




































©2024 Ada Academica https://adac.eeEur. J. Math. Anal. 4 (2024) 17doi: 10.28924/ada/ma.4.17
Tensorial Simpson 18 Type Inequalities for Convex Functions of Selfadjoint Operators in

Hilbert Space

Vuk Stojiljković1,∗, Sever Silvestru Dragomir2
1Faculty of Science, University of Novi Sad, Trg Dositeja Obradovića 3, 21000 Novi Sad, Serbia

vuk.stojiljkovic999@gmail.com
2Mathematics, College of Sport Health and Engineering, Victoria University Melbourne City, VIC 8001,

Australia
Sever.Dragomir@vu.edu.au

∗Correspondence: vuk.stojiljkovic999@gmail.com

Abstract. Several Simpson 1
8

tensorial type inequalities for selfadjoint operators have been obtainedwith variation depending on the conditions imposed on the function f∣∣∣∣∣∣∣∣18
[
f (A)⊗ 1 + 6f

(
A⊗ 1 + 1⊗B

2

)
+ 1⊗ f (B)

]
−
∫ 1
0

f (λ1⊗B+ (1− λ)A⊗ 1)dλ
∣∣∣∣∣∣∣∣

≤ 5 ‖1⊗B− A⊗ 1‖
32

∥∥f ′∥∥
I,+∞ .

1. Introduction and preliminaries
The concept we now call a "tensor" wasn’t originally named that way. When Josiah Willard Gibbsfirst described the idea in the late 19th century, he used the term "dyadic." Today, mathematiciansdefine a tensor as the mathematical embodiment of Gibbs’ initial concept. Tensors and inequalitiesare natural partners, thanks to the widespread use of inequalities in mathematics. These mathemat-ical statements about comparisons have a profound impact on various scientific disciplines. Whilemany types of inequalities exist, some of the most significant ones include Jensen’s, Ostrowski’s,Hermite-Hadamard’s, and Minkowski’s inequalities. For those interested in delving deeper, refer-ences [17] and [18] provide more details about inequalities and their fascinating history. Regardingthe generalizations of the aforementioned inequalities, numerous studies have been published; foradditional information, check the following and the references therein [1–5,7–9,21–23].Classical inequalities of Simpson type have been given by Hezenci et al. [15] and Sarikaya etal. [19]. To enhance the presentation of this work, we will demonstrate new developments in the
Received: 12 Apr 2024.
Key words and phrases. Tensorial product, Selfadjoint operators, Convex functions.1

https://adac.ee
https://doi.org/10.28924/ada/ma.4.17


Eur. J. Math. Anal. 10.28924/ada/ma.4.17 2theory of inequalities in Hilbert spaces. One such development is the Dragomir’s inequality fornormal operators given by the following [10]:
Theorem 1. Let (H ; 〈., .〉) be a Hilbert space and T : H → H a normal linear operator on H .
Then

‖Tx‖2 ≥
1

2

(
‖Tx‖2 + |〈T2x, x〉|

)
≥ |〈Tx, x〉|2,

for any x ∈ H, ‖x‖ = 1. The constant 12 is the best possible.

The Hermite-Hadamard inequality in the selfadjoint operator sense, as provided by Dragomir[11], is another intriguing conclusion.
Theorem 2. Let f : I → R be an operator convex function on the interval I . Then for any selfadjoint
operators A and B with spectra in I we have the inequality

f

(
A+B

2

)
≤ f

(
3A+B

4

)
+ f

(
A+ 3B

4

)
≤
∫ 1
0

f ((1− t)A+ tB)dt

≤
1

2

[
f

(
A+B

2

)
+
f (A) + f (B)

2

]
≤
f (A) + f (B)

2
.

The first paper related to tensorial inequalities in Hilbert space was written by Dragomir [13].In the paper, he proved the tensorial version of the Ostrowski type inequality given by the following.
Theorem 3. Assume that f is continuously differentiable on I with ‖f ′‖I,+∞ := supt∈I |f ′(t)| < +∞
and A,B are selfadjoint operators with Sp(A), Sp(B) ⊂ I . Then the following inequality holds:∥∥∥∥f ((1− λ)A⊗ 1 + λ1⊗B)− ∫ 1

0

f ((1− u)A⊗ 1 + u1⊗B)du
∥∥∥∥ (1)

≤
∥∥f ′∥∥

I,+∞

[
1

4
+

(
λ−
1

2

)2 ]
‖1⊗B− A⊗ 1‖

for λ ∈ [0, 1].

Recently, various inequalities in the same tensorial surrounding have been obtained. The fol-lowing result of Simpson type was obtained by Stojiljković [24].
Theorem 4. Assume that f is continuously differentiable on I and |f ′′| is convex and A,B are
selfadjoint operators with Sp(A), Sp(B) ⊂ I . Then the following inequality holds:∣∣∣∣∣∣∣∣16

(
f (A)⊗ 1 + 4f

(
A⊗ 1 + 1⊗B

2

)
+ 1⊗ f (B)

)

https://doi.org/10.28924/ada/ma.4.17


Eur. J. Math. Anal. 10.28924/ada/ma.4.17 3

−
1

2
α

(∫ 1
0

f

((
1− k
2

)
A⊗ 1 +

(
1 + k

2

)
1⊗B

)
kα−1dk

+

∫ 1
0

f

((
1−

k

2

)
A⊗ 1 +

k

2
1⊗B

)
(1− k)α−1dk

)∣∣∣∣∣∣∣∣
≤ ‖1⊗B− A⊗ 1‖2

(‖f ′′(A)‖+ ‖f ′′(B)‖)
(
3α2 + 8α+ 7

)
(α+ 2)(24α+ 24)

for α ≥ 0.

The following inequality has been recently obtained by the same author [25].
Theorem 5. Assume that f is continuously differentiable on I with ‖f ′‖I,+∞ := supt∈I |f ′(t)| < +∞
and A,B are selfadjoint operators with Sp(A), Sp(B) ⊂ I . Then the following inequality holds:∥∥∥∥∫ 1

0

f ((1− λ)A⊗ 1 + λ1⊗B)dλ− f
(
A⊗ 1 + 1⊗B

2

)∥∥∥∥
6 ‖1⊗B− A⊗ 1‖2

‖f ′′‖I,+∞
24

.

Recently, the following inequality of Ostrowski type was obtained by Stojiljković et al. [26] whichgeneralized the recently obtained results by Dragomir [13].
Theorem 6. The formulation is the same as the one given by Dragomir in his Ostrowski type
Theorem given above (1) with an exception that α > 0, then∣∣∣∣∣∣∣∣(λα + (1− λ)α)f ((1− λ)A⊗ 1 + λ1⊗B)

−α
(
(1− λ)α

∫ 1
0

f ((1− λ)(1− u)A⊗ 1 + (u + (1− u)λ)1⊗B)(1− u)α−1du

+λα
∫ 1
0

uα−1f (((1− u) + u(1− λ))A⊗ 1 + uλ1⊗B)du
)∣∣∣∣∣∣∣∣

6 ‖1⊗B− A⊗ 1‖
(
λα+1

α+ 1
+
(1− λ)α+1

α+ 1

)∥∥f ′∥∥
I,+∞ .

Stojiljković et al., [27] recently obtained a Trapezoid type tensorial inequality which is given by
Theorem 7. Assume that f is continuously differentiable on I with ‖f ′‖I,+∞ := supt∈I |f ′(t)| < +∞
and A,B are selfadjoint operators with Sp(A), Sp(B) ⊂ I . Then the following inequality holds:∣∣∣∣∣∣∣∣ (f (A)⊗ 1 + 1⊗ f (B)) (2)

−α
[ ∫ 1
0

(1− λ)α−1f (λ1⊗B+ (1− λ)A⊗ 1)dλ

+

∫ 1
0

λα−1f (λ1⊗B+ (1− λ)A⊗ 1)dλ
]∣∣∣∣∣∣∣∣

https://doi.org/10.28924/ada/ma.4.17


Eur. J. Math. Anal. 10.28924/ada/ma.4.17 4

≤ ‖1⊗B− A⊗ 1‖
1

1 + α

(
2− 21−α

) ∥∥f ′∥∥
I,+∞ .

In order to derive similar inequalities of the tensorial type, we need the following introductionand preliminaries.Let I1, ..., Ik be intervals from R and let f : I1 × ... × Ik → R be an essentially bounded realfunction defined on the product of the intervals. Let A = (A1, ...,Ak) be a k-tuple of boundedselfadjoint operators on Hilbert spaces H1, ..., Hk such that the spectrum of Ai is contained in Iifor i = 1, ..., k . We say that such a k-tuple is in the domain of f . If
Ai =

∫
Ii

λidEi(λi)

is the spectral resolution of Ai for i = 1, ..., k by following , we define
f (A1, ...,Ak) :=

∫
I1

...

∫
Ik

f (λ1, ..., λk)dE1(λ1)⊗ ...⊗ dEk(λk)

as bounded selfadjoint operator on the tensorial product H1 ⊗ ...⊗Hk .If the Hilbert spaces are of finite dimension, then the above integrals become finite sums, and wemay consider the functional calculus for arbitrary real functions. This construction [6] extends thedefinition of Koranyi [16] for functions of two variables and have the property that
f (A1, ...Ak) = f1(A1)⊗ ...⊗ fk(Ak),

whenever f can be separated as a product f (t1, ..., tk) = f1(t1)...fk(tk) of k functions each de-pending on only one variable.Recall the following property of the tensorial product
(AC)⊗ (B⊗D) = (A⊗B)(C⊗D)

that holds for any A,B,C,D ∈ B(H ).From the property we can deduce easily the following consequences
An ⊗Bn = (A⊗B)n, n > 0,

(A⊗ 1)(1⊗B) = (1⊗B)(A⊗ 1) = A⊗ B,

which can be extended, for two natural numbers m, n we have
(A⊗ 1)n(1⊗B)m = (1⊗B)m(A⊗ 1)n = An ⊗Bm.

For more information, consult the following book related to tensors [14]. The following Lemmawhich we require can be found in a paper of Dragomir [12].

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Eur. J. Math. Anal. 10.28924/ada/ma.4.17 5

Lemma 1. Assume A and B are selfadjoint operators with Sp(A) ⊂ I, Sp(B) ⊂ J and having the
spectral resolutions . Let f ; h be continuous on I, g, k continuous on J and φ and ψ continuous on
an interval K that contains the sum of the intervals f (I) + g(J); h(I) + k(J),then

φ(f (A)⊗ 1 + 1⊗ g(B))ψ(h(A)⊗ 1 + 1⊗ k(B))

=

∫
I

∫
J

φ(f (t) + g(s))ψ(h(t) + k(s))dEt ⊗ dFs .

In [20], Shuang, Wang and Qi used the following identity to obtain Simpson type inequalitiesand some applications.
Lemma 2. Let f : I ⊂ R→ R be a differentiable function on I◦, a,b ∈ I◦ with a < b. If f ′ ∈ L1[a, b],
then the following equality holds:

1

8

[
f (a) + 6f

(
a + b

2

)
+ f (b)

]
−
1

b − a

∫ b

a

f (x)dx (3)
=
b − a
4

(∫ 1
0

[(
3

4
− t
)
f ′
(
ta + (1− t)

a + b

2

)
+

(
1

4
− t
)
f ′
(
t
a + b

2
+ (1− t)b

)]
dt

)
.

This paper delves into a novel area of mathematics: tensorial inequalities of the Simpson type fordifferentiable functions within a tensorial Hilbert space. This field is young and ripe for exploration,and obtaining new bounds for various combinations of convex functions is crucial for its advancement.The paper is structured logically. The "Main Results" section unveils the key findings that contributeto the novelty of this work. Subsequently, the "Examples and Consequences" section showcasespractical applications of the obtained results. By leveraging known properties of the exponentialoperator and its integral, and by choosing specific convex functions, the authors generate numeroustensorial Simpson-type inequalities and bounds. Finally, the "Conclusion" section summarizes thepaper’s contributions and highlights its significance for the development of tensorial inequalities.In the following theorem, you’ll find a fundamental result that serves as the foundation for derivingfurther inequalities throughout the paper.
2. Main results

The following Lemma will be used crucial in obtaining the inequalities which follow.
Lemma 3. Assume that f is continuously differentiable on I, A and B are selfadjoint operators with
Sp(A), Sp(B) ⊂ I , then

1

8

[
f (A)⊗ 1 + 6f

(
A⊗ 1 + 1⊗B

2

)
+ 1⊗ f (B)

]
−
∫ 1
0

f (λ1⊗B+ (1− λ)A⊗ 1)dλ (4)
=
1⊗B− A⊗ 1

4

∫ 1
0

[(
3

4
− k
)
f ′
(
A⊗ 1

(
1 + k

2

)
+ 1⊗B

(
1− k
2

))

https://doi.org/10.28924/ada/ma.4.17


Eur. J. Math. Anal. 10.28924/ada/ma.4.17 6

+

(
1

4
− k
)
f ′
(
k

2
A⊗ 1 + 1⊗B

(
2− k
2

))]
dk.

Proof. We will start the proof with Lemma (3). Introducing the substitutions on the left hand sideand simplifying the fractional integral, then assuming that A and B have the spectral resolutions
A =

∫
tdE(t) and B =

∫
sdF (s).

If we take the integral ∫I ∫I over dEt ⊗ dFs , then we get∫
I

∫
I

(
1

8

[
f (t) + 6f

(
t + s

2

)
+ f (s)

]
−
∫ 1
0

f (λs + (1− λ)t)dλ
)
dEt ⊗ dFs

=

∫
I

∫
I

(
s − t
2

∫ 1
0

[(
3

4
− k
)
f ′
(
t

(
1 + k

2

)
+ s

(
1− k
2

))
+

(
1

4
− k
)
f ′
(
k

2
t + s

(
2− k
2

))]
dk

)
dEt ⊗ dFs .By utilizing the Fubinis Theorem and Lemma 1 for appropriate choices of the functions involved,we have successively ∫

I

∫
I

f

(
t + s

2

)
dEt ⊗ dFs = f

(
A⊗ 1 + 1⊗B

2

)
,

∫
I

∫
I

∫ 1
0

f (λs + (1− λ)t)dλdEt ⊗ dFs

=

∫ 1
0

∫
I

∫
I

f (λs + (1− λ)t)dλ
)
dEt ⊗ dFsdλ

=

∫ 1
0

f (λ1⊗B+ (1− λ)A⊗ 1)dλ,

∫
I

∫
I

s − t
2

∫ 1
0

(
3

4
− k
)
f ′
(
t

(
1 + k

2

)
+ s

(
1− k
2

))
dkdEt ⊗ dFs

=

∫ 1
0

(
3

4
− k
)∫

I

∫
I

s − t
2

f ′
(
t

(
1 + k

2

)
+ s

(
1− k
2

))
dEt ⊗ dFsdk

=
(1⊗B− A⊗ 1)

4

∫ 1
0

(
3

4
− k
)
f ′
(
A⊗ 1

(
1 + k

2

)
+ 1⊗B

(
1− k
2

))
dk.

Following the same principle for other terms, the equality follows.
�

Theorem 8. Assume that f is continuously differentiable on I with
‖f ′‖I,+∞ := supt∈I |f ′(t)| < +∞ and A,B are selfadjoint operators with Sp(A), Sp(B) ⊂ I , then∣∣∣∣∣∣∣∣18

[
f (A)⊗ 1 + 6f

(
A⊗ 1 + 1⊗B

2

)
+ 1⊗ f (B)

] (5)
−
∫ 1
0

f (λ1⊗B+ (1− λ)A⊗ 1)dλ
∣∣∣∣∣∣∣∣

https://doi.org/10.28924/ada/ma.4.17


Eur. J. Math. Anal. 10.28924/ada/ma.4.17 7

≤
5 ‖1⊗B− A⊗ 1‖

32

∥∥f ′∥∥
I,+∞ .

Proof. If we take the operator norm of the previously obtained Lemma (4) and use the triangleinequality, we get
∥∥∥∥18
[
f (A)⊗ 1 + 6f

(
A⊗ 1 + 1⊗B

2

)
+ 1⊗ f (B)

]
−
∫ 1
0

f (λ1⊗B+ (1− λ)A⊗ 1)dλ
∥∥∥∥

≤
‖1⊗B− A⊗ 1‖

2

∫ 1
0

∣∣∣∣34 − k
∣∣∣∣ ∥∥∥∥f ′(A⊗ 1(1 + k2

)
+ 1⊗B

(
1− k
2

))∥∥∥∥
+

∣∣∣∣14 − k
∣∣∣∣ ∥∥∥∥f ′(k2A⊗ 1 + 1⊗B

(
2− k
2

))∥∥∥∥ ]dk
Realize here that by Lemma 1,∣∣∣∣f ′(A⊗ 1(1 + k2

)
+ 1⊗B

(
1− k
2

)) ∣∣∣∣
=

∫
I

∫
I

∣∣∣∣f ′(t (1 + k2
)
+ s

(
1− k
2

)) ∣∣∣∣dEt ⊗ dFs .
Since ∣∣∣∣f ′(t (1 + k2

)
+ s

(
1− k
2

)) ∣∣∣∣ 6 ∥∥f ′∥∥I,+∞ .
Holds for all t, s ∈ I . If we take the integral ∫I ∫I over dEt ⊗ dFs , then we get∣∣∣∣f ′(A⊗ 1(1 + k2

)
+ 1⊗B

(
1− k
2

)) ∣∣∣∣
=

∫
I

∫
I

∣∣∣∣f ′(t (1 + k2
)
+ s

(
1− k
2

)) ∣∣∣∣dEt ⊗ dFs .
6
∥∥f ′∥∥

I,+∞

∫
I

∫
I

dEt ⊗ dFs =
∥∥f ′∥∥

I,+∞ .From which we get the following,∫ 1
0

∥∥∥∥34 − k
∥∥∥∥∥∥∥∥f ′(A⊗ 1(1 + k2

)
+ 1⊗B

(
1− k
2

))∥∥∥∥ dk
6
∥∥f ′∥∥

I,+∞

∫ 1
0

∥∥∥∥34 − k
∥∥∥∥ dk = 5 ‖f’‖I,+∞16Evaluation of the second part is analogous, summing everything up we obtain the desired equality.

�

https://doi.org/10.28924/ada/ma.4.17


Eur. J. Math. Anal. 10.28924/ada/ma.4.17 8

Theorem 9. Assume that f is continuously differentiable on I and |f ′| is convex and A,B are
selfadjoint operators with Sp(A), Sp(B) ⊂ I , then∣∣∣∣∣∣∣∣18

[
f (A)⊗ 1 + 6f

(
A⊗ 1 + 1⊗B

2

)
+ 1⊗ f (B)

] (6)
−
∫ 1
0

f (λ1⊗B+ (1− λ)A⊗ 1)dλ
∣∣∣∣∣∣∣∣

≤
5 ‖1⊗B− A⊗ 1‖

64
(
∥∥f ′(A)∥∥+ ∥∥f ′(B)∥∥).

Proof. Since |f ′| is convex on I , then we get∣∣∣∣f ′(t (1 + k2
)
+ s

(
1− k
2

)) ∣∣∣∣ 6 (1 + k2
)
|f ′(t)|+

(
1− k
2

)
|f ′(s)|

for all k ∈ [0, 1] and t, s ∈ I .If we take the integral ∫I ∫I over dEt ⊗ dFs , then we get∣∣∣∣f ′(A⊗ 1(1 + k2
)
+ 1⊗B

(
1− k
2

)) ∣∣∣∣
=

∫
I

∫
I

∣∣∣∣f’((1 + k2
)
t +

(
1− k
2

)
s

) ∣∣∣∣dEt ⊗ dFs
6
∫
I

∫
I

[(
1 + k

2

)
|f ′(t)|+

(
1− k
2

)
|f ′(s)|

]
dEt ⊗ dFs

=

(
1 + k

2

)
|f ′(A)| ⊗ 1 +

(
1− k
2

)
1⊗ |f ′(B)|

for all k ∈ [0, 1].If we take the norm in the inequality, we get the following∥∥∥∥f ′(A⊗ 1(1 + k2
)
+ 1⊗B

(
1− k
2

))∥∥∥∥
6

∥∥∥∥(1 + k2
)
|f ′(A)| ⊗ 1 +

(
1− k
2

)
1⊗ |f ′(B)|

∥∥∥∥
6

(
1 + k

2

)∥∥|f ′(A)| ⊗ 1∥∥+ (1− k
2

)∥∥1⊗ |f ′(B)|∥∥
=

(
1 + k

2

)∥∥f ′(A)∥∥+ (1− k
2

)∥∥f ′(B)∥∥ .Therefore, we obtain∫ 1
0

∥∥∥∥34 − k
∥∥∥∥∥∥∥∥f ′(A⊗ 1(1 + k2

)
+ 1⊗B

(
1− k
2

))∥∥∥∥ dk
6
∫ 1
0

∥∥∥∥34 − k
∥∥∥∥((1 + k2

)∥∥f ′(A)∥∥+ (1− k
2

)∥∥f ′(B)∥∥) dk
=
79 ‖f’(A)‖+ 41 ‖f’(B)‖

384
.Simplifying the other term and adding them, we obtain the desired inequality.

https://doi.org/10.28924/ada/ma.4.17


Eur. J. Math. Anal. 10.28924/ada/ma.4.17 9

�

We recall that the function f : I → R is quasi-convex, if
f ((1− λ)t + λs) 6 max(f (t), f (s)) =

1

2
(f (t) + f (s) + |f (s)− f (t)|)

holds for all t, s ∈ I and λ ∈ [0, 1].
Theorem 10. Assume that f is continuously differentiable on I with |f ′| is quasi-convex on I , A
and B are selfadjoint operators with Sp(A), Sp(B) ⊂ I , then∣∣∣∣∣∣∣∣18

[
f (A)⊗ 1 + 6f

(
A⊗ 1 + 1⊗B

2

)
+ 1⊗ f (B)

] (7)
−
∫ 1
0

f (λ1⊗B+ (1− λ)A⊗ 1)dλ
∣∣∣∣∣∣∣∣

≤
5 ‖1⊗B− A⊗ 1‖

64

∥∥|f ′(A)| ⊗ 1 + 1⊗ |f ′(B)|∥∥+ ∥∥|f ′(A)| ⊗ 1− 1⊗ |f ′(B∥∥ .
Proof. Since |f ′| is quasi-convex on I , then we get

|f ′(t(1 + k)/(2) + s(1− k)/(2))| ≤ 1/2(|f ′(t)|+ |f ′(s)|+ ||f ′(t)− f ′(s)||)

for all k ∈ [0, 1] and t, s ∈ I. If we take the integral ∫I ∫I over dEt ⊗ dFs , then we get∣∣∣∣f ′(A⊗ 1(1 + k2
)
+ 1⊗B

(
1− k
2

)) ∣∣∣∣
=

∫
I

∫
I

f ′(t(1 + k)/(2) + s(1− k)/(2))dEt ⊗ dFs

6
1

2

∫
I

∫
I

(|f ′(t)|+ |f ′(s)|+ ||f ′(t)| − |f ′(s)||)dEt ⊗ dFs

=
1

2
(|f ′(A)| ⊗ 1 + 1⊗ |f ′(B)|+ ||f ′(A)| ⊗ 1− 1⊗ |f ′(B)||)for all k ∈ [0, 1].If we take the norm, then we get∥∥∥∥f ′(A⊗ 1(1 + k2

)
+ 1⊗B

(
1− k
2

))∥∥∥∥
6

∥∥∥∥12(|f ′(A)| ⊗ 1 + 1⊗ |f ′(B)|+ ||f ′(A)| ⊗ 1− 1⊗ |f ′(B)||)
∥∥∥∥

6
1

2

(∥∥|f ′(A)| ⊗ 1 + 1⊗ |f ′(B)|∥∥+ ∥∥|f ′(A)| ⊗ 1− 1⊗ |f ′(B)|∥∥)Which when applied in our case, we get∫ 1
0

∥∥∥∥34 − k
∥∥∥∥∥∥∥∥f ′(A⊗ 1(1 + k2

)
+ 1⊗B

(
1− k
2

))∥∥∥∥ dk
6
∫ 1
0

∥∥∥∥34 − k
∥∥∥∥(12 (∥∥|f ′(A)| ⊗ 1 + 1⊗ |f ′(B)|∥∥+ ∥∥|f ′(A)| ⊗ 1− 1⊗ |f ′(B)|∥∥)

)
dk.

Which when simplified, we obtain the desired inequality. �

https://doi.org/10.28924/ada/ma.4.17


Eur. J. Math. Anal. 10.28924/ada/ma.4.17 103. Some examples and consequences
It is known that if U and V are commuting, that is UV = V U , then the exponential functionsatisfies the property

exp(U) exp(V ) = exp(V ) exp(U) = exp(U + V ).

Also, if U is invertible and a, b ∈ R and a < b then∫ b

a

exp(tU)dt = U−1[exp(bU)− exp(aU)].

Moreover, if U and V are commuting and V − U is invertible, then∫ 1
0

exp((1− k)U + kV )dk =
∫ 1
0

exp(k(V − U)) exp(U)dk

=

∫ 1
0

(exp(k(V − U))dk)exp(U)

= (V − U)−1[exp(V − U)− I] exp(U) = (V − U)−1[exp(V )− exp(U)].Since the operators U = A⊗ 1 and V = 1⊗B are commutative and if 1⊗B−A⊗ 1 is invertible,then ∫ 1
0

exp((1− k)A⊗ 1 + k1⊗B)dk

= (1⊗B− A⊗ 1)−1[exp(1⊗B)− exp(A⊗ 1)].In the following sequel we provide examples to the obtained Theorems in Main section. Examplesconsist of taking f to be an exponential operator and applying various conditions as given by theTheorems.
Corollary 1. If A,B are selfadjoint operators with Sp(A), Sp(B) ⊂ [m,M] and 1⊗B− A⊗ 1 is
invertible, then by (5), we get∣∣∣∣∣∣∣∣18

[
exp(A)⊗ 1 + 6 exp

(
A⊗ 1 + 1⊗B

2

)
+ 1⊗ exp(B)

] (8)
−(1⊗B− A⊗ 1)−1[exp(1⊗B)− exp(A⊗ 1)]

∣∣∣∣∣∣∣∣
≤
5 ‖1⊗B− A⊗ 1‖

32
exp(M).

Corollary 2. Since for f (t) = exp(t), t ∈ R, |f ′| is convex, then by (6)∣∣∣∣∣∣∣∣18
[
exp(A)⊗ 1 + 6 exp

(
A⊗ 1 + 1⊗B

2

)
+ 1⊗ exp(B)

] (9)
−(1⊗B− A⊗ 1)−1[exp(1⊗B)− exp(A⊗ 1)]

∣∣∣∣∣∣∣∣
≤
5 ‖1⊗B− A⊗ 1‖

64
(‖exp(A)‖+ ‖exp(B)‖).

https://doi.org/10.28924/ada/ma.4.17


Eur. J. Math. Anal. 10.28924/ada/ma.4.17 11

∣∣∣∣∣∣∣∣18
[
exp(A)⊗ 1 + 6 exp

(
A⊗ 1 + 1⊗B

2

)
+ 1⊗ exp(B)

] (10)
−(1⊗B− A⊗ 1)−1[exp(1⊗B)− exp(A⊗ 1)]

∣∣∣∣∣∣∣∣
≤
5 ‖1⊗B− A⊗ 1‖

64
(‖| exp(A)| ⊗ 1 + 1⊗ | exp(B)|‖+ ‖exp(A)| ⊗ 1− 1⊗ | exp(B‖).

4. Conclusion
Tensors have become important in various fields, for example in physics because they providea concise mathematical framework for formulating and solving physical problems in fields suchas mechanics, electromagnetism, quantum mechanics, and many others. As such inequalities arecrucial in numerical aspects. Reflected in this work is the tensorial Shuang’s Lemma, which asa consequence enabled us to obtain Simpson type inequalities in Hilbert space. New Simpsontype inequalities are given, examples of specific convex functions and their inequalities using ourresults are given in the section some examples and consequences. Plans for future research can bereflected in the fact that the obtained inequalities in this work can be sharpened or generalized byusing other methods. An interesting perspective can be seen in incorporating other techniques forHilbert space inequalities with the techniques shown in this paper. One direction is the techniqueof the Mond-Pecaric inequality, on which we will work on.

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	1. Introduction and preliminaries
	2. Main results
	3. Some examples and consequences
	4. Conclusion
	References

