International Journal of Analysis and Applications

Volume 16, Number 1 (2018), 75-82

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-16-2018-75

NEW SUBFAMILY OF MEROMORPHIC CONVEX FUNCTIONS IN CIRCULAR

DOMAIN INVOLVING q-OPERATOR

BAKHTIAR AHMAD∗ AND MUHAMMAD ARIF

Department of Mathematics, Abdul Wali Khan University Mardan, KP, Pakistan

∗Corresponding author: pirbakhtiarbacha@gmail.com

Abstract. The main object of the present paper is to investigate a number of useful properties such as

sufficiency criteria, distortion bounds, coefficient estimates, radius of starlikness and radius of convexity for

a new subclass of meromorphic convex functions, which are defined here by means of a newly defined q-linear

differential operator.

1. Introduction and Definitions

Quantum calculus (q-calculus), which is the study of classical calculus without the notion of limits, attracted

the researchers because of its applications in various branches of mathematics, physics and various other

branches of science, for details see [6, 7].The q-analogue of derivative and integral operators were introduced

by Jackson [13, 14] along with some applications of q-calculus. Later on Aral and Gupta [5–7] introduced

the q-Baskakov Durrmeyer operator by using q-beta function while the author’s in [4, 8, 9] discussed the q-

generalization of complex operators known as q-Picard and q-Gauss-Weierstrass singular integral operators.

Kanas and Răducanu [15] gave the q-analogue of Ruscheweyh differential operator using the concepts of

convolution and then studied some of its properties. More applications of this operator can be seen in the

paper [3].

Received 13th September, 2017; accepted 4th December, 2017; published 3rd January, 2018.

2010 Mathematics Subject Classification. 30C45, 30C50.

Key words and phrases. Meromorphic functions; Janowski functions; q-differential operator.

c©2018 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

75

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-16-2018-75


Int. J. Anal. Appl. 16 (1) (2018) 76

In this paper a q-differential operator for meromorphic functions using convolution is defined. We use this

operator to define and study some properties of a family of meromorphic convex functions associated with

circular domain.

Let A denote the family of all meromorphic functions f that are analytic in the punctured disc D =

{z ∈ C : 0 < |z| < 1} and satisfying the normalization

f(z) =
1

zp
+

∞∑
k=1

ak+pz
k+p, (z ∈ D) . (1.1)

Also let MS∗ (α) and MK(α) denote the well known families of meromorphic starlike and meromorphic

convex functions of order α (0 ≤ α < 1) respectively.

For f and g be two meromorphic functions that are analytic in D and have the form (1.1), then convolution

of these functions can be defined by

f(z) ∗g(z) =
1

zp
+

∞∑
k=1

ak+pbk+pz
k+p, (z ∈ D) .

For 0 < q < 1, the q-derivative of a function f is defined by

∂qf(z) =
f (qz) −f(z)
z (q − 1)

, (z 6= 0, q 6= 1) . (1.2)

Simple calculations yields that for n ∈ N := {1, 2, 3, . . .} and z ∈ D

∂q

{
∞∑
n=1

anz
n

}
=

∞∑
n=1

[n,q] anz
n−1, (1.3)

where

[n,q] =
1 −qn

1 −q
= 1 +

n∑
l=1

ql, [0,q] = 0.

For any non-negative integer n the q-number shift factorial is defined by

[n,q]! =


 1, n = 0,[1,q] [2,q] [3,q] · · · [n,q] , n ∈ N.

Also the q-generalized Pochhammer symbol for x ∈ R is given by

[x,q]n =


 1, n = 0,[x,q][x + 1,q] . . . [x + n− 1,q], n ∈ N,

and for x > 0, let q-gamma function is defined as

Γq (x + 1) = [x,q] Γq (t) and Γq (1) = 1.

We now define a function

Φp (q,µ; z) =
1

zp
+

∞∑
n=1

Λn+p z
n+p, (µ > −1, z ∈ D), (1.4)



Int. J. Anal. Appl. 16 (1) (2018) 77

with

Λn+p =
Γq (µ + n + p + 1)

Γq (µ + 1) [n + p,q]!
=

[µ + 1,q]n+p
[n + p,q]!

.

It is quite clear that the series defined in (1.4) is convergent absolutely in D. Using the function Φ (q,µ; z)

and definition of q-derivative along with the idea of convolutions, we now define the differential operator

Lµ+p−1q : Ap → Ap by

Lµ+p−1q f (z) = Φp (q,µ; z) ∗f(z) =
1

zp
+

∞∑
n=1

Λn+panz
n+p , (µ > −1, z ∈ D) . (1.5)

Also for more details on the q-analogue of differential operators see the work [1, 2, 17].

Motivated from the work studied in [10, 12, 18–20], we now define a subfamily MC∗q (p,µ,A,B) of Ap by

using the operator Lµq as follows;

Definition 1.1. Let −1 ≤ B < A ≤ 1 and 0 < q < 1. Then a function f ∈ Ap is in the class

MC∗q (p,µ,A,B) , if it satisfies

−qp∂q
(
z∂qLµ+p−1q f (z)

)
[p,q] ∂qL

µ+p−1
q f (z)

≺
1 + Az

1 + Bz
. (1.6)

where the notation ”≺” denotes the familiar subordinations.

Equivalently, a function f ∈ Ap is in the class MC∗q (p,µ,A,B) , if and only if∣∣∣∣∣∣
qp∂q

(
z∂qLµ+p−1q f (z)

)
+ [p,q] ∂qLµ+p−1q f (z)

A [p,q] ∂qL
µ+p−1
q f (z) + Bqp∂q

(
z∂qL

µ+p−1
q f (z)

)
∣∣∣∣∣∣ < 1. (1.7)

2. The Main Results and Their Consequences

Theorem 2.1. Let f ∈ Ap be of the form (1.1) and satisfy the inequality

∞∑
n=1

qp[n+p,q][µ+1,q]n+p
[n+p,q]!

(qp[n + p,q] (1 + B) + (1 + A) [p,q]) |an+p| ≤ [p,q]
2

(A−B) . (2.1)

Then the function f ∈MC∗q (p,µ,A,B) .

Proof. To show f ∈ MC∗q (p,µ,A,B) , we only need to prove the inequality (1.7). For this using (1.5), and

then with the help of (1.2) and (1.3) we have∣∣∣∣∣∣
qp∂q

(
z∂qLµ+p−1q f (z)

)
+ [p,q] ∂qLµ+p−1q f (z)

A [p,q] ∂qL
µ+p−1
q f (z) + Bqp∂q

(
z∂qL

µ+p−1
q f (z)

)
∣∣∣∣∣∣



Int. J. Anal. Appl. 16 (1) (2018) 78

=

∣∣∣∣∣∣
qp
(

[p,q]2

q2pzp+1
+
∑∞

n=1 Λn+p[n+p,q]
2an+pz

n+p−1
)

+[p,q]
(
− [p,q]

qpzp+1
+
∑∞

n=1 Λn+p[n+p,q]an+pz
n+p−1

)
A[p,q]

(
− [p,q]

qpzp+1
+
∑∞

n=1
Λn+p[n+p,q]an+pzn+p−1

)
+Bqp

(
[p,q]2

q2pzp+1
+
∑∞

n=1
Λn+p[n+p,q]

2an+pzn+p−1
)
∣∣∣∣∣∣

=

∣∣∣∣∣
∑∞

n=1 Λn+p[n+p,q](q
p[n+p,q]+[p,q])an+pz

n+p−1

−(A−B)[p,q]
2

qpzp+1
+
∑∞

n=1 Λn+p[n+p,q](A[p,q]+Bq
p[n+p,q])an+pzn+p−1

∣∣∣∣∣
=
∣∣∣ ∑∞n=1 qpΛn+p[n+p,q](qp[n+p,q]+[p,q])an+pzn+2p

(A−B)[p,q]2+
∑∞

n=1
qpΛn+p[n+p,q](A[p,q]+Bqp[n+p,q])an+pzn+2p

∣∣∣
≤

∑∞
n=1

qpΛn+p[n+p,q](q
p[n+p,q]+[p,q])|an+p|

(A−B)[p,q]2−
∑∞

n=1
qpΛn+p[n+p,q](A[p,q]+Bqp[n+p,q])|an+p|

< 1,

where we have used the inequality (2.1) and this completes the proof. �

Theorem 2.2. Let f ∈MC∗q (p,µ,A,B) and has the form (1.1) . Then for |z| = r

1

rp
− τ1rp ≤ |f(z)| ≤

1

rp
+ τ1r

p,

where

τ1 =
(A−B) [p,q]! [p,q]2

qp[µ + 1,q]p+1((1 + A) [p,q] + qp[p + 1,q] (1 + B))
.

Proof. Consider

|f(z)| =

∣∣∣∣∣ 1zp +
∞∑
n=1

an+p z
n+p

∣∣∣∣∣ ,
≤

1

|zp|
+

∞∑
n=1

|an+p| |z|
n+p

=
1

rp
+

∞∑
n=1

|an+p| rn+p

As |z| = r < 1 so rn+p < rp and

|f(z)| ≤
1

rp
+ rp

∞∑
n=1

|an+p| (2.2)

Similarly

|f(z)| ≥
1

rp
−rp

∞∑
n=1

|an+p| (2.3)

Since (2.1) implies that

∞∑
n=1

qp[n+p,q][µ+1,q]n+p
[n+p,q]!

(qp[n + p,q] (1 + B) + (1 + A) [p,q]) |an+p| ≤ [p,q]
2

(A−B) .

But

qp
[µ+1,q]p+1

[p,q]!
((1 + A) [p,q] + qp[p + 1,q] (1 + B))

∞∑
n=1

|an+p| ≤

∞∑
n=1

qp
[µ+1,q]n+p

[n+p,q]!
((1 + A) [p,q] + qp[n + p,q] (1 + B)) |an+p| .

Hence

qp
[µ+1,q]p+1

[p,q]!
((1 + A) [p,q] + qp[p + 1,q] (1 + B))

∞∑
n=1

|an+p| ≤ [p,q]
2

(A−B) ,



Int. J. Anal. Appl. 16 (1) (2018) 79

which gives
∞∑
n=1

|an+p| ≤
[p,q]2(A−B)[p,q]!

qp((1+A)[p,q]+qp[p+1,q](1+B))[µ+1,q]p+1

Now by putting this value in (2.2) and (2.3) we get the required result. �

Theorem 2.3. Let f ∈MC∗q (p,µ,A,B) and has the form (1.1) . Then for |z| = r

[p,q]m
qmp+ζrm+p

− τ2rp ≤
∣∣∂mq f(z)∣∣ ≤ [p,q]mqmp+ζrm+p + τ2rp.

where

τ2 =
[p,q]

2
(A−B) [p,q]!

((1 + A) [p,q] + qp[p + 1,q] (1 + B))
and ζ =

m∑
n=1

n.

Proof. By the virtue of (1.2) and (1.3) , we can write

∂mq f(z) =
(−1)m [p,q]m
qmp+ζzp+m

+

∞∑
n=1

[n + p− (m− 1) ,q]m+1ap+nzp+n−m.

Since |z| = r < 1 so rp+n−m ≤ rp for m ≤ n hence

∣∣∂mq f(z)∣∣ ≤ [p,q]mqmp+ζrm+p + rp
∞∑
n=1

[n + p− (m− 1) ,q]m+1 |ap+n| , (2.4)

and similarly ∣∣∂mq f(z)∣∣ ≥ [p,q]mqmp+ζrm+p −rp
∞∑
n=1

[n + p− (m− 1) ,q]m+1 |ap+n| . (2.5)

Now by using (2.1) and the following inequality

qp
((1+A)[p,q]+qp[p+1,q](1+B))

[p,q]!

∞∑
n=1

[µ + p,q]p+n |ap+n| ≤

∞∑
n=1

qp
[µ+p,q]n+p

[n+p,q]!
((1 + A) [p,q] + qp[p + n,q] (1 + B)) |an| ,

we have
∞∑
n=1

[µ + p,q]n+p |an+p| ≤
(A−B) [p,q]2[p,q]!

qp ((1 + A) [p,q] + qp[p + 1,q] (1 + B))
,

but certainly
∞∑
n=1

[n + p− (m− 1) ,q]m+1 |ap+n| ≤
∞∑
n=1

[µ + p,q]n+p |an+p| ,

which implies

∞∑
n=1

[n + p− (m− 1) ,q]m+1 |ap+n| ≤
(A−B) [p,q]2[p,q]!

qp ((1 + A) [p,q] + qp[p + 1,q] (1 + B))
.

Finally, using this in (2.4) and (2.5) we obtain the required result. �

Theorem 2.4. Let f ∈MC∗q (p,µ,A,B) . Then f ∈MCp (α) for |z| < r1, where

r1 =

(
p (p−α) qp ((1 + A) [p,q] + qp[p + n,q] (1 + B)) [µ + p,q]n+p

(p + n) (p + n + α) (A−B) [p,q]2 [n + p− 1,q]!

) 1
n+2p

.



Int. J. Anal. Appl. 16 (1) (2018) 80

Proof. Let f ∈MC∗q (p,µ,A,B). To prove f ∈MCp (α) , we only need to show∣∣∣∣ zf′′(z) + (p + 1) f′(z)zf′′(z) + (1 + 2α−p) f′(z)
∣∣∣∣ ≤ 1.

Using (1.1) along with some simple computation yields

∞∑
n=1

(p + n) (n + p + α)

p (p−α)
|an+p| |z|

n+2p ≤ 1. (2.6)

From (2.1) , we can easily obtain that

∞∑
n=1

qp[p+n,q][µ+p,q]n+p
[n+p,q]!

(
((1+A)[p,q]+qp[p+n,q](1+B))

(A−B)[p,q]2

)
|an+p| < 1.

Now inequality (2.6) will be true, if the following holds

∞∑
n=1

(p+n)(n+p+α)
p(p−α) |an+p| |z|

n+2p
<

∞∑
n=1

qp[µ+p,q]n+p
[n+p−1,q]!

(
((1+A)[p,q]+qp[p+n,q](1+B))

(A−B)[p,q]2

)
|an+p| ,

which implies that

|z|n+2p <
p (p−α) qp ((1 + A) [p,q] + qp[p + n,q] (1 + B)) [µ + p,q]n+p

(p + n) (p + n + α) (A−B) [p,q]2 [n + p− 1,q]!
,

and so

|z| <

(
p (p−α) qp ((1 + A) [p,q] + qp[p + n,q] (1 + B)) [µ + p,q]n+p

(p + n) (p + n + α) (A−B) [p,q]2 [n + p− 1,q]!

) 1
n+2p

= r1,

we get the required condition. �

Theorem 2.5. Let f ∈MC∗q (p,µ,A,B). Then f ∈MS
∗
p (α) for |z| < r2, where

r2 =

(
(p−α) qp ((1 + A) [p,q] + qp[p + n,q] (1 + B)) [µ + p,q]n+p

(p + n + α) (A−B) [p,q]2 [n + p− 1,q]!

) 1
n+2p

.

Proof. We know that f ∈MS∗p (α) , if and only if∣∣∣∣ zf′(z) + pf(z)zf′(z) − (p− 2α)f(z)
∣∣∣∣ ≤ 1.

Using (1.1) and upon simplification yields

∞∑
n=1

(
n + p + α

p−α

)
|an+p| |z|

n+2p ≤ 1. (2.7)

Now from (2.1) we can easily obtain

∞∑
n=1

qp[µ+p,q]n+p
[n+p−1,q]!

(
((1+A)[p,q]+qp[p+n,q](1+B))

(A−B)[p,q]2

)
|an+p| < 1.

For inequality (2.7) to be true it will be enough if

∞∑
n=1

(
n+p+α
p−α

)
|an+p| |z|

n+2p
<

∞∑
n=1

qp[µ+p,q]n+p
[n+p−1,q]!

(
((1+A)[p,q]+qp[p+n,q](1+B))

(A−B)[p,q]2

)
|an+p| .



Int. J. Anal. Appl. 16 (1) (2018) 81

This gives

|z|n+2p <
(p−α) qp ((1 + A) [p,q] + qp[p + n,q] (1 + B)) [µ + p,q]n+p

(p + n + α) (A−B) [p,q]2 [n + p− 1,q]!
,

and hence

|z| <

(
(p−α) qp ((1 + A) [p,q] + qp[p + n,q] (1 + B)) [µ + p,q]n+p

(p + n + α) (A−B) [p,q]2 [n + p− 1,q]!

) 1
n+2p

= r2.

Thus we obtain the required result. �

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	1. Introduction and Definitions
	2. The Main Results and Their Consequences
	References