EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 6064
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Fekete–Szegő Inequalities for a New Class of
Bi-Univalent Functions Defined via the Mittag-Leffler

Function

Mohammad Al-Ityan1, Ala Amourah2,∗, Abdullah Alsoboh 3,∗, Nidal Anakira2,
Mohammad Bani Raba’a 4, Suha Hammad5, Tala Sasa 6

1 Department of Mathematics, Faculty of Science, Al-Balqa Applied University, 19117,
Salt, Jordan
2 Mathematics Education Program, Faculty of Education and Arts, Sohar University,
Sohar 311, Oman
3 Department of Basic and Applied Sciences, College of Applied and Health Sciences,
A’Sharqiyah University, Post Box No. 42, Post Code No. 400, Ibra, Sultanate of Oman
4 Department of Mathematics, Faculty of Science and Technology, Irbid National University,
Irbid, Jordan
5 Department of Mathematics, College of Education for Pure Sciences University of Tikrit,
Iraq
6 Department of Mathematics, Faculty of Science, Applied Science Private University,
Amman, Jordan

Abstract. In this paper, we introduce a new subclass of analytic functions denoted byMp,q
Σ (∝, β),

where we use the subordination relationship between the Mittag-Leffler function and the (p, q)-
derivative of F(z) to define this new class. By employing the Taylor-Maclaurin series expansion,
we focus on estimating the bounds for the coefficients |a2| and |a3|. Moreover, we establish Fekete–
Szegő inequalities for functions within this class.

2020 Mathematics Subject Classifications: 30C45

Key Words and Phrases: Mittag-Leffler function, Fekete-Szegő inequalities, Analytic Functions,
(p, q)-derivative

∗Corresponding author.
∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6064

Email addresses: Mohammad65655vv22@gmail.com (M. EL-Ityan), AAmourah@su.edu.om (A. Amourah),
abdullah.alsoboh@asu.edu.om (A. Alsoboh),nanakira@su.edu.om (N. Anakira),
0779382684mohammad@gmail.com (M. Bani Raba’a), suhajumaa1987@tu.edu.iq (S. Hammad),
t sasa@asu.edu.jo (T. Sasa)

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



M. EL-Ityan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6064 2 of 12

1. Introduction

Quantum calculus has become an essential tool across various fields, including math-
ematics, physics, and computer science. A significant development in this area is the
(p, q)-calculus, which extends the concept of (p, q)-numbers. Since its inception in 1991, it
has garnered significant interest from researchers [1–4]. Notably, Fibonacci oscillators were
introduced in [1], and [2] explored the use of (p, q)-numbers to create a (p, q)-Harmonic
oscillator. In [3], this approach was used to generalize certain q-oscillator algebras, while
[4] utilized it in the calculation of (p, q)-Stirling numbers.

Let A denote the class of all functions F that are analytic within the open unit disk

Θ = {z : z ∈ C and |z| < 1}

and normalized by the conditions:

F(0) = 0 and F′(0) = 1.

Thus, the function F ∈ A has the following Taylor-Maclaurin series representation:

F(z) = z +

∞∑
n=2

anz
n, (z ∈ Θ). (1)

For two functions F,G ∈ A, the function F is said to be subordinate to the function G
in Θ, denoted by

F(z) ≺ G(z) (z ∈ Θ), (2)

if there exists a function w ∈ B0 := {w : w ∈ A, w(0) = 0 and |w(z)| < 1 (z ∈ Θ)} such
that

F(z) = G(w(z)) (z ∈ Θ). (3)

In the case when the function G is univalent in Θ, the following equivalence is estab-
lished:

F(z) ≺ G(z) (z ∈ Θ) ⇔ F(0) = G(0) and F(Θ) ⊂ G(Θ). (4)

It is well known that every univalent function F has an inverse F−1, defined by

F−1(F(z)) = z = F(F−1(z)) (z ∈ Θ),

and

F(F−1(w)) = w

(
|w| < r0(F); r0(F) ≥

1

4

)
,

where
F−1(w) = w − a2w

2 + (2a22 − a3)w
3 − (5a23 − 5a2a3 + a4)w

4 + · · · (5)

A function F ∈ A is said to be bi-univalent in the domain Θ if both F and its inverse
F−1 are univalent in Θ. The set of such functions is represented by Σ. The influential study



M. EL-Ityan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6064 3 of 12

conducted by Srivastava et al. [5] has significantly renewed interest in the exploration of
various subclasses within the analytic and bi-univalent function class Σ. Following this
foundational work [6], a considerable number of research papers have focused on defining
and analyzing different subclasses of the bi-univalent class Σ, as evidenced in several
contributions (see, for instance, [5–26]).

One of the interesting problems in Geometric Function Theory is the Fekete–Szegő
problem. This problem deals with the coefficients of functions F ∈ S, and in [27], Fekete
and Szegő established the following sharp result for such functions:

∣∣a3 − ςa22
∣∣ ≤


4ς − 3, ς ≥ 1,

1 + 2e
−2ς
1−ς , 0 ≤ ς < 1,

3− 4ς, ς < 0.

The fundamental inequality
∣∣a3 − ςa22

∣∣ ≤ 1 is achieved when ς → 1. The combination
Fς(F) = a3 − ςa22 plays an important role in the theory, and finding sharp bounds for
|Fς(F)| is a notable maximization problem.

In geometric function theory, a wide range of analytic function subclasses has been
explored through diverse analytical approaches. One of the essential frameworks facili-
tating this investigation is fractional q-calculus, which has emerged as a valuable tool in
understanding the structure and properties of these function classes. The integration of
q-calculus into geometric function theory notably began with the incorporation of basic
(or q-) hypergeometric functions, as first introduced in a foundational work by Srivastava
(see [28]).

The framework of univalent function theory is particularly well-suited for formulation
using concepts from (q)-calculus. Recently, various researchers have utilized fractional
(q)-integral and fractional (q)-differential operators to define and explore new subclasses
of analytic functions (see [28, 29]). In this paper, we summarize the key ideas and operator
definitions from (q)-calculus that are relevant to our analysis.

Unless otherwise specified, we assume that 0 < q < p ≤ 1. The definitions for fractional
q-calculus operators, applicable to complex-valued functions F(z), are given in alignment
with the notation adopted in [30].

Definition 1. ([31]). The (p, q)-derivative of the function F, given by (1.1), is defined as:

Dp,qF(z) =

{
F(pz)−F(qz)

(p−q)z , z ̸= 0,

F′(0), z = 0, provided F′(0) exists.
(6)

From Definition 1.1, we deduce that:

Dp,qF(z) = 1 +

∞∑
n=2

[n]p,qanz
n−1, (7)



M. EL-Ityan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6064 4 of 12

where the symbol [n]p,q denotes the so-called (p, q)-bracket or twin-basic number:

[n]p,q =
pn − qn

p− q
.

It is clear that:
Dp,qz

n = [n]p,qz
n−1.

Note also that for p = 1, the Jackson (p, q)-derivative reduces to the Jackson q-
derivative given by (see [32]):

DqF(z) =

{
F(z)−F(qz)

(1−q)z , z ̸= 0,

F′(0), z = 0.
(8)

The twin-basic number is a natural generalization of the q-number, that is:

lim
p→1

[n]p,q = [n]q =
1− qn

1− q
, q ̸= 1.

The familiar Mittag-Leffler function Eג(z), introduced by Mittag-Leffler [33], and its
generalization Eג,ℶ(z), introduced by Wiman (see [34, 35]), are defined as follows:

Eג(z) =

∞∑
n=0

zn

Γ(גn+ 1)
= E1,ג(z), (9)

and

Eג,ℶ(z) =
∞∑
n=0

zn

Γ(גn+ ℶ)
. (10)

where ,ℶ,ג z ∈ C;ℜ(ג) > 0 These functions appear in various fields, including solutions
to fractional differential equations, random walks, super-diffusive transport problems, and
studies of complex systems.

Several properties of the Mittag-Leffler functions Eג(z) and Eג,ℶ(z), along with their
generalizations, can be found in a number of recent works (see [36], [37],[38], and [39–41]).

Since the Mittag-Leffler function Eג,ℶ(z) does not belong to the class A, The following
normalized form of the Mittag-Leffler function is considered see [39] :

Ξג,ℶ(z) = Γ(ℶ)zEג,ℶ(z) = z +
∞∑
n=2

Γ(ℶ)zn

Γ(ג(n− 1) + ℶ)
.

= z +
Γ(ℶ)z2

Γ(ג+ ℶ)
+

Γ(ℶ)z3

Γ(2ג+ ℶ)
+ .... (11)

where ,ℶ,ג z ∈ C;ℜ(ג) > 0;ℶ ̸= 0,−1,−2, ... Whilst the definition (3) holds true for
complex-valued parameters ג and ℶ and z ∈ C, yet (for the purpose of this paper) we shall
restrict our attention to the case of real-valued parameters ג and ℶ and z ∈ D.



M. EL-Ityan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6064 5 of 12

We observe that the normalized Mittag-Leffler function Ξג,ℶ in (3) contains such well-
known functions as its special cases given below:

Ξ2,1(z) = z cosh(
√
z), Ξ2,2(z) =

√
z sinh(

√
z),

Ξ2,3(z) = 2[cosh(
√
z)− 1], and Ξ2,4(z) =

6[sinh(
√
z)−

√
z]√

z
.

Definition 2. A function F ∈ Σ given by (1) is said to be in the class Mp,q
Σ ,(ℶ,ג) if the

following conditions are satisfied:

Dp,qF(z) ≺ Ξג,ℶ(z), (12)

and
Dp,qG(w) ≺ Ξג,ℶ(w) (13)

where z, w ∈ Θ, and G = F−1

We can derive the following corollaries:

For p = 1 on the class Mp,q
Σ ,(ℶ,ג) we have that M1,q

Σ :(ℶ,ג)

Corollary 1. A function F ∈ Σ given by (1) is said to be in the class M1,q
Σ ,(ℶ,ג) if the

following conditions are satisfied:

DqF(z) ≺ Ξג,ℶ(z), (14)

and
DqG(w) ≺ Ξג,ℶ(w) (15)

where z, w ∈ Θ, and G = F−1

For p = 1 and q = 1 on the class Mp,q
Σ ,(ℶ,ג) we have that MΣ(ג,ℶ):

Corollary 2. A function F ∈ Σ given by (1) is said to be in the class MΣ(ג,ℶ), if the
following conditions are satisfied:

F′(z) ≺ Ξג,ℶ(z), (16)

and
G′(w) ≺ Ξג,ℶ(w) (17)

where z, w ∈ Θ, and G = F−1

Lemma 1.5 [42] If h ∈ H, where H represents all analytic functions in Θ and satisfy
ℜ(h(z)) > 0, where

h(z) = 1 + h1z + h2z
2 + . . . , (18)

then |hi| ≤ 2 for each index i.



M. EL-Ityan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6064 6 of 12

2. Estimating Coefficients for the Class Mp,q
Σ (ℶ,ג)

This section is devoted to deriving coefficient bounds for the class Mp,q
Σ .(ℶ,ג) We

present several estimates for the initial coefficients and establish related results. In the
concluding part of this section, some of these results are highlighted as special cases in the
form of corollaries.

Theorem 1. Suppose F defined by (1) is in the class Mp,q
Σ (ℶ,ג) , where z, w ∈ Θ, ℶ,ג ∈

C;ℜ(ג) > 0;ℶ ̸= 0,−1,−2, ..., 0 < q < p ≤ 1 Then:

|a2| ≤
Γ(ℶ)

√
2Γ(2ג+ ℶ)√

|2
(
[3]p,qΓ(ℶ)Γ(2ג+ ℶ)− [2]2p,q (Γ(ג+ ℶ))2

)
Γ(ג+ ℶ)|

and

|a3| ≤
2Γ(ℶ)

|[3]p,qΓ(ג+ ℶ)|
+

4 (Γ(ℶ))2

|[2]2p,q (Γ(ג+ ℶ))2 |

Proof : Suppose F ∈ Mp,q
Σ (ℶ,ג) and let G be the analytic extension of F−1 to Θ. Then,

there exist two functions s and t, which are analytic in Θ, satisfying s(0) = t(0) = 0,
|s(z)| < 1, and |t(w)| < 1 for all z, w ∈ Θ, such that:

Dp,qF(z) = Ξג,ℶ(s(z)) (19)

Dp,qG(w) = Ξג,ℶ(t(w)). (20)

Next, let the functions s and t be defined as:

s(z) = s1z + s2z
2 + · · ·

and
t(w) = t1w + t2w

2 + · · ·

By combining equations (19) (20) :

1 + [2]p,qa2z + [3]p,qa3z
2 + ... = Ξα,ℶ(s1z + s2z

2 + · · · )

1+[2]p,qa2z+[3]p,qa3z
2+... = 1+

Γ(ℶ)
Γ(ג+ ℶ)

s1z+

(
Γ(ℶ)

Γ(ג+ ℶ)
s2 +

Γ(ℶ)
Γ(2ג+ ℶ)

s21

)
z2+.... (21)

and similarly

1− [2]p,qa2w + [3]p,q(2a
2
2 − a3)w

2 + ... = Ξα,ℶ(t1w + t2w
2 + · · · )

1−[2]p,qa2w+[3]p,q(2a
2
2−a3)w

2+... = 1+
Γ(ℶ)

Γ(ג+ ℶ)
t1w+

(
Γ(ℶ)

Γ(ג+ ℶ)
t2 +

Γ(ℶ)
Γ(2ג+ ℶ)

t21

)
w2+....

(22)



M. EL-Ityan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6064 7 of 12

Based on (21) and (22), it is obtained that:

[2]p,qa2 =
Γ(ℶ)

Γ(ג+ ℶ)
s1, (23)

[3]p,qa3 =
Γ(ℶ)

Γ(ג+ ℶ)
s2 +

Γ(ℶ)
Γ(2ג+ ℶ)

s21, (24)

−[2]p,qa2 =
Γ(ℶ)

Γ(ג+ ℶ)
t1, (25)

[3]p,q(2a
2
2 − a3) =

Γ(ℶ)
Γ(ג+ ℶ)

t2 +
Γ(ℶ)

Γ(2ג+ ℶ)
t21. (26)

From equations (23) and (25), it is derived that:

s1 = −t1, (27)

2[2]2p,qa
2
2 =

(
Γ(ℶ)

Γ(ג+ ℶ)

)2 (
s21 + t21

)
. (28)

By adding (24) to (26), it is obtained that:

2[3]p,qa
2
2 =

Γ(ℶ)
Γ(ג+ ℶ)

(s2 + t2) +
Γ(ℶ)

Γ(2ג+ ℶ)
(
s21 + t21

)
. (29)

Substituting (28) into equation (29), it has been found that:

a22 =
(Γ(ℶ))2Γ(2ג+ ℶ) (s2 + t2)

2
(
[3]p,qΓ(ℶ)Γ(2ג+ ℶ)− [2]2p,q (Γ(ג+ ℶ))2

)
Γ(ג+ ℶ)

, (30)

From Lemmas 1.5 and (30) we get:

|a2| ≤
Γ(ℶ)

√
2Γ(2ג+ ℶ)√

|2
(
[3]p,qΓ(ℶ)Γ(2ג+ ℶ)− [2]2p,q (Γ(ג+ ℶ))2

)
Γ(ג+ ℶ)|

The result of subtracting (24) from (26) is:

2[3]p,q(a3 − a22) =
Γ(ℶ)

Γ(ג+ ℶ)
(s2 − t2) +

Γ(ℶ)
Γ(2ג+ ℶ)

(
s21 − t21

)
. (31)

In view of (27) and (28), it is obtained that (31):

a3 =
Γ(ℶ) (s2 − t2)

2[3]p,qΓ(ג+ ℶ)
+ a22, (32)

a3 =
Γ(ℶ) (s2 − t2)

2[3]p,qΓ(ג+ ℶ)
+

(Γ(ℶ))2
(
s21 + t21

)
2[2]2p,q (Γ(ג+ ℶ))2

. (33)



M. EL-Ityan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6064 8 of 12

As indicated by Lemma 1.5:

|a3| ≤
2Γ(ℶ)

|[3]p,qΓ(ג+ ℶ)|
+

4 (Γ(ℶ))2

|[2]2p,q (Γ(ג+ ℶ))2 |

We can derive the following corollaries:
For p = 1 on the class Mp,q

Σ ,(ℶ,ג) we have that M1,q
Σ :(ℶ,ג)

Corollary 3. Suppose F defined by (1) is in the class Mp,1
Σ (ℶ,ג) , where z, w ∈ Θ, ℶ,ג ∈

C;ℜ(ג) > 0;ℶ ̸= 0,−1,−2, ..., 0 < q < 1 Then:

|a2| ≤
Γ(ℶ)

√
2Γ(2ג+ ℶ)√

|2
(
[3]qΓ(ℶ)Γ(2ג+ ℶ)− [2]2q (Γ(ג+ ℶ))2

)
Γ(ג+ ℶ)|

and

|a3| ≤
2Γ(ℶ)

|[3]qΓ(ג+ ℶ)|
+

4 (Γ(ℶ))2

|[2]2q (Γ(ג+ ℶ))2 |

For p = 1 and q = 1 on the class Mp,q
Σ ,(ℶ,ג) wehavethatMΣ(ג,ℶ):

Corollary 4. Suppose F defined by (1) is in the class MΣ(ג,ℶ) , where z, w ∈ Θ, ℶ,ג ∈
C;ℜ(ג) > 0;ℶ ̸= 0,−1,−2, ... Then:

|a2| ≤
Γ(ℶ)

√
2Γ(2ג+ ℶ)√

|2
(
3Γ(ℶ)Γ(2ג+ ℶ)− 4 (Γ(ג+ ℶ))2

)
Γ(ג+ ℶ)|

and

|a3| ≤
2Γ(ℶ)

|3Γ(ג+ ℶ)|
+

4 (Γ(ℶ))2

|4 (Γ(ג+ ℶ))2 |

3. Fekete–Szegő Inequalities for the Function Class Mp,q
Σ (ℶ,ג)

In this section, we focus on the Fekete–Szegő inequalities for the function classMp,q
Σ .(ℶ,ג)

We also present some special cases in the form of corollaries.

Theorem 2. Suppose F, as defined by equation (1), belongs to the class Mp,q
Σ ,(ℶ,ג) where

ℶ,ג ∈ C;ℜ(ג) > 0;ℶ ̸= 0,−1,−2, ..., 0 < q < p ≤ 1, ς ∈ R. Then:

|a3 − ςa22| ≤


2Γ(ℶ)

[3]p,qΓ(α+ℶ) for |h(ς)| ≤ 1
2[3]p,q

4|h(ς)| for |h(ς)| ≥ 1
2[3]p,q

(34)



M. EL-Ityan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6064 9 of 12

Proof :
From equations (30) and (32), it is derived that:

a3 − ςa22 =
Γ(ℶ) (s2 − t2)

2[3]p,qΓ(ג+ ℶ)
+ (1− ς)a22

Also,

a3 − ςa22 =
Γ(ℶ) (s2 − t2)

2[3]p,qΓ(ג+ ℶ)
+

(Γ(ℶ))2Γ(2ג+ ℶ)(1− ς) (s2 + t2)

2
(
[3]p,qΓ(ℶ)Γ(2ג+ ℶ)− [2]2p,q (Γ(ג+ ℶ))2

)
Γ(ג+ ℶ)

Simplify to:

a3 − ηa22 =
Γ(ℶ)

Γ(ג+ ℶ)

[(
h(ς) +

1

2[3]p,q

)
s2 +

(
h(ς)− 1

2[3]p,q

)
t2

]
(35)

where

h(ς) =
Γ(ℶ)Γ(2ג+ ℶ)(1− ς)

2
(
[3]p,qΓ(ℶ)Γ(2ג+ ℶ)− [2]2p,q (Γ(ג+ ℶ))2

) (36)

We can derive the following corollaries:

For p = 1 on the class Mp,q
Σ ,(ℶ,ג) wehavethatM1,q

Σ :(ℶ,ג)

Corollary 5. Suppose F, as defined by equation (1), belongs to the class Mp,1
Σ ,(ℶ,ג) where

ℶ,ג ∈ C;ℜ(ג) > 0;ℶ ̸= 0,−1,−2, ..., 0 < q < p ≤ 1, ς ∈ R. Then:

|a3 − ςa22| ≤


2Γ(ℶ)

[3]qΓ(α+ℶ) for | Γ(ℶ)Γ(2ג+ℶ)(1−ς)

2([3]qΓ(ℶ)Γ(2ג+ℶ)−[2]2q(Γ(ג+ℶ))2)
| ≤ 1

2[3]q

4| Γ(ℶ)Γ(2ג+ℶ)(1−ς)

2([3]qΓ(ℶ)Γ(2ג+ℶ)−[2]2q(Γ(ג+ℶ))2)
| for | Γ(ℶ)Γ(2ג+ℶ)(1−ς)

2([3]qΓ(ℶ)Γ(2ג+ℶ)−[2]2q(Γ(ג+ℶ))2)
| ≥ 1

2[3]q

(37)

For p = 1 and q = 1 on the class Mp,q
Σ ,(ℶ,ג) wehavethatMΣ(ג,ℶ):

Corollary 6. Suppose F, as defined by equation (1), belongs to the class MΣ(ג,ℶ), where
ℶ,ג ∈ C;ℜ(ג) > 0;ℶ ̸= 0,−1,−2, ..., ς ∈ R. Then:

|a3 − ςa22| ≤


2Γ(ℶ)

3Γ(α+ℶ) for | Γ(ℶ)Γ(2ג+ℶ)(1−ς)

2(3Γ(ℶ)Γ(2ג+ℶ)−4(Γ(ג+ℶ))2)
| ≤ 1

6

4| Γ(ℶ)Γ(2ג+ℶ)(1−ς)

2(3Γ(ℶ)Γ(2ג+ℶ)−4(Γ(ג+ℶ))2)
| for | Γ(ℶ)Γ(2ג+ℶ)(1−ς)

2(3Γ(ℶ)Γ(2ג+ℶ)−4(Γ(ג+ℶ))2)
| ≥ 1

6

(38)

In conclusion, we have estimated the basic Taylor coefficients |a2| and |a3|, and derived
upper bounds for the Fekete–Szegő problem |a3 − ςa22|. Additionally, we presented some
results as special cases.



M. EL-Ityan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6064 10 of 12

4. Conclusions

In this paper, we introduced a new subclass of analytic functions defined by Mp,q
Σ .(ℶ,ג)

A key element of our approach is the use of the Mittag-Leffler function, which plays a fun-
damental role in defining and analyzing the functions in this class. The sharp bounds
we obtained for the Fekete-Szegő functional within this subclass yield improved results
and pave the way for further exploration in this area. The subclass offers a robust frame-
work for studying bi-univalent functions, particularly in relation to coefficient estimates,
distortion theorems, and other central topics in geometric function theory. The construc-
tions and methodologies we presented can be employed by researchers to investigate more
complex problems associated with coefficient bounds and to expand the applicability of
Mittag-Leffler functions in practical contexts. Ultimately, these results may contribute
to the formation of new subclasses, enhancing our understanding of bi-univalent function
properties and providing a solid foundation for novel developments and applications in
this vibrant area of research.

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