New Subclasses of Analytic Function Associated with q-Difference Operator

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

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 developq-integral andq-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 andq-integral equations and inq-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 theseq-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)

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 Adenote the class of all analytic functions f(z) of the form

f(z) =z+

∞

X

n=2

anzn, (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 tog(z), written asf ≺ginUorf(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 functiong(z) is univalent inU, the above subordination is equivalence holds (see [11] and [5])

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

For functionf ∈ Agiven by (1) and 0< q <1, theq-derivative of a functionf is defined by (see [6, 7])

Dqf(z) =

f(qz)−f(z)

(q−1)z (z6= 0), (2)

Dqf(0) =f0(0) andD2qf(z) =Dq(Dqf(z)). From (2), we deduce that

Dqf(z) = 1 + ∞

X

k=2

[k]q akzk−1, (3)

where

[k]q=

1−qk

1−q . (4)

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

Dq(h(z)) = Dq

zk

= 1−q

k

1−q z

k−1

= [k]q zk−1,

lim

q→1−(Dq(h(z))) = _qlim_→1−

[k]qzk−1

=kzk−1 =h0(z),

whereh0 is the ordinary derivative.

As a right inverse, Jackson [7] introduced the q-integral

Z z

0

f(t)dqt=z(1−q) ∞

X

k=0

qkf

zqk

provided that the series converges. For a function h(z) =zk_{, we observe that} Z z

0

h(t)dqt = Z z

0

tkdqt=

zk+1

[k+ 1]q

(k6=−1)

lim

q→1−

Z z

0

h(t)dqt = lim q→1−

zk+1

[k+ 1]q

= z

k+1

k+ 1 =

Z z

0

h(t)dt,

whereR₀zh(t)dt is the ordinary integral.

Making use of the q-derivative Dqf(z), the subclassesSq∗(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)

C_q(α) =

f ∈ A:Re

Dq(zDqf(z))

Dqf(z)

≥α, z∈_U

. (6)

We note that

f ∈ C_q(α)⇔zDqf ∈ Sq∗(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 Rˇaducanu in [8] used the Ruscheweyh q -differential operator to introduce and study some properties of (q, k) uniformly starlike functions of orderα. It is clear thatDqf(z)→f0(z) asq→ 1−. This difference operator

helps us to generalize the class of starlike functions S∗ analytically.

By making use of theq-derivative of a function f ∈ Aand 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 diskUonto a region in the right half-plane and is symmetric with respect to the

real axis, with

φ(0) = 1 and φ0(0)≥0.

A functionf ∈ Ais said to be in the class M_q,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)

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] Ifp(z) = 1 +c1z+c2z2+· · · 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.

Whenν <0 orν >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 +z

2

1−z2 or one of its

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

p(z) =

1 2 +

1 2η

1 +z

1−z +

1 2 −

1 2η

1−z

1 +z, (0≤η≤1)

or one of its rotations. Ifν = 1, the equality holds true if and only ifp(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+c2z2+· · · 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

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) andz∈U.

By making use of Lemma 1, we first prove the Fekete-Szeg¨o 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+B2z2+B3z3+. . ., where the coefficients Bn are real withB1>0 andB2≥0.

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

|a3−µa22| ≤



        

        

1 2ξ

2B2−

ρ2+ 2µξ−τ ρ2

B₁2

if µ≤σ1,

B1

ξ if σ1 ≤µ≤σ2,

1 2ξ

−2B2+

ρ2_{+ 2µξ−τ}

ρ2

B₁2

if µ≥σ2,

(9)

where, for convenience,

σ1 :=

2ρ2(B2−B1)−(ρ2−τ)B12

2ξB2 1

, (10)

σ2 :=

2ρ2(B2+B1)−(ρ2−τ)B12

2ξB2 1

, (11)

σ3 :=

2ρ2B2−(ρ2−τ)B12

2ξB2₁ . (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]2_q−1

α+ [2]2_q−1 + 2[2]2_qλ+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, ifσ3≤µ≤σ2, then

|a3−µa22|+

ρ2 ξB1

1 +B2

B1 −

ρ2+ 2µξ−τ

2ρ2

B1

|a2|2 ≤

B1

ξ . (17)

Proof. Iff(z)∈ Mq,α,β,λ(φ), then there exists a Schwarz functionw(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 functionp1(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+b2z2+· · ·, (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

2B1c1 and b2 = 1 2B1

c2−

1 2c

2 1

+1 4B2c

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]a3z2+

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]a22z2+· · ·

which, in view of (20), yields

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 hα

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 h

(([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]2_q−1α+ [2]2_q−1 + 2[2]2_qλ+λ2β.

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:

K_φn(z) (n∈_N\{1};N:={1,2,3,· · · }), with

K_φn(0) = 0 =K0_φ

n(0)−1,

by

zK0_φ

n(z)

K_φn(z)

!α"

(1−λ) zK

0 φn(z)

K_φn(z)

!

+λ K

0 φn(zK

0 φn(z))

K0 φn(z)

!#β

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

F_η(0) = 0 =F_η0(0)−1 and G_η(0) = 0 =G_η0(0)−1 by

_zF0 η(z)

Fη(z) α

(1−λ)

_zF0 η(z)

Fη(z)

+λ

_F0

η(zFη0(z))

F0 η(z)

β

=φ

z(z+η) 1 +ηz

and

_zG0 η(z)

G_η(z)

α

(1−λ)

_zG0 η(z)

G_η(z)

+λ

_G0

η(zG0η(z))

G0 η(z)

β

=φ

−z(z+η)

1 +ηz

respectively. Then, clearly, the functions K_φn,F_η,G_η ∈ M_q,α,β,λ(φ). Also we write

K_φ:=K_φ2.

If µ < σ1 orµ > σ2, then the equality in Theorem 1 holds true if and only if f is Kφ

or one of its rotations. Whenσ1 ≤µ≤σ2, then the equality holds true if and only if f is K_φ3 or one of its rotations. Ifµ=σ1, then the equality holds true if and only iff isFη or

one of its rotations. If µ=σ2 , then the equality holds true if and only iff is Gη or one

of its rotations.

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

Theorem 2. Let 0 ≤ µ ≤ 1, 0 ≤ α ≤ 1, 0 ≤ β ≤ 1 and 0 ≤ λ ≤ 1. Also let φ(z) = 1 +B1z+B2z2+B3z3+. . ., where the coefficients Bn are real withB1>0 andB2≥0.

If f(z) given by (1)belongs to the function class M_q,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+B2z2+B3z3+. . ., where the coefficients Bn are real with B1 > 0

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

B₁2

if µ≤σ1,

B1

2ξ if σ1 ≤µ≤σ2,

1 4ξ

−B2+

ρ2_{+ 4µξ−τ}

ρ2

B₁2

if µ≥σ2,

where, for convenience,

σ1 :=

2ρ2_(B

2−B1)−(ρ2−τ)B12

4ξB2₁ ,

σ2 :=

2ρ2(B2+B1)−(ρ2−τ)B12

4ξB2 1

,

σ3 :=

2ρ2B2−(ρ2−τ)B12

4ξB2 1

.

ρ = α+ (1 +λ)β,

ξ = α+ (1 + 2λ)β,

τ = (3)α+ 3 + 8λ+λ2β.

If σ1≤µ≤σ3, then

|a3−µa22|+

ρ2

2ξB1

1−B2

B1

+

ρ2+ 4µξ−τ

2ρ2

B1

|a2|2 ≤

B1

2ξ.

Furthermore, ifσ3≤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 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 orderδ is defined, for a function f(z), by

D_z−δf(z) = 1 Γ(δ)

z Z

0

f(ζ)

(z−ζ)1−δdζ (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 thatlog(z−ζ)

to be real when z−ζ >0.

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

D_zδf(z) = 1 Γ(1−δ)

z Z

0

f(ζ)

(z−ζ)δdζ (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 ordern+δ

is defined, for a function f(z), by

Dn_z+δf(z) = d

n

dzn

D_zδf(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 → Adefined by}

In terms of the Owa-Srivastava operator Ωδ _{defined by (26), we now introduce the}

function classMδ

q,α,β,λ(φ) in the following way:

Mδ_q,α,β,λ(φ) :={f : f ∈ A and Ωδf ∈ M_q,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+

∞

X

n=2

Γ (n+ 1) Γ (2−δ) Γ (n+ 1−δ) z

n_{. (28)}

Suppose now that

g(z) =z+

∞

X

n=2

gnzn (gn>0).

Then, since

f(z) =z+

∞

X

n=2

anzn∈ Mg_q,α,β,λ(φ)⇐⇒(f∗g)(z) =z+ ∞

X

n=2

gnanzn∈ 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 classM_q,α,β,λ(φ). By applying Theorem 1 to the following Hadamard product (or convolution):

(f∗g)(z) =z+g2a2z2+g3a3z3+· · · ,

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

Theorem 3. Let 0 ≤ µ ≤ 1, 0 ≤ α ≤ 1, 0 ≤ β ≤ 1 and 0 ≤ λ ≤ 1. Also let φ(z) = 1 +B1z+B2z2+B3z3+. . ., where the coefficients Bn are real withB1>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−

B2₁ ρ2γ2

if µ≤σ4,

B1

ξg3

if σ4 ≤µ≤σ5,

1 2ξg3

−2B2+

B₁2 ρ2γ2

if µ≥σ5,

where, for convenience,

σ4 :=

g3

g₂2

2ρ2(B2−B1)−(ρ2−τ)B21

2ξB₁2

σ5 :=

g3

g2 2

2ρ2(B2+B1)−(ρ2−τ)B21

2ξB2 1

,

and

γ2:=

ρ2+2µξg3

g2 2

−τ

(30)

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

Since, by (1) and the definition 4,

(Ωδf)(z) =z+

∞

X

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+B2z2+B3z3+. . ., where the coefficientsBn 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

ρ2γ3

if µ≤σ4,

(2−δ) (3−δ)

6ξ B1 if σ4≤µ≤σ5,

(2−δ) (3−δ) 12ξ

−2B2+

B₁2 ρ2 γ3

if µ≥σ5,

where, for convenience,

σ4:=

2 (3−δ) 3 (2−δ)

2ρ2(B2−B1)−(ρ2−τ)B21

2ξB2 1

,

σ5:=

2 (3−δ) 3 (2−δ)

2ρ2(B2+B1)−(ρ2−τ)B21

2ξB₁2

,

and

γ3 :=

ρ2+ 2µξ2 (3−δ)

3 (2−δ)−τ

(34)

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].

References

[1] R. P. Agarwal. Certain fractional q-integrals and q-derivatives. Proc. Cambridge Philos. Soc., 66:365–370, 1969.

[2] Waleed A. Al-Salam. Some fractional q-integrals and q-derivatives. Proc. Edinburgh Math. Soc. (2), 15:135–140, 1966/1967.

[3] R. M. Ali, S. K. Lee, V. Ravichandran, and S. Supramaniam. The Fekete-Szego coefficient functional for transforms of analytic functions. Bull. Iranian Math. Soc., 35(2):119–142, 276, 2009.

[4] Ali Aral, Vijay Gupta, and Ravi P. Agarwal. Applications of q-calculus in operator theory. Springer, New York, 2013.

[5] T. Bulboacˇa.Differential Subordinations and Superordinations, Recent Results. House of Scientific Book Publ., Cluj-Napoca, 2005.

[6] F. H. Jackson. On q-functions and a certain difference operator. Transactions of the Royal Society of Edinburgh, 46:253–281, 1908.

[7] F. H. Jackson. On q-definite integrals. Quarterly J. Pure Appl. Math., 41:193–203, 1910.

[8] Stanis l awa Kanas and Dorina R˘ aducanu. Some class of analytic functions related to conic domains. Math. Slovaca, 64(5):1183–1196, 2014.

[9] Anatoly A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo. Theory and applications of fractional differential equations, volume 204 ofNorth-Holland Mathematics Studies. Elsevier Science B.V., Amsterdam, 2006.

[10] Anatoly A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo. Theory and applications of fractional differential equations, volume 204 ofNorth-Holland Mathematics Studies. Elsevier Science B.V., Amsterdam, 2006.

[11] Sanford S. Miller and Petru T. Mocanu. Differential subordinations, volume 225 of

Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, Inc., New York, 2000. Theory and applications.

[12] Aabed Mohammed and Maslina Darus. A generalized operator involving the q -hypergeometric function. Mat. Vesnik, 65(4):454–465, 2013.

[14] Shigeyoshi Owa and H. M. Srivastava. Univalent and starlike generalized hypergeo-metric functions. Canad. J. Math., 39(5):1057–1077, 1987.

[15] Igor Podlubny. Fractional differential equations, volume 198 of Mathematics in Sci-ence and Engineering. Academic Press, Inc., San Diego, CA, 1999. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications.

[16] S. D. Purohit and R. K. Raina. Fractional q-calculus and certain subclasses of uni-valent analytic functions. Mathematica, 55(78)(1):62–74, 2013.

[17] Sunil Dutt Purohit. A new class of multivalently analytic functions associated with fractionalq-calculus operators. Fract. Differ. Calc., 2(2):129–138, 2012.

[18] Sunil Dutt Purohit and Ravinder Krishna Raina. Certain subclasses of analytic functions associated with fractional q-calculus operators. Math. Scand., 109(1):55– 70, 2011.

[19] C. Ramachandran, D. Kavitha, and T. Soupramanien. Certain bound for q-starlike and q-convex functions with respect to symmetric points. Int. J. Math. Math. Sci., pages Art. ID 205682, 7, 2015.

[20] C. Ramachandran, S. Sivasubramanian, H. M. Srivastava, and A. Swaminathan. Co-efficient inequalities for certain subclasses of analytic functions and their applications involving the Owa-Srivastava operator of fractional calculus. Math. Inequal. Appl., 12(2):351–363, 2009.

[21] V. Ravichandran, Maslina Darus, M. Hussain Khan, and K. G. Subramanian. Fekete-Szego inequality for certain class of analytic functions. Aust. J. Math. Anal. Appl., 1(2):Art. 4, 7, 2004.

[22] V. Ravichandran, A. Gangadharan, and Maslina Darus. Fekete-Szego inequality for certain class of Bazilevic functions. Far East J. Math. Sci. (FJMS), 15(2):171–180, 2004.

[23] Malcolm I. S. Robertson. On the theory of univalent functions. Ann. of Math. (2), 37(2):374–408, 1936.

[24] T. M. Seoudy and M. K. Aouf. Coefficient estimates of new classes of q-starlike and

q-convex functions of complex order. J. Math. Inequal., 10(1):135–145, 2016.

[25] H. M. Srivastava. Some families of fractional derivative and other linear operators associated with analytic, univalent, and multivalent functions. In Analysis and its applications (Chennai, 2000), pages 209–243. Allied Publ., New Delhi, 2001.

[26] H. M. Srivastava and Shigeyoshi Owa. An application of the fractional derivative.