EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 6107
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

A Specific Category Of Meromorphic Functions With
Positive Coefficients

Omar Alnajar1,∗, K. Alshammari2, Ala Amourah3, Maslina Darus1,
Tala Sasa4

1 Department of Mathematical Sciences, Faculty of Science and Technology,
Universiti Kebangsaan Malaysia, Bangi 43600, Malaysia
2 Department of Mathematics, College of Sciences, Faculty of Science and Technology,
University of Ha’il, Ha’il 55425 , Saudi Arabia
3 Mathematics Education Program, Faculty of Education and Arts,
Sohar University, Sohar 311, Oman
4 Department of Mathematics, Faculty of Science, Applied Science Private University,
Amman, Jordan

Abstract. This work presents and investigates a new class of meromorphically uniformly convex
functions with positive coefficients that a differential operator defines. It also derives properties
such as coefficient bounds, distortion properties, δ-neighborhoods, convex linear combination, and
convolution properties.

2020 Mathematics Subject Classifications: 30C45.

Key Words and Phrases: Uniformly convex, Uniformly starlike, Coefficient estimates

1. Introduction

Let’s say that Ψ represents the class of functions f of this form:

f(z) =
1

z
+

∞∑
n=1

anz
n, (1)

they have a single pole at the origin with residue 1 and are regular in the domain U∗ =
{z ∈ C : 0 < |z| < 1}. Let the univalent, meromorphically starlike (of order Π), and

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6107

Email addresses: P117246@siswa.ukm.edu.my (O. Alnajar),
AAmourah@su.edu.om (A. Amourah), KHMO.ALSHAMMARI@uoh.edu.sa (K. Alshammari),
maslina@ukm.edu.my (M. Darus), t sasa@asu.edu.jo (T. Sasa)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6107 2 of 12

meromorphically convex (of order Π) subclasses of Ψ be denoted by Ψs and Ψ∗(Π) and
Ψk(Π), 0 ≤ Π < 1, f(z) of the form (1) is analytically contained in Ψ∗(Π) if and only if

Re

{
−zf ′(z)

f(z)

}
> Π, z ∈ U∗.

Likewise, f ∈ Ψk(Π) if and only if f(z) has the form (1) and fulfills

Re

{
−
(
1 +

zf ′′(z)

f ′(z)

)}
> Π, z ∈ U∗.

It is recognized that, in the case of Π = 1, the only function that is both Ψ∗(1) and
Ψk(1) is f(z) = 1

z . Looking for a subclass of Ψs with properties resembling those of
Ψ∗(Π) makes sense because the work in the meromorphic univalent scenario has partly
mirrored that of the regular univalent case. Juneja and Reddy [1] introduced the class Ψp

of functions of the sort.

f(z) =
1

z
+

∞∑
n=1

anz
n, an ≥ 0,

Ψ∗
p(Π) = Ψp ∩Ψ∗(Π).

(2)

A linear operator Am
ξ is defined for functions f(z) in the class Ψp in the following way.

A0
ξf(z) = f(z),

A1
ξf(z) =

(
1 +

µ+ ξ

α+ µ

)
f(z) +

µ+ ξ

α+ µ
zf ′(z),

...

Am
ξ f(z) = A(Am−1

ξ f(z)) =
1

z
+

∞∑
n=1

[
1 +

(µ+ ξ)(1 + n)

α+ µ

]m
anz

n (3)

for m ∈ N0 = 0, 1, 2, . . ..

Definition 1. Let φp(Π,Λ, ξ, µ, α) be the subclass of Ψp that consists of the form (2) and
satisfies the analytical criteria for −1 ≤ Π < 1, ξ > 0 and Λ ≥ 1.

Re

{
Am+1

ξ f(z)

Am
ξ f(z)

−Π

}
> Λ

∣∣∣∣∣A
m+1
ξ f(z)

Am
ξ f(z)

− 1

∣∣∣∣∣ , (4)

Am
ξ f(z) is given by (3).

Well-known classes of meromorphic uniformly linear function with positive coefficients
are unified by the function class φp(Π,Λ, ξ, µ, α).

Differential or integral operators with normalized analytic univalent functions are now
popular in the study of Geometric function theory. The classes φp(Π,Λ, ξ, µ, α) and various



O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6107 3 of 12

other subclasses of Ψ were studied rather extensively by Clunie [2] and also see ([3], [4], [5],
[6], [7], [8], [9], [10], [11], [12], [13], [14], [15]). Motivated by the works of Madhavi et al.
[16], we define the following a new subclass φp(Π, v, ξ). In this paper, we introduce and
study the subclass φp(Π,Λ, ξ, µ, α) of meromorphic functions with positive coefficients
generalization of a differential operator, including the Madhavi et al. operator [16] for
functions in φp(Π,Λ, ξ, µ, α).

2. Coefficient Inequalities

The coefficient bounds of function f(z) for the class φp(Π,Λ, ξ, µ, α) are obtained in
this section.

Theorem 1. A funcion f(z) of the form (2) is in φp(Π,Λ, ξ, µ, α) if

∞∑
n=1

[
1 +

(µ+ ξ)(1 + n)

α+ µ

]m [(
(µ+ ξ)(1 + n)

α+ µ

)
(1 + Λ) + 1−Π

]
|an| ≤ (1−Π).

Proof. It suffices to demonstrate that

Λ

∣∣∣∣∣A
m+1
ξ f(z)

Am
ξ f(z)

− 1

∣∣∣∣∣− Re

{
Am+1

ξ f(z)

Am
ξ f(z)

− 1

}
≤ (1−Π).

We have

Λ

∣∣∣∣∣A
m+1
ξ f(z)

Am
ξ f(z)

− 1

∣∣∣∣∣− Re

{
Am+1

ξ f(z)

Am
ξ f(z)

− 1

}
≤ (Λ + 1)

∣∣∣∣∣A
m+1
ξ f(z)

Am
ξ f(z)

− 1

∣∣∣∣∣
≤

(Λ + 1)
∑∞

n=−1

[
1 + (µ+ξ)(1+n)

α+µ

]m (
(µ+ξ)(1+n)

α+µ

)
|an| |zn|

1
|z| −

∑∞
n=−1

[
1 + (µ+ξ)(1+n)

α+µ

]m
|an| |zn|

.

By letting z → 1 move along the real axis, we can get

(Λ + 1)
∑∞

n=−1

[
1 + (µ+ξ)(1+n)

α+µ

]m (
(µ+ξ)(1+n)

α+µ

)
|an|

1−
∑∞

n=−1

[
1 + (µ+ξ)(1+n)

α+µ

]m
|an|

.

The boundary of the above formula is (1−Π) if

∞∑
n=1

[
1 +

(µ+ ξ)(1 + n)

α+ µ

]m [(
(µ+ ξ)(1 + n)

α+ µ

)
(1 + Λ) + 1−Π

]
|an| ≤ (1−Π).

This completes the theorem.



O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6107 4 of 12

Corollary 1. Let the function f(z) defined by (2) be in the class φp(Π,Λ, ξ, µ, α). Then

an ≤ (1−Π)∑∞
n=1

[
1 + (µ+ξ)(1+n)

α+µ

]m [(
(µ+ξ)(1+n)

α+µ

)
(1 + Λ) + 1−Π

] , n ≥ 1. (5)

Equality holds for the function of the form

fn(z) =
1

z
+

(1−Π)[
1 + (µ+ξ)(1+n)

α+µ

]m [(
(µ+ξ)(1+n)

α+µ

)
(1 + Λ) + 1−Π

]zn. (6)

Remark 1. For the choice of α, ξ = 1 and µ = 0, in Theorem 1 and Corollary 1, we
observed that the coefficient estimates for the functions of the class,

|an| ≤
(1−Π)

[n+ 2]m[(1 + Λ)(n+ 1) + 1−Π]

is coincide with [16].

3. Distortion Theorems

In this section, we obtain the sharp for the distortion theorems of the form (2).

Theorem 2. Let the function f(z) defined by (2) be in the class φp(Π,Λ, ξ, µ, α). Then
for 0 < |z| = r < 1,

1

r
− (1−Π)[

1 + 2(µ+ξ)
α+µ

]m [(
2(µ+ξ)
α+µ

)
(1 + Λ) + 1−Π

]r (7)

≤ |f(z)| ≤ 1

r
+

(1−Π)[
1 + 2(µ+ξ)

α+µ

]m [(
2(µ+ξ)
α+µ

)
(1 + Λ) + 1−Π

]r,
with equality for the function,

f(z) =
1

z
+

(1−Π)[
1 + 2(µ+ξ)

α+µ

]m [(
2(µ+ξ)
α+µ

)
(1 + Λ) + 1−Π

]z, at z = r, ir. (8)

Proof. Suppose f(z) is in φp(Π,Λ, ξ, µ, α). In view of Theorem 1, we have[
1 +

2(µ+ ξ)

α+ µ

]m [(
2(µ+ ξ)

α+ µ

)
(1 + Λ) + 1−Π

] ∞∑
n=1

an

≤
∞∑
n=1

[
1 +

(µ+ ξ)(1 + n)

α+ µ

]m [(
(µ+ ξ)(1 + n)

α+ µ

)
(1 + Λ) + 1−Π

]
≤ (1−Π)



O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6107 5 of 12

which evidently yields

∞∑
n=1

an ≤ (1−Π)[
1 + 2(µ+ξ)

α+µ

]m [(
2(µ+ξ)
α+µ

)
(1 + Λ) + 1−Π

] .
Consequently, we obtain

f(z) =

∣∣∣∣∣1z +

∞∑
n=1

anz
n

∣∣∣∣∣ ≤
∣∣∣∣1z

∣∣∣∣+ ∞∑
n=1

an |z|n ≤ 1

r
+ r

∞∑
n=1

an

≤ 1

r
+

(1−Π)[
1 + 2(µ+ξ)

α+µ

]m [(
2(µ+ξ)
α+µ

)
(1 + Λ) + 1−Π

]r.
Also,

f(z) =

∣∣∣∣∣1z +

∞∑
n=1

anz
n

∣∣∣∣∣ ≥
∣∣∣∣1z

∣∣∣∣− ∞∑
n=1

an |z|n ≥ 1

r
− r

∞∑
n=1

an

≥ 1

r
− (1−Π)[

1 + 2(µ+ξ)
α+µ

]m [(
2(µ+ξ)
α+µ

)
(1 + Λ) + 1−Π

]r.
Hence the result (7) follows.

Theorem 3. Let the function f(z) defined by (2) be in the class φp(Π,Λ, ξ, µ, α). Then
for 0 < |z| = r < 1,

1

r2
− (1−Π)[

1 + 2(µ+ξ)
α+µ

]m [(
2(µ+ξ)
α+µ

)
(1 + Λ) + 1−Π

]
≤ |f ′

(z)| ≤ 1

r2
+

(1−Π)[
1 + 2(µ+ξ)

α+µ

]m [(
2(µ+ξ)
α+µ

)
(1 + Λ) + 1−Π

] .
The outcome is sharp, with the shape of the extremal function being (1).

Proof. From Theorem 1, we have[
1 +

2(µ+ ξ)

α+ µ

]m [(
2(µ+ ξ)

α+ µ

)
(1 + Λ) + 1−Π

] ∞∑
n=1

nan

≤
∞∑
n=1

[
1 +

(µ+ ξ)(1 + n)

α+ µ

]m [(
(µ+ ξ)(1 + n)

α+ µ

)
(1 + Λ) + 1−Π

]
≤ (1−Π)

which evidently yields

∞∑
n=1

nan ≤ (1−Π)[
1 + 2(µ+ξ)

α+µ

]m [(
2(µ+ξ)
α+µ

)
(1 + Λ) + 1−Π

] .



O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6107 6 of 12

Consequently, we obtain∣∣∣f ′
(z)

∣∣∣ ≤ ∣∣∣∣∣ 1r2 +

∞∑
n=1

nanr
n−1

∣∣∣∣∣ ≤ 1

r2
+

∞∑
n=1

nan

≤ 1

r2
+

(1−Π)[
1 + 2(µ+ξ)

α+µ

]m [(
2(µ+ξ)
α+µ

)
(1 + Λ) + 1−Π

] .
Also, ∣∣∣f ′

(z)
∣∣∣ ≥ ∣∣∣∣∣1z −

∞∑
n=1

nanr
n−1

∣∣∣∣∣ ≥ 1

r2
−

∞∑
n=1

nan

≥ 1

r2
− (1−Π)[

1 + 2(µ+ξ)
α+µ

]m [(
2(µ+ξ)
α+µ

)
(1 + Λ) + 1−Π

] .
This completes the proof.

Remark 2. For the choice of α, ξ = 1 and µ = 0, in Theorems 2 and 3, we observed
that the sharp for the distortion theorems for the functions of the class are coincide with
[16].

4. The class φp(Π,Λ, ξ, µ, α, γ) and its neighborhoods

In this section, we obtain neighborhoods from class φp(Π,Λ, ξ, µ, α, γ).

Definition 2. A function f ∈ Ψp is said to in the class φp(Π,Λ, ξ, µ, α, γ) if there exists
a function φp(Π,Λ, ξ, µ, α) such that∣∣∣∣f(z)g(z)

− 1

∣∣∣∣ < 1− γ, z ∈ U∗, (0 ≤ γ < 1).

Following the earlier works on neighborhoods of analytic functions by [17] univalent
and [18], we define the δ-neighborhood of a function f ∈ Ψp by

Nδ(f) :=

{
g ∈ Ψp : g(z) =

1

z
+

∞∑
n=1

bnz
n :

∞∑
n=1

n |an − bn| ≤ δ

}
(9)

Theorem 4. If g ∈ φp(Π,Λ, ξ, µ, α) and

γ = 1−
δ
[(

2(µ+ξ)
α+µ

)
(1 + Λ) + 1−Π

] [
1 + 2(µ+ξ)

α+µ

]m[
1 + 2(µ+ξ)

α+µ

]m [(
2(µ+ξ)
α+µ

)
(1 + Λ) + 1−Π

]
− 1 + Π

(10)

Then
Nδ(g) ⊂ φp(Π,Λ, ξ, µ, α, γ).



O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6107 7 of 12

Proof. Let f ∈ Nδ(g). Then we find from (9) that

∞∑
n=1

n |an − bn| ≤ δ

which implies the coefficient inequality

∞∑
n=1

|an − bn| ≤ δ, (n ∈ N).

Since g ∈ φp(Π,Λ, ξ, µ, α), we have

∞∑
n=1

bn ≤ (1−Π)[
1 + 2(µ+ξ)

α+µ

]m [(
2(µ+ξ)
α+µ

)
(1 + Λ) + 1−Π

] .
So that∣∣∣∣f(z)g(z)

− 1

∣∣∣∣ ≤ ∑∞
n=1 |an − bn|

1−
∑∞

n=1 bn
≤

δ
[(

2(µ+ξ)
α+µ

)
(1 + Λ) + 1−Π

] [
1 + 2(µ+ξ)

α+µ

]m[
1 + 2(µ+ξ)

α+µ

]m [(
2(µ+ξ)
α+µ

)
(1 + Λ) + 1−Π

]
− 1 + Π

= 1−γ

provided γ is given by (10). Hence, by Definition 2 , f ∈ φp(Π,Λ, ξ, µ, α, γ) for γ given by
(10), which completes the proof.

5. Convex linear combinations and convolution properties

In this section, we obtain sharp for f(z) is meromorphically convex of order δ and
necessary and sufficient condition for f(z) is in the class φp(Π,Λ, ξ, µ, α). And also proved
that convolution is in the class φp(Π,Λ, ξ, µ, α).

Theorem 5. If the function f(z) = 1
z +

∑∞
n=1 anz

n is in φp(Π,Λ, ξ, µ, α) then f(z) is
meromorphically convex of order δ(0 ≤ δ < 1) in |z| < r = r(Π,Λ, δ, µ, α), where

r(Π,Λ, δ, µ, α) = inf
m≥1

(1− δ)
[
1 + (µ+ξ)(1+n)

α+µ

]m [(
(µ+ξ)(1+n)

α+µ

)
(1 + Λ) + 1−Π

]
(1−Π)n(n+ 2− δ)


1

n+1

.

The result is sharp.
Proof. Let f(z) be in φp(Π,Λ, ξ, µ, α). Then, by Theorem 1 , we have

∞∑
n=1

[
1 +

(µ+ ξ)(1 + n)

α+ µ

]m [(
(µ+ ξ)(1 + n)

α+ µ

)
(1 + Λ) + 1−Π

]
|an| ≤ (1−Π). (11)



O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6107 8 of 12

It is sufficient to show that ∣∣∣∣2 + zf ′′(z)

f ′(z)

∣∣∣∣ ≤ (1− δ),

for |z| < r = r(Π,Λ, δ, ξ, µ, α), where r(Π,Λ, δ, ξ, µ, α) is specified in the statement of
the theorem. Then∣∣∣∣2 + zf ′′(z)

f ′(z)

∣∣∣∣ =
∣∣∣∣∣
∑∞

n=1 n(n+ 1)anz
n−1

−1
z2

+
∑∞

n=1 nanz
n−1

∣∣∣∣∣ ≤
∑∞

n=1 n(n+ 1)am|z|n+1

1−
∑∞

n=1 nan|z|n+1
.

This will be bounded by (1− δ) if

∞∑
n=1

n(n+ 2− δ)

1− δ
an|z|n+1 ≤ 1. (12)

By (11), it follows that (12) is true if

n(n+ 2− δ)

1− δ
|z|n+1 ≤

[
1 + (µ+ξ)(1+n)

α+µ

]m [(
(µ+ξ)(1+n)

α+µ

)
(1 + Λ) + 1−Π

]
1−Π

|an| , n ≥ 1

or |z| ≤

(1− δ)
[
1 + (µ+ξ)(1+n)

α+µ

]m [(
(µ+ξ)(1+n)

α+µ

)
(1 + Λ) + 1−Π

]
(1−Π)n(n+ 2− δ)


1

n+1

.

(13)

Setting |z| = r(Π,Λ, δ, ξ, µ, α) in (13), the result follows. The result is sharp for the
function.

fn(z) =
1

z
+

(1−Π)[
1 + (µ+ξ)(1+n)

α+µ

]m [(
(µ+ξ)(1+n)

α+µ

)
(1 + Λ) + 1−Π

]zn, n ≥ 1.

Theorem 6. Let f0(z) =
1
z and

fn(z) =
1

z
+

(1−Π)[
1 + (µ+ξ)(1+n)

α+µ

]m [(
(µ+ξ)(1+n)

α+µ

)
(1 + Λ) + 1−Π

]zn, n ≥ 1.

Then f(z) = 1
z+

∑∞
n=1 anz

n is in the class φp(Π,Λ, ξ, µ, α) if and only if it can be expressed
in the form

f(z) = ϑ0f0(z) +

∞∑
n=1

ϑnfn(z),

where ϑ0 ≥ 0, ϑn ≥ 0, n ≥ 1 and ϑ0 +
∑∞

n=1 ϑn = 1.



O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6107 9 of 12

Proof. Let f(z) = ϑ0f0(z) +
∑∞

n=1 ϑnfn(z) with ϑ0 ≥ 0, ϑn ≥ 0, n ≥ 1 and

ϑ0 +
∞∑
n=1

ϑn = 1.

Then

f(z) = ϑ0f0(z)+

∞∑
n=1

ϑnfn(z) =
1

z
+

∞∑
n=1

ϑn
(1−Π)[

1 + (µ+ξ)(1+n)
α+µ

]m [(
(µ+ξ)(1+n)

α+µ

)
(1 + Λ) + 1−Π

]zn.
Since

∞∑
n=1

[
1 + (µ+ξ)(1+n)

α+µ

]m [(
(µ+ξ)(1+n)

α+µ

)
(1 + Λ) + 1−Π

]
1−Π

ϑn
(1−Π)[

1 + (µ+ξ)(1+n)
α+µ

]m [(
(µ+ξ)(1+n)

α+µ

)
(1 + Λ) + 1−Π

]
=

∞∑
n=1

ϑn = 1− ϑ0 ≤ 1.

By Theorem 1, f(z) is in the class φp(Π,Λ, ξ, µ, α). Conversely suppose that the function
f(z) is in the class φp(Π,Λ, ξ, µ, α), since

an ≤ (1−Π)[
1 + (µ+ξ)(1+n)

α+µ

]m [(
(µ+ξ)(1+n)

α+µ

)
(1 + Λ) + 1−Π

] , n ≥ 1.

ϑn =
∞∑
n=1

[
1 + (µ+ξ)(1+n)

α+µ

]m [(
(µ+ξ)(1+n)

α+µ

)
(1 + Λ) + 1−Π

]
1−Π

an,

and ϑ0 = 1 −
∑∞

n=1 ϑn, it follows that f(z) = ϑ0f0(z) +
∑∞

n=1 ϑnfn(z). This completes
the proof of the theorem.

For the functions f(z) = 1
z +

∑∞
n=1 anz

n and g(z) = 1
z +

∑∞
n=1 bnz

n belongs to Ψp, we
denoted by (f ∗ g)(z) the convolution of f(z) and g(z) and defined as

(f ∗ g)(z) = 1

z
+

∞∑
n=1

anbnz
n

Theorem 7. If the function f(z) = 1
z +

∑∞
n=1 anz

n and g(z) = 1
z +

∑∞
n=1 bnz

n are in the
class φp(Π,Λ, ξ, µ, α) then (f ∗ g)(z) is in the class φp(Π,Λ, ξ, µ, α).

Proof. Suppose f(z) and g(z) are in φp(Π,Λ, ξ, µ, α). By Theorem 1, we have

∞∑
n=1

[
1 + (µ+ξ)(1+n)

α+µ

]m [(
(µ+ξ)(1+n)

α+µ

)
(1 + Λ) + 1−Π

]
1−Π

an ≤ 1



O. Alnajar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6107 10 of 12

and
∞∑
n=1

[
1 + (µ+ξ)(1+n)

α+µ

]m [(
(µ+ξ)(1+n)

α+µ

)
(1 + Λ) + 1−Π

]
1−Π

bn ≤ 1.

Since f(z) and g(z) are regular are in U∗, so is (f ∗ g)(z). Furthermore

∞∑
n=1

[
1 + (µ+ξ)(1+n)

α+µ

]m [(
(µ+ξ)(1+n)

α+µ

)
(1 + Λ) + 1−Π

]
1−Π

anbn

≤
∞∑
n=1


[
1 + (µ+ξ)(1+n)

α+µ

]m [(
(µ+ξ)(1+n)

α+µ

)
(1 + Λ) + 1−Π

]
1−Π


2

anbn

≤

 ∞∑
n=1

[
1 + (µ+ξ)(1+n)

α+µ

]m [(
(µ+ξ)(1+n)

α+µ

)
(1 + Λ) + 1−Π

]
1−Π

an


 ∞∑

n=1

[
1 + (µ+ξ)(1+n)

α+µ

]m [(
(µ+ξ)(1+n)

α+µ

)
(1 + Λ) + 1−Π

]
1−Π

bn


≤ 1.

Hence, by Theorem 1, (f ∗ g)(z) is in the class φp(Π,Λ, ξ, µ, α).

Remark 3. For the choice of α, ξ = 1 and µ = 0, in Theorems 5, 6 and 7, we observed
that the the results are coincide with [16].

6. Summary

By utilizing the new differential operator Am
ξ f(z) for meromorphic functions, we intro-

duced a new subclass φp(Π,Λ, ξ, µ, α). The important results for these subclasses include
coefficient bounds, distortion properties, δ-neighbourhoods, convex linear combinations,
and convolution properties. Furthermore, this study develops the classes utilised in [19–31]
by using the new operator for future studies.

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