EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5725
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

New Study of Prabhakar Operators Associated With
Inequalities and Its Significant Applications With

Different Convexity

Rana Safdar Ali1,∗, Nazia Yaseen1, Gauhar Rahman2, Ahmad Aloqaily3,
Nabil Mlaiki3

1 Department of Mathematics and Statistics, Faculty of Science, The University of Lahore,
Sargodha Campus, Sargodha, Punjab, Pakistan
2 Department of Mathematics and Statistics, Faculty of Science, Hazara University, Mansehra,
KPK, Pakistan
3 Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586,
Saudi Arabia

Abstract. Convexity plays a dominant role in the modification of fractional inequalities. Most
fractional inequalities are proved based on different types of convexity and fractional operators,
which have immense applications in various areas of mathematics. This article aims to investigate
the Hermite-Hadamard type inequalities with a different kind of convexity by the implementation of
Prabhakar fractional operators. Moreover, we discuss the behavior of trapezoidal type inequalities
for the h-Godunova-Levin pre-invex function through Prabhakar fractional operators. Additionally,
we present a comparison of our findings with existing literature, which are summarized through
corollaries.
2020 Mathematics Subject Classifications: 26A51, 26D10,2 6A33, 26D20
Key Words and Phrases: Convexity, Hermite-Hadamard inequalities, Prabhaker fractional
integral operators, trapezoid inequalities

1. Introduction

Fractional integrals and inequalities define an essential area of research in the field of
mathematical analysis and its great applications [1, 16–19, 21–23]. These notions are a
generalization of classical integral inequalities and provide important new ideas with wide
areas of eventual application. With regard to the integral type of inequalities, it is also
observed that an impressive development has been achieved in this area of study as these
inequalities are becoming more significant to pure and applied mathematics [6, 14].

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5725

Email addresses: rsafdar0@gmail.com (R. S. Ali),
naziaimran145@gmail.com (N. Yaseen), gauhar55uom@gmail.com (G. Rehman),
maloqaily@psu.edu.sa (A. Aloqaily), nmlaiki@psu.edu.sa; nmlaiki2012@gmail.com (N. Mlaiki)

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



R. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5725 2 of 14

The development of the theory of convexity, which is very closely linked to inequality,
has progressed considerably. It is worth noting that convex functions are very important in
the study and derivation of many integral inequalities [3, 7, 8]. One such is the well-known
Hermite-Hadamard inequality. Formulated by Charles Hermite in 1881 and modernized
by Jacques Hadamard in 1893, the inequality has been a part of fundamental concepts of
convexity . The historical background of the Hermite-Hadamard inequality would suggest
that it thrived long before the mid-1970s revival by Dragoslav Mitrinović, who published
several books on the history of mathematics in general and convex inequalities in particular
[9, 15].

Recently, various generalizations have turned the studies in inequality areas to another
range. The classical structure of differentiable convex functions was broadened by the term
invexity, introduced in 1981 by Robert Hanson [13], and so new optimization and analysis
areas could be developed . Building on this, Mond [28] and Weir [27] further developed
the notion of preinvexity, which has been helpful in perfecting optimization theory .

Further advancements have been inclusive of various generalized concepts of convexity
that were defined. Dragomir [7] presented the s-Godunova-Levin type convexity, which
has been the focus of many studies in the subsequent period [20]. Moreover, the produc-
tive concept of h-convexity by Varošanec [26] and also h-Godunova-Levin convexity and
preinvexity by Almutari [4] opened new avenues and methods in this area .

This paper study h-Godunova-Levin types of convex and preinvex functions and the
fractional integral operators, including the Mittag-Leffler functions, to achieve the new
fractional Hermite-Hadamard and the trapezoid inequalities.

2. Preliminaries

In this section, we discuss the basic definitions which help to understand our main
results.

Definition 1. [2] A function § : I → R is termed convex if it satisfies the following
condition:

§[tos̊+ (1− to)æ] ≤ to§(̊s) + (1− to)§(æ),

for all to ∈ [0, 1], s̊,æ ∈ I.
Building upon this concept of convexity, we can establish the Hermite-Hadamard (H-H)
type inequality as:

§
(
s̊+æ

2

)
≤ 1

æ− s̊

∫ æ

s̊
§(to) dto ≤

§(̊s) + §(æ)
2

. (1)

Numerous related results are presented in [25], assuming s̊,æ ∈ I ⊆ Rand̊s < æ.

Definition 2. [24] Consider an invex set I ⊆ R defined in relation to a bifunction § :
I × I → R. For æ, s̊ ∈ I and λ ∈ [0, 1], we define:

s̊+ λ§(æ, s̊) ∈ I



R. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5725 3 of 14

Definition 3. [24] A function § : I → R is called preinvex for æ, s̊ ∈ I and to ∈ [0, 1] if:

§(̊s+ toζ(æ, s̊)) ≤ to§(æ) + (1− to)§(̊s),

where I is an invex set relative to the binary function ζ.

Definition 4. [10] A function § : I ⊆ R → R, which takes only positive values, is known
as a Godunova-Levin function if, for all æ, s̊ ∈ I and to ∈ (0, 1), the following inequality
holds:

§(toæ+ (1− to)̊s) ≤
§(æ)
to

+
§(̊s)
1− to

,

for all æ, s̊ ∈ I, to ∈ (0, 1).

Definition 5. [4] Assume h : (0, 1) → R is a non-negative function. We say a function
§ : I → R is h-Godunova-Levin if, for any æ, s̊ ∈ I and to ∈ (0, 1), the following inequality
holds:

§(toæ+ (1− to)̊s) ≤
§(æ)
h(to)

+
§(̊s)

h(1− to)
.

Definition 6. [4] A function § : I → R is called h-Godunova-Levin preinvex with respect
to ζ if, for any æ, s̊ ∈ I and to ∈ (0, 1), the inequality

§(æ + toζ (̊s,æ)) ≤
§(æ)

h(1− to)
+

§(̊s)
h(to)

,

is satisfied.

Definition 7. [12] Let § ∈ L1[æ, s̊], then the Riemann-Liouville left and right fractional
integrals are defined as follows:

Iαæ+§(z) =
1

Γ(α)

∫ z

æ
(z − u)α−1§(u) du, z > æ,

Iαs̊−§(z) =
1

Γ(α)

∫ s̊

z
(u− z)α−1§(u) du, z < s̊.

Definition 8. [29] The gamma function is defined by the following integral representation:

Γ(z) =

∫ +∞

0
uz−1e−udu,

for ℜ(z) > 0.

Definition 9. [29] The Pochhammer symbol is defined as follows:

(z)k =

{
1, for k = 0, z ̸= 0,

z(z + 1) · · · (z + k − 1), for k ≥ 1.

For k ∈ N and z ∈ C.

(z)k =
Γ(z + k)

Γ(z)
,

where Γ is the gamma function.



R. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5725 4 of 14

Definition 10. [5] The Mittag-Leffler function of three parameters:

Eα
β,γ(w; p) =

+∞∑
n=0

(γ)n
Γ(βn+ α)

wn

n!
, (w,α, β, γ ∈ C,ℜ(β) > 0).

Definition 11. [11] Let α, β, γ ∈ C,ℜ(α) > 0,ℜ(β) > 0. Let § ∈ L1[æ, s̊] and x ∈ [æ, s̊].
Then the left-sided and the right-sided Prabhakar fractional operators Jα,γ

β;æ+§ and Jα,γ
β ;̊s−§,

are defined by (
Jα,γ
β;æ+§

)
(x; r) =

∫ x

æ
(x− t)β−1Eα

β,γ (ω(x− t)α; r) §(t)dt,(
Jα,γ
β ;̊s−§

)
(x; r) =

∫ s̊

x
(t− x)β−1Eα

β,γ (ω(t− x)α; r) §(t)dt.

In this work, the following notations will be used:(
Jæ

+

s̊,β

)
(ω, §) =

(
Jα,γ
β;æ+§

)
(̊s, p)(

Js̊
−
æ,β

)
(ω, §) =

(
Jα,γ
β ;̊s−§

)
(æ; p)

3. Fractional Analysis of the Hermite-Hadamard (H-H) type
Inequalities via h-Godunova-Levin convexity (h-GL).

This section focuses on deriving Hermite-Hadamard type inequalities for h-Godunova-
Levin convex functions by means of the Fractional Function Operator, which is detailed
below.

Theorem 1. Let § : [æ, s̊] → R be an h-Godunova-Levin convex function, where 0 < æ < s̊
and § ∈ L1[æ, s̊]. Assume h : (0, 1) → R is a positive function with h(to) ̸= 0 ; then,

h(1/2)

2
§
(
æ+ s̊

2

)(
Js̊

−
æ,β

) (
ω′, 1

)
≤ 1

2

[(
ℑs̊−
æ,β

) (
ω′, §

)
+

(
Jæ

+

s̊,β

) (
ω′, §

)]
≤ §(æ) + §(̊s)

2

∫ 1

0

[
1

h(to)
+

1

h(1− to)

]
(1− to)

β−1Jαβ,γ (ω(1− to)
α; p) dto,

ω′ =
ω

(̊s− æ)α
.

Proof. Using the h -Godunova-Levin convexity of § on [æ, s̊] , let m,n ∈ [æ, s̊], and we
obtain

§((η)m+ (1− η)n) ≤ §(m)

h(η)
+

§(n)
h(1− η)

(2)

For putting the values m = toæ + (1 − to)̊s, n = (1 − to)æ + tos̊ and η = 1
2 in equation

(2), we have

§
(
æ+ s̊

2

)
≤ 1

h(1/2)
[§(toæ+ (1− to)̊s) + §((1− to)æ + tos̊)] (3)



R. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5725 5 of 14

Multiplying each side by (1−to)
β−1Jαβ,γ (ω(1− to)

α; p) and integrate the resultant inequal-
ity on [0, 1] in terms of to in the equation (3), we have

§
(
æ+ s̊

2

)∫ 1

0
(1− to)

β−1Jαβ,γ (ω(1− to)
α; p) dto ≤

1

h(12)
×[ ∫ 1

0
(1− to)

β−1Jαβ,γ (ω(1− to)
α; p) §(toæ+ (1− to)̊s)dto

+

∫ 1

0
(1− to)

β−1Jαβ,γ (ω(1− to)
α; p) §((1− to)æ + tos̊)dto

]
§
(
æ+ s̊

2

) +∞∑
n=0

(γ)n
Γ(βn+ α)

wn

n!

∫ 1

0
(1− to)

αn+β−1dto ≤
1

h(1/2)

+∞∑
n=0

(γ)n
Γ(βn+ α)

wn

n!

×
[∫ 1

0
(1− to)

αn+β−1§(toæ+ (1− to)̊s)dto +

∫ 1

0
(1− to)

αn+β−1§((1− to)æ + tos̊)dto

]
(4)

By evaluating the integrals in inequality (4), we obtain

h(1/2)

2
§
(
æ+ s̊

2

)(
Js̊

−
æ,β

(
ω′, 1

))
≤ 1

2

[(
Jæ

+

s̊,β

) (
ω′; §

)
+
(
Js̊

−
æ,β

) (
ω′; §

)]
(5)

For the second half of the inequality, we similarly employ the h-Godunova-Levin con-
vexity of §, we have

§(toæ+ (1− to)̊s) ≤
§(æ)
h(to)

+
§(̊s)

h(1− to)

§((1− to)æ + tos̊) ≤
§(æ)

h(1− to)
+

§(̊s)
h(to)

After adding the above inequalities, we have

§(toæ+ (1− to)̊s) + §((1− to)æ + tos̊) ≤ (§(æ) + §(̊s))
[

1

h(to)
+

1

h(1− to)

]
(6)

Multiplying both sides by (1 − to)
β−1Jαβ,γ (ω(1− to)

α; p) and integrating the resultant
inequality on [0, 1] with respect to to in equation (6), we obtain[∫ 1

0
(1− to)

β−1Jαβ,γ (ω(1− to)
α; p) §(toæ+ (1− to)̊s)dto

]
+

[∫ 1

0
(1− to)

β−1Jαβ,γ (ω(1− to)
α; p) §((1− to)æ + tos̊)dto

]
≤ (§(æ) + §(̊s))

∫ 1

0

[
1

h(to)
+

1

h(1− to)

]
(1− to)

β−1Jαβ,γ (ω(1− to)
α; p) dto (7)



R. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5725 6 of 14

After solving the equation (7), we have

1

2

[(
Jæ

+

s̊,β

)
(ω′, §) +

(
Js̊

−
æ,β

) (
ω′; §

)]
≤ §(æ) + §(̊s)

2

∫ 1

0

[
1

h(to)
+

1

h(1− to)

]
(1−to)

β−1Jαβ,γ (ω(1− to)
α; p) dto

(8)
Combining the equations (5) and (8), we obtain

h(1/2)

2
§
(
æ+ s̊

2

)(
Js̊

−
æ,β

) (
ω′, 1

)
≤ 1

2

[(
ℑs̊−
æ,β

) (
ω′, §

)
+
(
Jæ

+

s̊,β

) (
ω′, §

)]
≤ §(æ) + §(̊s)

2

∫ 1

0

[
1

h(to)
+

1

h(1− to)

]
(1− to)

β−1Jαβ,γ (ω(1− to)
α; p) dto

Corollary 1. Taking h(to) = tso in Theorem (1), we derive an inequality of the (H-H)
type for s-Godunova-Levin (GL) type convex functions:

(1/2)s

2
§
(
æ+ s̊

2

)(
Js̊

−
æ,β

) (
ω′, 1

)
≤ 1

2

[(
Js̊

−
æ,β

) (
ω′, §

)
≤ §(æ) + §(̊s)

2

∫ 1

0

[
1

tso
+

1

(1− to)s

]
(1− to)

β−1Jαβ,γ (ω(1− to)
α; p) dto

Corollary 2. By selecting h(to) = 1 in Theorem (1), we derive an inequality of the (H-H)
type for the p function:

1
2§

(
æ+s̊
2

) (
Js̊

−
æ,β

)
(ω′, 1) ≤ 1

2

[(
ℑs̊−
æ,β

)
(ω′, §) +

(
ℑæ+

s̊,β

)
(ω′, §)

]
≤ (§(æ) + §(̊s))

(
Jæ

+

s̊,β

)
(ω′, 1)

Corollary 3. Selecting h(to) = 1/to in Theorem (1), an inequality of the (H-H) type is
derived for functions that are convex:

§
(
æ+s̊
2

) (
Js̊

−
æ,β

)
(ω′, 1) ≤ 1

2

[(
ℑs̊−
æ,β

)
(ω′, §) +

(
ℑæ+

s̊,β

)
(ω′, §)

]
≤ §(æ)+§(̊s)

2

(
Jæ

+

s̊,β

)
(ω′, 1)

Corollary 4. By choosing h(to) = to in Theorem (1), we obtain an (H-H) type inequality
for (G-L) functions:

1
4§

(
æ+s̊
2

) (
Js̊

−
æ,β

)
(ω′, 1) ≤ 1

2

[(
Js̊

−
æ,β

)
(ω′, §) +

(
Jæ

+

s̊,β

)
(ω′, §)

]
≤ §(æ)+§(̊s)

2

∫ 1
0

[
(1−to)β−2

1−to

]
Jαβ,γ (ω(1− to)

α; p) dto

Corollary 5. Choosing h(to) = 1/tso in Theorem (1), we obtain an (H-H) type inequality
for s-convex functions:

2s−1§
(
æ+s̊
2

) (
ℑs̊−
æ,β

)
(ω′, 1) ≤ 1

2

[(
ℑs̊−
æ,β

)
(ω′, §) +

(
Jæ

+

s̊,β

)
(ω′, §)

]
≤ §(æ)+§(̊s)

2

∫ 1
0 [tso + (1− to)

s] (1− to)
β−1Jαβ,γ (ω(1− to)

α; p) dto



R. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5725 7 of 14

4. On Trapezoidal-Type Inequalities for Prabhakar Functions with
Preinvexity Properties of the h-Godunova-Levin Type

In this section, we prove a lemma concerning Prabhakar fractional operators that
possess the h-Godunova-Levin preinvexity property. This lemma is crucial for supporting
the derivation of our main results.

Lemma 1. Let § : I = [æ,æ+ ζ (̊s,æ)] → R be a differentiable function, and let I be a set
that is invex with respect to ζ : I × I → R, where ζ (̊s,æ) > 0 for all s̊,æ ∈ I. Then

§(æ) + §(æ + ζ (̊s,æ))

2
ℑα
β,γ(ω; p)−

1

2ζ (̊s,æ)β−1
(9)

×
[(

Jæ
+

æ+ζ (̊s,æ),β−1

) (
ω′; §

)
+
(
J
(æ+ζ (̊s,æ))−

æ,β−1

) (
ω′; §

)]
=

ζ (̊s,æ)

2
I

where

I =

∫ 1

0
(1− to)

β−1Jαβ,γ (ω(1− to)
α; p) §′(æ + toζ (̊s,æ))dto +

∫ 1

0
(−to)

β−1Jαβ,γ (ω(to)
α; p) §′(æ + toζ (̊s,æ))dto,

and ω′ = (ω/ζ (̊s,æ)α).
Proof. Consider the integral

I =

∫ 1

0
(1− to)

β−1Jαβ,γ (ω(1− to)
α; p) §′(æ + toζ (̊s,æ))dto

+

∫ 1

0
(−to)

β−1Jαβ,γ (ω(to)
α; p) §′(æ + toζ (̊s,æ))dto (10)

Let
I = I1 + I2

First, we take the fractional integral I1, we have

I1 =
+∞∑
s̊=0

(γ)n
Γ(βn+ α)

wn

n!

∫ 1

0
(1− to)

β−1+αn§′(æ + toζ (̊s,æ))dto

Taking the integration by parts, we have

I1 =
+∞∑
n=0

(γ)n
Γ(βn+ α)

wn

n!
×[

(1− to)
β+αn−1 §(æ + toζ (̊s,æ))

ζ (̊s,æ)

∣∣∣∣1
0

−β + αn− 1

ζ (̊s,æ)

∫ 1

0
(1− to)

β+αn−2§(æ + toζ (̊s,æ))dto

]

I1 =
+∞∑
n=0

(γ)n
Γ(βn+ α)

wn

n!



R. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5725 8 of 14

×
[
(æ + ζ (̊s,æ))

ζ (̊s,æ)
− β + αn− 1

ζ (̊s,æ)

∫ 1

0
(1− to)

β+αn−2§(æ + toζ (̊s,æ))dto

]

I1 =
§(æ + ζ (̊s,æ))

ζ (̊s,æ)
ℑα
β,γ(ω; p)−

1

(ζ (̊s,æ))β

(
Jæ

+

æ+ζ (̊s,æ),β−1

) (
ω′, §

)
Continuing in the same manner, we obtain

I2 =
§(æ)
ζ (̊s,æ)

Jαβ,γ(ω; p)−
1

(ζ (̊s,æ))β

(
J
æ+ζ (̊s,æ)−

æ,β−1

) (
ω′, §

)
I =

§(æ) + §(æ + ζ (̊s,æ))

ζ (̊s,æ)
Jαβ,γ(ω; p)−

1

(ζ (̊s,æ))β

×
[(

J
(æ+ζ (̊s,æ))−

æ,β−1

) (
ω′, §

)
+
(
Jæ

+

æ+ζ (̊s,æ),β−1

) (
ω′, §

)]
By multiplying by ζ (̊s,æ)/2, we obtain

§(æ) + §(æ + ζ (̊s,æ))

2
ℑα
β,γ(ω; p)−

1

2ζ (̊s,æ)β−1

×
[(

Jæ
+

æ+ζ (̊s,æ),β−1

) (
ω′; §

)
+
(
J
(æ+ζ (̊s,æ))−

æ,β−1

) (
ω′; §

)]
=

ζ (̊s,æ)

2
I

By Lemma 1, we present the following theorem.

Theorem 2. Consider a function §: I=[æ, æ+ ζ (̊s,æ)] −→ (0,+∞)withI ∈ R, and let it
be a differentiable function on I. Also, suppose that |§′| is a h-Godunova-Levin preinvex
function on I; then,

§(æ) + §(æ + ζ (̊s,æ))

2
Jαβ,γ(ω; p)−

1

2ζ (̊s,æ)β−1

×
[(

Jæ
+

æ+ζ (̊s,æ),β−1

) (
ω′, §

)
+
(
J
(æ+ζ (̊s,æ))−

æ,β−1

) (
ω′, §

)]
≤ ζ (̊s,æ)

2

(∣∣§′(æ)∣∣+ ∣∣§′(̊s)∣∣) ∫ 1

0

+∞∑
n=0

∣∣∣∣ (γ)n
Γ(βn+ α)

wn

n!

∣∣∣∣
×
∣∣∣∣(1− to)

β+αn−1 − (to)
β+αn−1

h(to)

∣∣∣∣ dto.
Proof.∣∣∣∣ §(æ) + §(æ + ζ (̊s,æ))

2
Jαβ,γ(ω; p)−

1

2ζ (̊s,æ)β−1

×
[(

ℑæ+

æ+ζ (̊s,æ),β−1

) (
ω′, §

)
+
(
ℑ(æ+ζ (̊s,æ))−

æ,β−1

) (
ω′, §

)]
|



R. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5725 9 of 14

=

∣∣∣∣ζ (̊s,æ)2
I

∣∣∣∣
≤ ζ (̊s,æ)

2

+∞∑
n=0

∣∣∣∣ (γ)n
Γ(βn+ α)

wn

n!

∣∣∣∣ ∫ 1

0

∣∣∣(1− to)
β+αn−1 − (to)

β+αn−1
∣∣∣ ∣∣§′(æ + toζ (̊s,æ))

∣∣dto
≤ ζ (̊s,æ)

2

+∞∑
n=0

∣∣∣∣ (γ)n
Γ(βn+ α)

wn

n!

∣∣∣∣
×
∫ 1

0

∣∣∣(1− to)
β+αn−1 − (to)

β+αn−1
∣∣∣ ∣∣∣∣§′(æ)h(to)

+
§′(̊s)

h(1− to)

∣∣∣∣ dto
≤ ζ (̊s,æ)

2

+∞∑
n=0

∣∣∣∣ (γ)n
Γ(βn+ α)

wn

n!

∣∣∣∣
×
[∣∣§′(æ)∣∣ ∫ 1

0

∣∣∣(1− to)
β+αn−1 − (to)

β+αn−1
∣∣∣ 1

h(to)
dto + |§′(̊s)

∣∣∣∣∫ 1

0

∣∣∣(1− to)
β+αn−1 − (to)

β+αn−1
∣∣∣ 1

h(1− to)
dto

=
ζ (̊s,æ)

2

(∣∣§′(æ)∣∣+ ∣∣§′(̊s)∣∣) ∫ 1

0

+∞∑
n=0

∣∣∣∣ (γ)n
Γ(βn+ α)

wn

n!

∣∣∣∣
×
∣∣∣∣(1− to)

β+αn−1 − (to)
β+αn−1

h(to)

∣∣∣∣dto.
Corollary 6. Taking ζ (̊s,æ) = s̊−æ in Theorem (2), we derive the following inequality:

§(æ) + §(̊s)
2

Jαβ,γ(ω; p)−
1

2(̊s− æ)β−1

×
[(

Jæ
+

æ+ζ (̊s,æ),β−1

) (
ω′, §

)
+
(
J
(æ+ζ (̊s,æ))−

æ,β−1

) (
ω′, §

)]
≤ (̊s− æ)

2

(∣∣§′(æ)∣∣+ ∣∣§′(̊s)∣∣) ∫ 1

0

+∞∑
n=0

∣∣∣∣ (γ)n
Γ(βn+ α)

wn

n!

∣∣∣∣
×
∣∣∣∣(1− to)

β+αn−1 − (to)
β+αn−1

h(to)

∣∣∣∣ dto.
Theorem 3. Consider the function § : I = [æ,æ+ζ (̊s,æ)] −→ (0,+∞), where I ∈ R, and
assume it is differentiable on I. Additionally, let |§′|q be an h-Godunova-Levin preinvex
function on I, with p > 1 and q = p

p−1 ; then.∣∣∣∣ §(æ) + §(æ + ζ (̊s,æ))

2
Jαβ,γ(ω; p)−

1

2ζ (̊s,æ)β−1

×
[(

ℑæ+

æ+ζ (̊s,æ),β−1

) (
ω′, §

)
+
(
ℑ(æ+ζ (̊s,æ))−

æ,β−1

) (
ω′, §

)]
|



R. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5725 10 of 14

≤ ζ (̊s,æ)

2

(∣∣§′(æ)∣∣q + ∣∣§′(̊s)∣∣q)1/q
×
(∫ 1

0

∣∣∣(1− to)
β−1Jαβ,γ (ω(1− to)

α; p)− (to)
β−1Jαβ,γ (ω(to)

α; p)
∣∣∣p dto)1/p

×
(∫ 1

0

1

h(to)
dto

)1/q

.

Proof. Using Lemma 1, we have:∣∣∣∣ §(æ) + §(æ + ζ (̊s,æ))

2
Jαβ,γ(ω; p)−

1

2ζ (̊s,æ)β−1

×
[(

ℑæ+

æ+ζ (̊s,æ),β−1

) (
ω′, §

)
+
(
ℑ(æ+ζ (̊s,æ))−

æ,β−1

) (
ω′, §

)]
|

=

∣∣∣∣ζ (̊s,æ)2
I

∣∣∣∣
≤ ζ (̊s,æ)

2

∫ 1

0

∣∣∣(1− to)
β−1Jαβ,γ (ω(1− to)

α; p)− (to)
β−1Jαβ,γ (ω(to)

α; p)
∣∣∣

×
∣∣§′(æ + toζ (̊s,æ))

∣∣dto.
Using Holder integral inequality, we have

≤ ζ (̊s,æ)

2

(∫ 1

0

∣∣∣(1− to)
β−1Jαβ,γ (ω(1− to)

α; p)− (to)
β−1Jαβ,γ (ω(to)

α; p)
∣∣∣p dto)1/p

×
(∫ 1

0

∣∣§′(æ + toζ (̊s,æ))
∣∣q dto)1/q

. (11)

Since (1/p) + (1/q) = 1, and because |§′|q is an (h-GL) preinvex function, we obtain:∫ 1

0

∣∣§′(æ + toζ (̊s,æ))
∣∣q dto ≤ ∫ 1

0

(
|§′(æ)|q

h(to)
+

|§′(̊s)|q

h(1− to)

)
dto

≤
(∣∣§′(æ)∣∣q + ∣∣§′(̊s)∣∣q) ∫ 1

0

1

h(to)
dto. (12)

Using (12) in (11), we have the required result.

Theorem 4. With the assumptions of Theorem 3, we get the following inequality related
to the Hermite-Hadamard inequality:∣∣∣∣§(æ) + §(æ + ζ (̊s,æ))

2
Jαβ,γ(ω; p)−

1

2ζ (̊s,æ)β−1

×
[(

ℑæ+

æ+ζ (̊s,æ),β−1

) (
ω′, §

)
+
(
ℑ(æ+ζ (̊s,æ))−

æ,β−1

) (
ω′, §

)]
≤ ζ (̊s,æ)

21/q
(∣∣§′(æ)∣∣q + ∣∣§′(̊s)∣∣q)1/q [Jαβ,γ(ω; p)− (

1

2

)β−1

Jαβ,γ

(
ω

(
1

2

)α

; p

)]1−(1/q)



R. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5725 11 of 14

×

∫ 1

0

∣∣∣(1− to)
β−1Jαβ,γ (ω(1− to)

α; p)− (to)
β−1Jαβ,γ (ω(to)

α; p)
∣∣∣

h(to)
dto


1
q

.

where β, α ∈ R+.
Proof. According to Lemma 1, we have∣∣∣∣ §(æ) + §(æ + ζ (̊s,æ))

2
Jαβ,γ(ω; p)−

1

2ζ (̊s,æ)β−1

×
[(

ℑæ+

æ+ζ (̊s,æ),β−1

) (
ω′, §

)
+
(
ℑ(æ+ζ (̊s,æ))−

æ,β−1

) (
ω′, §

)]
|

=

∣∣∣∣ζ (̊s,æ)2
I

∣∣∣∣
≤ ζ (̊s,æ)

2

∫ 1

0

∣∣∣(1− to)
β−1Jαβ,γ (ω(1− to)

α; p)− (to)
β−1Jαβ,γ (ω(to)

α; p)
∣∣∣ ∣∣§′(æ + toζ (̊s,æ))

∣∣dto
Applying the power mean inequality, we derive:∣∣∣∣ §(æ) + §(æ + ζ (̊s,æ))

2
Jαβ,γ(ω; p)

− 1

2ζ (̊s,æ)β

[(
Jæ

+

æ+ζ (̊s,æ),β−1

) (
ω′, §

)
+
(
J
(æ+ζ (̊s,æ))−

æ,β−1

) (
ω′, §

)]∣∣∣∣
≤ ζ (̊s,æ)

2

(∫ 1

0

∣∣∣(1− to)
β−1Jαβ,γ (ω(1− to)

α; p)− (to)
β−1Jαβ,γ (ω(to)

α; p)
∣∣∣dto)1−(1/q)

×
(∫ 1

0

∣∣∣(1− to)
β−1Jαβ+1,γ (ω(1− to)

α; p)− (to)
β−1Jαβ+1,γ (ω(to)

α; p)
∣∣∣

×
∣∣§′(æ + toζ (̊s,æ))

∣∣q dto
)1/q

.

Since |§′|q is (h-GL) preinvex, we have∫ 1

0

∣∣∣(1− to)
β−1Jαβ,γ (ω(1− to)

α; p)− (to)
β−1Jαβ,γ (ω(to)

α; p)
∣∣∣ ∣∣§′(æ + toζ (̊s,æ))

∣∣q dto

≤
∫ 1

0

∣∣∣(1− to)
β−1Jαβ,γ (ω(1− to)

α; p)− (to)
β−1Jαβ,γ (ω(to)

α; p)
∣∣∣

h(to)

∣∣§′(æ)∣∣q dto

+

∫ 1

0

∣∣∣(1− to)
β−1Jαβ,γ (ω(1− to)

α; p)− (to)
β−1Jαβ,γ (ω(to)

α; p)
∣∣∣

h(1− to)

∣∣§′(̊s)∣∣q dto

=
(∣∣§′(æ)∣∣q + ∣∣§′(̊s)∣∣q) ∫ 1

0

∣∣∣(1− to)
β−1Jαβ,γ (ω(1− to)

α; p)− (to)
β−1Jαβ,γ (ω(to)

α; p)
∣∣∣

h(to)
dto.



R. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5725 12 of 14

Now consider,

∫ 1

0

∣∣∣(1− to)
βJαβ,γ (ω(1− to)

α; p)− (to)
βJαβ,γ (ω(to)

α; p)
∣∣∣ dto

=
+∞∑
n=0

∣∣∣∣ (γ)n
Γ(βn+ α)

wn

n!

∣∣∣∣ ∫ 1

0

∣∣∣(1− to)
β+αs̊−1 − (to)

β+αs̊−1
∣∣∣ dto

=
+∞∑
n=0

∣∣∣∣ (γ)n
Γ(βn+ α)

wn

n!

∣∣∣∣
×

[∫ 1/2

0

∣∣∣(1− to)
β+αs̊−1 − (to)

β+αs̊−1
∣∣∣dto + ∫ 1

1/2

∣∣∣(1− to)
β+αs̊−1 − (to)

β+αs̊−1
∣∣∣dto]

= 2

[
Jαβ+1,γ(ω; p)−

(
1

2

)β−1

Jαβ+1,γ

(
ω

(
1

2

)α

; p

)]
.

5. Conclusion

In this work, we discussed the refinements of some well known inequalities for different
convexity through prabhaker fractional operators. Using the Prabhakar fractional integral
operators, Hermite-Hadamard fractional inequalities and trapezoidal inequalities for h Go-
dunova Levin convex and preinvex functions are developed. To obtained some other well
known inequalities, and presented in the form of corollaries, which shows the straightened
of our main results. Various fractional versions of other recognized inequalities can be
derived for h-Godunova-Levin convex and preinvex functions, contributing to significant
advancements in the theory of fractional inequalities.

Acknowledgements

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

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