International Journal of Analysis and Applications

Volume 18, Number 6 (2020), 1056-1065

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

DOI: 10.28924/2291-8639-18-2020-1056

ON MEROMORPHIC FUNCTIONS DEFINED BY A NEW CLASS OF

LIU-SRIVASTAVA INTEGRAL OPERATOR

SYED GHOOS ALI SHAH1, SAIMA NOOR2, MASLINA DARUS3,†,∗, WASIM UL HAQ4,

SAQIB HUSSAIN1

1Department of Mathematics, COMSATS University Islamabad, Abbottabad Campus 22060, Pakistan

2Department of Basic Sciences, Preparatory year deanship, King Faisal University, Hofuf 31982 Al Ahsa,

Saudia Arabia

3Department of Mathematical Sciences, Universiti Kebangsaan Malaysia, Bangi 43600, Selangor, Malaysia

4Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22010,

Pakistan

∗Corresponding author: maslina@ukm.edu.my

Abstract. In this work, we introduce and explore certain new subclasses of meromorphic functions. We

aim to study some important properties such as coefficient estimates, growth rate and partial sums for these

newly defined subclasses. It is important to mentioned that our results are generalization of number of

existing results.

1. Introduction

Let
∑
p denote the class of p-valent meromorphic function of the form:

(1.1) λ (ω) =
1

ωp
+

∞∑
t=p

atω
t,

Received September 5th, 2020; accepted September 30th, 2020; published October 28th, 2020.

2010 Mathematics Subject Classification. Primary 30C45, Secondary 30C50.

Key words and phrases. integral operator; meromorphic function; starlike function.

†This author is supported by the grant number GUP-2019-032.

©2020 Authors retain the copyrights

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

1056

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-18-2020-1056


Int. J. Anal. Appl. 18 (6) (2020) 1057

which are analytic in the punctured open unit disc U∗ = {ω : ω ∈ C and 0 < {ω} < 1} = U −{0}, where

U = U∗ ∪{0}.

Here we are listing some important subclasses of meromorphic functions which will be used in our subsequal

work. In 1936, Roberston [23] introduced the classes of meromorphic starlike and meromorphic convex

functions of order α. By
∑MS

(α) we mean the subclass of
∑

1 consisting of all meromorphic starlike functions

of order α. Analytically

(1.2) λ (ω) ∈
MS∑

(α) ⇔<

(
ωλ

′
(ω)

λ (ω)

)
< −α, (0 ≤ α < 1; ω ∈ U∗).

A closely related class of meromorphic convex functions of order α is denoted by
∑MC

(α) and defined as:

(1.3) λ (ω) ∈
MC∑

(α) ⇔−ωλ
′
(ω) ∈

MS∑
(α).

In 1952, Kaplan [16] introduced and studied an important class of analytic functions in the open unit disc

U known as close-to-convex functions. A function λ belongs to
∑

1 is in class
∑MK

(α,β), of meromorphic

close-to-convex functions of order α and type β, if there exist δ (ω) ∈
∑MS

(β) and

(1.4) <

(
ωλ

′

(ω)

δ (ω)

)
< −α.

Many differential and integral operators can be written in terms of convolution of certain holomorphic

functions. Let δ (ω) ∈
∑
p and having series representation of the form

(1.5) δ (ω) =
1

ωp
+

∞∑
t=0

btω
t,

then convolution (Hadamard product) is denoted by λ∗ δ and defined as:

(1.6) (λ∗ δ) (ω) =
1

ωp
+

∞∑
t=0

atbtω
t = (δ ∗λ) (ω) ,

where λ (ω) as given by (1.1).

Following the current work of Liu and Srivastava [18] (see also [1]- [6]), now we defined the integral

operator given below

(1.7) Mmp (a,b)λ (ω) =
1

ωp
+

∞∑
t=p

[
a

a + b(p + t)

]m
atω

t (ab > 0; p ∈ N) .

The above integral operator converts into the following operator when p = 1

(1.8) Mm1 (a,b)λ (ω) =
1

ω
+

∞∑
t=p

[
a

a + b(1 + t)

]m
atω

t (a > 0,b ≥ 0,m ∈ N) .

It can be easily verified from (1.8)

(1.9) λ (ω) (Mm1 (a,b)λ (ω))
′

= aMm1 (a,b)λ (ω) − (a + b)M
m+1
1 (a,b)λ (ω) (b > 0).

For more details see [7–9, 12, 15, 20, 21, 24].



Int. J. Anal. Appl. 18 (6) (2020) 1058

Definition 1.1. A function λ (ω) is subordinate to δ (ω) in U and written as: λ (ω) ≺ δ (ω) , if there exists a

Schwarz function k(ω), which is holomorphic in U∗ with |k(ω)| < 1, such that λ (ω) = δ(k (ω)). Furthermore,

if the function δ (ω) is univalent in U∗, then we have the following equivalence (see [22]):

(1.10) λ (ω) ≺ δ (ω) and λ (U) ⊂ δ (U) .

Further, λ (ω) is quasi-subordinate to δ (ω) in U∗ and written as:

λ (ω) ≺q δ (ω) ( ω ∈ U∗) ,

if there exist two analytic functions ϕ (ω) and k (ω) in U∗ such that
λ(ω)
ϕ(ω)

is analytic in U∗ and

|ϕ (ω)| ≤ 1 and k (ω) ≤ |ω| < 1 ω ∈ U∗,

satisfying

(1.11) λ (ω) = ϕ (ω) δ (k (ω)) ω ∈ U∗.

Definition 1.2. For −1 ≤ S < T ≤ 1 the function λ ∈
∑
p is in the class N

m
p (a,b; d,S,T) if it satisfies

the inequality

1 −
1

d


ω(Mmp (a,b)λ (ω))′

Mmp (a,b)λ (ω)
+ 1


 ≺ 1 + S (ω)

1 + T (ω)
,

or, equivalently to:

(1.12)

∣∣∣∣∣∣∣
ω(Mmp (a,b)λ(ω))

′

Mmp (a,b)λ(ω)
+ 1

T
ω(Mmp (a,b)λ(ω))

′

Mmp (a,b)λ(ω)
+ [|d|(S −T) + T ]

∣∣∣∣∣∣∣ < 1.
Let

∑∗
p denote the subclass of functions

∑
p consisting of functions of the form:

(1.13) λ (ω) =
1

ωp
+

∞∑
t=p

|at|ωt ( p ∈ N = {1, 2, ...}) .

Now, we define the class N∗mp (a,b; d,S,T) = N
m
p (a,b; d,S,T) ∩

∑∗
p . For recent work on meromorphic

functions we refer [10, 11, 13, 14, 17, 19]. Motivated, from the above cited work we obtained the following

results.

2. Main Results

In this section, in present the work to acquire sufficient conditions in which (1.13) gives the function λ (ω)

within the class N∗mp (a,b; d,S,T), as well as demonstrates that this condition is required for function which

belong to this class. In our first theorem, we begin with the necessary and sufficient condition for function

λ in N∗mp (a,b; d,S,T). We also prove some other related theorems.



Int. J. Anal. Appl. 18 (6) (2020) 1059

Theorem 2.1. Let the function λ (ω) is of the form (1.1). Then λ (ω) ∈ N∗mp (a,b; d,S,T) if and only if

(2.1)

∞∑
t=p

[
a

a + b(p + t)

]m
|at| [1 + t−{|d|(S −T) + (1 + t) T}] ≤ (1 −p) (T − 1) + |d|(S −T).

Proof. Assuming that (2.1) holds true, we obtain∣∣∣∣∣∣∣
ω(Mmp (a,b)λ(ω))

′

Mmp (a,b)λ(ω)
+ 1

T
ω(Mmp (a,b)λ(ω))

′

Mmp (a,b)λ(ω)
+ [|d|(S −T) + T ]

∣∣∣∣∣∣∣
=

∣∣∣∣∣∣
ω
(
Mmp (a,b)λ (ω)

)′
+ Mmp (a,b)λ (ω)

Tω
(
Mmp (a,b)λ (ω)

)′
+ [|d|(S −T) + T ] Mmp (a,b)λ (ω)

∣∣∣∣∣∣
=

∣∣∣∣∣∣∣∣
(1 −p) 1

ωp
+
∞∑
t=p

[
a

a+b(p+t)

]m
atω

t

[(1 −p) T + |d|(S −T)] ωp +
∑∞
t=1 ((t + 1) T + |d|(S −T)) |at|ωt

∣∣∣∣∣∣∣∣ < 1.
Then, by maximum modulus theorem, we have λ (ω) ∈ N∗mp (a,b; d,S,T).

Conversely, assume that λ (ω) is in the class N∗mp (a,b; d,S,T) with λ (ω) of the form (1.13), then we find

from (1.12) that ∣∣∣∣∣∣
ω
(
Mmp (a,b)λ (ω)

)′
+ Mmp (a,b)λ (ω)

Tω
(
Mmp (a,b)λ (ω)

)′
+ [|d|(S −T) + T ] Mmp (a,b)λ (ω)

∣∣∣∣∣∣ =∣∣∣∣∣∣∣∣
(1 −p) 1

ωp
+
∞∑
t=p

[
a

a+b(p+t)

]m
atω

t

[(1 −p) T + |d|(S −T)] ωp +
∑∞
t=1 ((t + 1) T + |d|(S −T)) |at|ωt

∣∣∣∣∣∣∣∣ < 1,
since the above inequality is genuine for all ω ∈ U, let the value of ω on the real axis. Letting ω −→ 1−

through real values, we get

∞∑
t=p

[
a

a + b(p + t)

]m
|at| [1 + t−{|d|(S −T) + (1 + t) T}] ≤ (1 −p) (T − 1) + |d|(S −T).

Which complete the proof. �

Corollary 2.1. If the function λ (ω) is of the form (1.1) is in the class N∗mp (a,b; d,S,T) then

|at| ≤
(1 −p) (T − 1) + |d|(S −T)

∞∑
t=p

[
a

a+b(p+t)

]m
[1 + t−{|d|(S −T) + (1 + t) T}]

, (t ≥ 1).

The result is sharp for the function

(2.2) λ (ω) =
1

ωp
+


 (1 −p) (T − 1) + |d|(S −T)∞∑
t=p

[
a

a+b(p+t)

]m
[1 + t−{|d|(S −T) + (1 + t) T}]


ωt.

Growth and distortion bounds for functions belonging to the class N∗mp (a,b; d,S,T) will be given in the

following result:



Int. J. Anal. Appl. 18 (6) (2020) 1060

Theorem 2.2. If a function λ (ω) given by (1.1) is in the class N∗mp (a,b; d,S,T) then for |ω| = r, we have:

1

rp
−


 (1 −p) (T − 1) + |d|(S −T)[

a
a+2bp

]m
[2 −{|d|(S −T) + 2T}]


r ≤ |λ (ω)|

≤
1

rp
+


 (1 −p) (T − 1) + |d|(S −T)[

a
a+2bp

]m
[2 −{|d|(S −T) + 2T}]


r,(2.3)

and

−p
|r|P+1

−


 (1 −p) (T − 1) + |d|(S −T)[

a
a+2bp

]m
[2 −{|d|(S −T) + 2T}]




≤
∣∣∣λ′ (ω)∣∣∣ ≤ −p

|r|p+1
+


 (1 −p) (T − 1) + |d|(S −T)[

a
a+2bp

]m
[2 −{|d|(S −T) + 2T}]


(2.4)

Proof. In view of Theorem 2.2, we have[
a

a + 2bp

]m
[2 −{|d|(S −T) + 2T}]

∞∑
t=p

|at|

≤
∞∑
t=p

[
a

a + b(p + t)

]m
|at| [1 + t−{|d|(S −T) + (1 + t) T}]

≤ (1 −p) (T − 1) + |d|(S −T),

which yield
∞∑
t=p

|at| ≤
(1 −p) (T − 1) + |d|(S −T)[
a

a+2bp

]m
[2 −{|d|(S −T) + 2T}]

(t ∈ N).

Therefore,

(2.5) |λ (ω)| ≤
1

|ω|p
+ |ω|

∞∑
t=p

|at| ≤
1

|ω|p
+ |ω|

(1 −p) (T − 1) + |d|(S −T)[
a

a+2bp

]m
[2 −{|d|(S −T) + 2T}]

,

and

(2.6) |λ (ω)| ≥
1

|ω|p
−|ω|

∞∑
t=p

|at| ≤
1

|ω|p
−|ω|

(1 −p) (T − 1) + |d|(S −T)[
a

a+2bp

]m
[2 −{|d|(S −T) + 2T}]

.

Now, by differentiating(1.13), we have

(2.7)
∣∣∣λ′ (ω)∣∣∣ ≤ −p

|ω|p+1
+

∞∑
t=p

|at| ≤
−p
|ω|p+1

+
(1 −p) (T − 1) + |d|(S −T)[
a

a+2bp

]m
[2 −{|d|(S −T) + 2T}]

,

and

(2.8)
∣∣∣λ′ (ω)∣∣∣ ≥ −p

|ω|p+1
−
∞∑
t=p

|at| ≥
−p
|ω|p+1

−
(1 −p) (T − 1) + |d|(S −T)[
a

a+2bp

]m
[2 −{|d|(S −T) + 2T}]

.



Int. J. Anal. Appl. 18 (6) (2020) 1061

We have thus completed the proof. �

Theorem 2.3. Let the function λ (ω) given by (1.13) is in the class N∗mp (a,b; d,S,T). Then we have

(i) λ is meromorphically starlike of order q in the disc |ω| < r3, that is

<

(
−
ωλ

′
(ω)

λ (ω)

)
> q (|ω| < r3, 0 ≤ q < 1),

where

(2.9) r3 = inf
t≥1


−

∞∑
t=p

[
a

a+b(p+t)

]m
[1 + t−{|d|(S −T) + (1 + t) T}]

(1 −p) (T − 1) + |d|(S −T)




1
t+p

.

(ii) λ is meromorphically convex of order q in the disc |ω| < r4, that is

<

{
−

(
1 +

ωλ
′′

(ω)

λ
′
(ω)

)}
> q (|ω| < r4, 0 ≤ q < 1),

where

(2.10) r4 = inf
t≥1



∞∑
t=p

[
a

a+b(p+t)

]m
[1 + t−{|d|(S −T) + (1 + t) T}] p (1 −q)

(1 −p) (T − 1) + |d|(S −T) [t (1 + q)]




1
t+p

.

Proof. (i) In order to the inequality (2.9), we set∣∣∣∣∣∣
ωλ

′
(ω)

λ(ω)
+ 1

ωλ
′
(ω)

λ(ω)
− 1 + 2q

∣∣∣∣∣∣ ≤
(1 −p) +

∑∞
t=p(t + 1) |at| |ω|

t+p

(2q −p− 1) +
∑∞
t=1(2q − 1 + t) |at| |ω|

t+p
.

Then we have ∣∣∣∣∣∣
ωλ

′
(ω)

λ(ω)
+ 1

ωλ
′
(ω)

λ(ω)
− 1 + 2q

∣∣∣∣∣∣ ≤ 1 (0 ≤ q < 1),
if

(2.11)

∞∑
t=1

|at| |ω|
t+p ≤−1.

Thus, by Theorem 2.1, the inequality (2.11) will be true if

|ω|t+p ≤−

∞∑
t=p

[
a

a+b(p+t)

]m
[1 + t−{|d|(S −T) + (1 + t) T}]

(1 −p) (T − 1) + |d|(S −T)
,

then

|ω| =


−

∞∑
t=p

[
a

a+b(p+t)

]m
[1 + t−{|d|(S −T) + (1 + t) T}]

(1 −p) (T − 1) + |d|(S −T)




1
t+p

.

The last inequality leads us immediately to the disc |ω| < r3, where r3 is given by (2.9).



Int. J. Anal. Appl. 18 (6) (2020) 1062

(ii) in order to prove the second affirmation of Theorem 2.3, we find from (1.1) that:

∣∣∣∣∣∣∣
ωλ

′′
(ω)

λ
′
(ω)

+ 2

ωλ
′′
(ω)

λ
′
(ω)

+ 2q

∣∣∣∣∣∣∣ ≤
p (p− 1) +

∑∞
t=p t(t + 1) |at| |ω|

t+p

p (p + 1 − 2q) |ω|p−1 +
∑∞
t=p t(t− 1 − 2q) |at| |ω|

t+p
.

Thus we have desired inequality:

∣∣∣∣∣∣∣
ωλ

′′
(ω)

λ
′
(ω)

+ 2

ωλ
′′
(ω)

λ
′
(ω)

+ 2q

∣∣∣∣∣∣∣ ≤ 1 (0 ≤ q < 1),
if

(2.12)

∞∑
t=1

(
t (1 + q)

p (1 −q)

)
|at| |ω|

t+1 ≤ 1.

Thus, by Theorem 2.1, the inequality (2.12) will be true if

(
t (1 + q)

p (1 −q)

)
|ω|t+p ≤

∞∑
t=p

[
a

a+b(p+t)

]m
[1 + t−{|d|(S −T) + (1 + t) T}]

(1 −p) (T − 1) + |d|(S −T)
,

then

|ω| =



∞∑
t=p

[
a

a+b(p+t)

]m
[1 + t−{|d|(S −T) + (1 + t) T}] p (1 −q)

(1 −p) (T − 1) + |d|(S −T) [t (1 + q)]




1
t+p

.

The last inequality readily yields the disc |ω| < r4, where r4 is given by (2.10), which complete the

proof. �

Theorem 2.4. The class N∗mp (a,b; d,S,T), is closed under convex linear combinations.

Proof. Let the function

λi (ω) =
1

ωp
+

∞∑
t=p

|at,i|ωt (i = 1, 2) ,

are in N∗mp (a,b; d,S,T), it suffices to show that the function h defined by

h (ω) = (1 − c)λ1 (ω) + cλ2 (ω) (0 ≤ c ≤ 1) ,

is in the class N∗mp (a,b; d,S,T). Since

h (ω) =
1

ωp
+

∞∑
t=p

[(1 − c) |at,1| + c |at,2|] ωt (0 ≤ c ≤ 1) .



Int. J. Anal. Appl. 18 (6) (2020) 1063

In view of Theorem 2.1, we have

∞∑
t=p

[
a

a + b(p + t)

]m
[1 + t−{|d|(S −T) + (1 + t) T}] [(1 − c) |at,1| + c |at,2|]

=

∞∑
t=p

[
a

a + b(p + t)

]m
[1 + t−{|d|(S −T) + (1 + t) T}] (1 − c) |at,1|

+

∞∑
t=p

[
a

a + b(p + t)

]m
[1 + t−{|d|(S −T) + (1 + t) T}] c |at,2|

≤ (1 − c) [(1 −p) (T − 1) + |d|(S −T)] + c [(1 −p) (T − 1) + |d|(S −T)]

= [(1 −p) (T − 1) + |d|(S −T)] ,

which show that h (ω) ∈ N∗mp (a,b; d,S,T), which is required. �

Theorem 2.5. Let λ0 (ω) =
1
ωp

and

λt (ω) =
1

ω
+


 (1 −p) (T − 1) + |d|(S −T)∞∑
t=p

[
a

a+b(p+t)

]m
[1 + t−{|d|(S −T) + (1 + t) T}]


ωt t ≥ 1,

then λ ∈ N∗mp (a,b; d,S,T). If and only if it can be expressed in the form

(2.13) λ (ω) =

∞∑
t=p

vtλt (ω) ,

where vt ≥ 0, and
∞∑
t=p

vt = 1.

Proof. Let the function λ (ω) be expressed in the form given by (2.13), then

λ (ω) =
1

ω
+


vt (1 −p) (T − 1) + |d|(S −T)∞∑

t=p

[
a

a+b(p+t)

]m
[1 + t−{|d|(S −T) + (1 + t) T}]


ωt,

and for this function, we have

∞∑
t=p

[
a

a + b(p + t)

]m
[1 + t−{|d|(S −T) + (1 + t) T}]

×vt
(1 −p) (T − 1) + |d|(S −T)

∞∑
t=p

[
a

a+b(p+t)

]m
[1 + t−{|d|(S −T) + (1 + t) T}]

ωt

=

∞∑
t=p

vt (1 −p) (T − 1) + |d|(S −T)

= [1 −v0] (1 −p) (T − 1) + |d|(S −T) ≤ (1 −p) (T − 1) + |d|(S −T),



Int. J. Anal. Appl. 18 (6) (2020) 1064

the condition (2.1) is satisfied. Thus, λ ∈ N∗mp (a,b; d,S,T). Conversely, we suppose that λ ∈

N∗mp (a,b; d,S,T). Since

|at| ≤
(1 −p) (T − 1) + |d|(S −T)

∞∑
t=p

[
a

a+b(p+t)

]m
[1 + t−{|d|(S −T) + (1 + t) T}]

, (t ≥ 1).

we set

vt =

∞∑
t=p

[
a

a+b(p+t)

]m
[1 + t−{|d|(S −T) + (1 + t) T}]

(1 −p) (T − 1) + |d|(S −T)
|at| (t ≥ 1),

and

v0 = 1 −
∞∑
vt

t=p

,

so it follows that

λ (ω) =

∞∑
t=p

vtλt (ω) .

This completes the assertion of Theorem 2.5. �

3. Conclusion

In our current investigation, we have presented and studied thoroughly some new subclasses of p−valent

functions related with meromorphic convex and meromorphic starlike functions, in connection with the

integral operator given by (1.7). We have obtained sufficient and necessary conditions in relation to these

classes, including growth and distortion theorem along with a radius problem. The technique and ideas of

this paper may stimulate further research in the theory of multivalent meromorphic functions.

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

of this paper.

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	1. Introduction
	2. Main Results
	3. Conclusion
	References