EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5790
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Gradient Descent and Twice Differentiable
Simpson-Type Inequalities via K-Riemann-Liouville

Fractional Operators in Function Spaces

Waqar Afzal1, Mujahid Abbas2,3, Jorge E. Maćıas-Dı́az4,5,∗, Mutum Zico Meetei6,
Mehreen S. Khan7, Armando Gallegos8

1 Abdus Salam School of Mathematical Sciences, Government College University, 68-B,
New Muslim Town, Lahore 54600, Pakistan
2 Department of Mechanical Engineering Sciences, Faculty of Engineering and the Built
Environment, Doornfontein Campus, University of Johannesburg, South Africa
3 Department of Medical Research, China Medical University, Taichung 406040, Taiwan
4 Department of Mathematics and Didactics of Mathematics,
Tallinn University, Tallinn 10120 , Estonia
5 Department of Mathematics and Physics, Autonomous University of
Aguascalientes, Aguascalientes 20100, Mexico
6 Department of Mathematics, College of Science, Jazan University, P.O. Box 114,
Jazan 45142, Saudi Arabia
7 Department of Mathematics, Faculty of Science, Jazan University, Jazan 45142,
Saudi Arabia
8 University Center of Los Lagos, University of Guadalajara, Jalisco 47460, Mexico

Abstract. This paper investigates novel properties of Hilbert spaces through tensor operations
and establishes new bounds for Simpson-type inequalities using fractional integral operators. The
results contribute to advancing the theoretical understanding of these mathematical structures and
their applications in functional analysis and related fields.

2020 Mathematics Subject Classifications: 11B73, 11B83
Key Words and Phrases: Simpson, Hilbert spaces, generalized convex mappings

1. Introduction

The concept of convexity is integral to numerous disciplines within mathematics and
applied sciences, including optimization, machine learning, and energy systems. Recent
advancements have highlighted the utility of convex functions in addressing complex real-
world problems. For instance, machine learning techniques leverage convexity for im-
proving market surveillance and customer relationship management systems to estimate

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5790

Email address: jemacias@correo.uaa.mx (J. E. Maćıas-Dı́az)

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



W. Afzal et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5790 2 of 30

error terms using inequalities [9]. Convexity principles are also applied to transportation
systems, such as in fatigue detection for drivers using adaptive fuzzy classifiers [44]. In
industrial optimization, convexity plays a critical role in designing squeeze casting pa-
rameters with neural networks [43]. Moreover, it contributes to the efficient design of
perovskite solar cells by enabling specific material configurations [42]. In the field of
control systems, convex optimization is employed to achieve robust stability designs for
inverters [51]. For autonomous systems, convexity-based methods improve docking con-
trol for underwater vehicles through adaptive reinforcement learning [64]. Lastly, convex
functions have enabled advancements in biotechnology, such as efficient co-production of
hydrogen and methane through microbial regulation [63]. These applications demonstrate
the extensive and varied impact of convexity across multiple domains [33, 60, 65].

Recent years have witnessed remarkable progress in fractional calculus, influencing di-
verse domains of mathematics and applied sciences [26, 29, 35, 52]. The development of
novel definitions for fractional integrals and derivatives, extending classical approaches,
has attracted significant research interest. These new definitions have become a cen-
tral focus in mathematical analysis, paving the way for innovative methodologies. The
adaptability of fractional calculus [23, 24, 61] has enabled researchers to establish convex
integral inequalities, which play a crucial role in approximation theory. Inequalities such
as Jensen’s [49], Simpson’s [37], Ostrowski’s [8], Hermite–Hadamard’s [48], and trapezoidal
[1] inequalities are frequently employed to derive error bounds for numerical integration
methods. To construct such inequalities, researchers utilize different ways, including maps,
operators, relations, and other analytical techniques, demonstrating the profound impact
of fractional calculus in modern mathematical analysis.

For instance, in [5], the creators used symmetric curved composed capabilities to foster
Hermite-Hadamard imbalances. In [55], partial Riemann-Liouville integrals were utilized
to determine Newton-type disparities for summed up arched capabilities. The work in
[53] introduced Simpson-type results utilizing different classes of convexities, while [66]
presented Bullen-type results utilizing various novel convex mappings. The authors of [19]
upgraded Young’s disparity by giving interesting limits and applications, while [30] broad-
ened Hölder’s type result by addressing defer differential conditions through mean con-
gruity and demonstrating their uniqueness. Moreover, in [16], Ostrowski-type imbalances
were created utilizing differentiable s-arched mappings, and [54] presented trapezoidal-type
disparities utilizing quantum integrals.

Simpson’s disparity is pivotal as it not just lays out a hypothetical reason for surveying
the accuracy of mathematical combination procedures yet in addition helps scientists in
choosing ideal techniques in view of the particular properties of the capabilities being scru-
tinized, especially in the domains of quadrature mistake assessment and complex distinct
integrals. Beginning from crafted by eighteenth century mathematician Thomas Simpson,
Simpson’s standard supports this disparity. The standard approximates a capability uti-
lizing a quadratic polynomial, giving a viable means to gauge its vital. In particular, for a
capability ℑ that is ceaseless more than the span [ν1, ν2], Simpson’s 3

8
rule approximates

the basic as observes:



W. Afzal et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5790 3 of 30

∫ ν2

ν1

ℑ(W)dW ≈ ν2 − ν1
8

[
ℑ(ν1) + 3ℑ

(
2ν1 + ν2

3

)
+ 3ℑ

(
ν1 + 2ν2

3

)
+ ℑ(ν2)

]
.

The most frequently utilized Simpson-type disparity has the accompanying definition.

Theorem 1 (See [36]). Assume ℑ : [ν1, ν2] → R be a real-valued convex map, and suppose
that

∥∥ℑ(4)
∥∥
∞ = supW∈(ν1,ν2)

∣∣ℑ(4)(W)
∣∣ < ∞. Then, we have∣∣∣16 [ℑ(ν1) + 4ℑ

(
ν1+ν2

2

)
+ ℑ(ν2)

]
− 1

ν2−ν1

∫ ν2
ν1

ℑ(W)dW
∣∣∣ ≤ 1

2880

∥∥ℑ(4)
∥∥
∞ (ν2 − ν1)

4.

Scholars have utilized various approaches to investigate Simpson’s inequality. For
example, the authors of [12] established several novel inequalities using bi-dimensional
convex functions using quantum integral operators. In [21], researchers employed vari-
ous non-integer integral operators to, uncovering numerous enhanced bounds. The work
in [15] focused on refinements and reversals of Simpson’s inequality by utilizing preinvex
mappings and quantum calculus. Similarly, the authors of [11] explored the concept of
tempered fractional integral operators, while [45] employed multiplicative calculus to de-
rive a range of bounds and reversals for such inequalities. For further insights into these
related developments, readers are encouraged to consult [6, 10, 18, 20, 38, 39, 47] and the
associated references.

Self-adjoint operators, a crucial concept in mathematics and physics, enable the ex-
tension of classical numerical inequalities to linear operators on Hilbert spaces. These
operators, generalizing Hermitian matrices, are characterized by their symmetry, ensuring
real eigenvalues and orthogonal eigenvectors. The study of such inequalities has signifi-
cant applications in areas such as functional analysis, quantum mechanics, operator theory,
and optimization. Recently, researchers have focused on adapting classical inequalities like
Hermite–Hadamard, Jensen, and Hölder inequalities to the operator framework, deepen-
ing their applicability in quantum physics, matrix theory, and variational methods within
Hilbert space settings. For instance, in [46], operators within Hilbert spaces were utilized
to derive numerical-type inequalities, highlighting their significance in functional analy-
sis and optimization. Similarly, [31] introduced various means inequalities for bounded
operators, further enriching the theoretical framework of operator inequalities in Hilbert
spaces. In [27], Hölder form inequalities involving power series were proposed, revealing
intriguing applications within Hilbert space settings. Additionally, [59] investigated vari-
ational problems linked to inequalities and graph structures in Hilbert spaces, showcasing
their utility in diverse mathematical and physical contexts. For additional insights and
related results, readers are referred to the references in [4, 7, 14, 17, 34, 40, 62, 67].

Dragomir [28] introduces various innovative modifications and refinements of the fol-
lowing double inequality within the tensorial framework.

Theorem 2 (See [28]). Let D be a complete inner product space. Suppose the operators
(adjoint) U and D satisfy the condition 0 < S1 ≤ U ,D ≤ S2, where S1 and S2 are positive



W. Afzal et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5790 4 of 30

constants. Then

0 ≤ S1

S2
2

W(1−W)

(
U2 ⊗ 1 + 1⊗D2

2
− U ⊗D

)
≤ (1−W)U ⊗ 1 +W1⊗D − U1−W ⊗DW

≤ S2

S2
1

W(1−W)

(
U2 ⊗ 1 + 1⊗D2

2
− U ⊗D

)
.

Theorem 3 (See [2]). Assume that U and D are operators operators (adjoint) with cor-
responding spectra SP(U),SP(D) ⊂ ∆. Let ℑ is a continous function on ∆, then we
have

2min{κ, 1− κ}
[
ℑ(U)⊗ 1 + 1⊗ℑ(D)

2
−ℑ

(
2U ⊗ 1⊗D ⊗ 1

U ⊗ 1 + 1⊗D

)]
≤ κℑ(U)⊗ 1 + (1− κ)1⊗ℑ(D)−ℑ(κU ⊗ 1 + (1− κ)1⊗D)

≤ 2min{κ, 1− κ}
[
ℑ(U)⊗ 1 + 1⊗ℑ(D)

2
−ℑ

(
2U ⊗ 1⊗D ⊗ 1

U ⊗ 1 + 1⊗D

)]
.

The author of [56] used standard operators and differentiable mappings to generate
the following type of double inequalities.

Theorem 4 (See [2, 56]). Assume that U and D are selfadjoint operators with associated
sepctrums
SP(U),SP(D) ⊂ ∆. Let ℑ is a continous function on ∆, then we have∫ 1

0
ℑ((1−W)U ⊗ 1 +W1⊗D)dW −ℑ

(
U ⊗ 1 + 1⊗D

2

)
=
(1⊗D − U ⊗ 1)2

16

[∫ 1

0
W2ℑ′′ ((1−W)U ⊗ 1 +W1⊗D) dW

+

∫ 1

0
(W − 1)2ℑ′′

((
1−W

2

)
U ⊗ 1 +

(
1 +W

2

)
1⊗D

)
dW

]
.

The following double inequality was derived by the authors using positive semidefinite
operators on a Hilbert space.

Theorem 5 (See [25]). Let U and D be positive and semidefinite operators, with associated
spectrums SP(U),SP(D) ⊂ ∆. Then

(U#D)⊗ (U#D) ⩽
1

2

{
(UσD)⊗

(
Uσ⊥D

)
+
(
Uσ⊥D

)
⊗ (UσD)

}
⩽

1

2
{(U ⊗ D) + (D ⊗ U)}.

Significance of the study

Tensor inequalities extend the concepts of scalar and matrix inequalities to higher-
dimensional spaces, offering a framework to handle multi-dimensional data. his generaliza-
tion allows researchers to study and model problems in more complex settings where scalar



W. Afzal et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5790 5 of 30

or matrix representations are insufficient. They often play a role in estimating operator
norms and proving convergence results in functional spaces. In optimization, tensor in-
equalities provide constraints and bounds that facilitate the solution of multi-dimensional
problems, including those involving tensor decompositions or eigenvalue problems. Ten-
sor inequalities provide bounds for quantum information measures, such as entanglement
entropy and mutual information. Tensor inequalities are vital in understanding geometric
structures like curvature in Riemannian geometry, where tensors such as the Riemann
curvature tensor play a central role.

Our motivation to develop an advanced and novel version of various inequalities in
tensor Hilbert spaces is largely inspired by the works of [2, 22, 57, 68]. By incorporating
innovative techniques and perspectives that have been explored in only a limited number
of studies, this work aims to significantly expand and enhance the existing theory of
inequalities. The paper is organized into five sections. Section 2 presents a summary
of fundamental concepts associated with Hilbert spaces and their basic operations. In
Section 3, we derive various new bounds for numerical integral inequalities. Section 4
explores non-trivial examples and general observations. Finally, Section 5 discusses the
main findings and potential directions for future research related to these results.

2. Preliminaries

In this section, we revisit fundamental concepts related to Hilbert spaces and extended
convex mappings. For further details and additional insights, readers are encouraged to
consult [14].

Definition 1 (See [41]). A pre-Hilbert space on R is defined as

(·, ·) : R× R → C,

for all W1,W2,W3 ∈ R and λ ∈ C, we have

⟨W1 +W2,W3⟩ = ⟨W1,W3⟩+ ⟨W2 +W3⟩
⟨λW1,W2⟩ = λ⟨W1,W2⟩
⟨W1,W2⟩ = ⟨W2,W1⟩
⟨W1,W1⟩ ≥ 0, ⟨W1,W1⟩ = 0 ⇐⇒ W1 = 0.

Definition 2 (See [41]). Assume ℑ : U ×D → R be a function. The corresponding product
of U with D on Hilbert space R is denoted as

• the space R be a collections of all vectors ℑ(ν1, ν2)(ν1 ∈ U , ν2 ∈ D) such that R is
generated;

• (ℑ (ν1, ν2) | ℑ (ν3, ν4)) = (ν1 | ν2) (ν3 | ν4) for ν1, ν2 ∈ U , ν3, ν4 ∈ D. If (K,ℑ) is
taken to be product of U and D, it is write ν1 ⊗ ν2 inplace of ℑ(ν1, ν2). A product



W. Afzal et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5790 6 of 30

U ⊗ D and a mapping (ν1, ν2) 7→ ν1 ⊗ ν2 of U × D into U ⊗ D, holds

(ν1 + ν2)⊗ ν2 = ν1 ⊗ ν2 + ν2 ⊗ ν2

(λν1)⊗ ν2 = λ(ν1 ⊗ ν2)

ν1 ⊗ (ν3 + ν4) = ν1 ⊗ ν3 + ν1 ⊗ ν3

ν1 ⊗ (λν2) = λ(ν1 ⊗ ν2),

where λ ∈ R.

Assume ℑ : ∆1 × . . . × ∆p → R be mapping defined over intervals. Assume that
P =

(
P1, . . . ,Pp

)
be a collections of operators with E1, . . . , Ep are associated Hilbert spaces.

Then

Pi =

∫
∆i

WidEi (Wi)

is the spectra of possible operators for i = 1, . . . , p; following [25], we define Pi as follows:

ℑ
(
P1, . . . ,Pp

)
:=

∫
∆1

. . .

∫
∆p

ℑ
(
W1, . . . ,Wp

)
dE1 (W1)⊗ . . .⊗ dEp

(
Wp

)
.

Integrating processes into finite summations can significantly reduce the complexity
of many complex processes when the dimensions of Hilbert spaces are finite. In [68], the
author elaborates on the construction [25] and defines it as follows:

ℑ
(
P1, . . . ,Pp

)
= ℑ1 (P1)⊗ . . .⊗ℑp

(
Pp

)
,

where ℑ is a product of one variable mapping and can be split. ℑ
(
a1, . . . , ap

)
= ℑ1 (a1) . . .ℑp

(
ap
)
.

If ℑ is sub(super)-multiplicative across the interval ∆, then

ℑ(ν1ν2) ≥ (≤)ℑ(ν1)ℑ(ν2) for all ν1ν2 ∈ [0,∞).

If ℑ is continuous on the interval [0,∞), then

ℑ(U ⊗ D) ≥ (≤)ℑ(U)⊗ℑ(D) for all U ,D ≥ 0.

This leads to the conclusion that, if

U =

∫
[0,∞)

ν1dE(ν1) and D =

∫
[0,∞)

ν2dF(ν2)

are the assocaited spectrums.

ℑ(U ⊗ D) =

∫
[0,∞)

∫
[0,∞)

ℑ(ν1ν2)dE(ν1)⊗ dF(ν2).

The geometric property of linear bounded operator U ,D > 0 is defined as follows.



W. Afzal et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5790 7 of 30

U#pD := U1/2
(
U−1/2DU−1/2

)p
U1/2,

where p ∈ [0, 1] and

U#D := U1/2
(
U−1/2DU−1/2

)1/2
U1/2.

By the definitions of # and ⊗ we have

U#D = D#U and (U#D)⊗ (D#U) = (U ⊗ D)#(D ⊗ U).

Consider the eventually similar to the tensorial product :

(Uβ)⊗ (DS) = (U ⊗ D)(U ⊗ S),

that holds ∀U ,D, β,S ∈ B(ν2). If we take β = U and S = D, then we get

U2 ⊗D2 = (U ⊗ D)2.

Through induction, we have

Up ⊗Dp = (U ⊗ D)p for natural number σ ≥ 0.

Specifically
Uκ ⊗ 1 = (U ⊗ 1)κ and 1⊗Dκ = (1⊗D)κ

for all κ ≥ 0. Additionally, we note that the 1⊗D and U ⊗ 1 are commutative with each
other

(U ⊗ 1)(1⊗D) = (1⊗D)(U ⊗ 1) = U ⊗ D.

Moreover, for any two natural numbers κ1, κ2

(U ⊗ 1)κ1(1⊗D)κ2 = (1⊗D)κ1(U ⊗ 1)κ2 = Uκ2 ⊗Dκ1 .

Definition 3 (See [3]). A function ℑ : ∆ → R is known as convex over ∆, if

ℑ(Wν1 + (1−W)ν2) ≤ (≥)Wℑ(ν1) + (1−W)ℑ(ν2)

valid for all ν1, ν2 ∈ ∆ and W ∈ [0, 1].

Definition 4 (See [3]). A mapping ℑ : ∆ → R is known as convex in quasi sense, if

ℑ((1−W)ν1 +Wν2) ≤ max{ℑ(ν2),ℑ(ν1)} =
1

2
(ℑ(ν2) + ℑ(ν1) + |ℑ(ν2)−D(ν1)|)

for all ν1, ν2 ∈ ∆ and W ∈ [0, 1].



W. Afzal et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5790 8 of 30

Some Needed Technical Lemmas

Lemma 1. Let U and D be self-adjoint operators such that the spectra of U and D are
contained within the set ∆ (i.e., SP(U) ⊂ ∆ and SP(D) ⊂ ∆). If ℑ is a convex and
differentiable function defined on ∆, the following inequality holds:

(
ℑ′(U)⊗ 1

)
(U ⊗ 1− 1⊗D) ≤ ℑ(U)⊗ 1− 1⊗ℑ(D)

≤ (U ⊗ 1− 1⊗D)
(
1⊗ℑ′(D)

)
. (2.1)

Proof. Taking into account th following double inequality for ℑ on ∆, we derive

ℑ′(κ)(κ− γ) ≤ ℑ(κ)−ℑ(γ) ≤ ℑ′(γ)(κ− γ)

for all κ, γ ∈ ∆. Since

U =

∫
∆
κdE(κ) and D =

∫
∆
γdF(γ).

This imply that∫
∆

∫
∆
ℑ′(κ)(κ− γ)dEκ ⊗ dEγ ≤

∫
∆

∫
∆
(ℑ(κ)−ℑ(γ))dEκ ⊗ dEγ

≤
∫
∆

∫
∆
ℑ′(γ)(κ− γ)dEκ ⊗ dEγ . (2.2)

Observe that ∫
∆

∫
∆
ℑ′(κ)(κ− γ)dEκ ⊗ dEγ

=

∫
∆

∫
∆

(
ℑ′(κ)κ−ℑ′(κ)γ

)
dEκ ⊗ dEγ

=

∫
∆

∫
∆
ℑ′(κ)tdEκ ⊗ dEγ −

∫
∆

∫
∆
ℑ′(κ)γdEκ ⊗ dEγ

=
(
ℑ′(U)U

)
⊗ 1−ℑ′(U)⊗D∫

∆

∫
∆
(ℑ(κ)−ℑ(γ))dEκ ⊗ dEγ = ℑ(U)⊗ 1− 1⊗ℑ(D) (2.3)

and ∫
∆

∫
∆
ℑ′(γ)(κ− γ)dEκ ⊗ dEγ

=

∫
∆

∫
∆

(
κℑ′(γ)−ℑ′(γ)γ

)
dEκ ⊗ dEγ

=

∫
∆

∫
∆
κℑ′(γ)dEκ ⊗ dEγ −

∫
∆

∫
∆
ℑ′(γ)γdEκ ⊗ dEγ

= U ⊗ ℑ′(D)− 1⊗
(
ℑ′(D)D

)



W. Afzal et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5790 9 of 30

and by (2.3) we derive the inequality of interest

Lemma 2. Let U and D be adjoint operators, with their spectra denoted by σ1 and σ2,
respectively. Suppose the functions ℑ and ϑ are defined on σ1, while D and Q are defined
on σ2, and let φ be convex on σ. Then, the set ϑ(σ1)+ℑ(σ2) satisfies the following equality:

(ℑ(U)⊗ 1 + 1⊗D(D))φ(ϑ(U)⊗ 1 + 1⊗ℑ(D))

=

∫
σ1

∫
σ2

(ℑ(ν2) +D(ν1))φ(ϑ(ν2) +Q(ν1))dEν2 ⊗ dFν1 , (2.4)

where U and D have the spectral resolutions

U =

∫
σ1

ν2dE(ν2) and D =

∫
σ2

ν1dF(ν1).

Proof. According to Stone-Weierstrass, any continuous function can be represented in
terms of polynomial sequence, hence simply checking its equivalence is adequate. Let

D(µ) =

{
1
2µ

2σ1, −σ ≤ µ ≤ σ

σ
(
|µ|σ1 − σ

2

)
, σ > µ > −σ.

If σ1 and σ2 are integers, and |µ| ≤ σ, then the following holds:

ℑ :=

∫
σ1

∫
σ2

(ℑ(ν2) +D(ν1))
1

2
(ϑ(ν2) +Q(ν1))

2σ1dEν2 ⊗ dFν1

=

∫
σ1

∫
σ2

(ℑ(ν2) +D(ν1))

σ2∑
σ1=0

Cσ1
σ2

1

2
[ϑ(ν2)]

2σ1 [Q(ν1)]
2σ2−2σ1dEν2 ⊗ dFν1

=

σ2∑
σ1=0

Cσ1
σ2

∫
σ1

∫
σ2

(ℑ(ν2) +D(ν1))
1

2
[ϑ(ν2)]

2σ1 [Q(ν1)]
2σ2−2σ1dEν2 ⊗ dFν1

=

σ2∑
σ1=0

Cσ1
σ2

[∫
σ1

∫
σ2

ℑ(ν2)
1

2
[ϑ(ν2)]

2σ1 [Q(ν1)]
2σ2−2σ1dEν2 ⊗ dFν1

+

∫
σ1

∫
σ2

D(ν1)
1

2
[ϑ(ν2)]

2σ1 [Q(ν1)]
2σ2−2σ1dEν2 ⊗ dFν1

]
.

Observe that∫
σ1

∫
σ2

ℑ(ν2)
1

2
[ϑ(ν2)]

2σ1 [Q(ν1)]
2σ2−2σ1dEν2 ⊗ dFν1

= ℑ(U)1
2
[ϑ(U)]2σ1 ⊗ [ℑ(D)]2σ2−2σ1 = (ℑ(U)⊗ 1)

(
[ϑ(U)]2σ2 ⊗ [ℑ(D)]2σ2−2σ1

)
= (ℑ(U)⊗ 1)

1

2

(
[ϑ(U)]2σ2 ⊗ 1

)(
1⊗ [ℑ(D)]2σ2−2σ1

)
= (ℑ(U)⊗ 1)

1

2
(ϑ(U)⊗ 1)2σ1(1⊗ℑ(D))2σ2−2σ1



W. Afzal et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5790 10 of 30

and ∫
σ1

∫
σ2

1

2
[ϑ(ν2)]

2σ1D(ν1)[Q(ν1)]
2σ2−2σ1dEν2 ⊗ dFν1

=
1

2
[ϑ(U)]2σ1 ⊗

(
D(D)[ℑ(D)]2σ2−2σ1

)
= (1⊗D(D))

1

2

(
[ϑ(U)]2σ2 ⊗ [ℑ(D)]2σ2−2σ1

)
= (1⊗D(D))

1

2

(
[ϑ(U)]2σ2 ⊗ 1

)(
1⊗ [ℑ(D)]2σ2−2σ1

)
= (1⊗D(D))

1

2
(ϑ(U)⊗ 1)2σ1(1⊗ℑ(D))2σ2−2σ1

where 1
2(ϑ(U)⊗ 1) and 1

2(1⊗ℑ(D)) are commutative, so we have

ℑ = (ℑ(U)⊗ 1 + 1⊗D(D))

σ2∑
σ1=0

Cσ1
σ2

1

2
(ϑ(U)⊗ 1)2σ1(1⊗ℑ(D))2σ2−2σ1

= (ℑ(U)⊗ 1 + 1⊗D(D))
1

2
(ϑ(U)⊗ 1 + 1⊗ℑ(D))2σ2 .

Taking into account: if |µ| > σ, then

ℑ :=

∫
σ1

∫
σ2

(ℑ(ν2) +D(ν1))σ
(
|(ϑ(ν2) +Q(ν1))|σ1 − σ

2

)
dEν2 ⊗ dFν1

=

∫
σ1

∫
σ2

(ℑ(ν2) +D(ν1))

σ2∑
σ1=0

Cσ1
σ2
σ
(
|(ϑ(ν2))|σ1 − σ

2

)
σ
(
|(Q(ν1))|σ2−σ1 − σ

2

)
dEν2 ⊗ dFν1

=

σ2∑
σ1=0

Cσ1
σ2

∫
σ1

∫
σ2

(ℑ(ν2) +D(ν1))σ
(
|(ϑ(ν2))|σ1 − σ

2

)
σ
(
|(Q(ν1))|σ2−σ1 − σ

2

)
dEν2 ⊗ dFν1

=

σ2∑
σ1=0

Cσ1
σ2

[∫
σ1

∫
σ2

ℑ(ν2)σ
(
|(ϑ(ν2))|σ1 − σ

2

)
σ
(
|(Q(ν1))|σ2−σ1 − σ

2

)
dEν2 ⊗ dFν1

+

∫
σ1

∫
σ2

D(ν1)σ
(
|(ϑ(ν2))|σ1 − σ

2

)
σ
(
|(Q(ν1))|σ2−σ1 − σ

2

)
dEν2 ⊗ dFν1

]
.

Observe that∫
σ1

∫
σ2

ℑ(ν2)σ
(
|(ϑ(ν2))|σ1 − σ

2

)
σ
(
|(Q(ν1))|σ2−σ1 − σ

2

)
dEν2 ⊗ dFν1

= ℑ(U)σ
(
|(ϑ(U))|σ1 − σ

2

)
⊗ σ

(
|(ℑ(D))|σ2−σ1 − σ

2

)
= (ℑ(U)⊗ 1)

[
σ
(
|(ϑ(U))|σ1 − σ

2

)
⊗ σ

(
|(ℑ(D))|σ2−σ1 − σ

2

)]
= (ℑ(U)⊗ 1)

[
σ
(
|(ϑ(U))|σ1 ⊗ 1− σ

2

)
⊗ σ

(
1⊗ |(ℑ(D))|σ2−σ1 − σ

2

)]
= (ℑ(U)⊗ 1)

[
σ
(
(|(ϑ(U))| ⊗ 1)σ1 − σ

2

)
⊗ σ

(
(1⊗ |(ℑ(D))|)σ2−σ1 − σ

2

)]



W. Afzal et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5790 11 of 30

and ∫
σ1

∫
σ2

D(ν1)σ
(
|(ϑ(ν2))|σ1 − σ

2

)
σ
(
|(Q(ν1))|σ2−σ1 − σ

2

)
dEν2 ⊗ dFν1

= D(ν1)σ
(
|(ϑ(U))|σ1 − σ

2

)
⊗ σ

(
|(ℑ(D))|σ2−σ1 − σ

2

)
= (1⊗D(ν1))

[
σ
(
|(ϑ(U))|σ1 − σ

2

)
⊗ σ

(
|(ℑ(D))|σ2−σ1 − σ

2

)]
= (1⊗D(ν1))

[
σ
(
|(ϑ(U))|σ1 ⊗ 1− σ

2

)
⊗ σ

(
1⊗ |(ℑ(D))|σ2−σ1 − σ

2

)]
= (1⊗D(ν1))

[
σ
(
(|(ϑ(U))| ⊗ 1)σ1 − σ

2

)
⊗ σ

(
(1⊗ |(ℑ(D))|)σ2−σ1 − σ

2

)]
where σ

(
|(ϑ(U))| ⊗ 1)− σ

2

)
and σ

(
1⊗ |(ℑ(D))|)− σ

2

)
are commute with each other.

Therefore

ℑ = (ℑ(U)⊗ 1 + 1⊗D(D))

σ2∑
σ1=0

Cσ1
σ2
σ
(
(|(ϑ(U))| ⊗ 1)σ1 − σ

2

)
σ
(
(1⊗ |(ℑ(D))|)σ2−σ1 − σ

2

)
= (ℑ(U)⊗ 1 + 1⊗D(D))

[
σ
(
(|(ϑ(U))| ⊗ 1)− σ

2

)
+ σ

(
(1⊗ |(ℑ(D))|)− σ

2

)]σ2

Lemma 3. Let U and D be adjoint operators, with their spectra lying within the sets ∆1

and ∆2, respectively. Assume that the functions ℑ and ϑ are continuous on ∆1, while D
and ℑ are continuous on ∆2, and φ is convex on ∆. Then, the product of the intervals
ϑ(∆1) + ℑ(∆2) satisfies the following equality:

φ(ℑ(U)⊗D(D))χ(ϑ(U)⊗ℑ(D)) =

∫
∆1

∫
∆2

φ(ℑ(ν2)D(ν1))χ(ϑ(ν2)ℑ(ν1))dEν1 ⊗ dFν2

(2.5)

where U and D have the spectral resolutions

U =

∫
∆1

ν2dE(ν2) and D =

∫
∆2

ν1dF(ν1).

Proof. According to Weierstrass, any real-valued differentiable mapping can be repre-
sented in terms of polynomial sequence, hence simply checking its equivalence is adequate.
Let two non-negative mappings φ(µ) = eµσ1 , χ(µ) = eµσ2 with σ1 and σ2 for each natural
numbers, one has

∫
∆1

∫
∆2

(eν1eν2)σ2(eν1eν2)σ1dEν1 ⊗ dFν2 =

∫
∆1

∫
∆2

[eν1 ]σ2 [eν2 ]σ2 [eν1 ]σ1 [eν2 ]σ1dEν1 ⊗ dFν2

=

∫
∆1

∫
∆2

[eν1 ]σ2 [eν1 ]σ1 [eν2 ]σ2 [eν2 ]σ1dEν1 ⊗ dFν2 =
(
[eU ]σ2 [eU ]σ1

)
⊗
(
[eD]σ2 [eD]σ1

)
=
(
[eU ]σ2 ⊗ [eD]σ2

) (
[eU ]σ1 ⊗ [eD]σ1

)
= (eU ⊗ eD)σ2(eU ⊗ eD)σ1

and the equality (2.5) is proven.



W. Afzal et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5790 12 of 30

3. The main results

First, we recall the following fractional operators that play a key role in our main
findings:

Definition 5 (See [22]). Let ℑ : [ν1, ν2] → R be a continuous mapping on [ν1, ν2]. For
S > 0 the associated integrals are defined as:

JSν1+,kℑ(℘) =
1

Γ(S)

∫ ℘

ν1

(℘− ε)
S−k
k ℑ(ε)dε

for ν1 < ℘ ⩽ ν2 and

JSν2−,kℑ(℘) =
1

Γ(S)

∫ ν2

℘
(ε− ℘)

S−k
k ℑ(ε)dε

for ν1 ⩽ ℘ < ν2, where Γ is the gamma function.

We now have defined thek-Riemann-Liouville fractional integral operators and their
corresponding generic identities.

Definition 6 (See [22]). Let ℑ : [ν1, ν2] → R be a real-valued mapping on [ν1, ν2]. For
k,S > 0 the associated k-Riemann-Liouville integrals are represented as:

JSν1+,kℑ(℘) =
1

kΓk(S)

∫ ℘

ν1

(℘− ε)
S−k
k ℑ(ε)dε

for ν1 < ℘ ⩽ ν2 and

JSν2−,kℑ(℘) =
1

kΓk(S)

∫ ν2

℘
(ε− ℘)

S−k
k ℑ(ε)dε

for ν1 ⩽ ℘ < ν2, where Γk is the k-Gamma function.

Lemma 4. Let ℑ : [ν1, ν2] → R be a real-valued mapping on [ν1, ν2]. For any ℘ ∈ (ν1, ν2)
we have

J S
ν1+,kℑ(℘) + JSν2−,kℑ(℘)

=
1

kΓk(S + k)

[
(℘− ν1)

S
k ℑ(ν1) + (ν2 − ℘)

S
k ℑ(ν2)

]
+

1

kΓk(S + k)

[∫ ℘

ν1

(℘− ε)
S
k ℑ′(ε)dε−

∫ ν2

℘
(ε− ℘)

S
k ℑ′(ε)dε

]
. (3.1)

Proof. Let ℑ : [ν1, ν2] → R be a continuous function. In this case, the symmetry of
the integrals is given by:∫ ℘

ν1

(℘− ε)
S
k ℑ′(ε)dε and

∫ ν2

℘
(ε− ℘)

S
k ℑ′(ε)dε,



W. Afzal et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5790 13 of 30

applying integration this follows

1

kΓk(S + k)

∫ ℘

ν1

(℘− ε)
S
k ℑ′(ε)dε

=
1

Γ(S)

∫ ℘

ν1

(℘− ε)
S−k
k ℑ(ε)dε− 1

kΓk(S + k)
(℘− ν1)

S
k ℑ(ν1)

= JSν1+,kℑ(℘)−
1

kΓk(S + k)
(℘− ν1)

S
k ℑ(ν1) (3.2)

for ν1 < ℘ ⩽ ν2 and

1

kΓk(S + k)

∫ ν2

℘
(ε− ℘)

S
k ℑ′(ε)dε

=
1

kΓk(S + k)
(ν2 − ℘)

S
k ℑ(ν2)−

1

Γ(S)

∫ ν2

℘
(ε− ℘)

S−k
k ℑ(ε)dε

=
1

kΓk(S + k)
(ν2 − ℘)

S
k ℑ(ν2)− JSν2−,kℑ(℘) (3.3)

for ν1 ⩽ ℘ < ν2. From (3.2), one has

JSν1+,kℑ(℘) =
1

kΓk(S + k)
(℘− ν1)

S
k ℑ(ν1) +

1

kΓk(S + k)

∫ ℘

ν1

(℘− ε)
S
k ℑ′(ε)dε

for ν1 < ℘ ⩽ ν2 and from (3.3), one has

JSν2−,kℑ(℘) =
1

kΓk(S + k)
(ν2 − ℘)

S
k ℑ(ν2)−

1

kΓk(S + k)

∫ ν2

℘
(ε− ℘)

S
k ℑ′(ε)dε.

Now, for any ℘ ∈ (ν1, ν2) we have

J S
℘−,kℑ(ν1) + JS℘+,kℑ(ν2)

=
1

kΓk(S + k)

[
(℘− ν1)

S
k + (ν2 − ℘)

S
k

]
ℑ(℘)

+
1

kΓk(S + k)

[∫ ν2

℘
(ν2 − ε)

S
k ℑ′(ε)dε−

∫ ℘

ν1

(ε− ν1)
S
k ℑ′(ε)dε

]
.

Proof. Since we have

JS℘+,kℑ(ν2) =
1

kΓ(S)

∫ ν2

℘
(ν2 − ε)

S−k
k ℑ(ε)dε

for ν1 ⩽ ℘ < ν2 and

JS℘−,kℑ(ν1) =
1

Γ(S)

∫ ℘

ν1

(ε− ν1)
S−k
k ℑ(ε)dε



W. Afzal et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5790 14 of 30

for ν1 < ℘ ⩽ ν2. Since ℑ : [ν1, ν2] → R be an continuous function [ν1, ν2], then the integrals∫ ℘

ν1

(ε− ν1)
Sℑ′(ε)dε and

∫ ν2

℘
(ν2 − ε)Sℑ′(ε)dε,

holds, and by performing the integration, we obtain:

1

kΓk(S + k)

∫ ℘

ν1

(ε− ν1)
Sℑ′(ε)dε

=
1

kΓk(S + k)
(℘− ν1)

S
k ℑ(℘)− 1

kΓ(S)

∫ ℘

ν1

(ε− ν1)
S−k
k ℑ(ε)dε

=
1

kΓk(S + k)
(℘− ν1)

S
k ℑ(℘)− JS℘−,kℑ(ν1) (3.4)

for ν1 < ℘ ⩽ ν2 and

1

kΓk(S + k)

∫ ν2

℘
(ν2 − ε)

S
k ℑ′(ε)dε

=
1

kΓ(S)

∫ ν2

℘
(ν2 − ε)

S−k
k ℑ(ε)dε− 1

kΓk(S + k)
(ν2 − ℘)

S
k ℑ(℘)

= JS℘+,kℑ(ν2)−
1

kΓk(S + k)
(ν2 − ℘)

S
k ℑ(℘) (3.5)

for ν1 ⩽ ℘ < ν2. From (3.4) we have

JS℘−,kℑ(ν1) =
1

kΓk(S + k)
(℘− ν1)

S
k ℑ(℘)− 1

kΓk(S + k)

∫ ℘

ν1

(ε− ν1)
S
k ℑ′(ε)dε

for ν1 < ℘ ⩽ ν2 and from (3.5)

JS℘+,kℑ(ν2) =
1

kΓk(S + k)
(ν2 − ℘)

S
k ℑ(℘) + 1

kΓk(S + k)

∫ ν2

℘
(ν2 − ε)

S
k ℑ′(ε)dε.

3.1. Some new fractional identities

We draw inspiration from section 2.1 in this section, which enables us to construct
fractional identities at the interval’s midpoint, which we utilize in important findings.

Lemma 5. Let ℑ : [ν1, ν2] → R be a real-valued mapping on [ν1, ν2]. We have the following
double equality

J S
ν1+,kℑ

(
ν1 + ν2

2

)
+ JSν2−,kℑ

(
ν1 + ν2

2

)
=

1

2
S−k
k kΓk(S + k)

Q(ν1) + ℑ(ν2)
2

+
1

kΓk(S + k)

[∫ ν1+ν2
2

ν1

(
ν1 + ν2

2
− ε

)S
k

ℑ′(ε)dε−
∫ ν2

ν1+ν2
2

(
ε− ν1 + ν2

2

)S
k

ℑ′(ε)dε

]



W. Afzal et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5790 15 of 30

and

J S
ν1+ν2

2
−,k

ℑ(ν1) + JSν1+ν2
2

+,k
ℑ(ν2)

=
1

2
S−k
k kΓk(S + k)

ℑ
(
ν1 + ν2

2

)
(ν2 − ν1)

S
k

+
1

kΓk(S + k)

[∫ ν2

ν1+ν2
2

(ε− ν2)
S
k ℑ′(ε)dε−

∫ ν1+ν2
2

ν1

(ε− ν1)
S
k ℑ′(ε)dε

]
, (3.6)

for ν1 ⩽
ν1+ν2

2 < ν2. From (3.6) we have

J S
ν1+ν2

2
−,k

ℑ(ν1) =
1

2
S−k
k kΓk(S + k)

ℑ
(
ν1 + ν2

2

)
(ν2 − ν1)

S
k

− 1

kΓk(S + k)

[∫ ν1+ν2
2

ν1

(ε− ν1)
S
k ℑ′(ε)dε

]
=

1

2
S−k
k kΓk(S + k)

ℑ
(
ν1 + ν2

2

)
(ν2 − ν1)

S
k

− W
S
k (ν2 − ν1)

S+k
k

2
S+k
k kΓk(S + k)

[∫ 1

0
ℑ′
(
(1−W)ν1 +

(
ν1 + ν2

2

)
W
)
dW

]
, (3.7)

for ν1 <
ν1+ν2

2 ⩽ ν2 and from (3.6) we have

J S
ν1+ν2

2
+,k

ℑ(ν2) =
1

2
S−k
k kΓk(S + k)

ℑ
(
ν1 + ν2

2

)
(ν2 − ν1)

S
k

+
1

kΓk(S + k)

[∫ ν2

ν1+ν2
2

(ν2 − ε)Sℑ′(ε)dε

]
=

1

2
S−k
k kΓk(S + k)

ℑ
(
ν1 + ν2

2

)
(ν2 − ν1)

S
k

− (1−W)
S
k (ν2 − ν1)

S+k
k

2
S+k
k kΓk(S + k)

[∫ 1

0
ℑ′
(
(1−W)

(
ν1 + ν2

2

)
+ ν2W

)
dW

]
. (3.8)

Lemma 6. Assume ℑ is a convex mapping on ∆, and U , D are adjoint operators whose
spectra SP(U),SP(D) ⊂ ∆, then

[
1

6
(ℑ(U)⊗ 1) +

2

3
ℑ
(
U ⊗ 1 + 1⊗D

2

)
+

1

6
(1⊗ℑ(D))

]
−

[
ℑ
(
U ⊗ 1 + 1⊗D

2

)
− W

S
k (ν2 − ν1)

4

[∫ 1

0
ℑ′
(
(1−W)U ⊗ 1 +

(
W1⊗D

2

))
dW

]

+ ℑ
(
U ⊗ 1 + 1⊗D

2

)
− (1−W)

S
k (ν2 − ν1)

4

[∫ 1

0
ℑ′
((

1−W
2

)
U ⊗ 1 +

(
1 +W

2

)
1⊗D

)
dW

]



W. Afzal et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5790 16 of 30

=
(1⊗D − U ⊗ 1)2

6

[∫ 1
2

0

(
W − 3k.2

S
k W

S+k
k

S + k

)[
ℑ′′ (U ⊗ 1W + 1⊗ U (1−W)) dW

]
+

∫ 1

1
2

(
(1−W)− 3k.2

S
k (1−W)

S+k
k

S + k

)
ℑ′′ (U ⊗ 1W + 1⊗ U (1−W)) dW

]
. (3.9)

Proof. Considering the following result [13]. Let mapping ℑ : [ν1, ν2] → R be defined
over interval (ν1, ν2) such that ℑ′′ ∈ L([ν1, ν2]). Then, we have

1

6

[
ℑ(ν1) + 4ℑ

(
ν1 + ν2

2

)
+ ℑ(ν2)

]
− 2

S−k
k Γ(S + k)

(ν2 − ν1)
S
k

[
JSν1+ν2

2
−,k

ℑ(ν1) + JSν1+ν2
2

+,k
ℑ(ν2)

]

=
(ν2 − ν1)

2

6

[∫ 1
2

0

(
W − 3k.2

S
k W

S+k
k

S + k

)[
ℑ′′ (ν2W + (1−W) ν2)

]
dW

+

∫ 1

1
2

(
(1−W)− 3k.2

S
k (1−W)

S+k
k

S + k

)[
ℑ′′ (ν2W + (1−W) ν2)

]
dW

]
. (3.10)

By applying substitution from equations (3.7) and (3.8), we obtain:

1

6

[
ℑ(ν1) + 4ℑ

(
ν1 + ν2

2

)
+ ℑ(ν2)

]
− 2

S−k
k kΓk(S + k)

(ν2 − ν1)
S
k

[
ℑ
(
ν1+ν2

2

)
(ν2 − ν1)

S
k

2
S−k
k kΓk(S + k)

− W
S
k (ν2 − ν1)

S+k
k

2
S+k
k kΓk(S + k)

[∫ 1

0
ℑ′
(
(1−W)ν1 +

(
ν1 + ν2

2

)
W
)
dW

]

+
ℑ
(
ν1+ν2

2

)
(ν2 − ν1)

S
k

2
S−k
k kΓk(S + k)

− (1−W)
S
k (ν2 − ν1)

S+k
k

2
S+k
k kΓk(S + k)

[∫ 1

0
ℑ′
(
(1−W)

(
ν1 + ν2

2

)
+ ν2W

)
dW

]]

=
(ν2 − ν1)

2

6

[∫ 1
2

0

(
W − 3k · 2

S
k W

S+k
k

S + k

)[
ℑ′′ (ν2W + (1−W)ν2)

]
dW

+

∫ 1

1
2

(
(1−W)− 3k · 2

S
k (1−W)

S+k
k

S + k

)[
ℑ′′ (ν2W + (1−W)ν2)

]
dW

]
. (3.11)

By making several simplifications, we may have

1

6

[
ℑ(ν1) + 4ℑ

(
ν1 + ν2

2

)
+ ℑ(ν2)

]
−

[
ℑ
(
ν1 + ν2

2

)
− W

S
k (ν2 − ν1)

4

[∫ 1

0
ℑ′
(
(1−W) ν1 +

(ν2
2

)
W
)
dW

]



W. Afzal et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5790 17 of 30

+ℑ
(
ν1 + ν2

2

)
− (1−W)

S
k (ν2 − ν1)

4

[∫ 1

0
ℑ′
((

1−W
2

)
ν1 +

((
1 +W

2

)
ν2

)
dW

]]

=
(ν2 − ν1)

2

6

[∫ 1
2

0

(
W − 3k.2

S
k W

S+k
k

S + k

)[
ℑ′′ (ν2W + (1−W) ν2)

]
dW

+

∫ 1

1
2

(
(1−W)− 3k.2

S
k (1−W)

S+k
k

S + k

)[
ℑ′′ (ν2W + (1−W) ν2)

]
dW

]
. (3.12)

Assume that the spectral resolutions of U and D

U =

∫
∆
ν2dE(ν2) and D =

∫
∆
ν1dF(ν1).∫

∆

∫
∆ over dEν1 ⊗ dFν2 in (3.12), then we get∫

∆

∫
∆

1

6

[
ℑ(ν1) + 4ℑ

(
ν1 + ν2

2

)
+ ℑ(ν2)

]
dEν1 ⊗ dFν2

−

[∫
∆

∫
∆

(
ℑ
(
ν1 + ν2

2

)
− W

S
k (ν2 − ν1)

4

[∫ 1

0
ℑ′
(
(1−W) ν1 +

(ν2
2

)
W
)
dW

)]
dEν1 ⊗ dFν2

+

∫
∆

∫
∆

(
ℑ
(
ν1 + ν2

2

)
− (1−W)

S
k (ν2 − ν1)

4

×
[∫ 1

0
ℑ′
((

1−W
2

)
ν1 +

((
1 +W

2

)
ν2

)
dW

]
dEν1 ⊗ dFν2

]
=
(ν2 − ν1)

2

6

∫
∆

∫
∆

[∫ 1
2

0

(
W − 3k.2

S
k W

S+k
k

S + k

)[
ℑ′′ (ν2W + (1−W) ν2)

]
dW

+

∫ 1

1
2

(
(1−W)− 3k.2

S
k (1−W)

S+k
k

S + k

)[
ℑ′′ (ν2W + (1−W) ν2)

]
dW

]
dEν1 ⊗ dFν2 .

(3.13)

Taking into account Lemma 2 and Fubini’s theorem, we obtain:

∫
∆

∫
∆
ℑ(ν2)dEν1 ⊗ dFν2 = (ℑ(U)⊗ 1),∫

∆

∫
∆
ℑ
(
ν1 + ν2

2

)
dEν1 ⊗ dFν2 = ℑ

(
U ⊗ 1 + 1⊗D

2

)
,∫

∆

∫
∆
ℑ(ν1)dEν1 ⊗ dFν2 = (1⊗ℑ(D)),∫

∆

∫
∆

∫ 1

0
ℑ′
(
(1−W) ν1 +

(ν2
2

)
W
)
dWdEν1 ⊗ dFν2



W. Afzal et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5790 18 of 30

=

∫ 1

0

∫
∆

∫
∆
ℑ′
(
(1−W) ν1 +

(ν2
2

)
W
)
dEν1 ⊗ dFν2dW

=

∫ 1

0
ℑ′
(
(1−W)U ⊗ 1 +

(
W1⊗D

2

))
dW,∫

∆

∫
∆

∫ 1

0
ℑ′
((

1−W
2

)
ν1 +

(
1 +W

2

)
ν2

)
dWdEν1 ⊗ dFν2

=

∫ 1

0

∫
∆

∫
∆
ℑ′
((

1−W
2

)
ν1 +

(
1 +W

2

)
ν2

)
dEν1 ⊗ dFν2dW

=

∫ 1

0

∫
∆

∫
∆
ℑ′
((

1−W
2

)
U ⊗ 1 +

(
1 +W

2

)
1⊗D

)
dW,

ℑ′′
(
ν1W + ν1 (1−W)

)
dWdEν1 ⊗ dFν2 = ℑ′′ (U ⊗ 1W + 1⊗ U (1−W)) dW. (3.14)

A same technique has been taking into consideration we have∫
∆

∫
∆

(ν2 − ν1)
2

6

[∫ 1
2

0

(
W − 3k.2

S
k W

S+k
k

S + k

)[
ℑ′′ (ν2W + (1−W) ν2)

]
dW

+

∫ 1

1
2

(
(1−W)− 3k.2

S
k (1−W)

S+k
k

S + k

)[
ℑ′′ (ν2W + (1−W) ν2)

]
dW

]
dEν1 ⊗ dFν2

=
(1⊗D − U ⊗ 1)2

6

[∫ 1
2

0

(
W − 3k.2

S
k W

S+k
k

S + k

)[
ℑ′′ (U ⊗ 1W + 1⊗ U (1−W)) dW

]
+

∫ 1

1
2

(
(1−W)− 3k.2

S
k (1−W)

S+k
k

S + k

)
ℑ′′ (U ⊗ 1W + 1⊗ U (1−W)) dW

]
. (3.15)

The desired result is obtained by incorporating (3.14) and (3.15) into (3.13).

Remark 1. • Choosing S and k = 1 in Lemma 6 leads to a refinement of Lemma
2.1, in [57].

• By choosing S and k = 1 in Lemma 6, we obtain a refined version of Lemma 2.3, in
[2].

• By setting S and k = 1 in Lemma 6, we enhance Lemma 3, in [58].

Theorem 6. Assume ℑ is a convex mapping on ∆, and U , D are adjoint operators whose
spectra SP(U),SP(D) ⊂ ∆, then

∥∥∥∥∥
[
1

6
(ℑ(U)⊗ 1) +

2

3
ℑ
(
U ⊗ 1 + 1⊗D

2

)
+

1

6
(1⊗ℑ(D))

]



W. Afzal et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5790 19 of 30

−

[
ℑ
(
U ⊗ 1 + 1⊗D

2

)
− W

S
k (ν2 − ν1)

4

[∫ 1

0
ℑ′
(
(1−W)U ⊗ 1 +

(
W1⊗D

2

))
dW

]

+ ℑ
(
U ⊗ 1 + 1⊗D

2

)
− (1−W)

S
k (ν2 − ν1)

4

[∫ 1

0
ℑ′
((

1−W
2

)
U ⊗ 1 +

(
1 +W

2

)
1⊗D

)
dW

] ∥∥∥∥∥
≤ ∥(1⊗D − U ⊗ 1)2∥

6

[(
S(S + k)

a+2k
S + 3

S+2k
a k

2S+2k
S

4 · 3
2k
S k

2k
S (S + k)(S + 2k)

− 1

8

)∥∥∥ ∣∣ℑ′′ (U)
∣∣+ ∣∣ℑ′′ (D)

∣∣ ∥∥∥] .
Proof.
As |ℑ′′| is convex over ∆, one has

∣∣ℑ′′ (ν1W + ν1 (1−W))
∣∣ ≤ W

∣∣ℑ′′ (ν1)
∣∣+ (1−W)

∣∣ℑ′′(ν1)
∣∣

Similarly, we get∣∣ℑ′′ (ν2W + ν2 (1−W))
∣∣ ≤ W

∣∣ℑ′′ (ν2)
∣∣+ ((1−W))

∣∣ℑ′′(ν2)
∣∣

for all for τ ∈ [0, 1] and ν1, ν2 ∈ ∆.
Taking

∫
∆

∫
∆ over dEν1 ⊗ dFν2 , then we get∣∣ℑ′′ (1⊗ UW + 1⊗ U (1−W))

∣∣ = ∫
∆

∫
∆

∣∣ℑ′′ (ν1W + ν1 (1−W))
∣∣ dEν1 ⊗ dFν2

≤
∫
∆

∫
∆
W
∣∣ℑ′′ (ν1)

∣∣+ (1−W)
∣∣ℑ′′(ν1)

∣∣ dEν1 ⊗ dFν2

≤ W1⊗
∣∣ℑ′′ (U)

∣∣+ (1−W)
∣∣ℑ′′(U)

∣∣⊗ 1. (3.16)

If we apply norm in (3.16), then we have

∥∥ℑ′′ (1⊗ UW + 1⊗ U (1−W))
∥∥

≤
∥∥W1⊗

∣∣ℑ′′ (U)
∣∣+ (1−W)

∣∣ℑ′′(U)
∣∣⊗ 1

∥∥ ≤ W
∥∥ℑ′′(U)

∥∥+ (1−W)
∥∥ℑ′′(U)

∥∥ .
Similarly, we get∥∥ℑ′′ (1⊗DW + 1⊗D (1−W))

∥∥
≤
∥∥W1⊗

∣∣ℑ′′ (D)
∣∣+ ((1−W))

∣∣ℑ′′(D)
∣∣⊗ 1

∥∥ ≤ W
∥∥ℑ′′(D)

∥∥+ ((1−W))
∥∥ℑ′′(D)

∥∥ .
Using the norm in (3.12) and considering triangle inequality, we have

∥∥∥∥[16(ℑ(U)⊗ 1) +
2

3
ℑ
(
U ⊗ 1 + 1⊗D

2

)
+

1

6
(1⊗ℑ(D))

]
−

[
ℑ
(
U ⊗ 1 + 1⊗D

2

)
− W

S
k (ν2 − ν1)

4

[∫ 1

0
ℑ′
(
(1−W)U ⊗ 1 +

(
W1⊗D

2

))
dW

]



W. Afzal et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5790 20 of 30

+ ℑ
(
U ⊗ 1 + 1⊗D

2

)
− (1−W)

S
k (ν2 − ν1)

4

[∫ 1

0
ℑ′
((

1−W
2

)
U ⊗ 1 +

(
1 +W

2

)
1⊗D

)
dW

] ∥∥∥∥∥
≤ ∥(1⊗D − U ⊗ 1)2∥

6

(∥∥∥∥∥
∫ 1

2

0

(
W − 3k.2

S
k W

S+k
k

S + k

)[
ℑ′′ (U ⊗ 1W + 1⊗ U (1−W)) dW

]
+

∫ 1

1
2

(
(1−W)− 3k.2

S
k (1−W)

S+k
k

S + k

)
ℑ′′ (U ⊗ 1W + 1⊗ U (1−W))

∥∥∥∥∥
)

≤ ∥(1⊗D − U ⊗ 1)2∥
6

(∥∥∥∥∥
∫ 1

2

0

(
W − 3k.2

S
k W

S+k
k

S + k

)[
W1⊗

∣∣ℑ′′ (U)
∣∣+ (1−W)

∣∣ℑ′′(U)
∣∣⊗ 1

]
dW

+

∫ 1

1
2

(
(1−W)− 3k.2

S
k (1−W)

S+k
k

S + k

)
W1⊗

∣∣ℑ′′ (D)
∣∣+ (1−W)

∣∣ℑ′′(D)
∣∣⊗ 1

∥∥∥∥∥
)

≤ ∥(1⊗D − U ⊗ 1)2∥
6

(∥∥∥∥∥
∫ 1

2

0

(
W2 − 3.2SWS+2

S + 1

)
⊗
∣∣ℑ′′ (U)

∣∣
+

∫ 1

1
2

(
(W −W2)− 3.2SW(1−W)S+1

S + 1

)
⊗
∣∣ℑ′′ (D)

∣∣ ∥∥∥∥∥
)
dW

≤ ∥(1⊗D − U ⊗ 1)2∥
6

[(
k

4 (S + 2 k)

(
S
k

(
S + k

3 k

) 2 k
S

+
3 k

S + k

)
− 1

8

)∥∥∥ ∣∣ℑ′′ (U)
∣∣+ ∣∣ℑ′′ (D)

∣∣ ∥∥∥]

=
∥(1⊗D − U ⊗ 1)2∥

6

[(
S(S + k)

a+2k
S + 3

S+2k
a k

2S+2k
S

4 · 3
2k
S k

2k
S (S + k)(S + 2k)

− 1

8

)∥∥∥ ∣∣ℑ′′ (U)
∣∣+ ∣∣ℑ′′ (D)

∣∣ ∥∥∥] .
(3.17)

Remark 2. • Setting S and k = 1, and the tensor operations are vanished in Theorem
6, then Theorem 6 reduces to Theorem 2.2 in [50].

• If the norm structure and tensor operations are vanished in Theorem 6, then Theorem
6 simplifies to Corollary 2 in [13].

• If the tensor operations are vanished in Theorem 6, then Theorem 6 simplifies to
Theorem 2.3 in [32].

• Setting S, k = 1 in Theorem 6, it strained Theorem 2.3 in [57].

• Setting S, k = 1 in Theorem 6, it strained Theorem 2.3 in [2].

• If we choose S, k = 1 in Theorem 6, it strained Theorem 9 in [58].

Theorem 7. Let U and D be adjoint operators whose spectra lies in ∆1 and ∆2 respec-
tively. Let ℑ be a continuous over ∆ with ∥ℑ′′∥∆,∞ := supS∈∆ |ℑ′′(S)| < ∞, then we



W. Afzal et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5790 21 of 30

have∥∥∥∥∥
[
1

6
(ℑ(U)⊗ 1) +

2

3
ℑ
(
U ⊗ 1 + 1⊗D

2

)
+

1

6
(1⊗ℑ(D))

]

−

[
ℑ
(
U ⊗ 1 + 1⊗D

2

)
− W

S
k (ν2 − ν1)

4

[∫ 1

0
ℑ′
(
(1−W)U ⊗ 1 +

(
W1⊗D

2

))
dW

]

+ ℑ
(
U ⊗ 1 + 1⊗D

2

)
− (1−W)

S
k (ν2 − ν1)

4

[∫ 1

0
ℑ′
((

1−W
2

)
U ⊗ 1 +

(
1 +W

2

)
1⊗D

)
dW

] ∥∥∥∥∥
≤ ∥(1⊗D − U ⊗ 1)2∥

6

[(
W2

2
+

6 ln−
S
k
−1 (2) kΓk (S + k, ln (2) (1−W))

S + k

+
W2

2
+W +

6 ln−S−1 (2) kΓk (S + k, ln (2) (1−W))

S + k

)∥∥ℑ′∥∥
∆,+∞

)]
.

Proof. Taking into account lemma 6 and applying the triangle result, we have∥∥∥∥[16(ℑ(U)⊗ 1) +
2

3
ℑ
(
U ⊗ 1 + 1⊗D

2

)
+

1

6
(1⊗ℑ(D))

]
−

[
ℑ
(
U ⊗ 1 + 1⊗D

2

)
− W

S
k (ν2 − ν1)

4

[∫ 1

0
ℑ′
(
(1−W)U ⊗ 1 +

(
W1⊗D

2

))
dW

]

+ ℑ
(
U ⊗ 1 + 1⊗D

2

)
− (1−W)

S
k (ν2 − ν1)

4

[∫ 1

0
ℑ′
((

1−W
2

)
U ⊗ 1 +

(
1 +W

2

)
1⊗D

)
dW

] ∥∥∥∥∥
≤ ∥(1⊗D − U ⊗ 1)2∥

6

∥∥∥∥∥
∫ 1

2

0

(
W − 3k.2

S
k W

S+k
k

S + k

)[
ℑ′′ (U ⊗ 1W + 1⊗ U (1−W)) dW

]
+

∫ 1

1
2

(
(1−W)− 3k.2

S
k (1−W)

S+k
k

S + k

)
ℑ′′ (U ⊗ 1W + 1⊗ U (1−W))

∥∥∥∥∥
≤ ∥(1⊗D − U ⊗ 1)2∥

6

(∥∥∥∥∥
∫ 1

2

0

(
W − 3k.2

S
k W

S+k
k

S + k

)[
ℑ′′ (U ⊗ 1W + 1⊗ U (1−W)) dW

]
+

∫ 1

1
2

(
(1−W)− 3k.2

S
k (1−W)

S+k
k

S + k

)
ℑ′′ (U ⊗ 1W + 1⊗ U (1−W))

∥∥∥∥∥
)
. (3.18)

Observe that, by Lemma 2

∣∣∣∣(ℑ′′ (U1⊗W + 1⊗ U (1−W))
∣∣ = ∫

∆

∫
∆

∣∣∣∣ (ℑ′′ (ν1W + ν1 (1−W))

∣∣∣∣dEν1 ⊗ dFν2 .

Since ∣∣∣(ℑ′′ (ν1W + ν1 (1−W))
∣∣ ⩽ ∥∥ℑ′∥∥

∆,+∞



W. Afzal et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5790 22 of 30

for all ν1, ν2 ∈ ∆.
Taking

∫
∆

∫
∆ over dEν1 ⊗ dFν2 , then we get

∣∣ℑ′′ (1⊗ UW + 1⊗ U (1−W))
∣∣ = ∫

∆

∫
∆

∣∣ℑ′′ (ν1W + ν1 (1−W))
∣∣ dEν1 ⊗ dFν2

⩽
∥∥ℑ′∥∥

∆,+∞

∫
∆

∫
∆
dEν1 ⊗ dFν2 =

∥∥ℑ′∥∥
∆,+∞ . (3.19)

Similarly, we get∣∣ℑ′′ (1⊗DW + 1⊗D (1−W))
∣∣ = ∫

∆

∫
∆

∣∣ℑ′′ (ν2W + ν2 (1−W))
∣∣ dEν1 ⊗ dFν2

⩽
∥∥ℑ′∥∥

∆,+∞

∫
∆

∫
∆
dEν1 ⊗ dFν2 =

∥∥ℑ′∥∥
∆,+∞ . (3.20)

Taking into account equation (3.18), this follows

∥(1⊗D − U ⊗ 1)2∥
6

(∥∥∥∥∥
∫ 1

2

0

(
W − 3k.2

S
k W

S+k
k

S + k

)[
ℑ′′ (U ⊗ 1W + 1⊗ U (1−W)) dW

]
+

∫ 1

1
2

(
(1−W)− 3k.2

S
k (1−W)

S+k
k

S + k

)
ℑ′′ (U ⊗ 1W + 1⊗ U (1−W))

∥∥∥∥∥
)

≤ ∥(1⊗D − U ⊗ 1)2∥
6

(∥∥∥∥∥
∫ 1

2

0

(
W − 3k.2

S
k W

S+k
k

S + k

)∥∥∥∥∥
∥∥∥∥∥ℑ′′′ (1⊗ UW + 1⊗ U (1−W))

∥∥∥∥∥
+

∥∥∥∥∥
∫ 1

1
2

(
(1−W)− 3k.2

S
k (1−W)

S+k
k

S + k

)∥∥∥∥∥
∥∥∥∥∥ℑ′′ (1⊗ UW + 1⊗ U (1−W)) dW

∥∥∥∥∥
)

≤ ∥(1⊗D − U ⊗ 1)2∥
6

(∥∥∥∥∥W2

2
− 6 ln−

S
k
−1 (2) kΓk (S + k, ln (2) (1−W))

S + k

∥∥∥∥∥
∥∥∥∥∥ℑ′′′ (1⊗ UW + 1⊗ U (1−W))

∥∥∥∥∥
+

∥∥∥∥∥− W2

2
+W − 6 ln−

S
k
−1 (2) kΓk (S + k, ln (2) (1−W))

S + k

∥∥∥∥∥
∥∥∥∥∥ℑ′′ (1⊗ UW + 1⊗ U (1−W)) dW

∥∥∥∥∥
)

≤ ∥(1⊗D − U ⊗ 1)2∥
6

(∥∥∥∥∥W2

2
+

6 ln−
S
k
−1 (2) kΓk (S + k, ln (2) (1−W))

S + k

∥∥∥∥∥∥∥ℑ′∥∥
∆,+∞

+

∥∥∥∥∥W2

2
+W +

6 ln−S−1 (2) kΓk (S + 1, ln (2) (1−W))

S + 1

∥∥∥∥∥∥∥ℑ′∥∥
∆,+∞

)

≤ ∥(1⊗D − U ⊗ 1)2∥
6

[(
W2

2
+

6 ln−
S
k
−1 (2) kΓk (S + 1, ln (2) (1−W))

S + 1

+
W2

2
+W +

6 ln−S−1 (2) kΓk (S + k, ln (2) (1−W))

S + k

)∥∥ℑ′∥∥
∆,+∞

)]
. (3.21)



W. Afzal et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5790 23 of 30

Using equation (3.21) in (3.18), we get required result.

Theorem 8. Let ℑ be a twice differentiable as well as quasi convex |ℑ′′| on ∆, then the
following inequality holds true:∥∥∥∥∥
[
1

6
(ℑ(U)⊗ 1) +

2

3
ℑ
(
U ⊗ 1 + 1⊗D

2

)
+

1

6
(1⊗ℑ(D))

]

−

[
ℑ
(
U ⊗ 1 + 1⊗D

2

)
− W

S
k (ν2 − ν1)

4

[∫ 1

0
ℑ′
(
(1−W)U ⊗ 1 +

(
W1⊗D

2

))
dW

]

+ ℑ
(
U ⊗ 1 + 1⊗D

2

)
− (1−W)

S
k (ν2 − ν1)

4

[∫ 1

0
ℑ′
((

1−W
2

)
U ⊗ 1 +

(
1 +W

2

)
1⊗D

)
dW

] ∥∥∥∥∥
≤ ∥(1⊗D − U ⊗ 1)2∥

6

(
k

(4S + 8 k)

(
S
k

(
S + k

3 k

) 2 k
S

+
3 k

S + k

)
− 1

8

)

×
∥∥∥∥12 (∣∣ℑ′′(U)

∣∣⊗ 1 + 1⊗
∣∣ℑ′′(D)

∣∣+ ||ℑ′′(U)| ⊗ 1− 1⊗ |ℑ′′(D)||
)∥∥∥∥ .

Proof.
As |ℑ′′| is convex in a quasi sense over ∆, then one has∣∣(ℑ′′ (ν1W + ν1 (1−W))−ℑ′′ (ν2W + ν2 (1−W))

)∣∣
≤
∣∣(ℑ′′ (ν1W + ν1 (1−W)) + ℑ′′ (ν2W + ν2 (1−W))

)∣∣
≤ 1

2

(∣∣ℑ′′(ν2)
∣∣+ ∣∣ℑ′′(ν1)

∣∣+ ||ℑ′′(ν2)| − |ℑ′′(ν1)||
)

∀ τ ∈ [0, 1] and ν1, ν2 ∈ ∆.
Taking

∫
∆

∫
∆ over dEν1 ⊗ dFν2 yields:

∣∣(ℑ′′ (1⊗ UW + 1⊗ U (1−W))−ℑ′′ (1⊗DW + 1⊗D (1−W))
)∣∣

=

∫
∆

∫
∆

∣∣∣∣ (ℑ′′ (ν1W + ν1 (1−W))−ℑ′′ (ν2W + ν2 (1−W))
) ∣∣∣∣dEν1 ⊗ dFν2

≤ 1

2

∫
∆

∫
∆

(∣∣ℑ′′(ν2)
∣∣+ ∣∣ℑ′′(ν1)

∣∣+ ||ℑ′′(ν2)| − |ℑ′′(ν1)||
)
dEν1 ⊗ dFν2

=
1

2

(∣∣ℑ′′(U)
∣∣⊗ 1 + 1⊗

∣∣ℑ′′(D)
∣∣+ ||ℑ′′(U)| ⊗ 1− 1⊗ |ℑ′′(D)||

)
.

Applying the norm this follows as

∥∥(ℑ′′ (1⊗ UW + 1⊗ U (1−W))−ℑ′′ (1⊗DW + 1⊗D (1−W))
)∥∥

≤
∥∥∥∥12 (∣∣ℑ′′(U)

∣∣⊗ 1 + 1⊗
∣∣ℑ′′(D)

∣∣+ ||ℑ′′(U)| ⊗ 1− 1⊗ |ℑ′′(D)||
)∥∥∥∥



W. Afzal et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5790 24 of 30

⩽
1

2

(∥∥∣∣ℑ′′(U)
∣∣⊗ 1 + 1⊗

∣∣ℑ′′(D)
∣∣∥∥+ ∥∥∣∣ℑ′′(U)

∣∣⊗ 1− 1⊗
∣∣ℑ′′(D)

∣∣∥∥) .
Involving the norm in (3.12) and, we have

∥∥∥∥[16(ℑ(U)⊗ 1) +
2

3
ℑ
(
U ⊗ 1 + 1⊗D

2

)
+

1

6
(1⊗ℑ(D))

]
−

[
ℑ
(
U ⊗ 1 + 1⊗D

2

)
− W

S
k (ν2 − ν1)

4

[∫ 1

0
ℑ′
(
(1−W)U ⊗ 1 +

(
W1⊗D

2

))
dW

]

+ ℑ
(
U ⊗ 1 + 1⊗D

2

)
− (1−W)

S
k (ν2 − ν1)

4

[∫ 1

0
ℑ′
((

1−W
2

)
U ⊗ 1 +

(
1 +W

2

)
1⊗D

)
dW

] ∥∥∥∥∥
≤ ∥(1⊗D − U ⊗ 1)2∥

6

(∥∥∥∥∥
∫ 1

2

0

(
W − 3k.2

S
k W

S+k
k

S + k

)[
ℑ′′ (U ⊗ 1W + 1⊗ U (1−W)) dW

]
+

∫ 1

1
2

(
(1−W)− 3k.2

S
k (1−W)

S+k
k

S + k

)
ℑ′′ (U ⊗ 1W + 1⊗ U (1−W))

∥∥∥∥∥
)

≤ ∥(1⊗D − U ⊗ 1)2∥
6

×

(∥∥∥∥∥
∫ 1

2

0

(
W − 3k.2

S
k W

S+k
k

S + k

)
dW 1

2

(∣∣ℑ′′(U)
∣∣⊗ 1 + 1⊗

∣∣ℑ′′(D)
∣∣+ ||ℑ′′(U)| ⊗ 1− 1⊗ |ℑ′′(D)||

)
+

∫ 1

1
2

(
(1−W)− 3k.2

S
k (1−W)

S+k
k

S + k

)
dW

× 1

2

(∣∣ℑ′′(U)
∣∣⊗ 1 + 1⊗

∣∣ℑ′′(D)
∣∣+ ||ℑ′′(U)| ⊗ 1− 1⊗ |ℑ′′(D)||

) ∥∥∥∥∥
)

≤ ∥(1⊗D − U ⊗ 1)∥
6

∥∥∥∥∥
(

1

(4S + 4)

(
S
(
S + 4

2

) 3
2S

+
3

2S + 2

)
− 1

8

)∥∥∥∥∥
×
∥∥∥∥12 (∣∣ℑ′′(U)

∣∣⊗ 1 + 1⊗
∣∣ℑ′′(D)

∣∣+ ||ℑ′′(U)| ⊗ 1− 1⊗ |ℑ′′(D)||
)∥∥∥∥

≤ ∥(1⊗D − U ⊗ 1)2∥
12

∥∥∥∥∥
(

k

(4S + 8 k)

(
S
k

(
S + k

3 k

) 2 k
S

+
3 k

S + k

)
+

1

8

)∥∥∥∥∥
×
∥∥(∣∣ℑ′′(U)

∣∣⊗ 1 + 1⊗
∣∣ℑ′′(D)

∣∣+ ||ℑ′′(U)| ⊗ 1 + 1⊗ |ℑ′′(D)||
)∥∥

=
∥(1⊗D − U ⊗ 1)2∥

12

(
S(S + k)

a+2k
S + 3

S+2k
a k

2S+2k
S

4 · 3
2k
S k

2k
S (S + k)(S + 2k)

+
1

8

)
×
∥∥(∣∣ℑ′′(U)

∣∣⊗ 1 + 1⊗
∣∣ℑ′′(D)

∣∣+ ||ℑ′′(U)| ⊗ 1 + 1⊗ |ℑ′′(D)||
)∥∥ . (3.22)

Remark 3. • Setting S = 1 in Theorem 8, then it strained Theorem 2.4 in [57].



W. Afzal et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5790 25 of 30

• Setting S = 1 in Theorem 8, then it strained Theorem 2.4 in [2].

• Setting S = 1 in Theorem 8, then it strained Theorem 10 in [58].

4. Examples and consequences

For an exponential function, if self-adjoint operators U and D are commute, then we
have

eUeD = eDeU = e(U+D).

Further, if U is invertible and ν1, ν2 ∈ R with ν1 < ν2, then∫ ν2

ν1

eWUdW =
[eν2U − eν1U ]

U
.

Further if D − U is invertible, then we have∫ 1

0
e((1−ν2)U+SD)dS =

∫ 1

0
e(S(D−U))eUdS =

(∫ 1

0
e(S(D−U))dS

)
eU

=
[e(D−U) − I]eU

D − U
=

[eD − eU ]

D − U
.

Corollary 4.1. Assume the identical hypothesis of Theorem 7 with ℑ(µ) = lnµ over ∆,
and S = 1

5 , k = 1
2 , then

∥∥∥∥∥
[
1

6
(lnµ(U)⊗ 1) +

2

3
lnµ

(
U ⊗ 1 + 1⊗D

2

)
+

1

6
(1⊗ lnµ(D))

]

−

[
lnµ

(
U ⊗ 1 + 1⊗D

2

)
− W

2
5 (ν2 − ν1)

4

[∫ 1

0
lnµ′

(
(1−W)U ⊗ 1 +

(
W1⊗D

2

))
dW

]

+ lnµ

(
U ⊗ 1 + 1⊗D

2

)
− (1−W)

2
3 (ν2 − ν1)

4

[∫ 1

0
lnµ′

((
1−W

2

)
U ⊗ 1 +

(
1 +W

2

)
1⊗D

)
dW

] ∥∥∥∥∥
≤ ∥(1⊗D − U ⊗ 1)2∥

6

[
1

5
(
1
5 + 2

) (1

5

(
1
3 + 1

3

)
2
2
5

+
3

1
5 + 1

)
− 1

8

)∥∥∥ ∣∣lnµ′′ (U)
∣∣+ ∣∣lnµ′′ (D)

∣∣ ∥∥∥.

5. Conclusion and future remarks

Tensor Hilbert spaces allow for the extension and decomposition of operators in higher
dimensions, useful in spectral theory. Through the use of tensor operations for continuous
differentiable mappings, we introduced gradient type result in the setup of function spaces,
offering improvements and extensions of earlier findings that take fractional operators into



W. Afzal et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5790 26 of 30

account. We also point out important implications and comments that relate our results
to previous research. In addition to promoting further study of Simpson-type inequal-
ities and other quantum, fractional, and stochastic integrals, this publication advances
mathematical inequality theory in tensor Hilbert spaces, an area that has received little
attention. operators.

Acknowledgements

The authors acknowledge the financial support from the program PROSNI of the
University of Guadalajara, Mexico.

References

[1] M. Abbas, W. Afzal, T. Botmart, and A. M. Galal. Jensen, ostrowski and hermite-
hadamard type inequalities for h-convex stochastic processes by means of center-
radius order relation. 2023.

[2] W. Afzal, M. Abbas, and O. M. Alsalami. Bounds of different integral operators in
tensorial hilbert and variable exponent function spaces. Mathematics, 12(16):1–33,
2024.

[3] W. Afzal, M. Abbas, D. Breaz, and L. I. Cot̂ırlă. Fractional hermite–hadamard,
newton–milne, and convexity involving arithmetic–geometric mean-type inequalities
in hilbert and mixed-norm morrey spaces with variable exponents. Fractal & Frac-
tional, 8(9), 2024.

[4] W. Afzal, M. Abbas, J. E. Maćıas-Dı́az, and S. Treanţă. Some h-godunova–levin
function inequalities using center radius (cr) order relation. Fractal and Fractional,
6(9):518, 2022.

[5] W. Afzal, D. Breaz, M. Abbas, L. I. Cot̂ırlă, Z. A. Khan, and E. Rapeanu. Hyers–ulam
stability of 2d-convex mappings and some related new hermite–hadamard, pachpatte,
and fejér type integral inequalities using novel fractional integral operators via totally
interval-order relations with open problem. Mathematics, 12(8):1238, 2024.

[6] W. Afzal, S. M. Eldin, W. Nazeer, and A. M. Galal. Some integral inequalities for
harmonical cr-h-godunova–levin stochastic processes. AIMS Mathematics, 8:13473–
13491, 2023.

[7] W. Afzal, W. Nazeer, T. Botmart, and S. Treanta. Some properties and inequalities
for generalized class of harmonical godunova-levin function via center radius order
relation. AIMS Mathematics, 8(1):1696–1712, 2023.

[8] W. Afzal, E. Y. Prosviryakov, S. M. El-Deeb, and Y. Almalki. Some new estimates
of hermite–hadamard, ostrowski and jensen-type inclusions for h-convex stochastic
process via interval-valued functions. Symmetry, 15(4):831, 2023.

[9] A. A. H. Ahmadini, W. Afzal, M. Abbas, and E. S. Aly. Weighted fejér, her-
mite–hadamard, and trapezium-type inequalities for (h1, h2)–godunova–levin prein-
vex function with applications and two open problems. Mathematics, 12:382, 2024.



W. Afzal et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5790 27 of 30

[10] A. Al e’damat, A. Verma, J. Younis, and H. Aydi. Some properties of incomplete first
appell hypergeometric matrix functions. Applied Mathematics, 17(3):429–435, 2023.

[11] M. A. Ali, M. Abbas, H. Budak, P. Agarwal, G. Murtaza, and Y.-M. Chu. New
quantum boundaries for quantum simpson’s and quantum newton’s type inequalities
for preinvex functions. Advances in Differential Equations, 2021:64, 2021.

[12] M. A. Ali, H. Budak, Z. Zhang, and H. Yildirim. Some new simpson’s type inequalities
for coordinated convex functions in quantum calculus. Mathematical Methods in
Applied Sciences, 44:4515–4540, 2021.

[13] M.A. Ali, H. Kara, J. Tariboon, S. Asawasamrit, H. Budak, and F. Hezenci. Some
new simpson’s-formula-type inequalities for twice-differentiable convex functions via
generalized fractional operators. Symmetry, 13:2249, 2021.

[14] Y. Almalki and W. Afzal. Some new estimates of hermite–hadamard inequalities
for harmonical cr-h-convex functions via generalized fractional integral operator on
set-valued mappings. Mathematics, 11(19):4041, 2023.

[15] A. A. Almoneef, A.-A. Hyder, F. Hezenci, and H. Budak. Simpson-type inequalities
by means of tempered fractional integrals. AIMS Mathematics, 8:29411–29423, 2023.

[16] M. Alomari, M. Darus, S. S. Dragomir, and P. Cerone. Ostrowski type inequalities
for functions whose derivatives are s-convex in the second sense. Applied Mathematics
Letters, 23:1071–1076, 2010.

[17] H. Araki and F. Hansen. Jensen’s operator inequality for functions of several variables.
Proc. Amer. Math. Soc., 7:2075–2084, 2000.

[18] M. G. Bin-Saad, A. M. Al-Hashami, and J. A. Younis. Some fractional calculus
properties of the bivariate mittag-leffler function. Journal of Fractional Calculus and
Applications, 14(1):214–227, 2023.

[19] R. P. Boas and M. B. Marcus. Generalizations of young’s inequality. Journal of
Mathematical Analysis and Applications, 46:36–40, 1974.

[20] A. Bouchenak, M. Al Horani, J. Younis, R. Khalil, and M. A. Abd El Salam. Frac-
tional laplace transform for matrix valued functions with applications. Arab Journal
of Basic and Applied Sciences, 29(1):330–336, 2022.

[21] H. Budak, F. Hezenci, H. Kara, and M. Z. Sarikaya. Bounds for the error in approx-
imating a fractional integral by simpson’s rule. Mathematics, 11:2282, 2023.

[22] H. Budak, H. Kara, and F. Hezenci. Fractional simpson-type inequalities for twice
differentiable functions. SCMA, 2023.

[23] Z. Chen, B. Li, and B. Wang. Robust stability design for inverters using phase lag in
proportional-resonant controllers. IEEE Transactions on Industrial Electronics, 2024.

[24] S. Chu, M. Lin, D. Li, R. Lin, and S. Xiao. Adaptive reward shaping based rein-
forcement learning for docking control of autonomous underwater vehicles. Ocean
Engineering, 318:120139, 2025.

[25] S. Chu, Z. Xie, P. K. Wong, P. Li, W. Li, and J. Zhao. Observer-based gain scheduling
path following control for autonomous electric vehicles subject to time delay. Vehicle
System Dynamics, 60(5):1602–1626, 2022.

[26] J. Deng, G. Liu, L. Wang, G. Liu, and X. Wu. Intelligent optimization design of
squeeze casting process parameters based on neural network and improved sparrow



W. Afzal et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5790 28 of 30

search algorithm. Journal of Industrial Information Integration, 39:100600, 2024.
[27] J. F. Dong, Y. C. Liu, Y. Xu, S. C. Yuan, Q. Y. Wang, Z. W. Guan, and H. K. Chai.

Investigating the structural behavior of double-skin steel tubes filled with basalt fibre
reinforced recycled aggregate concrete under high temperature. Journal of Building
Engineering, page 111782, 2025.

[28] S. Dragomir. Refinements and reverses of tensorial and hadamard product inequalities
for selfadjoint operators in hilbert spaces related to young’s result. Communications
in Advanced Mathematical Sciences, 7:56–70, 2024.

[29] D. Gao, S. Liu, Y. Gao, P. Li, H. Zhang, M. Wang, and Y. Zhang. A comprehensive
adaptive interpretable takagi-sugeuo-kang fuzzy classifier for fatigue driving detection.
IEEE Transactions on Fuzzy Systems, 2024.

[30] S. Q. Hasan. Holder’s inequality mean continuity for existence and uniqueness so-
lution of fractional multi-integrodifferential delay system. Journal of Mathematics,
2020:1–16, 2020.

[31] Y. Hu and Y. Sugiyama. Well-posedness of the initial-boundary value problem
for 1d degenerate quasilinear wave equations. Advances in Differential Equations,
30(3/4):177–206, 2025.

[32] S. Iftikhar, M.U. Awan, and H. Budak. Exploring quantum simpson-type inequalities
for convex functions: A novel investigation. Symmetry, 15:1312, 2023.

[33] X. Ji, P. Jiang, Y. Jiang, H. Chen, W. Wang, W. Zhong, and D. Zang. Toward
enhanced aerosol particle adsorption in never-bursting bubble via acoustic levitation
and controlled liquid compensation. Advanced Science, 10(19):2300049, 2023.

[34] X. Ji, P. Jiang, Y. Jiang, H. Chen, W. Wang, W. Zhong, and D. Zang. Toward
enhanced aerosol particle adsorption in never-bursting bubble via acoustic levitation
and controlled liquid compensation. Advanced Science, 10(19):2300049, 2023.

[35] S. Jia, Y. Li, C. Gao, G. Liu, Y. Ren, C. He, and X. T. An. Realization of p-type
ma-based perovskite solar cells based on exposure of the (002) facet. Applied Physics
Letters, 126(2), 2025.

[36] H. Kara, H. Budak, M. A. Ali, and F. Hezenci. On inequalities of simpson’s type
for convex functions via generalized fractional integrals. Communications Faculty
of Sciences University of Ankara Series A1 Mathematics and Statistics, 71:806–825,
2022.

[37] Z. A. Khan, W. Afzal, M. Abbas, J. Ro, and N. M. Aloraini. A novel fractional
approach to finding the upper bounds of simpson and hermite-hadamard-type in-
equalities in tensorial hilbert spaces by using differentiable convex mappings. AIMS
Mathematics, 9(12):35151–35180, 2024.

[38] Z. A. Khan, W. Afzal, M. Abbas, J. S. Ro, and A. A. Zaagan. Some well-known in-
equalities on two-dimensional convex mappings by means of pseudo lr interval order
relations via fractional integral operators having non-singular kernel. AIMS Mathe-
matics, 9(6):16061–16092, 2024.

[39] Z. A. Khan, W. Afzal, W. Nazeer, and J. K. Asamoah. Some new variants of
hermite–hadamard and fejér-type inequalities for godunova–levin preinvex class of
interval-valued functions. Journal of Mathematics, 2024(1):8814585, 2024.



W. Afzal et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5790 29 of 30

[40] J. Li, D. Han, T. H. Weng, H. Wu, K. C. Li, and A. Castiglione. A secure data
storage and sharing scheme for port supply chain based on blockchain and dynamic
searchable encryption. Computer Standards Interfaces, 91:103887, 2025.

[41] W. Li, Z. Xie, J. Zhao, P. K. Wong, and P. Li. Fuzzy finite-frequency output feed-
back control for nonlinear active suspension systems with time delay and output
constraints. Mechanical Systems and Signal Processing, 132:315–334, 2019.

[42] J. Liang, W. Xu, T. Wang, Q. Yang, and S. Zhang. Implementing nat holpunching
with quic. In 2024 IEEE 100th Vehicular Technology Conference (VTC2024-Fall),
pages 1–7. IEEE, October 2024.

[43] F. Meng, A. Pang, X. Dong, C. Han, and X. Sha. H optimal performance design of
an unstable plant under bode integral constraint. Complexity, 2018:4942906, 2018.

[44] S. Meng, F. Meng, F. Zhang, Q. Li, Y. Zhang, and A. Zemouche. Observer design
method for nonlinear generalized systems with nonlinear algebraic constraints with
applications. Automatica, 162:111512, 2024.

[45] A. Moumen, H. Boulares, B. Meftah, R. Shafqat, T. Alraqad, E. E. Ali, and Z. Khaled.
Multiplicatively simpson type inequalities via fractional integral. Symmetry, 15:460,
2023.

[46] M. H. M. Rashid and F. Bani-Ahmad. An estimate for the numerical radius of
the hilbert space operators and a numerical radius inequality. AIMS Mathematics,
8:26384–26405, 2023.

[47] T. Saeed, W. Afzal, M. Abbas, S. Treanţă, and M. De la Sen. Some new generaliza-
tions of integral inequalities for harmonical cr-(h, h)-godunova–levin functions and
applications. Mathematics, 10(23):4540, 2022.

[48] T. Saeed, W. Afzal, K. Shabbir, S. Treanţă, and M. De la Sen. Some novel estimates of
hermite–hadamard and jensen type inequalities for (h, h)-convex functions pertaining
to total order relation. Mathematics, 10(24):4777, 2022.

[49] S. K. Sahoo, P. O. Mohammed, D. O. Regan, M. Tariq, and K. Nonlaopon. New
hermite–hadamard type inequalities in connection with interval-valued generalized
harmonically (h1, h2)-godunova–levin functions. Symmetry, 14(10):1964, 2022.

[50] M.Z. Sarikaya, Erhan Set, and M.E. Ozdemir. On new inequalities of simpson’s type
for functions whose second derivatives absolute values are convex. Journal of Applied
Mathematics, Statistics and Informatics, 9:37–45, 2013.

[51] X. Shi, U. Ishtiaq, M. Din, and M. Akram. Fractals of interpolative kannan mappings.
Fractal and Fractional, 8(8):493, 2024.

[52] X. Shi, Y. Zhang, M. Yu, and L. Zhang. Revolutionizing market surveillance: cus-
tomer relationship management with machine learning. PeerJ Computer Science,
10:e2583, 2024.

[53] Y. Shi and Z. Liu. Some sharp simpson type inequalities and applications. Applied
Mathematics E-Notes, 9:205–215, 2009.

[54] S. Sitho, M. A. Ali, H. Budak, S. K. Ntouyas, and J. Tariboon. Trapezoid and
midpoint type inequalities for preinvex functions via quantum calculus. Mathematics,
9:1666, 2021.

[55] T. Sitthiwirattham, K. Nonlaopon, M. A. Ali, and H. Budak. Riemann–liouville



W. Afzal et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5790 30 of 30

fractional newton’s type inequalities for differentiable convex functions. Fractal and
Fractional, 6:175, 2022.

[56] V. Stojiljkovic. Twice differentiable ostrowski type tensorial norm inequality for
continuous functions of selfadjoint operators in hilbert spaces. Eur. J. Pure Appl.
Math., 16:1421–1433, 2023.

[57] V. Stojiljkovic. Generalized tensorial simpson type inequalities for convex functions
of selfadjoint operators in hilbert space. Maltepe Journal of Mathematics, 6:78–89,
2024.

[58] V. Stojiljković and S.S. Dragomir. Tensorial simpson 1/8 type inequalities for convex
functions of selfadjoint operators in hilbert space. Eur. J. Math. Anal., 4:17, 2024.

[59] L. Wang, G. Liu, G. Wang, and K. Zhang. M-pinn: A mesh-based physics-informed
neural network for linear elastic problems in solid mechanics. International Journal
for Numerical Methods in Engineering, 125(9):e7444, 2024.

[60] Z. Wang, Z. Liao, B. Zhou, G. Yu, and W. Luo. Swinurnet: Hybrid transformer-cnn
architecture for real-time unstructured road segmentation. IEEE Transactions on
Instrumentation and Measurement, 2024.

[61] H. Wu, H. Zhang, R. Yan, S. Li, X. Guo, L. Qiu, and Y. Yao. Limosilactobacillus
regulating microbial communities to overcome the hydrolysis bottleneck with efficient
one-step co-production of h2 and ch4. Advanced Science, 11(43):2406119, 2024.

[62] Y. Wu and T. Shen. A finite convergence criterion for the discounted optimal control
of stochastic logical networks. IEEE Transactions on Automatic Control, 63(1):262–
268, 2017.

[63] C. Yan, M. Feng, Z. Wu, Y. Guo, W. Dong, Y. Wang, and A. Mian. Discriminative
correspondence estimation for unsupervised rgb-d point cloud registration. IEEE
Transactions on Circuits and Systems for Video Technology, 2024.

[64] H. Zhang, Y. Chang, Y. Xu, C. Liu, X. Xiao, J. Li, and H. Guo. Design and fabrication
of a chalcogenide hollow-core anti-resonant fiber for mid-infrared applications. Optics
Express, 31(5):7659–7670, 2023.

[65] H. H. Zhang, J. B. Chao, Y. W. Wang, Y. Liu, H. M. Yao, Z. P. Zhao, and K. Niu. 5g
base station antenna array with heatsink radome. IEEE Transactions on Antennas
and Propagation, 2024.

[66] X. Zhang, K. Shabbir, W. Afzal, H. Xiao, and D. Lin. Hermite–hadamard and
jensen-type inequalities via riemann integral operator for a generalized class of go-
dunova–levin functions. Journal of Mathematics, 2022(1):3830324, 2022.

[67] X. Zhang, M. Usman, A. U. R. Irshad, M. Rashid, and A. Khattak. Investigating
spatial effects through machine learning and leveraging explainable ai for child mal-
nutrition in pakistan. ISPRS International Journal of Geo-Information, 13(9):330,
2024.

[68] J. Zhao, X. Wang, P. K. Wong, Z. Xie, J. Jia, and W. Li. Multi-objective frequency
domain-constrained static output feedback control for delayed active suspension sys-
tems with wheelbase preview information. Nonlinear Dynamics, 103(2):1757–1774,
2021.