EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 3, 2024, 1894-1907
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

A Specific Class of Harmonic Meromorphic Functions
Associated with the Mittag- Leffler Transformation

Sarah Ahmed1,2, Abdullah Alsoboh3,∗, Maslina Darus2

1 Department of Mathematics, College of Education for Pure Science, University of Kerbala,
Kerbala, Iraq
2 Department of Mathematical Sciences, Universiti Kebangsaan Malaysia, Selangor, Malaysia
3 Department of Mathematics, Faculty of Science, Philadelphia University,
Amman, Jordan.

Abstract. This article uses the Mittag-Leffler transformation to explore a specific category of
harmonic meromorphic functions. The Mittag-Leffler transformation is a crucial tool for analysing
meromorphic functions and provides essential properties and insights into their behavior. The main
focus of this study is on harmonic meromorphic functions that can be represented by the Mittag-
Leffler transformation. Furthermore, this research introduces an innovative derivative operator
that incorporates this transformation into the domain of harmonic meromorphic functions. The
Mittag-Leffler transformation is widely recognised as a powerful technique for analysing various
mathematical functions, especially those with fractional order derivatives. It improves our un-
derstanding of harmonic meromorphic functions and their inherent characteristics. The research
findings highlight the effectiveness of this new derivative operator in unraveling the complexities
of these functions. They provide valuable insights into their behavior and fundamental traits.
Additionally, the study offers coefficient inequalities, the distortion theorem, distortion bounds,
extreme points, convex combinations, and convolution analyses specifically tailored to functions
within this particular class.

2020 Mathematics Subject Classifications: 30C41, 30C50

Key Words and Phrases: Analytic functions, Meromorphic functions, Starlike functions, Mul-
tiplier transformation , Mittag-Leffler function, Hadamard products

1. Introduction

We begin by considering the open unit disk in the complex plane, denoted by ♢ =
{ς ∈ C : |ς| < 1}, and the class of Meromorphic functions Σ defined as follows:

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5259

Email addresses: p102354@siswa.ukm.edu.my (S. Ahmed),
aalsoboh@philadelphia.edu.jo (A. Alsoboh), maslina@ukm.edu.my (M. Darus)

https://www.ejpam.com 1894 © 2024 EJPAM All rights reserved.



S. Ahmed, A. Alsoboh, M. Darus / Eur. J. Pure Appl. Math, 17 (3) (2024), 1894-1907 1895

f(ς) =
1

ς
+

∞∑
µ=1

aµς
µ, aµ ∈ C. (1)

where f is analytic in the punctured open unit disk ♢∗ = ♢\ {0} = {ς ∈ C : 0 < |ς| <
1}. The class Σ was investigated and studied by Clunie [13].

For f ∈ Σ of the form (1) and g ∈ Σ given by

g(ς) =
1

ς
+

∞∑
µ=1

bµς
µ, bµ ∈ C

.
the Hadamard product or convolution of f and g is defined as follows (see [1, 3]):

(f ∗ g)(ς) = (g ∗ f)(ς) = 1

ς
+

∞∑
µ=1

aµbµς
µ.

The well-known multiplier transformation operator I1(r, λ) : Σ → Σ has been studied
by Cho and Srivastava [12], Cho and Kim [11], and recently by Atshan and Joudah [6]. It
is defined as follows:

I1(r, λ)f(ς) =
1

ς
+

∞∑
µ=1

(
µ+ δ

1 + δ

)r

aµς
µ (δ ≥ 0, ς ∈ ♢∗). (2)

For α, η ∈ C, Wiman [23] introduced the generalised Mittag–Leffler function Eα,η(ς) which
is given by:

Eα(ς) =

∞∑
µ=0

ςµ

Γ(αµ+ 1)
(3)

and

Eα,η(ς) =

∞∑
µ=0

ςµ

Γ(αµ+ η)
, ℜe{α, η} > 0. (4)

The function given by (4) is not within the class Σ. Based on this, the function is then
normalised as follows [14]:

Ωα,η(ς) = ς−1Γ(η)Eα,η(ς) =
1

ς
+

∞∑
µ=1

Γ(η)

Γ(α(µ+ 1) + η)
ςµ. (5)

In recent years, there has been a growing interest in Mittag–Leffler-type functions,
driven by their increasing range of applications in probability, applied problem-solving,
statistical analysis, and distribution theory, among other domains. More information
about the utilization of Mittag–Leffler functions can be found in references [3, 4, 7, 8, 16,
20, 22].

Much of our research involving Mittag–Leffler functions focuses on aspects of convexity,
close-to-convexity, and starlikeness. Recent studies on the Eα,η(ς) Mittag–Leffler function



S. Ahmed, A. Alsoboh, M. Darus / Eur. J. Pure Appl. Math, 17 (3) (2024), 1894-1907 1896

can be found in [9]. Additionally, [21] has presented findings related to partial sums for
Eα,η(ς).

Motivated by Challab and Darus [10], we define the linear derivative operator Sα
η [r, δ, λ] :

Σ → Σ by

Sα
η [r, δ, λ]f(ς) =(1− λ)(I1(r, δ)f(ς) ∗ Ωα,η(ς) + λς((I1(r, δ)f(ς) ∗ Ωα,η(ς))

′

=
1

ς
+

∞∑
µ=1

Γ(η)[1 + λ(µ− 1)]k

Γ(α(µ+ 1) + η)

(
µ+ δ

1 + δ

)r

aµς
µ,

(6)

where δ ≥ 0, r ∈ N , 0 ≤ λ ≤ 1, α, η ∈ C and I1(r, λ)f(ς) of the form (2).

Example 1. If r = 0, then Sα
η [r, δ, λ] is reduced to

Sα
η [0, δ, λ]f(ς) =

1

ς
+

∞∑
µ=1

Γ(η)[1 + λ(µ− 1)]k

Γ(α(µ+ 1) + η)
aµς

µ

introduced by Ghanim and Al-Janaby [15].

Example 2. If α = 0, then Sα
η [r, δ, λ] is reduced to

S0
η [r, δ, λ]f(ς) =

1

ς
+

∞∑
µ=1

[1 + λ(µ− 1)]k
(
µ+ δ

1 + δ

)r

aµς
µ

Example 3. If α = 0 and k = 0, then Sα
η [r, δ, λ] is reduced to

S0
η [r, δ, λ]f(ς) =

1

ς
+

∞∑
µ=1

(
µ+ δ

1 + δ

)r

aµς
µ

introduced by Atshan and Joudah [6].

Example 4. If α = 0, r = 0, then Sα
η [r, δ, λ] is reduced to

S0
η [r, δ, λ]f(ς) =

1

ς
+

∞∑
µ=1

[1 + λ(µ− 1)]kaµς
µ

introduced by Challab and Darus [10].

For ς ∈ ♢∗ = ♢\{0}, by MH, we denote the class of harmonic meromorphic functions
of the form

f(ς) = ℏ(ς) + g(ς) =
1

ς
+

∞∑
µ=1

aµς
µ +

∞∑
µ=1

bµςµ, (7)

which are harmonic in the punctured unit disk ♢\{0}, where h and g are analytic in ♢∗and
♢, respectively, and ℏ has a simple pole at the origin with residue 1 here. This class was



S. Ahmed, A. Alsoboh, M. Darus / Eur. J. Pure Appl. Math, 17 (3) (2024), 1894-1907 1897

firstly studied by Jahangiri and Silverman [19], followed by Jahangiri et al. [18], Ahuja
and Jahangiri [2] and others. We further denote by the subclass MH of MH consisting of
functions f of the form

f(ς) =
1

ς
+

∞∑
µ=1

|aµ|ςµ +
∞∑
µ=1

|bµ|ςµ, (z ∈ ♢∗). (8)

The function f = ℏ + g, defined by the equation (7), can be classified as a harmonic
meromorphic function that is locally univalent and sense-preserving in the region ♢∗ if

and only if

∣∣∣∣g′ (ς)ℏ′ (ς)

∣∣∣∣ < 1.

The investigation of harmonic meromorphic functions within this specific context has
been examined by Jahangiri and Silverman in their work [19].

A function f ∈ MH is said to be in the class MS∗
H of meromorphically harmonic

starlike functions in ♢∗, if it satisfies the condition

ℜe

{
ςℏ′

(ς)− ς g′(ς)

ℏ(ς) + g(ς)

}
> 0, (z ∈ ♢∗).

In other hand, a function f ∈ MH is said to be in the class MCH of meromorphically
harmonic convex functions in ♢∗, if it satisfies the condition

ℜe

{
ςℏ′′

(ς) + ℏ′
(ς)− ς g′′(ς) + g′(ς)

ℏ′(ς) + g′(ς)

}
> 0, (z ∈ ♢∗).

The classes MS∗
H and MCH, which consist of harmonic meromorphic starlike functions

and harmonic meromorphic convex functions, have been the subject of study by Jahangiri
and Silverman [19], Jahangiri [17], Atshan and Joudah [6], Ghanem and Al-Janaby [15],
Elhaddad and Darus [14] and Alsoboh et al. [5].

Motivated by Challab and Darus [10], we define the linear operator for the harmonic
meromorphic class of functions f ∈ MH, by letting k ≥ 0 and

Ik(Sα
η [r, δ, λ])f(ς) = Ik(Sα

η [r, δ, λ])ℏ(ς) + (−1)kIk(Sα
η [r, δ, λ])g(ς), (9)

where

Ik(Sα
η [r, δ, λ])ℏ(ς) =

(−1)k

ς
+

∞∑
µ=1

Γ(η)[1 + λ(µ− 1)]k

Γ(α(µ+ 1) + η)

(
µ+ δ

1 + δ

)r

aµς
µ, (10)

and

Ik(Sα
η [r, δ, λ])g(ς) =

∞∑
µ=1

Γ(η)[1 + λ(µ− 1)]k

Γ(α(µ+ 1) + η)

(
µ+ δ

1 + δ

)r

bµς
µ. (11)

Utilising the operator Ik(Sα
η [r, δ, λ]), we introduce a class of generalised harmonic Mero-

morphically starlike functions within the region ♢∗ through the following definition.



S. Ahmed, A. Alsoboh, M. Darus / Eur. J. Pure Appl. Math, 17 (3) (2024), 1894-1907 1898

Definition 1. For 0 ≤ γ < 1, the class MKH(k, r, α, η, δ, λ, γ) is used to denote har-
monic meromorphic functions f defined as in (1), and belonging to this class is upon the
satisfaction of the condition:

Re

{
−
ς(Ik(Sα

η [r, δ, λ])ℏ(ς))′ − ς(Ik(Sα
η [r, δ, λ])g(ς))

′

Ik(Sα
η [r, δ, λ])ℏ(ς) + Ik(Sα

η [r, δ, λ])g(ς)

}
> γ, (ς ∈ ♢∗). (12)

Furthermore, by T MKH(k, r, α, η, δ, λ, γ) ⊂ MKH(k, r, α, η, δ, λ, γ), we denote the
subclass of harmonic Meromorphic functions fk = ℏk + gk(ς) where ℏk and gk of the form

ℏk(ς) =
(−1)k

ς
+

∞∑
µ=1

|aµ|ςµ and gk(ς) = (−1)k
∞∑
µ=1

|bµ|ςµ, (z ∈ ♢∗) . (13)

2. Coefficient Inequalities

In our initial theorem, we establish the sufficient coefficient bounds applicable to func-
tions f within the class T MKH(k, r, α, η, δ, λ, γ).

Theorem 1. For 0 ≤ γ < 1, consider the function f = ℏ + g defined by (7), subject to
the condition

∞∑
µ=1

Γ(η)[1 + λ(µ− 1)]k

Γ(α(µ+ 1) + η)

(
µ+ δ

1 + δ

)r [
(µ+ γ) |aµ|+ (µ− γ) |bµ|

]
≤ 1− γ,

where µ ∈ 0, 1, 2, . . . . Then, f is harmonic univalent and sense-preserving in ♢∗, and
f ∈ T MKH(k, r, α, η, δ, λ, γ).

Proof. Consider the function f = ℏ+ g as defined in Equation (7), which satisfies the
inequality given in Equation (1). Assuming that 0 < |ς1| ≤ |ς2| < 1, we can conclude that

∣∣f(ς1)− f(ς2)
∣∣ ≥ |ς1 − ς2|

|ς1ς2|

1− |ς2|2
∞∑
µ=1

(|aµ|+ |bµ|)
|ς1µ − ς2

µ|
|ς1 − ς2|


≥ |ς1 − ς2|

|ς1ς2|

1− |ς2|2
∞∑
µ=1

(|aµ|+ |bµ|)
∣∣∣ςµ−1
1 + · · ·+ ςµ−1

2

∣∣∣


≥ |ς1 − ς2|
|ς1ς2|

1− |ς2|2
∞∑
µ=1

Γ(η)[1 + λ(µ− 1)]k

Γ(α(µ+ 1) + η)

(
µ+ δ

1 + δ

)r

µ(|aµ|+ |bµ|)


> |ς1 − ς2|

(
1−

∞∑
µ=1

Γ(η)[1+λ(µ−1)]k

Γ(α(µ+1)+η)

(
µ+δ
1+δ

)r {(
µ+γ
1−γ

)
|aµ|+

(
µ+γ
1−γ

)
|bµ|
} )

.



S. Ahmed, A. Alsoboh, M. Darus / Eur. J. Pure Appl. Math, 17 (3) (2024), 1894-1907 1899

By utilising condition (1), one can determine that the last expression is non-negative.
Consequently, it can be deduced that f is univalent in ♢∗. To establish that f is sense-
preserving in ♢∗, it suffices to demonstrate that |ℏ′(ς)| > |g′(ς)| using the ordinary deriva-
tive. For 0 < |ς| = r < 1, this can be inferred from the utilization of (1).

∣∣∣ℏ′
(ς)
∣∣∣ =

∣∣∣∣∣∣−1

ς2
+

∞∑
µ=1

µaµς
µ−1

∣∣∣∣∣∣ ≥
∣∣∣∣∣∣−1

ς2
+

∞∑
µ=1

µΓ(η)[1 + λ(µ− 1)]k

Γ(α(µ+ 1) + η)

(
µ+ δ

1 + δ

)r

aµς
µ−1

∣∣∣∣∣∣
≥
∣∣∣∣−1

ς2

∣∣∣∣− ∞∑
µ=1

µΓ(η)[1 + λ(µ− 1)]k

Γ(α(µ+ 1) + η)

(
µ+ δ

1 + δ

)r

|aµ||ς|µ−1

≥ 1

r

2

−
∞∑
µ=1

µΓ(η)[1 + λ(µ− 1)]k

Γ(α(µ+ 1) + η)

(
µ+ δ

1 + δ

)r

|aµ|rµ−1

≥ 1−
∞∑
µ=1

µΓ(η)[1 + λ(µ− 1)]k

Γ(α(µ+ 1) + η)

(
µ+ δ

1 + δ

)r

|aµ|

≥ 1−
∞∑
µ=1

Γ(η)[1 + λ(µ− 1)]k

Γ(α(µ+ 1) + η)

(
µ+ δ

1 + δ

)r (µ+ γ

1− γ

)
|aµ|

≥
∞∑
µ=1

Γ(η)[1 + λ(µ− 1)]k

Γ(α(µ+ 1) + η)

(
µ+ δ

1 + δ

)r (µ− γ

1− γ

)
|bµ|

>

∣∣∣∣∣∣−1

ς2
+

∞∑
µ=1

µ bµς
µ−1

∣∣∣∣∣∣ ≥
∣∣∣g′

(ς)
∣∣∣.

In order to prove that f ∈ T MKH(k, r, α, η, δ, λ, γ), it suffices to show that

Re

−
ς
(
Ik(Sα

η [r, δ, λ]ℏ(ς))
)′ − ς

(
Ik(Sα

η [r, δ, λ]g(ς)
)′

Ik
(
Sα
η [r, δ, λ]ℏ(ς)

)
+ Ik

(
Sα
η [r, δ, λ]g(ς)

) − γ

 > 0, (ς ∈ ♢∗).

Since, ℜe(ρ(ς)) > 0 if and only if
∣∣∣ρ(ς)−1
ρ(ς)+1

∣∣∣ < 1 for an analytic function ρ(ς) = 1 + c1ς +

c2ς
2 + . . . .

We let

M(ς) =

{
−ς(Ik(Sα

η [r, δ, λ]ℏ(ς)))
′
+ ς(Ik(Sα

η [r, δ, λ]g(ς))
′ − γIk(Sα

η [r, δ, λ]ℏ(ς)
−γIk(Sα

η [r, δ, λ]g(ς)

}
(14)

and

N(ς) = Ik(Sα
η [r, δ, λ]ℏ(ς) + Ik(Sα

η [r, δ, λ]g(ς). (15)

Then, we have to show that

Ξ(ς) = |M(ς) +N(ς)| − |M(ς)−N(ς)| > 0. (16)



S. Ahmed, A. Alsoboh, M. Darus / Eur. J. Pure Appl. Math, 17 (3) (2024), 1894-1907 1900

Now, by substituting equations (9) and (10) into the left-hand side of inequality (16), we
obtain

Ξ(ς) ≥


2−2γ

r − 2
∞∑
µ=1

Γ(η)[1+λ(µ−1)]k

Γ(α(µ+1)+η)

(
µ+δ
1+δ

)r (
µ+ γ

)
|aµ|rµ

−2
∞∑
µ=1

Γ(η)[1+λ(µ−1)]k

Γ(α(µ+1)+η)

(
µ+δ
1+δ

)r (
µ− γ

)
|bµ|rµ



≥ 2(1− γ)

 1−
∞∑
µ=1

Γ(η)[1+λ(µ−1)]k

Γ(α(µ+1)+η)

(
µ+δ
1+δ

)r (
µ+γ
1−γ

)
|aµ|rµ

−
∞∑
µ=1

Γ(η)[1+λ(µ−1)]k

Γ(α(µ+1)+η)

(
µ+δ
1+δ

)r (
µ−γ
1−γ

)
|bµ|rµ

 .

The positivity of this expression is assured by the fulfillment of condition (1), thereby
concluding the proof.

In the subsequent theorem, it is demonstrated that the condition (1) is necessary for
the inclusion of f in the class T MKH(k, r, α, η, δ, λ, γ).

Theorem 2. Let (0 ≤ γ < 1) and fk = ℏk + gk ∈ T MH is given by (13). Then fk ∈
T MKH(k, r, α, η, δ, λ, γ) if and only if the inequality

∞∑
µ=1

Γ(η)[1 + λ(µ− 1)]k

Γ(α(µ+ 1) + η)

(
µ+ δ

1 + δ

)r [
(µ+ γ) |aµ|+ (µ− γ) |bµ|

]
≤ 1− γ,

is satisfied.

Proof. Considering Theorem 1, it is adequate to demonstrate the validity of the ”if”
part. Assume that fk ∈ T MKH(k, r, α, η, δ, λ, γ). Then

Re

{
−
ς(Ik(Sα

η [r, δ, λ]ℏk(ς)))′ − (−1)kς(Ik(Sα
η [r, δ, λ]gk(ς)))

′ + γIk(Sα
η [r, δ, λ]ℏk(ς) + (−1)kγIk(Sα

η [r, δ, λ]gk(ς)

Ik(Sα
η [r, δ, λ]ℏk(ς) + (−1)kIk(Sα

η [r, δ, λ]gk(ς)

}

= Re


1−γ
ς

−
( ∞∑

µ=1

Γ(η)[1+λ(µ−1)]k

Γ(α(µ+1)+η)

(
µ+δ
1+δ

)r
(µ+ γ) aµςµ +

∞∑
µ=1

Γ(η)[1+λ(µ−1)]k

Γ(α(µ+1)+η)

(
µ+δ
1+δ

)r
(µ− γ) bµ ςµ

)
1
ς
+

∞∑
µ=1

Γ(η)[1+λ(µ−1)]k

Γ(α(µ+1)+η)

(
µ+δ
1+δ

)r
|aµ|ςµ −

∞∑
µ=1

Γ(η)[1+λ(µ−1)]k

Γ(α(µ+1)+η)

(
µ+δ
1+δ

)r
|bµ|ςs

 > 0.

(17)

The equation (17) must be satisfied for all ς ∈ ♢∗. When we select the value of ς on the
positive real axis, with 0 < ς = r < 1, we obtain 1− γ −

∞∑
µ=1

Γ(η)[1+λ(µ−1)]k

Γ(α(µ+1)+η)

(
µ+δ
1+δ

)r
(µ+ γ) |aµ|rµ+1

−
∞∑

µ=1

Γ(η)[1+λ(µ−1)]k

Γ(α(µ+1)+η)

(
µ+δ
1+δ

)r
(µ+ γ) |bµ|(µ+ γ)rµ+1




1 +
∞∑

µ=1
[1 + (µ− 1)λ]k

(
µ+δ
1+δ

)r
Ωµ+1(α, η)|aµ|rµ+1−

∞∑
µ=1

[1 + (µ− 1)λ]k
(

µ+δ
1+δ

)r
Ωµ+1(α, η)|bµ|rµ+1


> γ. (18)



S. Ahmed, A. Alsoboh, M. Darus / Eur. J. Pure Appl. Math, 17 (3) (2024), 1894-1907 1901

If condition (2) is not fulfilled, then as r → 1−, the numerator of (18) becomes negative.
As a result, there exists a value ς0 = r0 within the interval (0, 1) such that the left-hand side
of inequality (18) is negative. This contradicts the condition stated in (2), thus completing
the proof.

For k = 0 in Theorem 2, we have the following corollary.

Corollary 1. For f = h + g of the form (7). Then, f ∈ MKH(0, r, α, η, δ, λ, γ) if and
only if the inequality

∞∑
µ=1

Γ(η)

Γ(α(µ+ 1) + η)

(
µ+ δ

1 + δ

)r [
(µ+ γ) |aµ|+ (µ− γ) |bµ|

]
≤ 1− γ,

is satisfied.

Corollary 2. [17] For f = h + g of the form (7). Then, f ∈ MKH(0, r, α, η, δ, λ, γ) if
and only if the inequality

∞∑
µ=1

[
(µ+ γ) |aµ|+ (µ− γ) |bµ|

]
≤ 1− γ,

is satisfied.

The ensuing theorem establishes a growth property for the class T MKH(k, r, α, η, δ, λ, γ).

Theorem 3. Let fk(ς) = ℏk(ς) + gk(ς) ∈ T MKH(k, r, α, η, δ, λ, γ) of the form (13), then
we have for |z| = r < 1:

1

r
− Γ(3α+ η)(1− γ)r2

Γ(η)(1 + λ)k (2− γ)
(
2+δ
1+δ

)r ≤ |fk(ς)| ≤
1

r
+

Γ(3α+ η)(1− γ)r2

Γ(η)(1 + λ)k (2− γ)
(
2+δ
1+δ

)r .
Proof. Taking the absolute value for fk(ς) given by (13), we have

|fk(ς)| =

∣∣∣∣∣∣(−1)k

ς
+

∞∑
µ=1

aµς
µ + (−1)k

∞∑
µ=1

bµςµ

∣∣∣∣∣∣
≤ 1

r
+

∞∑
µ=1

(|aµ|+ |bk|)rµ

≤ 1

r
+

∞∑
µ=1

(|aµ|+ |bk|)r

≤ 1
r +

Γ(3α+η)(1−γ)

Γ(η)(1+λ)k( 2+δ
1+δ )

r
(2−γ)

{
∞∑
µ=1

Γ(η)[1+λ(µ−1)]k

Γ(α(µ+1)+η)

(
µ+δ
1+δ

)r [
µ+γ
1−γ |aµ|+ µ−γ

1−γ |bµ|
]}

r

≤ 1

r
+

Γ(3α+ η)(1− γ)

Γ(η)(1 + λ)k (2− γ)
(
2+δ
1+δ

)r .
The second inequality, is the same of first inequality, so the proof is omitted. This proves
the required result.



S. Ahmed, A. Alsoboh, M. Darus / Eur. J. Pure Appl. Math, 17 (3) (2024), 1894-1907 1902

3. Extreme Points

Subsequently, we identify the extreme points of the closed convex hulls of the class
T MKH(k, r, α, η, δ, λ, γ), designated as clcoT MKH.

Theorem 4. Consider a function fk = ℏk + gk in the form (13). Then, f ∈ clcoT MKH
if and only if fk,µ(ς) can be represented as

fk(ς) =
∞∑
µ=1

ϱµℏk,µ(ς) + Ψµgk,µ(ς),

where

ℏk,0(ς) =
(−1)k

ς
, ℏk,µ(ς) =

(−1)k

ς
+

Γ(α(µ+ 1) + η)(1 + δ)r(1− γ)

Γ(η)[1 + λ(µ− 1)]k(µ+ δ)r(µ+ γ)
ςµ,

and

gk,0(ς) =
(−1)k

ς
, gk,µ(ς) =

(−1)k

ς
+

Γ(α(µ+ 1) + η)(1 + δ)r(1− γ)

Γ(η)[1 + λ(µ− 1)]k(µ+ δ)r(µ− γ)
ςµ

where ϱµ ≥ 0, Ψµ ≥ 0, µ = 1, 2, . . . and
∞∑
µ=0

(ϱµ+Ψµ) = 1. The extreme points of the class

T MKH(k, r, α, η, δ, λ, γ) are {ℏk,µ} and {gk,µ}.

Proof. For f(ς) =
∞∑
µ=0

(ϱµhk,µ + Ψµgk,µ) where
∞∑
µ=0

(ϱµ + Ψµ) = 1, we have

fk(ς) = ϱ0ℏ0,µ + Ψ0g0,n +
∞∑
µ=1

(ϱµℏk,µ + Ψµgk,µ)

=
∞∑
µ=0

(−1)k(ϱµ+Ψµ)
ς +

∞∑
µ=1

ϱµ

(
Γ(α(µ+1)+η)(1+δ)r(1−γ)

Γ(η)[1+λ(µ−1)]k(µ+δ)r(µ+γ)

)
ςµ

+(−1)k
∞∑
µ=1

Ψµ

(
Γ(α(µ+1)+η)(1+δ)r(1−γ)

Γ(η)[1+λ(µ−1)]k(µ+δ)r(µ−γ)

)
ςµ

= (−1)k

ς +
∞∑
µ=1

(
Γ(α(µ+1)+η)(1+δ)r(1−γ)

Γ(η)[1+λ(µ−1)]k(µ+δ)r(µ+γ)

)
ϱµς

µ

+(−1)k
∞∑
µ=1

(
Γ(α(µ+1)+η)(1+δ)r(1−γ)

Γ(η)[1+λ(µ−1)]k(µ+δ)r(µ−γ)

)
Ψµς

µ.

This belong to T MKH(k, r, α, η, δ, λ, γ) because

∞∑
µ=1

Γ(η)[1 + λ(µ− 1)]k(µ+ δ)r(µ+ γ)

Γ(α(µ+ 1) + η)(1 + δ)r

(
Γ(α(µ+ 1) + η)(1 + δ)r(1− γ)

Γ(η)[1 + λ(µ− 1)]k(µ+ δ)r(µ+ γ)

)
ϱµ

∞∑
µ=1

Γ(η)[1 + λ(µ− 1)]k(µ+ δ)r(µ− γ)

Γ(α(µ+ 1) + η)(1 + δ)r

(
Γ(α(µ+ 1) + η)(1 + δ)r(1− γ)

Γ(η)[1 + λ(µ− 1)]k(µ+ δ)r(µ− γ)

)
Ψµ



S. Ahmed, A. Alsoboh, M. Darus / Eur. J. Pure Appl. Math, 17 (3) (2024), 1894-1907 1903

=

∞∑
µ=1

(1− γ)ϱµ + (1− γ)Ψµ = (1− γ)

∞∑
µ=1

(ϱµ + Ψµ)

= (1− γ)(1− ϱ0 − Ψ0) ≤ 1− γ.

Conversely, suppose that f ∈ clcoT MKH(k, r, α, η, δ, λ, γ). For µ = 1, 2, 3, . . . , set

ϱµ =
Γ(η)[1 + λ(µ− 1)]k(µ+ δ)r(µ+ γ)

Γ(α(µ+ 1) + η)(1 + δ)r(1− γ)
|aµ|, (0 ≤ ϱµ ≤ 1)

Ψµ =
Γ(η)[1 + λ(µ− 1)]k(µ+ δ)r(µ− γ)

Γ(α(µ+ 1) + η)(1 + δ)r(1− γ)
|bµ|, (0 ≤ Ψµ ≤ 1)

and

ϱ0 + Ψ0 = 1−
∞∑
µ=1

ϱµ −
∞∑
µ=1

Ψµ..

Therefore, fk can be written as

fk(ς) =
(−1)k

ς
+

∞∑
µ=1

|aµ|ςµ + (−1)k
∞∑
µ=1

|bµ|ςµ

=
(−1)k

ς
+

∞∑
µ=1

(
Γ(α(µ+ 1) + η)(1 + δ)r(1− γ)

Γ(η)[1 + λ(µ− 1)]k(µ+ δ)r(µ+ γ)

)
ϱµς

µ

+ (−1)k
∞∑
µ=1

(
Γ(α(µ+ 1) + η)(1 + δ)r(1− γ)

Γ(η)[1 + λ(µ− 1)]k(µ+ δ)r(µ− γ)

)
Ψµς

µ

=
ϱ0 + Ψ0

ς
+

∞∑
µ=1

(
hk,n(ς)−

(−1)k

ς

)
ϱµ +

∞∑
µ=1

(
gk,n(ς)−

(−1)k

ς

)
Ψµ

=
∞∑
µ=0

(ϱµℏk,µ + Ψµgk,µ).

4. Convex Combination and Convolution

Subsequently, we can establish that the class T MKH(k, r, α, η, δ, λ, γ) exhibits closure
properties in relation to both convolution and convex combination.

Theorem 5. For 0 ≤ β ≤ γ < 1. Let fk(ς) ∈ T MKH(k, r, α, η, δ, λ, γ) and ξk(ς) ∈
T MHMKH(k, r, α, η, δ, λ, β, n), then
(fk ∗ ξk)(ς) ∈ T MKH(k, r, α, η, δ, λ, γ) ⊆ T MKH(k, r, α, η, δ, λ, β).

Proof. The convolution, or the Hadamard product, of fk(ς) and ξk(ς) is expressed as

(fk ∗ βk)(ς) =
(−1)k

ς
+

∞∑
µ=1

|aµ||cµ|ςµ + (−1)k
∞∑
µ=1

|bµ||dµ|ςµ.



S. Ahmed, A. Alsoboh, M. Darus / Eur. J. Pure Appl. Math, 17 (3) (2024), 1894-1907 1904

We want to show that the coefficients of fk ∗ ξk satisfy condition (2). For ξk(ς) ∈
T MKH(k, r, α, η, δ, λ, β), we note that |cµ| ≤ 1 and |dµ| ≤ 1,

∞∑
µ=1

Γ(η)[1 + λ(µ− 1)]k(µ+ δ)r(µ+ β)

Γ(α(µ+ 1) + η)(1 + δ)r(1− β)
|aµ||cµ|+

∞∑
µ=1

Γ(η)[1 + λ(µ− 1)]k(µ+ δ)r(µ− β)

Γ(α(µ+ 1) + η)(1 + δ)r(1− β)
|bµ||dµ|

≤
∞∑

µ=1

Γ(η)[1 + λ(µ− 1)]k(µ+ δ)r(µ+ β)

Γ(α(µ+ 1) + η)(1 + δ)r(1− β)
|aµ|+

∞∑
µ=1

Γ(η)[1 + λ(µ− 1)]k(µ+ δ)r(µ− β)

Γ(α(µ+ 1) + η)(1 + δ)r(1− β)
|bµ|

≤
∞∑

µ=1

Γ(η)[1 + λ(µ− 1)]k(µ+ δ)r(µ+ γ)

Γ(α(µ+ 1) + η)(1 + δ)r(1− γ)
|aµ|+

∞∑
µ=1

Γ(η)[1 + λ(µ− 1)]k(µ+ δ)r(µ− γ)

Γ(α(µ+ 1) + η)(1 + δ)r(1− γ)
|bµ| ≤ 1,

since fk(ς) ∈ T MKH(k, r, α, η, δ, λ, γ) and 0 ≤ β ≤ γ < 1. Therefore, (f ∗ ξ)(ς) ∈
T MKH(k, r, α, η, δ, λ, γ) ⊆ T MKH(k, r, α, η, δ, λ, β).

Theorem 6. Let fm,k defined as

fm,k =
(−1)k

ς
+

∞∑
µ=1

|aµ,m|ςµ + (−1)k
∞∑
µ=1

|bµ,m|ςµ

be in class T MKH(k, r, α, η, δ, λ, γ) for every m = 1, 2, ..., l, then the function

ℑm(ς) =
l∑

m=1

cmfm,k(ς), (0 ≤ cm ≤ 1), (19)

are also in the class T MKH(k, r, α, η, δ, λ, γ), where
l∑

m=1
cm = 1.

Proof. According to the definition of ℑm(ς) given by (19), we can write

ℑm(ς) =
(−1)k

ς
+

∞∑
µ=1

(
l∑

m=1

cm|aµ,m|

)
ςµ + (−1)k

∞∑
µ=1

(
l∑

m=1

cm|bµ,m|

)
ςµ.

Furthermore, for every m = 1, 2, · · · , l, we have fm,k ∈ T MKH(k, r, α, η, δ, λ, γ). Then,
by (2), we have

I =


∞∑

µ=1

(
Γ(η)[1+λ(µ−1)]k(µ+δ)r(µ+γ)

Γ(α(µ+1)+η)(1+δ)r

){ l∑
m=1

cµ|aµ,m|
}
+

∞∑
µ=1

(
Γ(η)[1+λ(µ−1)]k(µ+δ)r(µ−γ)

Γ(α(µ+1)+η)(1+δ)r

){ l∑
m=1

cm|bµ,m|
}


=

l∑
m=1

cm

{ ∞∑
µ=1

(
Γ(η)[1+λ(µ−1)]k(µ+δ)r(µ+γ)

Γ(α(µ+1)+η)(1+δ)r

)
|aµ,m|+

∞∑
µ=1

(
Γ(η)[1+λ(µ−1)]k(µ+δ)r(µ−γ)

Γ(α(µ+1)+η)(1+δ)r

)
|bµ,m|

}

≤
l∑

m=1

cm(1− γ) ≤ 1− γ.

Therefore, ℑm(ς) ∈ T MKH(k, r, α, η, δ, λ, γ).

Corollary 3. The class T MKH(k, r, α, η, δ, λ, γ) is closed under convex combination.



REFERENCES 1905

Conclusion

In the current study, we have introduced and examined the coefficient issues related
to each of class MKH(k, r, α, η, δ, λ, γ), which consists of harmonic meromorphic starlike
functions. This class is utilised to describe a derivative operator that incorporates the
Mittag-Leffler function as a multiplier transformation. This study derives coefficient in-
equalities, the distortion theorem, distortion bounds, extreme points, convex combination,
and convolution for functions inside this particular class. The results obtained in this ar-
ticle can be generalised in the future using quantum calculus and other q-analogues of the
fractional derivative operator.

Acknowledgements

The second author express their gratitude to Philadelphia University-Jordan support-
ing this work, emphasising non-financial support in providing necessary resources.

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