivative
in a systematic way. The fractional q-difference calculus had its origin in the works by
Al.Salam [2] and Agarwal [1]. This great interest is due to its application in various
branches of mathematics and physics, as for example, in the areas of ordinary fractional
calculus, optimal control problems, q-difference and q-integral equations and in q-transform
analysis. The generalization q-Taylor’s formula in fractional q-calculus was introduced by
Purohit and Raina [18]. Mohammed and Darus [12] studied approximation and geometric
properties of these q-operators in some subclasses of analytic functions in compact disk.
Purohit and Raina recently in [18, 16] have used the fractional q-calculus operators in
investigating certain classes of functions which are analytic in the open disk and Purohit
[17] also studied these q-operators are defined by using convolution of normalized analytic

∗Corresponding author.

Email addresses: crjsp2004@yahoo.com (C. Ramachandran), soupramani@gmail.com (T. Soupramanien),
bafrasin@yahoo.com (B.A. Frasin)

http://www.ejpam.com 348 c© 2017 EJPAM All rights reserved.



C. Ramachandran, T. Soupramanien, B.A. Frasin / Eur. J. Pure Appl. Math, 10 (2) (2017), 348-362 349

functions and q-hypergeometric functions. A comprehensive study on applications of q-
calculus in operator theory may be found in[4]. Ramachandran et. al. [19] have used the
fractional q-calculus operators in investigating certain bound for q-starlike and q-convex
functions with respect to symmetric points.

Let A denote the class of all analytic functions f(z) of the form

f(z) = z +

∞∑
n=2

anz
n, (1)

defined on the open unit disk U = {z : z ∈ C : |z| < 1}.
If the functions f(z) and g(z) are analytic in U, we say that the function f(z) is

subordinate to g(z), written as f ≺ g in U or f(z) ≺ g(z) (z ∈ U), if there exists a Schwarz
function w(z), in U with w(0) = 0 and |w(z)| < 1 (z ∈ U) such that f(z) = g(w(z)).
Furthermore, if the function g(z) is univalent in U, the above subordination is equivalence
holds (see [11] and [5])

f(z) ≺ g(z) ⇐⇒ f(0) = g(0), and f(U) ⊂ g(U).

For function f ∈ A given by (1) and 0 < q < 1, the q-derivative of a function f is defined
by (see [6, 7])

Dqf(z) =
f(qz)− f(z)

(q − 1)z
(z 6= 0), (2)

Dqf(0) = f ′(0) and D2
qf(z) = Dq(Dqf(z)). From (2), we deduce that

Dqf(z) = 1 +
∞∑
k=2

[k]q akz
k−1, (3)

where

[k]q =
1− qk

1− q
. (4)

As q → 1−, [k]q → k. For a function h(z) = zk, we observe that

Dq(h(z)) = Dq

(
zk
)

=
1− qk

1− q
zk−1 = [k]q z

k−1,

lim
q→1−

(Dq(h(z))) = lim
q→1−

(
[k]qz

k−1
)

= kzk−1 = h′(z),

where h′ is the ordinary derivative.
As a right inverse, Jackson [7] introduced the q-integral∫ z

0
f(t)dqt = z(1− q)

∞∑
k=0

qkf
(
zqk
)
,



C. Ramachandran, T. Soupramanien, B.A. Frasin / Eur. J. Pure Appl. Math, 10 (2) (2017), 348-362 350

provided that the series converges. For a function h(z) = zk, we observe that∫ z

0
h(t)dqt =

∫ z

0
tkdqt =

zk+1

[k + 1]q
(k 6= −1)

lim
q→1−

∫ z

0
h(t)dqt = lim

q→1−

zk+1

[k + 1]q
=

zk+1

k + 1
=

∫ z

0
h(t)dt,

where
∫ z
0 h(t)dt is the ordinary integral.

Making use of the q-derivative Dqf(z), the subclasses S∗q (α) and Cq(α) of the class A
for 0 ≤ α ≤ 1 are introduced by

S∗q (α) =

{
f ∈ A : Re

(
zDqf(z)

f(z)

)
≥ α, z ∈ U

}
(5)

Cq(α) =

{
f ∈ A : Re

(
Dq(zDqf(z))

Dqf(z)

)
≥ α, z ∈ U

}
. (6)

We note that
f ∈ Cq(α)⇔ zDqf ∈ S∗q (α), (7)

and

lim
q→1−

S∗q (α) =

{
f ∈ A : lim

q→1−
Re

(
zDqf(z)

f(z)

)
≥ α, z ∈ U

}
= S∗(α),

lim
q→1−

Cq(α) =

{
f ∈ A : lim

q→1−
Re

(
Dq(zDqf(z))

Dqf(z)

)
≥ α, z ∈ U

}
= C(α),

where S∗(α) and C(α) are respectively, the classes of starlike of order α and convex of
order α in U (see Robertson [23]). Kanas and Rǎducanu in [8] used the Ruscheweyh q-
differential operator to introduce and study some properties of (q, k) uniformly starlike
functions of order α. It is clear that Dqf(z)→ f ′(z) as q → 1−. This difference operator
helps us to generalize the class of starlike functions S∗ analytically.

By making use of the q-derivative of a function f ∈ A and the principle of subordina-
tion, we now introduce the following classes

Definition 1. Let φ(z) be a univalent starlike function with respect to 1, which maps the
open unit disk U onto a region in the right half-plane and is symmetric with respect to the
real axis, with

φ(0) = 1 and φ′(0) ≥ 0.

A function f ∈ A is said to be in the class Mq,α,β,λ(φ) if(
zDqf(z)

f(z)

)α [
(1− λ)

(
zDqf(z)

f(z)

)
+ λ

(
Dq(zDqf(z))

Dqf(z)

)]β
≺ φ(z) (8)

where 0 ≤ β ≤ 1; 0 ≤ α ≤ 1; 0 ≤ λ ≤ 1.



C. Ramachandran, T. Soupramanien, B.A. Frasin / Eur. J. Pure Appl. Math, 10 (2) (2017), 348-362 351

We note that

(i) lim
q→1−

Mq,α,β,λ(φ) =Mα,β,λ(φ) (C. Ramachandran et al. [20])

(ii) lim
q→1−

Mq,0,1,λ(φ) =M(λ, φ) (Ali et al. [3])

(iii) lim
q→1−

Mq,α,β,1(φ) =Mα,β(φ) (V. Ravichandran et al. [21])

(iv) lim
q→1−

Mq,0,1,0(φ) = lim
q→1−

Mq,1,0,λ(φ) = S∗(φ) and lim
q→1−

Mq,0,1,1(φ) = C(φ) (Ma and

Minda [9])

2. Preliminary Results

In order to prove the main results we need the following lemmas.

Lemma 1. [9] If p(z) = 1 + c1z+ c2z
2 + · · · is an analytic function with positive real part

in U, then

∣∣c2 − νc21∣∣ ≤

−4ν + 2 if ν ≤ 0

2 if 0 ≤ ν ≤ 1

4ν + 2 if ν ≥ 1.

When ν < 0 or ν > 1, the equality holds if and only if p(z) =
1 + z

1− z
or one of its rotations.

If 0 < ν < 1, then the equality holds true if and only if p(z) =
1 + z2

1− z2
or one of its

rotations. If ν = 0, the equality holds if and only if

p(z) =

(
1

2
+

1

2
η

)
1 + z

1− z
+

(
1

2
− 1

2
η

)
1− z
1 + z

, (0 ≤ η ≤ 1)

or one of its rotations. If ν = 1, the equality holds true if and only if p(z) is the reciprocal
of one of the functions such that the equality holds true in the case when ν = 0.

Although the above upper bound is sharp, in the case when 0 < ν < 1, it can be further
improved as follows:

|c2 − νc21|+ ν|c1|2 ≤ 2

(
0 < ν ≤ 1

2

)
and

|c2 − νc21|+ (1− ν)|c1|2 ≤ 2

(
1

2
< ν ≤ 1

)
.

We also need the following result in our investigation.

Lemma 2. [22] If p1(z) = 1 + c1z + c2z
2 + · · · is a function with positive real part in U,

then
|c2 − νc21| ≤ 2 max{1, |2ν − 1|}.

The result is sharp for the function p1(z) =
1 + z2

1− z2
and p1(z) =

1 + z

1− z
.



C. Ramachandran, T. Soupramanien, B.A. Frasin / Eur. J. Pure Appl. Math, 10 (2) (2017), 348-362 352

3. Main Results

Unless otherwise mentioned, we assume throughout this paper that the function 0 <
q < 1, φ ∈ P, [k]q is given by (4) and z ∈ U.

By making use of Lemma 1, we first prove the Fekete-Szegö type inequalities asserted
by Theorem 1 below.

Theorem 1. Let 0 ≤ µ ≤ 1, 0 ≤ α ≤ 1, 0 ≤ β ≤ 1 and 0 ≤ λ ≤ 1. Also let φ(z) =
1 +B1z +B2z

2 +B3z
3 + . . . , where the coefficients Bn are real with B1 > 0 and B2 ≥ 0.

If f(z) given by (1) belongs to the function class Mq,α,β,λ(φ), then

|a3 − µa22| ≤



1

2ξ

(
2B2 −

(
ρ2 + 2µξ − τ

ρ2

)
B2

1

)
if µ ≤ σ1,

B1

ξ
if σ1 ≤ µ ≤ σ2,

1

2ξ

(
−2B2 +

(
ρ2 + 2µξ − τ

ρ2

)
B2

1

)
if µ ≥ σ2,

(9)

where, for convenience,

σ1 :=
2ρ2(B2 −B1)− (ρ2 − τ)B2

1

2ξB2
1

, (10)

σ2 :=
2ρ2(B2 +B1)− (ρ2 − τ)B2

1

2ξB2
1

, (11)

σ3 :=
2ρ2B2 − (ρ2 − τ)B2

1

2ξB2
1

. (12)

ρ = ([2]q − 1)α+ ([2]q − 1 + λ)β, (13)

ξ = ([3]q − 1)α+ ([3]q − 1 + λ ([3]q ([2]q − 1) + 1))β, (14)

τ =
(
[2]2q − 1

)
α+

(
[2]2q − 1 + 2[2]2qλ+ λ2

)
β. (15)

If σ1 ≤ µ ≤ σ3, then

|a3 − µa22|+
ρ2

ξB1

(
1− B2

B1
+

(
ρ2 + 2µξ − τ

2ρ2

)
B1

)
|a2|2 ≤

B1

ξ
. (16)

Furthermore, if σ3 ≤ µ ≤ σ2, then

|a3 − µa22|+
ρ2

ξB1

(
1 +

B2

B1
−
(
ρ2 + 2µξ − τ

2ρ2

)
B1

)
|a2|2 ≤

B1

ξ
. (17)

Each of these results is sharp.



C. Ramachandran, T. Soupramanien, B.A. Frasin / Eur. J. Pure Appl. Math, 10 (2) (2017), 348-362 353

Proof. If f(z) ∈Mq,α,β,λ(φ), then there exists a Schwarz function w(z), analytic in U
with

w(0) = 0 and |w(z)| < 1 (z ∈ U),

such that(
zDqf(z)

f(z)

)α [
(1− λ)

(
zDqf(z)

f(z)

)
+ λ

(
Dq(zDqf(z))

Dqf(z)

)]β
= φ (w(z)) . (18)

Define the function p1(z) by

p1(z) =
1 + w(z)

1− w(z)
= 1 + c1z + c2z

2 + · · · . (19)

Since w(z) is a Schwarz function, we see that

<(p1(z)) > 0 (z ∈ U) and p1(0) = 1.

Now, defining the function p(z) by

p(z) :=

(
zDqf(z)

f(z)

)α [
(1− λ)

(
zDqf(z)

f(z)

)
+ λ

(
Dq(zDqf(z))

Dqf(z)

)]β

= 1 + b1z + b2z
2 + · · · , (20)

we find from (18) and (19) that

p(z) = φ

(
p1(z)− 1

p1(z) + 1

)
. (21)

Thus, by using (19) and (21), we obtain

b1 =
1

2
B1c1 and b2 =

1

2
B1

(
c2 −

1

2
c21

)
+

1

4
B2c

2
1.

An easy computation would show that(
zDqf(z)

f(z)

)α [
(1− λ)

(
zDqf(z)

f(z)

)
+ λ

(
Dq(zDqf(z))

Dqf(z)

)]β
=

1 + [([2]q − 1)α+ ([2]q − 1 + λ)β] a2z +

[([3]q − 1)α+ ([3]q − 1 + λ ([3]q ([2]q − 1) + 1))β] a3z
2 +[

α

2
([2]q − 1) ((α− 1) ([2]q − 1)− 2) +

β (β − 1)

2
([2]q − 1 + λ)2 +

α ([2]q − 1)β ([2]q − 1 + λ)− (([2]q − 1) + ([2]q ([2]q − 1) + 1)λ)β] a22z
2 + · · ·

which, in view of (20), yields

b1 = [([2]q − 1)α+ ([2]q − 1 + λ)β] a2



C. Ramachandran, T. Soupramanien, B.A. Frasin / Eur. J. Pure Appl. Math, 10 (2) (2017), 348-362 354

and

b2 = [([3]q − 1)α+ ([3]q − 1 + λ ([3]q ([2]q − 1) + 1))β] a3 +[
α

2
([2]q − 1) ((α− 1) ([2]q − 1)− 2) +

β (β − 1)

2
([2]q − 1 + λ)2 +

α ([2]q − 1)β ([2]q − 1 + λ)− (([2]q − 1) + ([2]q ([2]q − 1) + 1)λ)β] a22.

Equivalently, we have

a2 =
B1c1

2 [([2]q − 1)α+ ([2]q − 1 + λ)β]

and

a3 =
B1

2 [([3]q − 1)α+ ([3]q − 1 + λ ([3]q ([2]q − 1) + 1))β]

[
c2 −

1

2

(
1− B2

B1
+ Λ0B1

)
c21

]
where

Λ0 =
1

[([2]q − 1)α+ ([2]q − 1 + λ)β]2

[α
2

([2]q − 1) ((α− 1) ([2]q − 1)− 2) +

β (β − 1)

2
([2]q − 1 + λ)2 + α ([2]q − 1)β ([2]q − 1 + λ)−

(([2]q − 1) + ([2]q ([2]q − 1) + 1)λ)β] .

Therefore, we obtain

a3 − µa22 =
B1

2 [([3]q − 1)α+ ([3]q − 1 + λ ([3]q ([2]q − 1) + 1))β]

(
c2 − νc21

)
(22)

where

ν =
1

2

(
1− B2

B1
+

B1

2 [([2]q − 1)α+ ([2]q − 1 + λ)β]2

[
(([2]q − 1)α+ ([2]q − 1 + λ)β)2

+2µ (([3]q − 1)α+ ([3]q − 1 + λ ([3]q ([2]q − 1) + 1))β)−((
[2]2q − 1

)
α+

(
[2]2q − 1 + 2[2]2qλ+ λ2

)
β
)])

.

The assertion of Theorem 1 now follows by an application of Lemma 1.
To show that the bounds asserted by Theorem 1 are sharp, we define the following

functions:
Kφn(z) (n ∈ N\{1};N := {1, 2, 3, · · · }),

with
Kφn(0) = 0 = K′φn(0)− 1,

by (
zK′φn(z)

Kφn(z)

)α [
(1− λ)

(
zK′φn(z)

Kφn(z)

)
+ λ

(
K′φn(zK′φn(z))

K′φn(z)

)]β
= φ(zn−1),



C. Ramachandran, T. Soupramanien, B.A. Frasin / Eur. J. Pure Appl. Math, 10 (2) (2017), 348-362 355

and the functions Fη and Gη (0 ≤ η ≤ 1) with

Fη(0) = 0 = F ′η(0)− 1 and Gη(0) = 0 = G′η(0)− 1

by (
zF ′η(z)
Fη(z)

)α [
(1− λ)

(
zF ′η(z)
Fη(z)

)
+ λ

(F ′η(zF ′η(z))
F ′η(z)

)]β
= φ

(
z(z + η)

1 + ηz

)
and (

zG′η(z)
Gη(z)

)α [
(1− λ)

(
zG′η(z)
Gη(z)

)
+ λ

(G′η(zG′η(z))
G′η(z)

)]β
= φ

(
−z(z + η)

1 + ηz

)
respectively. Then, clearly, the functions Kφn , Fη, Gη ∈Mq,α,β,λ(φ). Also we write

Kφ := Kφ2 .

If µ < σ1 or µ > σ2, then the equality in Theorem 1 holds true if and only if f is Kφ
or one of its rotations. When σ1 ≤ µ ≤ σ2, then the equality holds true if and only if f is
Kφ3 or one of its rotations. If µ = σ1, then the equality holds true if and only if f is Fη or
one of its rotations. If µ = σ2 , then the equality holds true if and only if f is Gη or one
of its rotations.

By making use of Lemma 2, we immediately obtain the following Fekete-Szegö type
inequality.

Theorem 2. Let 0 ≤ µ ≤ 1, 0 ≤ α ≤ 1, 0 ≤ β ≤ 1 and 0 ≤ λ ≤ 1. Also let φ(z) =
1 +B1z +B2z

2 +B3z
3 + . . . , where the coefficients Bn are real with B1 > 0 and B2 ≥ 0.

If f(z) given by (1) belongs to the function class Mq,α,β,λ(φ), then

|a3 − µa22| ≤
B1

ξ
max

{
1,

∣∣∣∣−B2

B1
+

(
ρ2 + 2µξ − τ

2ρ2

)
B1

∣∣∣∣} (µ ∈ C),

where ρ, ξ and τ are defined by (13), (14) and (15). The result is sharp.

Remark 1. The coefficient bounds for |a2| and |a3| are special cases of those asserted by
Theorem 1.

Remark 2. In its special case when lim
q→1−

, Theorem 1 reduces to the result obtained in

[20]. Note that there were few typographical errors in the assertion of [20, Theorem 1] and
the following result is the corrected one:

Corollary 1. [20, Theorem 1] Let 0 ≤ µ ≤ 1, 0 ≤ α ≤ 1, 0 ≤ β ≤ 1 and 0 ≤ λ ≤ 1. Also
let φ(z) = 1 + B1z + B2z

2 + B3z
3 + . . . , where the coefficients Bn are real with B1 > 0

and B2 ≥ 0.



C. Ramachandran, T. Soupramanien, B.A. Frasin / Eur. J. Pure Appl. Math, 10 (2) (2017), 348-362 356

If f(z) given by (1) belongs to the function class Mα,β,λ(φ), then

|a3 − µa22| ≤



1

4ξ

(
2B2 −

(
ρ2 + 4µξ − τ

ρ2

)
B2

1

)
if µ ≤ σ1,

B1

2ξ
if σ1 ≤ µ ≤ σ2,

1

4ξ

(
−B2 +

(
ρ2 + 4µξ − τ

ρ2

)
B2

1

)
if µ ≥ σ2,

where, for convenience,

σ1 :=
2ρ2(B2 −B1)− (ρ2 − τ)B2

1

4ξB2
1

,

σ2 :=
2ρ2(B2 +B1)− (ρ2 − τ)B2

1

4ξB2
1

,

σ3 :=
2ρ2B2 − (ρ2 − τ)B2

1

4ξB2
1

.

ρ = α+ (1 + λ)β,

ξ = α+ (1 + 2λ)β,

τ = (3)α+
(
3 + 8λ+ λ2

)
β.

If σ1 ≤ µ ≤ σ3, then

|a3 − µa22|+
ρ2

2ξB1

(
1− B2

B1
+

(
ρ2 + 4µξ − τ

2ρ2

)
B1

)
|a2|2 ≤

B1

2ξ
.

Furthermore, if σ3 ≤ µ ≤ σ2, then

|a3 − µa22|+
ρ2

2ξB1

(
1 +

B2

B1
−
(
ρ2 + 4µξ − τ

2ρ2

)
B1

)
|a2|2 ≤

B1

2ξ
.

Each of these results is sharp.

Remark 3. When lim
q→1−

Mq,α,β,1(φ) =Mα,β(φ), Theorem 1 reduces to the result obtained

by V. Ravichandran et al. [21].

Remark 4. Special case if Mq,0,1,0(φ) = Mq,1,0,λ(φ) = S∗q (φ) Theorem 1 reduces to
starlike function with q-difference operator and Mq,0,1,1(φ) = Cq(φ), Theorem 1 reduces to
convex function with q-difference operator which was obtained by Seoudy et al. [24].



C. Ramachandran, T. Soupramanien, B.A. Frasin / Eur. J. Pure Appl. Math, 10 (2) (2017), 348-362 357

Remark 5. Special case if lim
q→1−

Mq,0,1,0(φ) = lim
q→1−

Mq,1,0,λ(φ) = S∗(φ) Theorem 1 re-

duces to starlike function and lim
q→1−

Mq,0,1,1(φ) = C(φ), Theorem 1 reduces to convex

function which was obtained by Ma and Minda [9].

4. Applications to analytic functions defined by using fractional calculus
operators and convolution

During the past three decades, the subject of fractional calculus (that is, calculus of
integrals and derivatives of any arbitrary real or complex order) has gained considerable
popularity and importance. There are two most recent works on this subject of widespread
investigations, namely rather comprehensive treatises on the theory, applications of frac-
tional differential equations by Podlubny [15] and Kilbas et al. [10].

For the applications of the results given in the preceding sections, we first introduce the
class Mδ

q,α,β,λ(φ), which is defined by means of the Hadamard product (or convolution)
and a certain operator of fractional calculus, known as the Owa-Srivastava operator (see,
for details, [25] and [27]; see also [13], [14], and [26]).

Definition 2. The fractional integral of order δ is defined, for a function f(z), by

D−δz f(z) =
1

Γ(δ)

z∫
0

f(ζ)

(z − ζ)1−δ
dζ (δ > 0), (23)

where the function f(z) be analytic in a simply connected domain of the complex z-plane
containing the origin and the multiplicity of (z−ζ)δ−1 is removed by requiring that log(z−ζ)
to be real when z − ζ > 0.

Definition 3. The fractional integral of order δ is defined, for a function f(z), by

Dδzf(z) =
1

Γ(1− δ)

z∫
0

f(ζ)

(z − ζ)δ
dζ (0 ≤ δ < 1), (24)

where f(z) is constrained, and the multiplicity of (z− ζ)−δ is removed, as in Definition 2.

Definition 4. Under the hypotheses of Definition 3, the fractional derivative of order n+δ
is defined, for a function f(z), by

Dn+δz f(z) =
dn

dzn

(
Dδzf(z)

)
(0 ≤ δ < 1; n ∈ N0 = N ∪ {0}). (25)

Using Definitions 2, 3 and 4 of fractional derivatives and fractional integrals, Owa and
Srivatsava [14] introduced what is popularly referred to in the current literature as the
Owa-Srivastava operator Ωδ : A → A defined by

(Ωδf)(z) := Γ(2− δ)zδDδzf(z), (δ 6= 2, 3, 4 · · · ). (26)



C. Ramachandran, T. Soupramanien, B.A. Frasin / Eur. J. Pure Appl. Math, 10 (2) (2017), 348-362 358

In terms of the Owa-Srivastava operator Ωδ defined by (26), we now introduce the
function class Mδ

q,α,β,λ(φ) in the following way:

Mδ
q,α,β,λ(φ) := {f : f ∈ A and Ωδf ∈Mq,α,β,λ(φ)}. (27)

It is easily seen that the function class Mδ
q,α,β,λ(φ) is a special case of the function class

Mg
q,α,β,λ(φ)when

g(z) = z +
∞∑
n=2

Γ (n+ 1) Γ (2− δ)
Γ (n+ 1− δ)

zn. (28)

Suppose now that

g(z) = z +

∞∑
n=2

gnz
n (gn > 0).

Then, since

f(z) = z +
∞∑
n=2

anz
n ∈Mg

q,α,β,λ(φ)⇐⇒ (f ∗ g)(z) = z +
∞∑
n=2

gnanz
n ∈Mq,α,β,λ(φ) (29)

we can obtain the coefficient estimates for functions in the class Mg
q,α,β,λ(φ) from the

corresponding estimates for functions in the classMq,α,β,λ(φ). By applying Theorem 1 to
the following Hadamard product (or convolution):

(f ∗ g)(z) = z + g2a2z
2 + g3a3z

3 + · · · ,

we get Theorem 3 below after an obvious change of the parameter µ.

Theorem 3. Let 0 ≤ µ ≤ 1, 0 ≤ α ≤ 1, 0 ≤ β ≤ 1 and 0 ≤ λ ≤ 1. Also let φ(z) =
1 +B1z +B2z

2 +B3z
3 + . . . , where the coefficients Bn are real with B1 > 0, B2 ≥ 0 and

Bn > 0 (n ∈ N\{1, 2}).
If f(z) given by (1) belongs to the function class Mg

q,α,β,λ(φ), then

|a3 − µa22| ≤



1

2ξg3

(
2B2 −

B2
1

ρ2
γ2

)
if µ ≤ σ4,

B1

ξg3
if σ4 ≤ µ ≤ σ5,

1

2ξg3

(
−2B2 +

B2
1

ρ2
γ2

)
if µ ≥ σ5,

where, for convenience,

σ4 :=
g3
g22

(
2ρ2(B2 −B1)− (ρ2 − τ)B2

1

2ξB2
1

)
,



C. Ramachandran, T. Soupramanien, B.A. Frasin / Eur. J. Pure Appl. Math, 10 (2) (2017), 348-362 359

σ5 :=
g3
g22

(
2ρ2(B2 +B1)− (ρ2 − τ)B2

1

2ξB2
1

)
,

and

γ2 :=

(
ρ2 +

2µξg3
g22

− τ
)

(30)

and ρ, ξ and τ are defined as in (13), (14) and (15), respectively. These results are sharp.

Since, by (1) and the definition 4,

(Ωδf)(z) = z +
∞∑
n=2

Γ (n+ 1) Γ (2− δ)
Γ (n+ 1− δ)

anz
n, (31)

we readily obtain

g2 :=
Γ(3)Γ(2− δ)

Γ(3− δ)
=

2

2− δ
(32)

and

g3 :=
Γ(4)Γ(2− δ)

Γ(4− δ)
=

6

(2− δ)(3− δ)
, (33)

For g2 and g3 given by (32) and (33), respectively, Theorem 3 reduces to the following
interesting result.

Theorem 4. Let 0 ≤ µ ≤ 1, 0 ≤ α ≤ 1, 0 ≤ β ≤ 1 and 0 ≤ λ ≤ 1. Also let φ(z) =
1 +B1z +B2z

2 +B3z
3 + . . . , where the coefficients Bn are real with B1 > 0, B2 ≥ 0.

If f(z) given by (1) belongs to the function class Mg
q,α,β,λ(φ), then

|a3 − µa22| ≤



(2− δ) (3− δ)
12ξ

(
2B2 −

B2
1

ρ2
γ3

)
if µ ≤ σ4,

(2− δ) (3− δ)
6ξ

B1 if σ4 ≤ µ ≤ σ5,

(2− δ) (3− δ)
12ξ

(
−2B2 +

B2
1

ρ2
γ3

)
if µ ≥ σ5,

where, for convenience,

σ4 :=
2 (3− δ)
3 (2− δ)

(
2ρ2(B2 −B1)− (ρ2 − τ)B2

1

2ξB2
1

)
,

σ5 :=
2 (3− δ)
3 (2− δ)

(
2ρ2(B2 +B1)− (ρ2 − τ)B2

1

2ξB2
1

)
,

and

γ3 :=

(
ρ2 + 2µξ

2 (3− δ)
3 (2− δ)

− τ
)

(34)

and ρ, ξ and τ are defined as in (13), (14) and (15), respectively.



REFERENCES 360

Remark 6. In its special case when lim
q→1−

Mq,α,β,λ(φ) = Mα,β,λ(φ) Theorem 4 coincide

with the result obtained earlier by C. Ramachandran et al. [20].

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