EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 7027
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Partial Sums for Normalized Mittag-Leffler-Prabhakar
Function and Barnes-Mittag-Leffler Function

Shahid Khan1, Niaz Ali Shah1, Hijaz Ahmad2,3,4,∗, Alwaleed Kamel5,
Waleed Mohammed Abdelfattah6,7, Osama Oqilat8

1 Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad,
Pakistan
2 Sustainability Competence Centre, Széchenyi István University, Egyetem tér 1, H-9026
Győr, Hungary
3 Operational Research Center in Healthcare, Near East University, Nicosia/TRNC, 99138
Mersin 10, Turkey
4 Department of Mathematics, College of Science, Korea University, 145 Anam-ro,
Seongbuk-gu, Seoul 02841, South Korea
5 Department of Mathematics, Faculty of Science, Islamic University of Madinah,
Saudi Arabia
6 College of Engineering, University of Business and Technology, Jeddah 23435,
Saudi Arabia
7 Department of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig
University, P.O. 44519, Egypt
8 Department of Basic Sciences, Faculty of Arts and Science, Hourani Center for Applied
Scientific Research, Al-Ahliyya Amman University, Amman, Jordan

Abstract. Building on recent research that established partial sum and lower bounds for various
special functions, this paper extends the scope to investigate the normalized Le Roy-type Mittag-
Leffler-Prabhakar and Barnes-Mittag-Leffler functions. We aim to determine lower bounds for
these functions and their partial sums. We are also presenting some new consequences, lemmas,
and corollaries that highlight the significance of our findings. Our results are novel and enhance
existing knowledge in the field.

2020 Mathematics Subject Classifications: 30C45, 30C50, 30C80

Key Words and Phrases: Univalent functions, partial sums and lower bounds, Mittag-Leffler
and Barnes-Mittag-Leffler functions, Le Roy-type Mittag-Leffler function

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7027

Email address: hijaz.ahmad@neu.edu.tr (H. Ahmad)

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



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7027 2 of 23

1. Introduction

Let A be the set of analytic functions (AFs) in the disc U = {ξ ∈ C : |ξ| < 1}. Ev-
ery f ∈ A, and f is normalized if f(0) = 0 and f

′
(0) = 1 and have the Taylor series

representation

f(ξ) = ξ +
+∞∑
n=2

anξ
n. (1)

Additionally, we define S is subset of A composed of function f that are univalent (one-
to-one) in U . For any function f in A, we can form a partial sum, denoted as fm(ξ), by
its Taylor series expansion to the mth term:

fm(ξ) = ξ +
m∑

n=2
anξ

n. (2)

A function f ∈ A is considered starlike in U if its image f(U) is a star-shaped domain
and such type of starlike functions [1] denoted by S∗. These functions can be analytically
characterized by the condition:

Re

(
ξf

′
(ξ)

f(ξ)

)
> 0, ξ ∈ U.

Similarly, a function f in A is considered convex if its image f(U) is a convex [1] domain
and denoted by K. These functions can be analytically characterized by the condition:

Re

(
1 +

ξf
′′
(ξ)

f ′(ξ)

)
> 0, ξ ∈ U

if and only if
f ∈ K.

It is shown by Alexander in [2] that

ξf
′ ∈ S∗

if and only if
f ∈ K.

For a function f defined in U , the integral transformation I [f ] is given by the expression

I [f ] =
z∫
0

f(t)
t dt which can be expanded as:

I [f ] = ξ +
+∞∑
n=2

an
n
ξn. (3)

This transformation is known as the Alexander Transformation, named after Alexander
[2], who first introduced it. Alexander made a significant discovery, proving that this



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7027 3 of 23

integral transformation I [f ] provides a one-to-one correspondence between the class S∗

and K.
The generalized Pochhammer symbol for x > 0 is given by

(x)n =

1, n = 0,

x(x+ 1) · · · (x+ n− 1), n ∈ N.

and for x > 0, the Gamma function is defined as:

Γ(x+ 1) = xΓ(x), and Γ(1) = 1.

The theory of special functions (SFs) is the useful area of mathematics that has evolved
over the last three centuries, driven by the need to solve problems in classical mechan-
ics, hydrodynamics, and control theory. This development has led to extensive applica-
tions in both pure and applied mathematics, including geometric function theory, applied
mathematics, physics, and statistics (see literature [3–11] for further reading). One no-
table example of a SFs is the Mittag-Leffler function, which emerges naturally in solv-
ing fractional-order integral and differential equations [12–14]. Its relevance extends to
studying fractional generalizations of kinetic equations, random walks, Lévy flights, and
superdiffusive transport, particularly in complex systems. The Mittag-Leffler function
bridges the gap between exponential and power-law behaviors, characteristic of phenom-
ena governed by classical and fractional kinetic equations, as explored in the works of
Lang [15], Hilfer [16], and Saxena [17]. Building on these findings, the study of geometric
properties in analytic functions involving special functions remains a vibrant and ongoing
field of research, with notable contributions from researchers such as Aktas [18], Aktas
and Orhan [19], and Bansal and Prajapat [20].

The Swedish mathematician Mittag-Leffler, who originally introduced the one-parameter
version of Mittag-Leffler function (MLF) [21] as:

Ed (ξ) =
+∞∑
n=0

1

Γ (dn+ 1)
ξn, d, ξ ∈ C, Re (d) > 0.

The two-parameter MLF, denoted as Ed,v (ξ), is a mathematical function that is defined
by Wiman in [22]. The series form of two-parameter Mittag-Leffler function is given as:

Ed,v (ξ) =
+∞∑
n=0

1

Γ (dn+ v)
ξn, d, v, ξ ∈ C, Re (d) > 0.

The Le Roy function denoted by Eσ (ξ) , is defined by French mathematician Édouard Le
Roy (see [23]) as follows:

Eσ (ξ) =
+∞∑
n=0

ξn

(Γ (n+ 1))σ
=

+∞∑
n=0

1

(n!)σ
ξn, ξ ∈ C, (4)



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7027 4 of 23

where σ is a positive real number. In recent work, Gerhold [24] and Garra and Polito [25]
separately developed the Le Roy-type MLF, which is given by the following definition:

Eσ
d,v (ξ) =

+∞∑
n=0

1

(Γ (dn+ v))σ
ξn, d, v, σ > 0, ξ ∈ C.

The Barnes–Mittag-Leffler function Bb,s
d,v (ξ) is defined by [26] as follows:

Bb,s
d,v (ξ) =

+∞∑
n=0

1

(n+ b)s Γ (dn+ v)
ξn.

In 2017, Tomovski, Mehrez [27] considered the Mittag-Leffler Prabhakar functions (MLPF)
of Le Roy-type defined as:

Eσ,χ
d,v (ξ) =

+∞∑
n=0

Γ (χ+ n)

n!Γ (χ) (Γ (dn+ v))σ
ξn, d, v, σ, χ > 0, ξ ∈ C. (5)

A special case of the function Eσ,χ
d,v (ξ) arises when χ = d = v = 1, which simplifies to the

Le Roy-type function (LRFs) defined in [23] and further investigated by Mehrez and Das
in their work [28]. This function, denoted as Eσ (ξ), and defined in (4).

The MLPF of Le Roy type and their generalizations have found applications in frac-
tional calculus, as discussed in Pane’s work [29]. In geometric function theory, Mehraz
and Raza [30] determined the conditions under which Mittag-Leffler-Prabhakar functions
of Le Roy type possess key geometric properties, including starlikeness, convexity, and
pre-starlikeness, by imposing specific constraints on their parameters. For a deeper un-
derstanding of the advanced properties and applications of these functions, including their
fractional integration and differentiation formulas, solutions to differential equations, in-
tegral transforms, and other uses, we refer to recent publications [6, 7].

Now we demonstrate that Le Roy-type Mittag-Leffler-Prabhakar functions (denoted
by Fχ,σ

d,v (ξ)) belong to class A.

Fχ,σ
d,v (ξ) = ξ (Γ (v))σ Eσ,χ

d,v (ξ)

=
+∞∑
n=0

Γ (χ+ n) (Γ (v))σ

n!Γ (χ) (Γ (dn+ v))σ
ξn+1, ξ ∈ C, (6)

where d, v, σ, χ > 0. Now Fχ,σ
d,v (ξ) satisfies the normalization conditions, namely Fχ,σ

d,v (0) =

0 and
(
Fχ,σ
d,v

)′

(0) = 1. Using the Stirling asymptotic formula for the gamma function,

which applies to large values of ξ, we find that the series in equation (6) represents an
entire function ( meaning it converges absolutely for all complex numbers ξ if dσ > 0.

The functions given in equations (5) and (6) provide a extended version that encom-
passes many well-known special functions found in the mathematical literature. Interest-
ingly, these functions overlap with several well-known functions when specific parameter
values are applied. For instance, when χ = 1 in (6), then Fχ,σ

d,v (ξ) reduces to the normal-

ized version of Le Roy-type Mittag-Leffler function E1,σ
d,v (ξ) , which was first defined by



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7027 5 of 23

Garra and Polito in [25]. Further research on the related function E1,σ
d,v (ξ) was conducted

by authors in [28].
Additionally, when σ = 1 in (6), then, the function Fχ,σ

d,v (ξ) becomes the normalized

version of three-parameter Mittag-Leffler function Eχ,1
d,v (ξ), introduced by Prabhakar [8].

Recently, Garro and Gorropo [31] investigated the applications of the Prabhakar or three-
parameter Mittag-Leffler function, focusing on nonlinear heat conduction equations with
memory that involve Prabhakar derivatives. They derived exact solutions and analyzed
the asymptotic behavior of these equations.

Moreover, when χ = σ = 1 in (6), then, the function Fχ,σ
d,v (ξ) reduces to the nor-

malized version of two-parameter Mittag-Leffler function Ed,v (ξ) defined by Wiman [22],
and extensively studied in [9] (see [32]). Finally, using χ = v = σ = 1, in (6), then the
function Fχ,σ

d,v (ξ) becomes the normalized version of Mittag-Leffler function Ed (ξ), defined

and studied by Mittag-Leffler [21] in 1903. We also observed that the function Fχ,σ
d,v (ξ)

contains many well-known functions as its special case, for example

F1,1
3,1 (ξ) =

eξ
1
3

2
+ e−

1
2
ξ
1
3 cos

√
3

2
ξ

1
3 , F1,1

4,1 (ξ) =
cos ξ

1
4

2
+

cosh ξ
1
4

2
,

F1,1
2,1

(
ξ2
)
= cosh ξ, F1,1

2,1

(
−ξ2

)
= cos ξ, F1,1

1,1 (ξ) = eξ, F1,1
1,2 (ξ) =

eξ − 1

ξ
,

F1,1
2,2

(
ξ2
)
=

sinh ξ

ξ
, F1,1

2,2

(
−ξ2

)
=

sin ξ

ξ
.

Similarly, we perform the normalized Barnes–Mittag-Leffler function, denoted asBM b,s
d,v (ξ),

which is defined as follows:

BM b,s
d,v (ξ) = ξbsΓ (v)Bb,s

d,v (ξ)

=
+∞∑
n=0

bsΓ (v)

(n+ b)s Γ (dn+ v)
ξn+1, (7)

where ( d, v, b, s > 0, ξ ∈ C) .
Partial sums of analytic functions play a significant role in Geometric Function Theory,

particularly in finding the largest disk Ur = {ξ ∈ C : |ξ| < r} where the partial sum remains
one-to-one. In 1928, Szegö [33] proved that for functions in the class S, each partial sum,
fm(ξ), is one-to-one within the disk U 1

4
=
{
ξ ∈ C : |ξ| < 1

4

}
. However, this does not mean

that partial sums of functions in S are always one-to-one in U . For example, the convex
univalent function f(ξ) = ξ/1− ξ shows that this is not the case. Furthermore, the second
partial sum f2(ξ) = ξ + 2ξ2 of the Koebe function

k(ξ) =
ξ

(1− ξ)2

is one-to-one within U 1
4
, and radius 1

4 is the best possible.

The radius of starlikeness of the partial sum (fm (ξ)) of functions in the class S∗

was established by Robertson [34]. Moreover, several researchers have investigated the



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7027 6 of 23

lower bound of the real part of the ratio of the partial sum of analytic functions to their
infinite series sum. This concept was first introduced by Silvia in [35]. Silverman [36] later
developed more useful techniques to find the partial sum of starlike and convex functions.
Subsequent studies have extended these results to various subclasses of analytic functions,
as seen in references [37–40]. Recently, researchers have explored partial sum of special
functions, such as the normalized Struve functions [41], Dini functions [42], and Wright
functions [43] while Kazımoğlu [44] have made significant contributions to this area.

The sequence of partial sums of Fχ,σ
d,v (ξ) is defined as:

(
Fχ,σ
d,v (ξ)

)
m
(ξ) = ξ +

m∑
n=1

Γ (χ+ n) (Γ (v))σ

n!Γ (χ) (Γ (dn+ v))σ
ξn+1, m ∈ N. (8)

Similarly, the sequence of partial sums of BM b,s
d,v (ξ) is defined as:

(
BM b,s

d,v (ξ)
)
m
(ξ) = ξ +

m∑
n=1

bsΓ (v)

(n+ b)s Γ (dn+ v)
ξn+1, m ∈ N. (9)

If m = 0, we have
0∑

n=1

Γ (χ+ n) (Γ (v))σ

n!Γ (χ) (Γ (dn+ v))σ
ξn+1 = 0

and
0∑

n=1

bsΓ (v)

(n+ b)s Γ (dn+ v)
ξn+1 = 0.

In this paper, we investigate the ratio of a function, defined by (6) and (7), to its sequence
of partial sums, given by (8) and (9), and establish lower bounds for

Re

 Fχ,σ
d,v (ξ)(

Fχ,σ
d,v

)
m
(ξ)

 , Re


(
Fχ,σ
d,v

)
m
(ξ)

Fχ,σ
d,v (ξ)

 , (10)

Re


(
Fχ,σ
d,v (ξ)

)′

(
Fχ,σ
d,v

)′

m
(ξ)

 , Re


(
Fχ,σ
d,v

)′

m
(ξ)(

Fχ,σ
d,v (ξ)

)′

 , (11)

Re

 I
(
Fχ,σ
d,v

)
(ξ)(

I
(
Fχ,σ
d,v

))
m
(ξ)

 , Re


(
I
(
Fχ,σ
d,v

))
m
(ξ)

I
(
Fχ,σ
d,v

)
(ξ)

 (12)

and

Re

 BM b,s
d,v (ξ)(

BM b,s
d,v

)
m
(ξ)

 , Re


(
BM b,s

d,v

)
m
(ξ)

BM b,s
d,v (ξ)

 , (13)



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7027 7 of 23

Re


(
BM b,s

d,v (ξ)
)′

(
BM b,s

d,v

)′

m
(ξ)

 , Re


(
BM b,s

d,v

)′

m
(ξ)(

BM b,s
d,v (ξ)

)′

 , (14)

Re

 I
(
BM b,s

d,v

)
(ξ)(

I
(
BM b,s

d,v

))
m
(ξ)

 , Re


(
I
(
BM b,s

d,v

))
m
(ξ)

I
(
BM b,s

d,v

)
(ξ)

 . (15)

The paper is organized as follows. Section 1 provides a comprehensive introduction,
covering the historical background, preliminary concepts, Mittag-Leffler-Prabhakar func-
tions of Le Roy type and Barnes-Mittag-Leffler function. It also presents the normalization
of these functions within the unit disk and discusses some special cases in this section. Sec-
tion 2 introduces a set of known and new lemmas essential for proving the main results.
Section 3 is divided into two parts: the first part investigates Theorems 1-3 related to
the normalized Le Roy-type Mittag-Leffler-Prabhakar function defined in (6), along with
several specific cases, while the second part examines Theorems 4-6 related to the normal-
ized Barnes-Mittag-Leffler function defined in (7). Finally, the last section discusses the
conclusions and future directions of the main work.

2. Basic concepts or Preliminaries

The following lemmas are necessary to investigate the main results for Fχ,σ
d,v (ξ) and

BM b,s
d,v (ξ) defined in (6) and (7).

Lemma 1. ([30], proof of Theorem 3.1 on page 747). Assume that d, v, χ, and σ are
arbitrary positive numbers.
(i) If d ≥ 1, dσ ≥ 1 and v ≥ χ, then the sequence (bn)n≥1 defined by

bn (d, v, χ, σ) =
Γ (χ+ n)

Γ (χ)

(
Γ (v)

Γ (dn+ v)

)σ

is decreasing.
(ii)Also,

cn (d, v, χ, σ) = (n+ 1) bn (d, v, χ, σ)

is decreasing.

Lemma 2. Assume that d, v, χ, and σ are arbitrary positive numbers.
(i) If d ≥ 1, dσ ≥ 1 and v ≥ χ, then∣∣∣Fχ,σ

d,v (ξ)
∣∣∣ ≤ 1 +B1 (e− 1) , ξ ∈ U.

(ii) ∣∣∣∣(Fχ,σ
d,v (ξ)

)′∣∣∣∣ ≤ 1 + 2B1 (e− 1) , ξ ∈ U.



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7027 8 of 23

(iii) ∣∣∣I [(Fχ,σ
d,v (ξ)

)]
(ξ)
∣∣∣ ≤ 1 +B1 (e− 2) , ξ ∈ U,

where

B1 = χ

(
Γ (v)

Γ (d+ v)

)σ

. (16)

Proof. (i) If d ≥ 1, dσ ≥ 1 and v ≥ χ and let

Bn =
bn (d, v, χ, σ)

n!
and B0 = 1. (17)

Thus, according to the Lemma 1, we see that∣∣∣∣+∞∑
n=0

Bnξ
n+1

∣∣∣∣ = ∣∣∣∣ξ + +∞∑
n=1

bn (d, v, χ, σ)

n!
ξn+1

∣∣∣∣
≤ 1 + b1 (d, v, χ, σ)

+∞∑
n=1

1

n!

= 1 + (e− 1)B1. (18)

Now from (6) and using (18) we have∣∣∣Fχ,σ
d,v (ξ)

∣∣∣ = ∣∣∣∣+∞∑
n=0

1

n!

Γ (χ+ n)

Γ (χ)

(
Γ (v)

Γ (dn+ v)

)σ∣∣∣∣ = ∣∣∣∣+∞∑
n=0

Bnξ
n+1

∣∣∣∣
≤ 1 + (e− 1)B1.

(ii) If d ≥ 1, dσ ≥ 1 and v ≥ χ, and let

Cn =
cn (d, v, χ, σ)

n!
and C0 = 1.

Thus, according to the Lemma 1, we see that∣∣∣∣+∞∑
n=0

Cnξ
n+1

∣∣∣∣ = ∣∣∣∣ξ + +∞∑
n=1

cn (d, v, χ, σ)

n!
ξn+1

∣∣∣∣
≤ 1 + C1

+∞∑
n=1

1

n!

= 1 + 2B1 (e− 1) . (19)

Now from (6) and using (19) we have∣∣∣∣(Fχ,σ
d,v (ξ)

)′∣∣∣∣ = ∣∣∣∣+∞∑
n=0

(n+ 1)
1

n!

Γ (χ+ n)

Γ (χ)

(Γ (v))σ

(Γ (dn+ v))σ
ξn
∣∣∣∣

=

∣∣∣∣+∞∑
n=0

Cnξ
n

∣∣∣∣ ≤ 1 + 2 (e− 1)B1.



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(iii) If d ≥ 1, dσ ≥ 1 and v ≥ χ, then from (6) and using (18) we have∣∣∣I [Fχ,σ
d,v (ξ)

]
(ξ)
∣∣∣ = ∣∣∣∣ξ + +∞∑

n=1

1

(n+ 1)

1

n!

Γ (χ+ n)

Γ (χ)

(
Γ (v)

Γ (dn+ v)

)σ

ξn+1

∣∣∣∣
≤ 1 + b1 (d, v, χ, σ)

+∞∑
n=1

1

(n+ 1)!

= 1 +B1 (e− 2) .

Hence, proof of Lemma 2 have completed.

Lemma 3. ([45], proof of Theorem 4 on page 8). Assume that d, v, b, and s are arbitrary
positive numbers.
(i): If min(b, d, v) > 0 and s ≥ 0. Then the sequence (Ψn (d, v, b, s))n≥1 defined by

Ψn (d, v, b, s) = n!

(
bsΓ (v)

(n+ b)s Γ (dn+ v)

)
is decreasing.

(ii) If min(b, d, v) > 0 and s ≥ 0. Then the sequence
(
Ψ̂n (d, v, b, s)

)
n≥1

defined by

Ψ̂n (d, v, b, s) = (n+ 1)!

(
bsΓ (v)

(n+ b)s Γ (dn+ v)

)
is also decreasing.

Lemma 4. Assume that d, v, b, and s are arbitrary numbers.
(i) If min(b, d, v) > 0 and s ≥ 0, then∣∣∣BM b,s

d,v (ξ)
∣∣∣ ≤ 1 +D1 (e− 1) , ξ ∈ U.

(ii) ∣∣∣∣(BM b,s
d,v (ξ)

)′∣∣∣∣ ≤ 1 + 2D1 (e− 1) , ξ ∈ U.

(iii) ∣∣∣I [BM b,s
d,v (ξ)

]
(ξ)
∣∣∣ ≤ 1 +D1 (e− 2) , ξ ∈ U,

where

D1 =
bsΓ (v)

(1 + b)s Γ (d+ v)
. (20)

Proof. Arguments used in the proof of Lemma 2 work also in the frame of Lemma 4
with using the unified Lemma 3 instead of Lemma 1.



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7027 10 of 23

3. Main Results

In this section, we investigate Theorems related to the normalized Le Roy-type Mittag-
Leffler-Prabhakar function defined in (6).

Theorem 1. If d ≥ 1, dσ ≥ 1 and v ≥ χ and B1 (e− 1) ≤ 1, then

Re

 Fχ,σ
d,v (ξ)(

Fχ,σ
d,v

)
m
(ξ)

 ≥ 1−B1 (e− 1) , ξ ∈ U (21)

and

Re


(
Fχ,σ
d,v

)
m
(ξ)

Fχ,σ
d,v (ξ)

 ≥ 1

1 +B1 (e− 1)
, ξ ∈ U, (22)

where B1 is given by (16).

Proof. First, we recall the inequality (i) of Lemma 2, that is∣∣∣Fχ,σ
d,v (ξ)

∣∣∣ ≤ 1 +B1 (e− 1) , ξ ∈ U. (23)

Using (6) in (23), we get∣∣∣∣ξ + +∞∑
n=1

Γ (χ+ n) (Γ (v))σ

n!Γ (χ) (Γ (dn+ v))σ
ξn+1

∣∣∣∣ ≤ 1 +B1 (e− 1) , ξ ∈ U.

Further, we have

1 +

+∞∑
n=1

|Bn| ≤ 1 +B1 (e− 1) , ξ ∈ U.

Or equivalently

1

B1 (e− 1)

+∞∑
n=1

|Bn| ≤ 1,

where Bn is given by (17).
To establish the inequality (21), we set

1

B1 (e− 1)

 Fχ,σ
d,v (ξ)(

Fχ,σ
d,v

)
m
(ξ)

− (1−B1 (e− 1))


=

1 +
∑m

n=1Bnξ
n + 1

B1(e−1)

∑+∞
n=m+1Bnξ

n

1 +
∑m

n=1Bnξn
=

1 + h1 (ξ)

1 + h2 (ξ)
. (24)

where

h1(ξ) =

m∑
n=1

Bnξ
n +

1

B1 (e− 1)

+∞∑
n=m+1

Bnξ
n. (25)



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7027 11 of 23

and

h2(ξ) =
m∑

n=1

Bnξ
n. (26)

Now we consider
1 + h1 (ξ)

1 + h2 (ξ)
=

1 + u(ξ)

1− u(ξ)
.

After some simplification, we have

u(ξ) =
h1 (ξ)− h2 (ξ)

2 + h1 (ξ) + h2 (ξ)
.

Thus, clearly, we have

u(ξ) =

1
B1(e−1)

∑+∞
n=m+1Bnξ

n

2 + 2
∑m

n=1Bnξn + 1
B1(e−1)

∑+∞
n=m+1Bnξn

.

Thus, we have

|u(ξ)| ≤
1

B1(e−1)

∑+∞
n=m+1 |Bn|

2− 2
∑m

n=1 |Bn| − 1
B1(e−1)

∑+∞
n=m+1 |Bn|

.

A well-known fact states that the following equivalence is hold:

Re

(
1 + u(ξ)

1− u(ξ)

)
≥ 0, ξ ∈ U ⇔ |u(ξ)| ≤ 1, ξ ∈ U.

We can now see that |u(ξ)| ≤ 1 follows once we prove

1

B1 (e− 1)

+∞∑
n=m+1

|Bn| ≤ 1−
m∑

n=1

|Bn| .

This is equivalent to the inequality

m∑
n=1

|Bn|+
1

B1 (e− 1)

+∞∑
n=m+1

|Bn| ≤ 1. (27)

Our goal is to prove that the left-hand side of inequality (27) is bounded above by

1

B1 (e− 1)

+∞∑
n=1

|Bn| .

After some simple calculations, we have(
1

B1 (e− 1)
− 1

) m∑
n=1

|Bn| ≥ 0. (28)



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7027 12 of 23

Thus, by virtue of (28), the proof of the inequality in (21) is now complete.
Next, to prove (22), we set

(
1 +

1

B1 (e− 1)

)
(
Fχ,σ
d,v

)
m
(ξ)

Fχ,σ
d,v (ξ)

− 1

1 +B1 (e− 1)


=

1 +
∑m

n=1Bnξ
n + 1

B1(e−1)

∑+∞
n=m+1Bnξ

n

1 +
∑+∞

n=1Bnξn

=
1 + h1 (ξ)

1 + h2 (ξ)
.

After some simplification, we have

u(ξ) =
h1 (ξ)− h2 (ξ)

2 + h1 (ξ) + h2 (ξ)
.

Thus, clearly, we have

|u(ξ)| ≤

(
1 + 1

B1(e−1)

)∑+∞
n=m+1 |Bn|

2− 2
∑m

n=1 |Bn| −
(

1
B1(e−1) − 1

)∑+∞
n=m+1 |Bn|

.

We can now see that |u(ξ)| ≤ 1 follows once we prove

1

B1 (e− 1)

+∞∑
n=m+1

|Bn| ≤ 1−
m∑

n=1

|Bn| .

This is equivalent to the inequality

m∑
n=1

|Bn|+
1

B1 (e− 1)

+∞∑
n=m+1

|Bn| ≤ 1. (29)

Our goal is to prove that the left-hand side of inequality (29) is bounded above by

1

B1 (e− 1)

+∞∑
n=1

|Bn| .

Alternatively, (
1

B1 (e− 1)
− 1

) +∞∑
n=1

|Bn| ≥ 0. (30)

Thus, by virtue of (30), the proof of the inequality in (22) is now complete. Hence, this
completes the proof of the Theorem 1.



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7027 13 of 23

Theorem 2. If d ≥ 1, dσ ≥ 1 and v ≥ χ and 1 ≥ 2B1 (e− 1) , then

Re


(
Fχ,σ
d,v (ξ)

)′

(
Fχ,σ
d,v

)′

m
(ξ)

 ≥ 1− 2B1 (e− 1) , ξ ∈ U (31)

and

Re


(
Fχ,σ
d,v

)′

m
(ξ)(

Fχ,σ
d,v (ξ)

)′

 ≥ 1

1 + 2B1 (e− 1)
, ξ ∈ U, (32)

where B1 is defined by (16).

Proof. First, we recall the inequality (ii) of Lemma 2, that is∣∣∣∣(Fχ,σ
d,v (ξ)

)′∣∣∣∣ ≤ 1 + 2B1 (e− 1) , ξ ∈ U. (33)

Using (6) in (33), we have∣∣∣∣1 + +∞∑
n=1

(n+ 1)
Γ (χ+ n) (Γ (v))σ

n!Γ (χ) (Γ (dn+ v))σ
ξn
∣∣∣∣

≤ 1 +
+∞∑
n=1

(n+ 1) |Bn| ≤ 1 + 2B1 (e− 1) , ξ ∈ U.

Or equivalently

1

2B1 (e− 1)

+∞∑
n=1

(n+ 1) |Bn| ≤ 1,

where Bn is given (17). To establish the inequality (31), we set

1

2B1 (e− 1)


(
Fχ,σ
d,v (ξ)

)′

(
Fχ,σ
d,v

)′

m
(ξ)

− (1− 2B1 (e− 1))


=

1 +
∑m

n=1 (n+ 1)Bnξ
n + 1

2B1(e−1)

∑+∞
n=m+1 (n+ 1)Bnξ

n

1 +
∑m

n=1 (n+ 1)Bnξn

=
1 + h1 (ξ)

1 + h2 (ξ)
.

We can write

u(ξ) =

1
2B1(e−1)

∑+∞
n=m+1 (n+ 1)Bnξ

n

2 + 2
∑m

n=1 (n+ 1)Bnξn + 1
2B1(e−1)

∑+∞
n=m+1 (n+ 1)Bnξn

.



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7027 14 of 23

Thus, we get

|u(ξ)| ≤
1

2B1(e−1)

∑+∞
n=m+1 (n+ 1) |Bn|

2− 2
∑m

n=1 (n+ 1) |Bn| − 1
2B1(e−1)

∑+∞
n=m+1 (n+ 1) |Bn|

.

We can now see that |u(ξ)| ≤ 1 follows once we prove

1

2B1 (e− 1)

+∞∑
n=m+1

(n+ 1) |Bn| ≤ 1−
m∑

n=1

(n+ 1) |Bn| .

This is equivalent to the inequality

m∑
n=1

(n+ 1) |Bn|+
1

2B1 (e− 1)

+∞∑
n=m+1

(n+ 1) |Bn| ≤ 1. (34)

Our goal is to prove that the left-hand side of inequality (34) is bounded above by

1

2B1 (e− 1)

+∞∑
n=1

(n+ 1) |Bn| .

Alternatively, (
1

2B1 (e− 1)
− 1

) m∑
n=1

(n+ 1) |Bn| ≥ 0. (35)

Thus, by virtue of (35), the proof of the inequality in (31) is now complete. To establish
inequality (32), consider the expression

(
1 +

1

2B1 (e− 1)

)
(
Fχ,σ
d,v

)′

m
(ξ)(

Fχ,σ
d,v (ξ)

)′ − 1

1 + 2B1 (e− 1)


=

1 +
∑m

n=1Bn (n+ 1) ξn +
(
1 + 1

2B1(e−1)

)∑+∞
n=m+1 (n+ 1)Bnξ

n

1 +
∑m

n=1 (n+ 1)Bnξn

=
1 + h1 (ξ)

1 + h2 (ξ)
.

We can write

u(ξ) =

(
1 + 1

2B1(e−1)

)∑+∞
n=m+1 |Bn| (n+ 1)

2 + 2
∑m

n=1 (n+ 1) |Bn|+
(

1
2B1(e−1) − 1

)∑+∞
n=m+1 |Bn| (n+ 1)

.

Thus, we have

|u(ξ)| ≤

(
1 + 1

2B1(e−1)

)∑+∞
n=m+1 (n+ 1) |Bn|

2− 2
∑m

n=1 (n+ 1) |Bn| −
(

1
2B1(e−1) − 1

)∑+∞
n=m+1 (n+ 1) |Bn|

.



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7027 15 of 23

We can now see that |u(ξ)| ≤ 1 follows once we prove

m∑
n=1

(n+ 1) |Bn|+
1

2B1 (e− 1)

+∞∑
n=m+1

(n+ 1) |Bn| ≤ 1. (36)

Therefore, we can see that the left hand side of (36) is bounded above by

1

2B1 (e− 1)

+∞∑
n=1

(n+ 1) |Bn| .

Alternatively, (
1

2B1 (e− 1)
− 1

) m∑
n=1

(n+ 1) |Bn| ≥ 0. (37)

Thus, by virtue of (37), the proof of the inequality in (32) is now complete.

Remark 1. If m = 0 in (31), we find that Re
(
Fχ,σ
d,v (ξ)

)′

> 0. So by Noshiro-Warschawski

Theorem (see [46]), we conclude that the normalized Le Roy-type MLF is univalent in the
unit disk U for

1 ≥ 2B1 (e− 1) ,

where d ≥ 1, dσ ≥ 1 and v ≥ χ.

Theorem 3. If d ≥ 1, dσ ≥ 1 and v ≥ χ and 1 ≥ B1 (e− 2) , then

Re

 I
(
Fχ,σ
d,v

)
(ξ)(

I
(
Fχ,σ
d,v

))
m
(ξ)

 ≥ 1−B1 (e− 2) , ξ ∈ U (38)

and

Re


(
I
(
Fχ,σ
d,v

))
m
(ξ)

I
(
Fχ,σ
d,v

)
(ξ)

 ≥ 1

1 +B1 (e− 2)
, ξ ∈ U. (39)

Proof. To establish equation (38), first, we recall the inequality (iii) of Lemma 2, that
is ∣∣∣I [(Fχ,σ

d,v (ξ)
)]

(ξ)
∣∣∣ ≤ 1 +B1 (e− 2) , ξ ∈ U. (40)

Using (6) in (40), we get∣∣∣∣I [ξ + +∞∑
n=1

Γ (χ+ n) (Γ (v))σ

n!Γ (χ) (Γ (dn+ v))σ
ξn+1

]
(ξ)

∣∣∣∣
=

∣∣∣∣ξ + +∞∑
n=1

1

(n+ 1)

Γ (χ+ n) (Γ (v))σ

n!Γ (χ) (Γ (dn+ v))σ
ξn+1

∣∣∣∣



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7027 16 of 23

≤ 1 +
+∞∑
n=1

1

n+ 1
|Bn| ≤ 1 +B1 (e− 2) , ξ ∈ U.

Alternatively

1

B1 (e− 2)

+∞∑
n=1

1

n+ 1
|Bn| ≤ 1,

where Bn is given (17). Now, we set

1

B1 (e− 2)

 I
(
Fχ,σ
d,v

)
(ξ)(

I
(
Fχ,σ
d,v

))
m
(ξ)

− (1−B1 (e− 2))


=

1 +
∑m

n=1
1

n+1Bnξ
n + 1

B1(e−2)

∑+∞
n=m+1

1
n+1Bnξ

n

1 +
∑m

n=1
1

n+1Bnξn

=
1 + h1 (ξ)

1 + h2 (ξ)
.

We can write

u(ξ) =

1
B1(e−2)

∑+∞
n=m+1

1
n+1Bnξ

n

2 + 2
∑m

n=1
1

n+1Bnξn + 1
B1(e−2)

∑+∞
n=m+1

1
n+1Bnξn

,

thus, we have

|u(ξ)| ≤
1

B1(e−2)

∑+∞
n=m+1

1
n+1 |Bn|

2− 2
∑m

n=1
1

n+1 |Bn| − 1
B1(e−2)

∑+∞
n=m+1

1
n+1 |Bn|

.

We can now see that |u(ξ)| ≤ 1 follows once we prove

1

B1 (e− 2)

+∞∑
n=m+1

1

n+ 1
|Bn| ≤ 1−

m∑
n=1

1

n+ 1
|Bn| .

This gives us
m∑

n=1

1

n+ 1
|Bn|+

1

B1 (e− 2)

+∞∑
n=m+1

1

n+ 1
|Bn| ≤ 1. (41)

It is enough to demonstrate that the inequality (41) is bounded above by

1

B1 (e− 2)

+∞∑
n=1

1

n+ 1
|Bn| .

Alternatively, (
1

B1 (e− 2)
− 1

) m∑
n=1

1

n+ 1
|Bn| ≥ 0. (42)



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7027 17 of 23

Thus, by virtue of (42), the proof of the inequality in (38) is now complete.
To prove (39), we set

(
1 +

1

B1 (e− 2)

)
(
I
(
Fχ,σ
d,v

))
m
(ξ)

I
(
Fχ,σ
d,v

)
(ξ)

− 1

1 +B1 (e− 2)


=

1 +
∑m

n=1
1

n+1Bnξ
n +

(
1 + 1

B1(e−2)

)∑+∞
n=m+1

1
n+1Bnξ

n

1 +
∑m

n=1
1

n+1Bnξn
=

1 + h1 (ξ)

1 + h1 (ξ)
.

We can write

u(ξ) =

(
1 + 1

B1(e−2)

)∑+∞
n=m+1

1
n+1 |Bn|

2 + 2
∑m

n=1
1

n+1 |Bn|+
(

1
B1(e−2) − 1

)∑+∞
n=m+1

1
n+1 |Bn|

.

Thus, we have

|u(ξ)| ≤

(
1 + 1

B1(e−2)

)∑+∞
n=m+1 |Bn| 1

n+1

2− 2
∑m

n=1
1

n+1 |Bn| −
(

1
B1(e−2) − 1

)∑+∞
n=m+1 |Bn| 1

n+1

.

We can now see that |u(ξ)| ≤ 1 follows once we prove

m∑
n=1

1

n+ 1
|Bn|+

1

B1 (e− 2)

+∞∑
n=m+1

1

n+ 1
|Bn| ≤ 1. (43)

It is enough to demonstrate that the inequality (43) is bounded above by

1

B1 (e− 2)

+∞∑
n=1

1

n+ 1
|Bn| .

Alternatively, (
1

B1 (e− 2)
− 1

) m∑
n=1

1

n+ 1
|Bn| ≥ 0. (44)

Thus, by virtue of (44), the proof of the inequality in (39) is now complete.

By substituting the values m = 0, γ = 1, σ = 1, d = 3 and v = 1 into Theorem 1, we
arrive at the following consequence:

Example 1. We have the following inequalities:

Re

eξ
1
3 + 2e−

1
2
ξ
1
3 cos

(√
3
2 ξ

1
3

)
2

 ≥ 7− e

6



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7027 18 of 23

and

Re

 2

eξ
1
3 + 2e−

1
2
ξ
1
3 cos

(√
3
2 ξ

1
3

)
 ≥ 6

5 + e
.

By substituting the values m = 0, γ = 1, σ = 1, d = 3 and v = 1 in Theorem 2, we
arrive at the following consequence:

Example 2. We have the following inequalities:

Re

eξ
1
3 −

√
3e−

1
2
ξ
1
3 sin

(√
3
2 ξ

1
3

)
− e−

1
2
ξ
1
3 cos

(√
3
2 ξ

1
3

)
6τ

2
3

 ≥ 4− e

3

and

Re

 6τ
2
3

eξ
1
3 −

√
3e−

1
2
ξ
1
3 sin

(√
3
2 ξ

1
3

)
− e−

1
2
ξ
1
3 cos

(√
3
2 ξ

1
3

)
 ≥ 3

2 + e
.

By substituting the values m = 0, γ = 1, σ = 1, d = 1 and v = 1 in Theorem 3, we
arrive at the following consequence:

Example 3. For γ = 1, σ = 1, d = 1 and v = 1, then

F1,1
1,1 (ξ) = τeξ

and for m = 0, γ = 1, σ = 1, d = 1 and v = 1, then
(
F1,1
1,1

)
0
(ξ) = ξ. Thus

I
[
F1,1
1,1 (ξ)

]
=

ξ∫
0

etdt = eξ − 1

and

I
[(

F1,1
1,1

)
0

]
=

ξ∫
0

dt = ξ.

Therefore by Theorem 3, we have

Re

(
eξ − 1

ξ

)
≥ 3− e

and

Re

(
ξ

eξ − 1

)
≥ 1

e− 1
.

Remark 2. For χ = 1, then our results reduces to the known results proved in [47].



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7027 19 of 23

3.1. Normalized Barnes–Mittag-Leffler function

This section explores theorems involving the normalized Barnes-Mittag-Leffler function
defined in (7).

Theorem 4. If min(b, d, v) > 0, s ≥ 0 and 1 ≥ D1 (e− 1) , then

Re

 BM b,s
d,v (ξ)(

BM b,s
d,v

)
m
(ξ)

 ≥ 1−D1 (e− 1) (45)

and

Re


(
BM b,s

d,v

)
m
(ξ)

BM b,s
d,v (ξ)

 ≥ 1

1 +D1 (e− 1)
, (46)

where D1 is given by (20).

Proof. The proof of (45) and (46) are analogous to the proof of Theorem 1, so omitted.

Theorem 5. If min(b, d, v) > 0, s ≥ 0 and 1 ≥ 2D1 (e− 1) , then

Re


(
BM b,s

d,v (ξ)
)′

(
BM b,s

d,v

)′

m
(ξ)

 ≥ 1− 2D1 (e− 1) , ξ ∈ U (47)

and

Re


(
BM b,s

d,v

)′

m
(ξ)(

BM b,s
d,v (ξ)

)′

 ≥ 1

1 + 2D1 (e− 1)
, ξ ∈ U, (48)

where D1 is given by (20).

Proof. The proof of (47) and (48) are analogous to the proof of Theorem 2, so we
omitted.

Theorem 6. If min(b, d, v) > 0, s ≥ 0 and 1 ≥ D1 (e− 2) , then

Re

 I
(
BM b,s

d,v

)
(ξ)(

I
(
BM b,s

d,v

))
m
(ξ)

 ≥ 1−D1 (e− 2) , ξ ∈ U (49)

and

Re


(
I
(
BM b,s

d,v

))
m
(ξ)

I
(
BM b,s

d,v

)
(ξ)

 ≥ 1

1 +D1 (e− 2)
, ξ ∈ U, (50)

where D1 is given by (20).

Proof. The proof of (49) and (50) are analogous to the proof of Theorem 3, so we
omitted.



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7027 20 of 23

4. Conclusion

This study investigated the normalized Le Roy-type Mittag-Leffler-Prabhakar function
and the normalized Barnes-Mittag-Leffler function and determined the lower bounds for
our theorems. We also derived bounds for the real parts of specific quotient expressions
involving their Alexander transforms. Several examples are given to demonstrate the main
findings.

The functions defined in (6) and (7) may inspire researchers to explore new subclasses
of analytic functions, investigating properties like coefficients, distortion theorems, and ex-
treme points. This could also lead to introducing new subclasses of bi-univalent functions
and p-valent functions, and estimating their second and third Taylor-Maclaurin coeffi-
cients, as well as solving Fekete-Szegő problems in future work.

Author Contributions: Author Contributions: Supervision, S.K and H.A.; Concep-
tualization, S.K.; N.A.S; Methodology, S.K.; N.A.S.; A.K.; and W.M.; Validation, O.O.;
S.K. and M.A.; Formal Analysis, H.A.; A.K.; W.M.;and O.O.; Investigation, S.K.; N.A.S.;
and H.A.; Writing—Original Draft Preparation, S.K. and N.A.S.; Writing—Review and
Editing, S.K.; N.A.S. and H.A.; Project Administration, H.A.; Funding Acquisition, M.A.
All authors have read and agreed to the published version of the manuscript.

Conflicts of Interest: There are no competing interests to declare.

References

[1] P. L. Duren. Univalent functions, volume 259 of Grundlehren der Mathematischen
Wissenschaften. Springer-Verlag, New York, 1983.

[2] J. W. Alexander. Functions which map the interior of the unit circle upon simple
regions. Annals of Mathematics, 17:12–29, 1915.

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