International Journal of Analysis and Applications

Volume 19, Number 6 (2021), 826-835

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

DOI: 10.28924/2291-8639-19-2021-826

ON NEW SUBCLASS OF HARMONIC UNIVALENT FUNCTIONS ASSOCIATED

WITH MODIFIED Q-OPERATOR

SHUJAAT ALI SHAH1,∗, ASGHAR ALI MAITLO1, MUHAMMAD AFZAL SOOMRO1,

KHALIDA INAYAT NOOR2

1Department of Mathematics and Statistics, Quaid-i-Awam University of Engineering, Science and

Technology, Nawabshah, 67480 Sindh, Pakistan

2 Department of Mathematics, COMSATS University Islamabad, Islamabad, 45550, Pakistan

∗Corresponding author: shahglike@yahoo.com

Abstract. In this article, we introduce new subclasses of harmonic univalent functions associated with

the q-difference operator. The modified q-Srivastava-Attiya operator is defined and certain applications of

this operator are discussed. We investigate the sufficient condition, distortion result, extreme points and

invariance of convex combination of the elements of the subclasses.

1. Introduction

A real-valued function u (x,y) is said to be harmonic in a domain D if it has continuous second order

partial derivatives in D and satisfies

uxx + uyy = 0.

We say that a continuous f : Ω (⊂ C) → C defined by f (z) = u (x,y) +iv (x,y) is harmonic if both u (x,y)

and v (x,y) are real harmonic in Ω. It is observed that every harmonic function f in any simply connected

domain Ω can be written as f (z) = h(z) +g(z), where h and g are analytic in Ω, and are called, respectively,

the analytic and co-analytic parts of f.

Received August 25th, 2021; accepted October 11th, 2021; published October 28th, 2021.

2010 Mathematics Subject Classification. 30C45, 30C55.

Key words and phrases. univalent functions; harmonic function; q-difference operator; the q-Srivastava-Attiya operator.

©2021 Authors retain the copyrights

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

826

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-19-2021-826


Int. J. Anal. Appl. 19 (6) (2021) 827

We denote by H the class of complex-valued harmonic functions f = h + g defined in the open unit disc

U = {z : |z| < 1} and normalized by h(0) = g(0) = h′(0) − 1 = 0. Such mappings have the following power

series representation

(1.1) f(z) = z +

∞∑
n=2

anz
n +

∞∑
n=2

bnzn, |b1| < 1.

It is clear that, when g(z) is identically zero, the class H reduces to the class A of normalized analytic

functions in U.

Due to Lewy [14], A function f ∈H is locally univalent and sense-preserving in U if and only if

|h′ (z)| > |g′ (z)| , for z ∈U.

We denote by SH the subclass of H consisting of all sense-preserving univalent harmonic functions f.

Firstly, Clunie et al. [5] was discussed some geometric properties of the class SH and its subclasses. Later

on, several authors contributed in the study of subclasses of the class SH, for example, see [1,3,6,7,9,11,19,20].

The theory of q-calculus operators are used in describing and solving various problems in applied sci-

ence such as ordinary fractional calculus, optimal control, q-difference and q-integral equations, as well as

geometric function theory of complex analysis. The fractional q-calculus is the q-extension of the ordinary

fractional calculus and dates back to early 20th century (e.g. see [10] or [2]). For 0 < q < 1, the q-difference

operator was introduced by Jackson [10] and is defined by

(1.2) ∂qh(z) =
h(z) −h(qz)

(1 −q)z
; q 6= 1, z 6= 0,

for q ∈ (0, 1) and h ∈A with h(z) = z +
∑∞
n=2 anz

n.

It is clear that lim
q→1−

∂qh(z) = h
′(z), where h′(z) is the ordinary derivative of the function. It can easily

be seen that for n ∈ N = {1, 2, 3, ..} and z ∈U

∂q

{
∞∑
n=1

anz
n

}
=

∞∑
n=1

[n]q z
n−1,

where

[n]q =
1 −qn

1 −q
= 1 + q + q2 + · · · .

Recently, in [18], Shah and Noor introduced the q-analogue of Srivastava-Attiya operator Jsq,b : A→A by

(1.3) Jsq,bh(z) = z +

∞∑
n=2

(
[1 + b]q
[n + b]q

)s
anz

n,

where h ∈ A, s ∈ C and b ∈ C\Z−0 . It is noted that for q → 1
− in (1.3), then the integral operator studied

by the authors in [21] is deduced. Moreover, for particular choices of s and b, the operator Jsq,b reduces to

the q-Alexander, q-Libera and q-Bernardi operators defined in [17].



Int. J. Anal. Appl. 19 (6) (2021) 828

Jahangiri [12] was the first who introduced the q-analogue of complex harmonic functions and studied

various geometric properties. Nowadays, certain subclasses of SH associated with operators and q-operators

were discussed by the prominent researchers, like [8, 12, 13, 15, 16, 22]. In motivation of above said literature,

first we modify the q-Srivastava-Attiya operator and then we define some new subclasses of SH.

For f = h + g given by (1.1), we define

(1.4) Jsq,bf(z) = J
s
q,bh(z) + J

s
q,bg(z),

where Jsq,bh(z) is given by (1.3) and

Jsq,bg(z) =

∞∑
n=1

(
[1 + b]q
[n + b]q

)s
bnz

n.

It is observed that, if co-analytic part of f = h + g is identically zero, then the modified q-Srivastava-Attiya

operator defined by (1.4) turn out to be the q-Srivastava-Attiya operator introduced in [18].

For f = h + g ∈SH, we define a new class HSq (γ,λ,β) as follows:

Definition 1.1. Let f = h + g ∈SH and is given by (1.1). Then f ∈HSq (γ,λ,β) if

<
[
1 +

1

γ

{
(1 −λ)

f(z)

z
+ λ (∂qf(z)) − 1

}]
≥ β,

where β ∈ [0, 1), γ ∈ C\{0}, λ ∈ [0, 1] and q ∈ (0, 1).

Particularly, If q → 1−, then the class HSq (γ,λ,β) reduces to the class, denoted by HS (γ,λ,β), of

functions f ∈SH satisfies

<
[
1 +

1

γ

{
(1 −λ)

f(z)

z
+ λf′(z) − 1

}]
≥ β,

where β ∈ [0, 1), γ ∈ C\{0} and λ ∈ [0, 1]. Moreover, we denote by HS (1,λ,β) = HS (λ,β) and

HS (1, 1,β) = HS (β) the classes of functions f = h + g ∈SH satisfies

<
[
(1 −λ)

f(z)

z
+ λf′(z)

]
≥ β

and <(f′(z)) ≥ β, respectively.

We further define HSq (γ,λ,β) = HSq (γ,λ,β) ∩SH, where SH denote the subclass of SH consisting of

functions of the type f(z) = h(z) + g (z), where

(1.5) h(z) = z −
∞∑
n=2

|an|zn and g(z) = −
∞∑
n=1

|bn|zn.

Now, by using modified q-Srivastava-Attiya operator given by (1.4), we define the following.

Definition 1.2. Let f = h + g ∈SH, and is given by (1.1). Then, for s ∈ R, b > 1, β ∈ [0, 1), γ ∈ C\{0},

λ ∈ [0, 1] and q ∈ (0, 1). f ∈HSs,bq (γ,λ,β) if

(1.6) <
[
1 +

1

γ

{
(1 −λ)

Jsq,bf(z)

z
+ λ

(
Jsq,b∂qf(z)

)
− 1
}]
≥ β,



Int. J. Anal. Appl. 19 (6) (2021) 829

where Jsq,bf(z) is given by (1.4).

Also, we define HS
s,b

q (γ,λ,β) = HS
s,b
q (γ,λ,β) ∩SH, where SH denote the subclass of SH consisting of

functions given by (1.5).

It is noted that, for s = 0 we have HSs,bq (γ,λ,β) = HSq (γ,λ,β) and HS
s,b

q (γ,λ,β) = HSq (γ,λ,β).

2. Main Results

Theorem 2.1. Let a function f = h + g ∈SH given by (1.1) and satisfies

(2.1)

∞∑
n=1

{{
1 +

(
[n]q − 1

)
λ
}

[|an| + |bn|]
}( [1 + b]q

[n + b]q

)s
≤ 1 + (1 −β) γ,

where s ∈ R, b > 1, β ∈ [0, 1), γ ∈ C\{0}, λ ∈ [0, 1] and q ∈ (0, 1). Then f ∈HSs,bq (γ,λ,β). This result is

sharp.

Proof. We wish to show that f = h + g ∈SH satisfies (1.6), whenever the coefficients of f satisfies (2.1).

We use the fact that <(w) ≥ ξ if and only if |1 − ξ + w| ≥ |1 + ξ −w|. So it suffices to show that

∣∣∣∣1 −β + 1 + 1γ
{

(1 −λ)
Jsq,bf(z)

z
+ λ

(
Jsq,b∂qf(z)

)
− 1
}∣∣∣∣

≥
∣∣∣∣1 + β − 1 − 1γ

{
(1 −λ)

Jsq,bf(z)

z
+ λ

(
Jsq,b∂qf(z)

)
− 1
}∣∣∣∣ ,

or equivalently,

∣∣∣∣(2 −β) γ + (1 −λ) Jsq,bf(z)z + λ(Jsq,b∂qf(z))− 1
∣∣∣∣
−
∣∣∣∣βγ − (1 −λ) Jsq,bf(z)z −λ(Jsq,b∂qf(z))−z

∣∣∣∣ ≥ 0.
Using (1.3) and (1.4), from the left hand side, we get

∣∣∣∣∣∣∣
(2 −β) γ + (1 −λ)

{
1 +

∑∞
n=2

(
[1+b]

q

[n+b]
q

)s
anz

n−1 +
∑∞
n=1

(
[1+b]

q

[n+b]
q

)s
bnzn−1

}
+λ
{

1 +
∑∞
n=2

(
[1+b]q
[n+b]q

)s
[n]q anz

n−1 +
∑∞
n=1

(
[1+b]q
[n+b]q

)s
[n]q bnz

n−1
}
− 1

∣∣∣∣∣∣∣
−

∣∣∣∣∣∣∣
βγ + (1 −λ)

{
1 +

∑∞
n=2

(
[1+b]

q

[n+b]
q

)s
anz

n−1 +
∑∞
n=1

(
[1+b]

q

[n+b]
q

)s
bnzn−1

}
+λ
{

1 +
∑∞
n=2

(
[1+b]q
[n+b]q

)s
[n]q anz

n−1 +
∑∞
n=1

(
[1+b]

q

[n+b]q

)s
[n]q bnz

n−1
}
− 1

∣∣∣∣∣∣∣ ,



Int. J. Anal. Appl. 19 (6) (2021) 830

this implies ∣∣∣∣∣∣∣
(2 −β) γ +

∑∞
n=2

{
(1 −λ) + λ [n]q

}(
[1+b]q
[n+b]

q

)s
anz

n−1

+
∑∞
n=1

{
(1 −λ) + λ [n]q

}(
[1+b]

q

[n+b]
q

)s
bnzn−1

∣∣∣∣∣∣∣
−

∣∣∣∣∣∣∣
βγ +

∑∞
n=2

{
(1 −λ) + λ [n]q

}(
[1+b]q
[n+b]q

)s
anz

n−1

+
∑∞
n=1

{
(1 −λ) + λ [n]q

}(
[1+b]

q

[n+b]
q

)s
bnzn−1

∣∣∣∣∣∣∣
≥ (2 −β) γ −

∞∑
n=2

{
(1 −λ) + λ [n]q

}( [1 + b]q
[n + b]q

)s
|an| |z|

n−1

−
∞∑
n=1

{
(1 −λ) + λ [n]q

}( [1 + b]q
[n + b]q

)s
|bn| |z|

n−1

−βγ −
∞∑
n=2

{
(1 −λ) + λ [n]q

}( [1 + b]q
[n + b]q

)s
|an| |z|

n−1

−
∞∑
n=1

{
(1 −λ) + λ [n]q

}( [1 + b]q
[n + b]q

)s
|bn| |z|

n−1

= 2 (1 −β) γ




1 − 2
∑∞
n=2

{
1 +

(
[n]q − 1

)
λ
}(

[1+b]
q

[n+b]q

)s
|an| |z|

n−1

−2
∑∞
n=1

{
1 +

(
[n]q − 1

)
λ
}(

[1+b]
q

[n+b]
q

)s
|bn| |z|

n−1




The above expression is nonnegative by (2.1). Hence f ∈HSs,bq (α,β).

The coefficient bound, given by (2.1), is sharp for the harmonic function

f(z) = z +

∞∑
n=2

(1 −β) γ

1 +
(

[n]q − 1
)
λ

(
[n + b]q
[1 + b]q

)s
xnz

n

+

∞∑
n=1

(1 −β) γ

1 +
(

[n]q − 1
)
λ

(
[n + b]q
[1 + b]q

)s
ynz

n,

with
∑∞
n=2 |xn| +

∑∞
n=1 |yn| = 1. �

For different choices of parameters, we deduce certain new results as following.

If s = 0 in Theorem 2.1, then we have a following new result.

Corollary 2.1. Let a function f(z) = h(z) + g(z) ∈SH given by (1.1) and satisfies

∞∑
n=1

{{
1 +

(
[n]q − 1

)
λ
}

[|an| + |bn|]
}
≤ 1 + (1 −β) γ,

where β ∈ [0, 1), γ ∈ C\{0}, λ ∈ [0, 1] and q ∈ (0, 1). Then f ∈HSq (γ,λ,β). This result is sharp.

If q → 1−, then Corollary 2.1 reduces to a new result as follows:



Int. J. Anal. Appl. 19 (6) (2021) 831

Corollary 2.2. Let a function f(z) = h(z) + g(z) ∈SH given by (1.1) and satisfies
∞∑
n=1

{{1 + (n− 1) λ} [|an| + |bn|]}≤ 1 + (1 −β) γ,

where β ∈ [0, 1), γ ∈ C\{0} and λ ∈ [0, 1]. Then f ∈HS (γ,λ,β). This result is sharp.

If we take γ = 1 in Corollary 2.2 then we obtain.

Corollary 2.3. Let a function f(z) = h(z) + g(z) ∈SH given by (1.1) and satisfies
∞∑
n=1

{{1 + (n− 1) λ} [|an| + |bn|]}≤ 2 −β,

where β ∈ [0, 1) and λ ∈ [0, 1]. Then f ∈HS (λ,β). This result is sharp.

Moreover, when λ = 1 in Corollary 2.3, we get the sufficient condition for f in HS (β).

Now, we state and prove the necessary and sufficient conditions for the harmonic functions f = h + g to

be in HS
s,b

q (γ,λ,β) as following.

Theorem 2.2. Let f = h + g ∈SH given by (1.5). Then f ∈HS
s,b

q (γ,λ,β) if and only if

(2.2)

∞∑
n=1

{{
1 +

(
[n]q − 1

)
λ
}

[|an| + |bn|]
}( [1 + b]q

[n + b]q

)s
≤ 1 + (1 −β) γ,

where s ∈ R, b > 1, β ∈ [0, 1), γ ∈ C\{0}, λ ∈ [0, 1] and q ∈ (0, 1).

Proof. The sufficient condition is obvious from the Theorem 2.1, because HS
s,b

q (γ,λ,β) ⊂ HS
s,b
q (γ,λ,β).

We need to prove the necessary condition only, that is, if f ∈ HS
s,b

q (γ,λ,β), then the coefficients of the

function f = h + g satisfy the inequality 2.2.

Let f ∈HS
s,b

q (γ,λ,β). Then, by the definition of HST
s,b

q (γ,λ,β), we have

(2.3) <
[
1 +

1

γ

{
(1 −λ)

Jsq,bf(z)

z
+ λ

(
Jsq,b∂qf(z)

)
− 1
}
−β

]
≥ 0,

where s ∈ R, b > 1, β ∈ [0, 1), γ ∈ C\{0}, λ ∈ [0, 1] and q ∈ (0, 1).

Equivalently, we can write (2.3) as

(2.4) <
{

(1 −β) γ + (1 −λ)
Jsq,bf(z)

z
+ λ

(
Jsq,b∂qf(z)

)
− 1
}
≥ 0.

Substituting f = h + g in (2.4) and employing (1.4) along with (1.5), and also some computation yields

<


 (1 −β) γ + (1 −λ)

{
1 −

∑∞
n=2

(
[1+b]q
[n+b]q

)s
|an|zn−1 −

∑∞
n=1

(
[1+b]

q

[n+b]q

)s
|bn|zn−1

}
+λ
{

1 −
∑∞
n=2

(
[1+b]

q

[n+b]
q

)s
[n]q |an|z

n−1 −
∑∞
n=1

(
[1+b]

q

[n+b]
q

)s
[n]q |bn|z

n−1
}
− 1


 ≥ 0.

This implies

<

[
(1 −β) γ −

∞∑
n=2

{
1 +

(
[n]q − 1

)
λ
}
|an|zn−1 −

∞∑
n=1

{
1 +

(
[n]q − 1

)
λ
}
|bn|zn−1

]
≥ 0.



Int. J. Anal. Appl. 19 (6) (2021) 832

The above required condition must hold for all values of z in U. Upon choosing the values of z on the positive

real axis where 0 ≤ z = r < 1, we must have

(2.5) (1 −β) γ −
∞∑
n=2

{
1 +

(
[n]q − 1

)
λ
}
|an|rn−1 −

∞∑
n=1

{
1 +

(
[n]q − 1

)
λ
}
|bn|rn−1 ≥ 0.

If the inequality (2.2) does not hold, then the numerator in (2.5) is negative for r sufficiently close to 1.

Hence there exists z0 = r0 in (0, 1) for which the quotient in (2.5) is negative. This contradicts the required

condition for f ∈HST
s,b

q (γ,λ,β) and so the proof is complete. �

Next, we want to discuss the distortion bounds for the function f ∈HS
s,b

q (γ,λ,β), which yields a covering

result for this class.

Theorem 2.3. If f ∈HS
s,b

q (γ,λ,β) and |z| = r < 1, then

(1 −|b1|) r −Tr2 ≤ |f(z)| ≤ (1 + |b1|) r + Tr2,

with

(2.6) T =

(
[2 + b]q
[1 + b]q

)s  (1 −β) γ(
[2]q − 1

)
λ + 1

−
1(

[2]q − 1
)
λ + 1

|b1|


 .

Proof. Let f ∈HS
s,b

q (γ,λ,β). Taking absolute value of f, we get

|f(z)| ≤ (1 + |b1|) r +
∞∑
n=2

(|an| + |bn|) r2

≤ (1 + |b1|) r +
(1 −β) γ [2 + b]sq{(

[2]q − 1
)
λ + 1

}
[1 + b]

s
q

×
∞∑
n=2





{(

[2]q − 1
)
λ + 1

}
[1 + b]

s
q

(1 −β) γ [2 + b]sq


 (|an| + |bn|)


r2

≤ (1 + |b1|) r +
(1 −β) γ [2 + b]sq{(

[2]q − 1
)
λ + 1

}
[1 + b]

s
q

∞∑
n=2





{(

[n]q − 1
)
λ + 1

}
[1 + b]

s
q

(1 −β) γ [n + b]sq


 (|an| + |bn|)


r2

≤ (1 + |b1|) r +
(1 −β) γ [2 + b]sq{(

[2]q − 1
)
λ + 1

}
[1 + b]

s
q

×
{

1 −
|b1|

(1 −β) γ

}
r2, (by (2.2))

≤ (1 + |b1|) r + Tr2,

where T is given by (2.6). Hence this is the required right hand inequality. Similarly, one can easily prove

the required left hand inequality. �



Int. J. Anal. Appl. 19 (6) (2021) 833

By making use of the left hand inequality of the above theorem and letting r → 1, we obtain

Corollary 2.4. (Covering result) If f ∈HS
s,b

q (γ,λ,β), then

{w : |w| < (1 −L) − (1 −M) |b1|}⊂ f (E) ,

where L =
(1−β)γ[2+b]s

q

{([2]q−1)λ+1}[1+b]sq
and M =

[2+b]s
q

{([2]q−1)λ+1}[1+b]sq
.

In particular, we obtain the covering results for the subclasses of harmonic functions defined in Definition

1.1 and its special cases.

Now, our task is to determine the extreme points of closed convex hulls of HS
s,b

q (γ,λ,β) denoted by

clcoHS
s,b

q (γ,λ,β).

Theorem 2.4. A function f ∈HS
s,b

q (γ,λ,β) if and only if

(2.7) f(z) =

∞∑
n=1

(Xnhn (z) + Yngn (z)) ,

where h1(z) = z,

hn (z) = z −
(1 −β) γ

1 +
(

[n]q − 1
)
λ

(
[n + b]q
[1 + b]q

)s
zn; (n = 2, 3, ...)

and

gn(z) = z −
(1 −β) γ

1 +
(

[n]q − 1
)
λ

(
[n + b]q
[1 + b]q

)s
zn; (n = 1, 2, 3, ...) ,

with
∑∞
n=1 (Xn + Yn) = 1 and Xn,Yn ≥ 0. Particularly, {hn} and {gsn} are the extreme points of

HS
s,b

q (γ,λ,β).

Proof. We assume function f as given by (2.7)

f(z) =

∞∑
n=1

(Xnhn (z) + Yngn (z))

=

∞∑
n=1

(Xn + Yn) z −
∞∑
n=2

XnRnz
n −

∞∑
n=1

YnRnz
n,(2.8)

where Rn =
(1−β)γ[n+b]s

q

{([n]q−1)λ+1}[1+b]sq
.

Equating (2.8) with (1.5), we get

|an| = XnRn and |bn| = YnRn.

Now,

∞∑
n=2

{(
[n]q − 1

)
λ + 1

}
[1 + b]

s
q

(1 −β) γ [n + b]sq
|an| +

∞∑
n=1

{(
[n]q − 1

)
λ + 1

}
[1 + b]

s
q

(1 −β) γ [n + b]sq
|bn|

=

∞∑
n=1

(Xn + Yn) −X1 = 1 −X1 ≤ 1,



Int. J. Anal. Appl. 19 (6) (2021) 834

this implies
∞∑
n=1

[{(
[n]q − 1

)
λ + 1

}
[|an| + |bn|]

]( [1 + b]q
[n + b]q

)s
≤ 1 + (1 −β) γ

Thus, by Theorem 2.2, f ∈HS
s,b

q (γ,λ,β). Conversely, let f ∈HS
s,b

q (γ,λ,β). We take

Xn =

{(
[n]q − 1

)
λ + 1

}
[1 + b]

s
q

(1 −β) γ [n + b]sq
|an| ; (n = 2, 3, ...)

and

Yn =

{(
[n]q − 1

)
λ + 1

}
[1 + b]

s
q

(1 −β) γ [n + b]sq
|bn| ; (n = 1, 2, ...) ,

with
∑∞
n=1 (Xn + Yn) = 1.

We follows our required result by substituting the values of |an| and |bn| from the above relations in

(1.5). �

Finally, we wish to show that the class HS
s,b

q (γ,λ,β) is closed under the convex combination of its

elements.

Theorem 2.5. The class HS
s,b

q (γ,λ,β) is closed under the convex combination.

Proof. Let fis ∈HS
s,b

q (γ,λ,β), (i = 1, 2, ...), with

fi = z −
∞∑
n=2

|ai,n|zn −
∞∑
n=1

|bi,n|zn.

On using Theorem 2.2, we can write[
∞∑
n=1

{{
1 +

(
[n]q − 1

)
λ
}

[|an| + |bn|]
}]( [1 + b]q

[n + b]q

)s
≤ 1 + (1 −β) γ.

Now,

(2.9)

∞∑
i=1

κifi = z −
∞∑
n=2

(
∞∑
i=1

κi |ai,n|

)
zn −

∞∑
n=2

(
∞∑
i=1

κi |ai,n|

)
zn.

To prove our result, we use (2.8) and (2.9)

∞∑
n=2

{(
[n]q − 1

)
λ + 1

}
[1 + b]

s
q

(1 −β) γ [n + b]sq

(
∞∑
i=1

κi |ai,n|

)

+

∞∑
n=1

{(
[n]q − 1

)
λ + 1

}
[1 + b]

s
q

(1 −β) γ [n + b]sq

(
∞∑
i=1

κi |bi,n|

)

≤
∞∑
i=1

κi = 1.

Therefore
∑∞
i=1 κifis ∈HS

s,b

q (γ,λ,β). �

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication

of this paper.



Int. J. Anal. Appl. 19 (6) (2021) 835

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	1. Introduction
	2. Main Results
	References