EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5816
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Advancements in Ostrowski type fractional Integral
Inequalities via Applications of Jensen’s and Young’s

Inequalities

Gauhar Rahman1,∗, Muhammad Samraiz2, Çetin Yıldız3,∗, Maryam Ali Alghafli4,
Nabil Mlaiki4

1 Department of Mathematics and Statistics, Hazara University, Mansehra 21300, Pakistan
2 Department of Mathematics, University of Sargodha P.O. Box 40100, Sargodha, Pakistan
3 Deparment of Mathematics, K.K. Education Faculty, Atatürk University, 25240 Erzurum,
Turkey
4 Department of Mathematics and Sciences, Prince Sultan University, Riyadh, 11586,
Saudi Arabia

Abstract. Fractional integral operators and convexity have a close link due to their fascinating
properties in the mathematical sciences. In this paper, we first establish an integral identity involv-
ing the generalized Hattaf-fractional integral operators. By using the Jensen integral inequality,
Young’s inequality, power-mean inequality, and Hölder inequality, we then apply this identity to
provide some new generalizations of Ostrowski type inequality for the convexity of |ℵ|. Further-
more, we deduce several special cases from the main results. The results of this novel investigation
should lead to new discoveries in the area of fractional calculus and inequalities.

2020 Mathematics Subject Classifications: 26A33, 26A51, 26D10

Key Words and Phrases: Young inequality, convex function, power mean inequality, fractional
operators

1. Introduction

Fractional integrals and derivatives have attracted a lot of attention from researchers
nowadays. In many cases, fractional derivatives and fractional integrals provide more
accurate representations of the frameworks than ordinary derivatives and integrals. The
fractional calculus is currently widely employed in many scientific domains due to its nu-
merous applications. The interest of researchers in fractional derivative and fractional

∗Corresponding author.
∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5816

Email addresses: gauhar55uom@gmail.com, drgauhar.rahman@hu.edu.pk (G. Rahman),
muhammad.samraiz@uos.edu.pk;msamraizuos@gmail.com (M. Samraiz),
cetin@atauni.edu.tr (Ç. Yıldız), mghafli@psu.edu.sa (M. A. Alghafli),
nmlaiki@psu.edu.sa; nmlaiki2012@gmail.com (N. Mlaiki)

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



G. Rahman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5816 2 of 18

integration has grown recently due to their wide applications in diverse domain, for exam-
ple (see, [1–5]). Dumitru and Arran [6] have given a new formula for fractional derivatives
and integrals using the Mittag-Leffler kernel. While more theoretical ideas about fractional
operators with Mittag-Leffler kernels (Atangana-Baleanu operators) and the higher-order
case have been discussed in [7–9], the generalization to the generalized Mittag-Leffler ker-
nels to gain a semigroup property has recently been developed in [10, 11].
In the beginning, many scientists working in different areas of theory of inequalities em-
ployed fractional calculus as an essential tool, for example, [12–16]. Shuang and Qi [17]
proved a number of Hermite-Hadamard-type inequalities and examined specific methods
for a class of s-convex functions. Mehrez and Agarwal [18] proved new integral inequali-
ties and looked at specific cases of their discoveries with application to special means by
employing the conventional Hermite-Hadamard inequalities. Park et al. [19] researched
and used new generalized inequalities to stability analysis. By utilizing the local fractional
approach, fractional integral inequalities were generated by Sarikaya et al. [20], expanding
upon the findings found in the classical literature. In [21], Set et al. presented integral
inequalities for differentiable convex functions via Atangana-Baleanu fractional integral
operators. The various researchers examined a few noteworthy integral inequalities using
various fractional methods. We refer the readers to the research conducted by [22–25].

2. Preliminaries

It is clear that the convex function is essential for the study of mathematical inequal-
ities since it has several applications in the fields of pure and practical mathematics,
mechanics, probability and statistics theory, economics, engineering, and optimization
theory. Recently, there have been several mathematicians working on convexity’s theories,
variations, augmentations, generalizations, and refinements. For example, in a number of
scientific and mathematical domains, it is a useful tool for presenting a variety of chal-
lenges and demonstrating awareness [26–29].
The convexity property can be used to generalize a number of well-known inequalities, such
as the Opial type inequality, the Hermite–Hadamard inequality, the Ostrowski inequality,
the Simpson inequality, the Bullen type inequality, and many more. The Ostrowski type
inequality is one of the most extensively studied conclusions involving several kinds of
convexities.

Definition 1. [30] Let ℵ : I ⊆ R → R be a differentiable function ℵ ∈ L1[r, s] with
r < s ∈ I. If |ℵ′(t)| ≤ K, for t ∈ [r, s], then the Ostrowski type integral inequality is given
by

| ℵ(t)− 1

s− r

∫ s

r
ℵ(t)dt |≤ K(s− r)

[
1

4
+

(
t− r+s

2

)
(r + s)2

]
,

where 1
4 is the least possible value.

Mathematicians and scholars have been studying this inequality with significant at-
tention and effort in recent years. This inequality was studied in 1997 by Dragomir and



G. Rahman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5816 3 of 18

Wang [31, 32] with relation to the lower and upper bounds of the first derivative. It was
investigated by Barnett et al. and Cerone et al. [33, 34] that this inequality involving
twice differentiable convex functions involved.

Definition 2. [35] A function ℵ : [r, s] ⊆ R → R is said to be convex if

ℵ (ρu+ (1− ρ)v) ≤ ρℵ(u) + (1− ρ)ℵ(v),

for all u, v ∈ [r, s] and ρ ∈ [0, 1].

Convex functions are a concept that is frequently utilized in inequality theory. The
Hermite-Hadamard inequality, which derives upper and lower bounds from averages of the
mean value of a convex function, is as follows:

Definition 3. Given a convex mapping ℵ : I ⊆ R → R, let r < s be on the interval I of
R. Then the Hermite-Hadamard inequality is defined by

ℵ
(
r + s

2

)
≤ 1

s− r

∫ s

r
ℵ(x)dx ≤ ℵ(r) + ℵ(s)

2
.

Definition 4. [36] The ABC-fractional derivative is defined by

ABCDκ
r,ξℵ(ξ) =

M(κ)

1− κ

∫ ξ

r
ℵ′(η)Eκ

(
−κ(ξ − η)κ

1− κ

)
dη,

where 0 < κ < 1, ℵ′ ∈ L1(r, T ] and M(κ) is normalization function which satisfies the
condition M(0) = M(1) = 1.

Definition 5. [37, 38] The AB-fractional operator for ℵ ∈ L1(r, T ] and 0 < κ < 1 is
defined by

ABIκ
r,ξℵ(ξ) =

1− κ

M(κ)
ℵ(ξ) + κ

M(κ)Γ(κ)

∫ ξ

r
(ξ − η)κ−1 ℵ(η)dη. (1)

Definition 6. [39] The Hattaf fractional derivative is defined by

Dκ,σ,δ
r,ξ,ωℵ(ξ) =

M(κ)

1− κ

1

ω(ξ)

∫ ξ

r
Eσ

(
−κ(ξ − η)δ

1− κ

)
d

dt
(ωℵ)(η)dη,

where 0 < κ < 1, ℵ′ ∈ L1(r, T ] ω ∈ C1(a, b), ω, ω′ > 0 on [a, b].

Definition 7. [39] The left sided Hattaf fractional operator for ℵ ∈ L1(r, T ] and 0 < κ < 1
is defined by

Iκ,σ,ω
r,ξ ℵ(ξ) = 1− κ

M(κ)
ℵ(ξ) + κ

M(κ)Γ(σ)ω(ξ)

∫ ξ

r
(ξ − η)σ−1 ω(η)ℵ(η)dη. (2)



G. Rahman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5816 4 of 18

Definition 8. The right sided Hattaf fractional operator for ℵ ∈ L1(r, s) and 0 < κ < 1
is defined by

Iκ,σ,ω
s,ξ ℵ(ξ) = 1− κ

M(κ)
ℵ(ξ) + κ

M(κ)Γ(σ)ω(ξ)

∫ s

ξ
(η − ξ)σ−1 ω(η)ℵ(η)dη. (3)

Remark 1. i. If we consider σ = κ in (2) and (3), then we get AB-operator defined in
(1).
ii. If we consider σ = κ and ω = 1 in (2) and (3), then we get AB-operator defined in
(1).

Definition 9. The left sided Hattaf fractional operator for ω = 1, ℵ ∈ L1(r, T ] and
0 < κ < 1 is defined by

Iκ,σ
r,ξ ℵ(ξ) =

1− κ

M(κ)
ℵ(ξ) + κ

M(κ)Γ(σ

∫ ξ

r
(ξ − η)σ−1 ℵ(η)dη. (4)

Definition 10. The right sided Hattaf fractional operator for ω = 1, ℵ ∈ L1(r, s) and
0 < κ < 1 is defined by

Iκ,σ
s,ξ ℵ(ξ) =

1− κ

M(κ)
ℵ(ξ) + κ

M(κ)Γ(σ)

∫ s

ξ
(η − ξ)σ−1 ℵ(η)dη. (5)

This paper aims to derive an integral identity by incorporating the Hattaf fractional
integral operators (4) and (5) and uses them to prove the refinement of Ostrowski type
integral inequalities for differentiable convex functions.

3. Main Result

In this section, we first prove the following fractional integral identity which will be
used in our main findings.

Lemma 1. Assume that ℵ : [r, s] → R represents a differentiable function on (r, s), where
ℵ′ ∈ L1[r, s] and r < s. Next, for modified Hattaf-fractional integral operators, we have
the identity given below:[

(t− r)σ + (s− t)σ

s− r

]
ℵ (t) +

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)]

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ
r,t ℵ(r) + Iκ,σ

s,t ℵ(s)
]

=
(t− r)σ+1

s− r

∫ 1

0
ρσℵ′(ρt+ (1− ρ)r)dρ− (s− t)σ+1

s− r

∫ 1

0
ρϱℵ′(ρt+ (1− ρ)s)dρ,

where κ ∈ (0, 1], t ∈ [r, s].



G. Rahman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5816 5 of 18

Proof. For simplicity, let us consider

I =
(t− r)σ+1

s− r

∫ 1

0
ρσℵ′(ρt+ (1− ρ)r)dρ− (s− t)σ+1

s− r

∫ 1

0
ρσℵ′(ρt+ (1− ρ)s)dρ

=
(t− r)σ+1

s− r
I1 −

(s− t)σ+1

s− r
I2, (6)

where

I1 =

∫ 1

0
ρσℵ′(ρt+ (1− ρ)r)dρ

=
ρσℵ(ρt+ (1− ρ)r)

t− r
|10 +

σ

t− r

∫ 1

0
ρσ−1ℵ(ρt+ (1− ρ)r)dρ

By substituting ρ = u−r
t−r in the integral part, we get

I1 =
ℵ(t)
t− r

+
σ

(t− r)σ+1

∫ t

r
(u− r)σ−1ℵ (u) du. (7)

Similarly, one can get

I2 = − ℵ(t)
s− t

+
σ

(s− t)σ+1

∫ s

t
(s− u)σ−1ℵ (u) du. (8)

By substituting (7) and (8) in (6) and then after a simple computation, we get the desired
Lemma 1.

Remark 2. Applying Lemma 1 for σ = κ, we get the Lemma 1 given in [40].

Theorem 1. Assume that ℵ : [r, s] → R be a differentiable function on (r, s), with ℵ′ ∈
L1[r, s] and r < s. Then, the following inequality holds for Hattaf-fractional integral
operators (4) and (5) if |ℵ′| is a convex function∣∣∣ [(t− r)σ + (s− t)σ

s− r

]
ℵ (t) +

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)]

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ
r,t ℵ(r) + Iκ,σ

s,t ℵ(s)
]∣∣∣

≤(t− r)σ+1

s− r

{
ℵ′ (t)

σ + 2
+

ℵ′ (r)

(σ + 1)(σ + 2)

}
+

(s− t)σ+1

s− r

{
ℵ′ (t)

σ + 2
+

ℵ′ (s)

(σ + 1)(σ + 2)

}
,

where t ∈ [r, s], κ ∈ (0, 1].

Proof. By using Lemma 1, we have∣∣∣ [(t− r)σ + (s− t)σ

s− r

]
ℵ (t) +

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)]



G. Rahman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5816 6 of 18

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ
r,t ℵ(r) + Iκ,σ

s,t ℵ(s)
]∣∣∣

≤(t− r)σ+1

s− r

∫ 1

0
ρσ|ℵ′(ρt+ (1− ρ)r)|dρ− (s− t)σ+1

s− r

∫ 1

0
ρσ|ℵ′(ρt+ (1− ρ)s)|dρ.

By applying the convexity of |ℵ′|, we get∣∣∣ [(t− r)σ + (s− t)σ

s− r

]
ℵ (t) +

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)]

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ
r,t ℵ(r) + Iκ,σ

s,t ℵ(s)
]∣∣∣

≤(t− r)σ+1

s− r

∫ 1

0
ρσ[ρ

∣∣ℵ′(t)
∣∣+ (1− ρ)

∣∣ℵ′(r)
∣∣]dρ

+
(s− t)σ+1

s− r

∫ 1

0
ρσ[ρ

∣∣ℵ′(s)
∣∣+ (1− ρ)

∣∣ℵ′(t)
∣∣]dρ.

After solving the above integrals, we get∣∣∣ [(t− r)σ + (s− t)σ

s− r

]
ℵ (t) +

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)]

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ
r,t ℵ(r) + Iκ,σ

s,t ℵ(s)
]∣∣∣

≤(t− r)σ+1

s− r

{
1

(σ + 2)
|ℵ′(t)|+ 1

(σ + 1)(σ + 2)
|ℵ′(r)|

}
+
(s− t)σ+1

s− r

{
1

(σ + 2)
|ℵ′(t)|+ 1

(σ + 1)(σ + 2)
|ℵ′(s)|

}
,

which complete the desired proof.

Corollary 1. Applying Theorem 1 for |ℵ′| ≤ K where K > 0 , we get the following
inequality ∣∣∣ [(t− r)σ + (s− t)σ

s− r

]
ℵ (t) +

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)]

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ
r,t ℵ(r) + Iκ,σ

s,t ℵ(s)
]∣∣∣

≤ K

s− r

(
1

σ + 2
+

1

(σ + 1)(σ + 2)

){
(t− r)σ+1 + (s− t)σ+1

}
.

Corollary 2. Applying Corollary 1 for t = r+s
2 , we get the following inequality

∣∣∣(s− r)σ−1

2σ−1
ℵ
(
r + s

2

)
+

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)]



G. Rahman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5816 7 of 18

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ

r, r+s
2

ℵ(r) + Iκ,σ

s, r+s
2

ℵ(s)
]∣∣∣

≤K

(
1

σ + 1

)
(s− r)σ

2σ
.

Remark 3. Applying Theorem 1 for σ = κ, we get Theorem 1 proved by Ahmad et al.
[40].

Remark 4. Applying Corollary 2 for σ = κ, we get Corollary 2 proved earlier by Ahmad
et al. [40].

Theorem 2. Let ℵ : [r, s] → R be a differentiable function on (r, s) with r < s and
ℵ′ ∈ L1[r, s]. For Hattaf-fractional integral operators, we have the following inequality if
|ℵ′|q is a convex function.∣∣∣ [(t− r)σ + (s− t)σ

s− r

]
ℵ (t) +

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)]

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ
r,t ℵ(r) + Iκ,σ

s,t ℵ(s)
]∣∣∣

≤(t− r)σ+1

s− r

(
1

σp+ 1

) 1
p
[
| ℵ′(t) |q + | ℵ′(r) |q

2

] 1
q

+
(s− t)σ+1

s− r

(
1

σp+ 1

) 1
p
[
| ℵ′(s) |q + | ℵ′(t) |q

2

] 1
q

,

where 1
p + 1

q = 1, t ∈ [r, s], κ ∈ (0, 1] and M(κ) > 0.

Proof. By applying Lemma 1, we have∣∣∣ [(t− r)σ + (s− t)σ

s− r

]
ℵ (t) +

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)]

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ
r,t ℵ(r) + Iκ,σ

s,t ℵ(s)
]∣∣∣

≤(t− r)σ+1

s− t

∫ 1

0
ρσ | ℵ′(ρt+ (1− ρ)r) | dρ

+
(s− t)σ+1

s− t

∫ 1

0
ρσ | ℵ′(ρt+ (1− ρ)s) | dρ.

By employing Hölder inequality, we obtain∣∣∣ [(t− r)σ + (s− t)σ

s− r

]
ℵ (t) +

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)] (9)

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ
r,t ℵ(r) + Iκ,σ

s,t ℵ(s)
]∣∣∣



G. Rahman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5816 8 of 18

≤(t− r)σ+1

s− t

[(∫ 1

0
ρϱpdρ

) 1
p
(∫ 1

0
| ℵ′(ρt+ (1− ρ)r) |q dρ

) 1
q

]

+
(s− t)ϱ+1

s− t

[(∫ 1

0
ρσpdρ

) 1
p
(∫ 1

0
| ℵ′(ρt+ (1− ρ)s) |q dρ

) 1
q

]
. (10)

By employing the convexity of | ℵ′ |q, we have∫ 1

0
| ℵ′(ρt+ (1− ρ)r) |q dρ ≤

∫ 1

0
[ρ | ℵ′(t) |q +(1− ρ) | ℵ′(r) |q]dρ

=
| ℵ′(t) |q + | ℵ′(r) |q

2
(11)

and ∫ 1

0
| ℵ′(ρt+ (1− ρ)s) |q dρ ≤

∫ 1

0
[ρ | ℵ′(t) |q +(1− ρ) | ℵ′(s) |q]dρ

=
| ℵ′(t) |q + | ℵ′(s) |q

2
. (12)

Substituting (11) and (12) in (9) and then solving the integrals, we get the required
inequality.

Corollary 3. Applying Theorem 2 for |ℵ′| ≤ K where K > 0 , we get the following
inequality ∣∣∣ [(t− r)σ + (s− t)σ

s− r

]
ℵ (t) +

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)]

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ
r,t ℵ(r) + Iκ,σ

s,t ℵ(s)
]∣∣∣

≤ K

s− r

(
1

σp+ 1

) 1
p {

(t− r)σ+1 + (s− t)σ+1
}
.

Corollary 4. Applying Corollary 3 for t = r+s
2 , we get the following inequality∣∣∣(s− r)σ−1

2σ−1
ℵ
(
r + s

2

)
+

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)]

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ

r, r+s
2

ℵ(r) + Iκ,σ

s, r+s
2

ℵ(s)
]∣∣∣

≤K

(
1

σp+ 1

) 1
p (s− r)σ

2σ
.

Remark 5. Applying Theorem 2 for σ = κ, we get Theorem 2 proved by Ahmad et al.
[40].



G. Rahman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5816 9 of 18

Remark 6. Applying Corollary 4 for σ = κ, we get Corollary 4 proved earlier by Ahmad
et al. [40].

Theorem 3. Let ℵ : [r, s] → R be a differentiable function on (r, s), where ℵ′ ∈ L1[r, s]
and r < s. The following inequality holds for Hattaf-fractional integral operators (4) and
(5) if |ℵ′|q is a convex function.∣∣∣ [(t− r)σ + (s− t)σ

s− r

]
ℵ (t) +

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)]

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ
r,t ℵ(r) + Iκ,σ

s,t ℵ(s)
]∣∣∣

≤(t− r)σ+1

s− r

(
1

p(σp+ 1)
+

| ℵ′(t) |q + | ℵ′(r) |q

2q

)
+
(s− t)σ+1

s− t

(
1

p(σp+ 1)
+

| ℵ′(s) |q + | ℵ′(t) |q

2q

)
,

where 1
p + 1

q = 1, t ∈ [r, s], κ ∈ (0, 1] and M(κ) > 0.

Proof. By utilizing Lemma 1, we have∣∣∣ [(t− r)σ + (s− t)σ

s− r

]
ℵ (t) +

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)]

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ
r,t ℵ(r) + Iκ,σ

s,t ℵ(s)
]∣∣∣

≤(t− r)σ+1

s− r

∫ 1

0
ρσ | ℵ′(ρt+ (1− ρ)r) | dρ

+
(s− t)σ+1

s− r

∫ 1

0
ρσ | ℵ′(ρt+ (1− ρ)s) | dρ.

By employing Young inequality as uv ≤ up

p + vq

q in above, we have∣∣∣ [(t− r)σ + (s− t)σ

s− r

]
ℵ (t) +

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)] (13)

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ
r,t ℵ(r) + Iκ,σ

s,t ℵ(s)
]∣∣∣

≤(t− r)σ+1

s− r

[
1

p

∫ 1

0
ρσpdρ+

1

q

∫ 1

0
| ℵ′(ρt+ (1− ρ)r) |q dρ

]
+
(s− t)σ+1

s− r

[
1

p

∫ 1

0
ρσpdρ+

1

q

∫ 1

0
| ℵ′(ρt+ (1− ρ)s) |q dρ

]
. (14)

Since | ℵ′ |q is convex, so therefore we have∫ 1

0
| ℵ′(ρt+ (1− ρ)r) |q dρ ≤

∫ 1

0
[ρ | ℵ′(t) |q +(1− ρ) | ℵ′(r) |q]dρ



G. Rahman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5816 10 of 18

=
| ℵ′(t) |q + | ℵ′(r) |q

2
(15)

and ∫ 1

0
| ℵ′(ρt+ (1− ρ)s) |q dρ ≤

∫ 1

0
[ρ | ℵ′(t) |q +(1− ρ) | ℵ′(s) |q]dρ

=
| ℵ′(t) |q + | ℵ′(s) |q

2
. (16)

Substituting (15) and (16) in (13) and then by solving the integrals, we get the desired
proof.

Corollary 5. Applying Theorem 3 for |ℵ′| ≤ K where K > 0 , we get the following
inequality ∣∣∣ [(t− r)σ + (s− t)σ

s− r

]
ℵ (t) +

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)]

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ
r,t ℵ(r) + Iκ,σ

s,t ℵ(s)
]∣∣∣

≤
(

1

p(σp+ 1)
+

Kq

q

){
(t− r)σ+1

s− r
+

(s− t)σ+1

s− r

}
.

Corollary 6. Applying Corollary 5 for t = r+s
2 , we get the following inequality

∣∣∣(s− r)σ−1

2σ−1
ℵ
(
r + s

2

)
+

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)]

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ

r, r+s
2

ℵ(r) + Iκ,σ

s, r+s
2

ℵ(s)
]∣∣∣

≤
(

1

p(σp+ 1)
+

Kq

q

){
(s− r)σ

2σ

}
.

Remark 7. Applying Theorem 3 for σ = κ, we get Theorem 3 proved by Ahmad et al.
[40].

Remark 8. Applying Corollary 6 for σ = κ, we get Corollary 6 proved earlier by Ahmad
et al. [40].

Theorem 4. Let ℵ : [r, s] → R be a differentiable function on (r, s), where ℵ′ ∈ L1[r, s]
and r < s. The following inequality holds for Hattaf-fractional integral operators if |ℵ′|q
is a convex function

∣∣∣ [(t− r)σ + (s− t)σ

s− r

]
ℵ (t) +

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)]



G. Rahman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5816 11 of 18

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ
r,t ℵ(r) + Iκ,σ

s,t ℵ(s)
]∣∣∣

≤(t− r)σ+1

s− r

(
1

σ + 1

)1− 1
q
[
| ℵ′(t) |q

(σ + 2)
+

| ℵ′(r) |q

(σ + 1)(σ + 2)

] 1
q

+
(s− t)σ+1

s− r

(
1

σ + 1

)1− 1
q
[

| ℵ′(s) |q

(σ + 1)(σ + 2)
+

| ℵ′(t) |q

(σ + 2)

] 1
q

,

where q ≥ 1, t ∈ [r, s], κ ∈ (0, 1] and M(κ) > 0.

Proof. By utilizing Lemma 1, we have∣∣∣ [(t− r)σ + (s− t)σ

s− r

]
ℵ (t) +

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)]

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ
r,t ℵ(r) + Iκ,σ

s,t ℵ(s)
]∣∣∣

≤(t− r)σ+1

s− r

∫ 1

0
ρσ | ℵ′(ρt+ (1− ρ)r) | dρ

+
(s− t)σ+1

s− r

∫ 1

0
ρσ | ℵ′(ρt+ (1− ρ)s) | dρ.

By employing power mean inequality, we obtain∣∣∣ [(t− r)σ + (s− t)σ

s− r

]
ℵ (t) +

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)]

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ
r,t ℵ(r) + Iκ,σ

s,t ℵ(s)
]∣∣∣

≤(t− r)σ+1

s− r

(∫ 1

0
ρσdρ

)1− 1
q
(∫ 1

0
ρσ | ℵ′(ρt+ (1− ρ)r) |q dρ

) 1
q

+
(s− t)σ+1

s− r

(∫ 1

0
ρσdρ

)1− 1
q
(∫ 1

0
ρσ | ℵ′(ρt+ (1− ρ)s) |q dρ

) 1
q

.

Now, by utilizing the convexity of | ℵ′ |q, we get∣∣∣ [(t− r)σ + (s− t)σ

s− r

]
ℵ (t) +

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)]

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ
r,t ℵ(r) + Iκ,σ

s,t ℵ(s)
]∣∣∣

≤(t− r)σ+1

s− r

(∫ 1

0
ρσdρ

)1− 1
q
(∫ 1

0
ρσ[ρ | ℵ′(t) |q +(1− ρ) | ℵ′(r) |q]dρ

) 1
q

+
(s− t)σ+1

s− r

(∫ 1

0
ρσdρ

)1− 1
q
(∫ 1

0
ρσ[ρ | ℵ′(t) |q +(1− ρ) | ℵ′(s) |q]dρ

) 1
q



G. Rahman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5816 12 of 18

≤(t− r)σ+1

s− r

(
1

σ + 1

)1− 1
q
[
| ℵ′(t) |q

(σ + 2)
+

| ℵ′(r) |q

(σ + 1)(σ + 2)

] 1
q

+
(s− t)σ+1

s− r

(
1

σ + 1

)1− 1
q
[
| ℵ′(t) |q

(σ + 2)
+

| ℵ′(s) |q

(σ + 1)(σ + 2)

] 1
q

,

which complete the proof.

Corollary 7. Applying Theorem 4 for |ℵ′| ≤ K where K > 0 , we get the following
inequality ∣∣∣ [(t− r)σ + (s− t)σ

s− r

]
ℵ (t) +

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)]

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ
r,t ℵ(r) + Iκ,σ

s,t ℵ(s)
]∣∣∣

≤ K

s− r

(
1

σ + 1

){
(t− r)σ+1 + (s− t)σ+1

}
.

Corollary 8. Applying Corollary 7 for t = r+s
2 , we get the following inequality

∣∣∣ [(s− r)σ−1

2σ−2

]
ℵ
(
r + s

2

)
+

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)]

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ

r, r+s
2

ℵ(r) + Iκ,σ

s, r+s
2

ℵ(s)
]∣∣∣

≤ K

s− r

(
1

σ + 1

)
(s− r)σ

2σ
.

Remark 9. Applying Theorem 4 for σ = κ, we get Theorem 4 proved by Ahmad et al.
[40].

Remark 10. Applying Corollary 8 for σ = κ, we get Corollary 8 proved earlier by Ahmad
et al. [40].

Theorem 5. Let ℵ : [r, s] → R be a differentiable function on (r, s), where ℵ′ ∈ L1[r, s]
and r < s. For Hattaf-fractional integral operators (4) and (5), we have the following
inequality if |ℵ′| is a concave function∣∣∣ [(t− r)σ + (s− t)σ

s− r

]
ℵ (t) +

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)]

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ
r,t ℵ(r) + Iκ,σ

s,t ℵ(s)
]∣∣∣

≤(t− r)σ+1

s− r

(
1

σ + 1

)
| ℵ′

(
(σ + 1)t+ r

σ + 2

)
| +(s− t)σ+1

s− r

(
1

σ + 1

)
| ℵ′

(
(σ + 1)t+ s

σ + 2

)
|,

where κ ∈ (0, 1] and M(κ) > 0.



G. Rahman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5816 13 of 18

Proof. By utilizing Lemma 1, we have∣∣∣ [(t− r)σ + (s− t)σ

s− r

]
ℵ (t) +

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)]

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ
r,t ℵ(r) + Iκ,σ

s,t ℵ(s)
]∣∣∣

≤(t− r)σ+1

s− r

∫ 1

0
ρσ | ℵ′(ρt+ (1− ρ)r) | dρ

+
(s− t)σ+1

s− r

∫ 1

0
ρσ | ℵ′(ρt+ (1− ρ)s) | dρ.

By employing the Jensen integral inequality, we get∣∣∣ [(t− r)σ + (s− t)σ

s− r

]
ℵ (t) +

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)]

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ
r,t ℵ(r) + Iκ,σ

s,t ℵ(s)
]∣∣∣

≤(t− r)σ+1

s− r

(∫ 1

0
ρσdρ

)
| ℵ′

(∫ 1
0 ρσ(ρt+ (1− ρ)r)dρ∫ 1

0 ρσdρ

)
|

+
(s− t)σ+1

s− r

(∫ 1

0
ρσdρ

)
| ℵ′

(∫ 1
0 ρσ(ρt+ (1− ρ)s)dρ∫ 1

0 ρσdρ

)
| .

After simple calculation of above integrals, we get the required inequality.

Corollary 9. Applying Theorem 5 for |ℵ′| ≤ K where K > 0 , we get the following
inequality ∣∣∣ [(t− r)σ + (s− t)σ

s− r

]
ℵ (t) +

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)]

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ
r,t ℵ(r) + Iκ,σ

s,t ℵ(s)
]∣∣∣

≤ K

s− r

(
1

σ + 1

){
(t− r)σ+1 + (s− t)σ+1

}
.

Remark 11. Applying Theorem 5 for κ = σ, we get Theorem 5 proved by Ahmad et al.
[40].

Remark 12. Applying Corollary 9 for κ = σ, we get Corollary 9 proved earlier by Ahmad
et al. [40].

Theorem 6. Assume that the function Ψ : [r, s] → R has a continuous derivative on [r, s]
and is strictly increasing and positive. Let ℵ : [r, s] → R be a differentiable function on



G. Rahman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5816 14 of 18

(r, s), where ℵ′ ∈ L1[r, s] and r < s. For Hattaf-fractional integral operators, we have the
following inequality if |ℵ′|q is a concave function∣∣∣ [(t− r)σ + (s− t)σ

s− r

]
ℵ (t) +

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)]

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ
r,t ℵ(r) + Iκ,σ

s,t ℵ(s)
]∣∣∣

≤(t− r)σ+1

s− r

(
1

σp+ 1

) 1
p

| ℵ′
(
t+ r

2

)
| +(s− t)σ+1

s− r

(
1

σp+ 1

) 1
p

| ℵ′
(
s+ t

2

)
|,

where 1
p + 1

q = 1, q > 1, t ∈ [r, s], κ ∈ (0, 1] and M(κ) > 0.

Proof. By utilizing Lemma 1, we have∣∣∣ [(t− r)σ + (s− t)σ

s− r

]
ℵ (t) +

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)]

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ
r,t ℵ(r) + Iκ,σ

s,t ℵ(s)
]∣∣∣

≤(t− r)σ+1

s− r

∫ 1

0
ρσ | ℵ′(ρt+ (1− ρ)r) | dρ

+
(s− t)σ+1

s− r

∫ 1

0
ρσ | ℵ′(ρt+ (1− ρ)s) | dρ.

By employing the Hölder integral inequality, we obtain∣∣∣ [(t− r)σ + (s− t)σ

s− r

]
ℵ (t) +

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)]

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ
r,t ℵ(r) + Iκ,σ

s,t ℵ(s)
]∣∣∣

≤(t− r)σ+1

s− r

(∫ 1

0
ρσpdρ

) 1
p
(∫ 1

0
| ℵ′(ρt+ (1− ρ)r) |q dρ

) 1
q

+
(s− t)σ+1

s− r

(∫ 1

0
ρσpdρ

) 1
p
(∫ 1

0
| ℵ′(ρt+ (1− ρ)s) |q dρ

) 1
q

.

Now, by employing the convexity of | ℵ′ |q and Jensen integral inequality, we obtain∣∣∣ [(t− r)σ + (s− t)σ

s− r

]
ℵ (t) +

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)] (17)

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ
r,t ℵ(r) + Iκ,σ

s,t ℵ(s)
]∣∣∣

≤(t− r)σ+1

s− r

(∫ 1

0
ρσpdρ

) 1
p
(∫ 1

0
| ℵ′(ρt+ (1− ρ)r) |q dρ

) 1
q



G. Rahman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5816 15 of 18

+
(s− t)σ+1

s− r

(∫ 1

0
ρσpdρ

) 1
p
(∫ 1

0
| ℵ′(ρt+ (1− ρ)s) |q dρ

) 1
q

. (18)

Now, since∫ 1

0
| ℵ′(ρt+ (1− ρ)r) |q dρ ≤

∫ 1

0
ρ0 | ℵ′(ρt+ (1− ρ)r) |q dρ

≤
(∫ 1

0
ρ0dρ

)
| ℵ′

(
1∫ 1

0 ρ0dρ

∫ 1

0
(ρt+ (1− ρ)r)dρ

)
|q

≤ | ℵ′
(
t+ r

2

)
|q . (19)

Similarly ∫ 1

0
| ℵ′(ρt+ (1− ρ)s) |q dρ ≤| ℵ′

(
t+ s

2

)
|q . (20)

By substituting (20) and (19) in (17) and then solving the integrals, we get the desire
assertion.

Corollary 10. Applying Theorem 6 for t = r+s
2 , we get the following inequality∣∣∣(s− r)σ−1

2σ−1
ℵ
(
r + s

2

)
+

σ(1− κ)

κ(s− r)
Γ(σ) [ℵ (r) + ℵ (s)]

−σM(κ)Γ(σ)

κ(s− r)

[
Iκ,σ

r, r+s
2

ℵ(r) + Iκ,σ

s, r+s
2

ℵ(s)
]∣∣∣

≤
(

1

σp+ 1

) 1
p 1

s− r

{
(s− t)σ+1 | ℵ′

(
3r + s

4

)
| +(s− t)σ+1 | ℵ′

(
r + 3s

4

)
|
}
.

Remark 13. Applying Theorem 6 for σ = κ, we get Theorem 6 proved by Ahmad et al.
[40].

Remark 14. Applying Corollary 10 for σ = κ, we get Corollary 10 proved earlier by
Ahmad et al. [40].

4. Concluding Remarks

In this paper, we developed Ostrowski-type inequalities for convex functions containing
the Hattaf fractional integral operators. The results presented in this paper are original
to the best of our knowledge. Given the widespread applications of convex functions in
numerous scientific fields, our unique advancements are believed to be applicable to sev-
eral special functions, including convexity, interval analysis, quantum calculus, fractional
calculus, and coordinates. The inequalities in term of AB operators will be restored if
we put κ = σ and classical inequalities if we put κ = σ = 1. One can obtain Grüss
type inequalities, Chebyshev type inequalities, Reverse Minkowski’s type inequalities and
certain other type inequalities by using Hattaf fractional integral operators.



G. Rahman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5816 16 of 18

Acknowledgements

The authors M. A. Alghafli and N. Mlaiki would like to thank Prince Sultan University
for paying the publication fees for this work through TAS LAB.

Declaration

Competing Interests

All authors declare no conflict of interest.

Author’s Contributions

The authors have worked equally when writing this paper. All authors read and
approved the final manuscript.

References

[1] M. Samraiz, Z. Perveen, T. Abdeljawad, S. Iqbal, and S. Naheed. On certain fractional
calculus operators and their applications in mathematical physics. Physica Scripta,
95(11):115210, 2020.

[2] M. Samraiz, Z. Perveen, G. Rahman, K. S. Nisar, and Devendra Kumar. On (k, s)-
Hilfer Prabhakar fractional derivative with applications in mathematical physics.
Frontiers in Physics, 8:1–9, 2020.

[3] A. Nazir, G. Rahman, A. Ali, S. Naheed, K. S. Nisar, W. Albalawi, and H. Y.
Zahran. On generalized fractional integral with multivariate Mittag-Leffler function
and its applications. Alexandria Engineering Journal, 61:9187–9201, 2022.

[4] M. Samraiz, A. Mehmood, S. Naheed, G. Rahman, A. Kashuri, and K. Nonlaopon. On
novel fractional operators involving the multivariate Mittag–Leffler function. Mathe-
matics, 10(21):3991, 2022.

[5] M. Samraiz, M. Umer, T. Abdeljawad, S. Naheed, G. Rahman, and K. Shah. On
Riemann-type weighted fractional operators and solutions to Cauchy problems. Com-
puter Modeling in Engineering and Sciences, 36(1):901–918, 2023.

[6] D. Baleanu and A. Fernandez. On some new properties of fractional derivatives
with Mittag-Leffler kernel. Communications in Nonlinear Science and Numerical
Simulation, 59:444–462, 2018.

[7] T. Abdeljawad and D. Baleanu. Integration by parts and its applications of a new
nonlocal fractional derivative with Mittag-Leffler nonsingular kernel. Journal of Non-
linear Sciences and Applications, 10(3):1098–1107, 2017.

[8] T. Abdeljawad. A Lyapunov type inequality for fractional operators with nonsingular
Mittag-Leffler kernel. Journal of Inequalities and Applications, 2017:130, 2017.

[9] W. H. Huang, M. Samraiz, A. Mehmood, D. Baleanu, G. Rahman, and S. Naheed.
Modified Atangana-Baleanu fractional operators involving generalized Mittag-Leffler
function. Alexandria Engineering Journal, 75:639–648, 2023.



G. Rahman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5816 17 of 18

[10] T. Abdeljawad and D. Baleanu. On fractional derivatives with generalized Mittag-
Leffler kernels. Advances in Difference Equations, 2018:468, 2018.

[11] T. Abdeljawad. Fractional operators with generalized Mittag-Leffler kernels and their
iterated differintegrals. Chaos: An Interdisciplinary Journal of Nonlinear Science,
29:2, 2019.

[12] S. S. Dragomir. Ostrowski type inequalities for Riemann–Liouville fractional inte-
grals of absolutely continuous functions in terms of norms. RGMIA Research Report
Collection, 20:49, 2017.

[13] M. Z. Sarikaya, E. Set, H. Yaldiz, and N. Başak. Hermite–Hadamard inequalities for
fractional integrals and related fractional inequalities. Mathematical and Computer
Modelling, 57:2403–2407, 2013.

[14] M. Z. Sarikaya and H. Yildirim. On Hermite–Hadamard type inequalities for
Riemann-Liouville fractional integrals. Miskolc Mathematical Notes, 17:1049–1059,
2017.

[15] H. M. Srivastava, Z. H. Zhang, and Y. D. Wu. Some further refinements and exten-
sions of the Hermite–Hadamard and Jensen inequalities in several variables. Mathe-
matical and Computer Modelling, 54:2709–2717, 2011.

[16] A. Rafiq, N. A. Mir, and F. Ahmad. Weighted Chebyshev–Ostrowski type inequalities.
Applied Mathematics and Mechanics, 28:901–906, 2007.

[17] Y. Shuang and F. Qi. Integral inequalities of Hermite–Hadamard type for extended
s-convex functions and applications. Mathematics, 6(11):223, 2018.

[18] K. Mehrez and P. Agarwal. New Hermite–Hadamard type integral inequalities for
convex functions and their applications. Journal of Computational and Applied Math-
ematics, 350:274–285, 2019.

[19] M. J. Park, O. M. Kwon, and J. H. Ryu. Generalized integral inequality: application
to time-delay systems. Applied Mathematics Letters, 77:6–12, 2018.

[20] M. Z. Sarikaya, T. Tunc, and H. Budak. On generalized some integral inequalities for
local fractional integrals. Applied Mathematics and Computation, 276:316–323, 2016.

[21] E. Set, S. I. Butt, A. O. Akdemir, A. Karaoğlan, and T. Abdeljawad. New integral in-
equalities for differentiable convex functions via Atangana-Baleanu fractional integral
operators. Chaos, Solitons and Fractals, 143:110554, 2021.

[22] G. Rahman, K. S. Nisar, and F. Qi. Some new inequalities of the Grüss type for
conformable fractional integrals. AIMS Mathematics, 3(4):575–583, 2018.

[23] G. Rahman, K. S. Nisar, A. Ghaffar, and F. Qi. Some inequalities of the Grüss
type for conformable k-fractional integral operators. Revista de la Real Academia de
Ciencias Exactas, F́ısicas y Naturales. Serie A. Matemáticas, 114:9, 2020.

[24] G. Rahman, Z. Ullah, A. Khan, E. Set, and K. S. Nisar. Certain Chebyshev type
inequalities involving fractional conformable integral operators. Mathematics, 7:364,
2019.

[25] G. Rahman, T. Abdeljawad, F. Jarad, and K. S. Nisar. Bounds of generalized propor-
tional fractional integrals in general form via convex functions and their applications.
Mathematics, 8:113, 2020.

[26] T. Toplu, M. Kadkal, and İ. İşcan. On n-polynomial convexity and some related



G. Rahman et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5816 18 of 18

inequalities. AIMS Mathematics, 5:1304–1318, 2020.
[27] M. Kadkal and İ. İşcan. Exponential type convexity and some related inequalities.

Journal of Inequalities and Applications, 2009:82, 2020.
[28] S. I. Butt, M. Tariq, A. Aslam, H. Ahmad, and T. A. Nofel. Hermite–Hadamard

type inequalities via generalized harmonic exponential convexity. Journal of Function
Spaces, 2021:5533491, 2021.

[29] M. Tariq. New Hermite–Hadamard type inequalities via p-harmonic exponential type
convexity and applications. Universal Journal of Mathematics and Applications, 4:59–
69, 2021.

[30] D. S. Mitrinović, J. Pečarić, and A. M. Fink. Inequalities Involving Functions and
Their Integrals and Derivatives, volume 53. Springer Science and Business Media,
Dordrecht, The Netherlands, 2012.

[31] S. S. Dragomir and S. Wang. A new inequality of Ostrowski type in L1 norm and
applications to some special means and to some numerical quadrature rules. Tamkang
Journal of Mathematics, 28:239–244, 1997.

[32] S. S. Dragomir and S. Wang. Applications of Ostrowski’s inequality to the estimation
of error bounds for some special means and for some numerical quadrature rules.
Applied Mathematics Letters, 11:105–109, 1998.

[33] N. S. Barnett and S. S. Dragomir. An Ostrowski type inequality for double integrals
and applications for cubature formulae. Soochow Journal of Mathematics, 27:109–114,
2001.

[34] P. Cerone, S. S. Dragomir, and J. Roumeliotis. An inequality of Ostrowski type
for mappings whose second derivatives are bounded and applications. East Asian
Mathematical Journal, 15:1–9, 1999.

[35] Z. Retkes. An extension of the Hermite–Hadamard inequality. Acta Scientiarum
Mathematicarum (Szeged), 74(1):95–106, 2008.

[36] K. M. Owolabi. Modelling and simulation of a dynamical system with the Atangana–
Baleanu fractional derivative. European Physical Journal Plus, 133(1):15, 2018.

[37] D. Kumar, J. Singh, and D. Baleanu. Analysis of regularized long-wave equation
associated with a new fractional operator with Mittag-Leffler type kernel. Physica A:
Statistical Mechanics and its Applications, 492:155–167, 2018.

[38] Z. Jianke, W. Gaofeng, Z. Xiaobin, and Z. Chang. Generalized Euler–Lagrange equa-
tions for fuzzy fractional variational problems under gH-Atangana–Baleanu differen-
tiability. Journal of Function Spaces, 2018:2740678, 2018.

[39] K. Hattaf. A new generalized definition of fractional derivative with non-singular
kernel. Computation, 8:49, 2020.

[40] H. Ahmad, M. Tariq, S. K. Sahoo, S. Askar, A. E. Abouelregal, and K. M. Khed-
her. Refinements of Ostrowski type integral inequalities involving Atangana–Baleanu
fractional integral operator. Symmetry, 13:2059, 2021.