EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 3, 2023, 1421-1433
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Twice Differentiable Ostrowski Type Tensorial Norm
Inequalities for Continuous Functions of Selfadjoint

Operators in Hilbert Spaces

Vuk Stojiljković

University of Novi Sad, Novi Sad, Serbia, Serbia

Abstract. In this paper several tensorial norm inequalities for continuous functions of selfadjoint
operators in Hilbert spaces have been obtained. Multiple inequalities are obtained with variations
due to the convexity properties of the mapping f∥∥∥∥(1⊗B −A⊗ 1)−1[exp(1⊗B)− exp(A⊗ 1)]− exp

(
A⊗ 1 + 1⊗B

2

)∥∥∥∥
⩽ ∥1⊗B −A⊗ 1∥2

∥f ′′∥I,+∞

24
.

2020 Mathematics Subject Classifications: 26D05, 26D07, 26D20

Key Words and Phrases: Tensorial product, Selfadjoint operators, Convex functions

1. Introduction and Preliminaries

The notion of a tensor has its origin in the 19th century, where it was formulated by
Gibbs, though he didn’t formally use the word tensor but a dyadic. In modern language, it
can be seen as the origin of the tensor definition and its introduction to the mathematics.
Interplay of inequalities in mathematics is vast, and as such it has applications in tensors
as well. Mathematics and other scientific fields are highly influenced by inequalities. Many
types of inequalities exist, but those involving Jensen, Ostrowski, Hermite–Hadamard, and
Minkowski hold particular significance among them. More about inequalities and its his-
tory can be found in these books [21, 23]. Multiple papers have been published concerning
the generalizations of the said inequalities, see the following and references therein for
more information [1–5, 7–9, 25–29].
Since our paper is about tensorial Ostrowski type inequalities, we give the brief introduc-
tion to the topic. In 1938, A. Ostrowski [22] , proved the following inequality concerning

the distance between the integral mean 1
b−a

∫ b
a f(t)dt and the value f(x), x ∈ [a, b].

DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4843

Email address: vuk.stojiljkovic999@gmail.com (V. Stojiljković)

https://www.ejpam.com 1421 © 2023 EJPAM All rights reserved.



V. Stojiljković / Eur. J. Pure Appl. Math, 16 (3) (2023), 1421-1433 1422

Theorem 1. Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, b) such
that f ′ : (a, b) → R is bounded on (a, b) and ∥f ′∥∞ := supt∈(a,b) |f ′(t)| < +∞. Then

∣∣∣∣f(x)− 1

b− a

∫ b

a
f(t)dt

∣∣∣∣ ⩽ [14 +

(
x− a+b

2

b− a

)2 ] ∥∥f ′∥∥∞ (b− a),

for all x ∈ [a, b] and the constant 1
4 is the best possible.

If we take x = a+b
2 we get the midpoint inequality∣∣∣∣f (a+ b

2

)
− 1

b− a

∫ b

a
f(t)dt

∣∣∣∣ ⩽ 1

4

∥∥f ′∥∥∞ (b− a),

with 1
4 as best possible constant.

In order to derive similar inequalities of the tensorial type, we need the following intro-
duction and preliminaries.
Let I1, ..., Ik be intervals from R and let f : I1 × ... × Ik → R be an essentially bounded
real function defined on the product of the intervals. Let A = (A1, ..., An) be a k-tuple of
bounded selfadjoint operators on Hilbert spaces H1, ...,Hk such that the spectrum of Ai

is contained in Ii for 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 [6] , 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 we may consider the functional calculus for arbitrary real functions. This construction
extends the definition of Kornyi [20] 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
depending on only one variable.

Since we will be using tensorial products, we will define in the following what tensors
and tensorial products are in short, for more consult the following book [17].
Let U, V and W be vector spaces over the same field F . A mapping Φ : U × V → W
is called a bilinear mapping if it is linear in each variable separately. Namely, for all
u, u1, u2 ∈ U , v, v1, v2 ∈ V and a, b ∈ F ,
Φ(au1 + bu2, v) = aΦ(u1, v) + bΦ(u2, v),
Φ(u, av1 + bv2) = aΦ(u, v1) + bΦ(u, v2). If W = F , a bilinear mapping Φ : U × V → F is



V. Stojiljković / Eur. J. Pure Appl. Math, 16 (3) (2023), 1421-1433 1423

called a bilinear function.
Let ⊗ : U × V → W be a bilinear mapping. The pair (W,⊗) is called a tensor product
space of U and V if it satisfies the following conditions:
1. Generating property < Im⊗ >=W ;
2. Maximal span property dim < Im⊗ >= dimU · dimV .
The member w ∈W is called a tensor, but not all tensors in W are products of two vectors
of the form u⊗ v.
The notation < Im⊗ > denotes the span.
Example
Let u = (x1, .., xm) ∈ Rm and v = (y1, ..., yn) ∈ Rn. We can view u and v as column
vectors. Namely,

u =

x1...
xm

 , v =

y1...
yn


are m× 1 and n× 1 matrices respectively.
We define ⊗ : Rm × Rn →Mm,n,

u⊗ v = uvt =

 x1y1 · · ·x1yn...
xmy1 · · ·xmyn

 ,
an m× n matrix with entries Aij = xiyj . (Mm,n,⊗) is a tensor product space of Rm and
Rn. Tensors do not need to be matrices. This is just one model given. For more consult
the following book [17].

Recall the following property of the tensorial product

(AC)⊗ (BD) = (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)n(A⊗ 1)m = An ⊗Bm.

The current research concerning tensorial inequalities can be seen in the following papers,
[10–14, 16]. The following Lemma which we require can be found in a paper of Silvestru
[15].



V. Stojiljković / Eur. J. Pure Appl. Math, 16 (3) (2023), 1421-1433 1424

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 the paper written by Ozdemir et al. [19] , the authors used the following Lemma.
We will utilize it to produce results in the tensorial setting.

Lemma 2. Let f : I ⊂ R → R be a differentiable mapping on I0 where a, b ∈ I with
a < b. If f ′′ ∈ L[a, b] then the following equality holds:

1

b− a

∫ b

a
f(x)dx− f

(
a+ b

2

)
=

(b− a)2

16

[ ∫ 1

0
l2f ′′

(
l
a+ b

2
+ (1− l)a

)
dl

+

∫ 1

0
(l − 1)2f ′′

(
lb+ (1− l)

a+ b

2

)
dl

]
.

In the following Theorem, we give a fundamental result which we will use in our paper
to produce inequalities.

2. Main results

Theorem 2. Assume that f is continuously differentiable on I, A and B are selfadjoint
operators with Sp(A), Sp(B) ⊂ I, then∫ 1

0
f((1− λ)A⊗ 1 + λ1⊗B)dλ− f

(
A⊗ 1 + 1⊗B

2

)

=
(1⊗B −A⊗ 1)2

16

[ ∫ 1

0
l2f ′′

((
1− l

2

)
A⊗ 1 +

l

2
1⊗B

)
dl

+

∫ 1

0
(l − 1)2f ′′

((
1− l

2

)
A⊗ 1 +

(
1 + l

2

)
1⊗B

)
dl

]
Proof. We start with Lemma 2. Introducing the substitution x = λb+ (1− λ)a on the

left hand side integral. Then we assume that A and B have the spectral resolutions

A =

∫
I
tdEt, B =

∫
I
sdFs.

If we take the integral
∫
I

∫
I dEt ⊗ dFs, then we get∫

I

∫
I

∫ 1

0

(
f((1− λ)t+ λs)dλ− f

(
s+ t

2

))
dEt ⊗ dFs



V. Stojiljković / Eur. J. Pure Appl. Math, 16 (3) (2023), 1421-1433 1425

=

∫
I

∫
I

(
(s− t)2

16

[ ∫ 1

0
l2f ′′

((
1− l

2

)
t+

l

2
s

)
dl

+

∫ 1

0
(l − 1)2f ′′

((
1− l

2

)
a+

(
1 + l

2

)
s

)
dl

])
dEt ⊗ dFs.

By utilizing Fubini’s Theorem for the left and right hand side with Lemma 1 for appropriate
choices of the functions involved, we have successively∫

I

∫
I

∫ 1

0
f((1− λ)t+ λs)dλdEt ⊗ dFs

=

∫ 1

0

∫
I

∫
I
f((1− λ)t+ λs)dEt ⊗ dFsdλ

=

∫ 1

0
f((1− λ)A⊗ 1 + λ1⊗B)dλ,∫

I

∫
I

(
(s− t)2

16

∫ 1

0
l2f ′′

((
1− l

2

)
t+

l

2
s

)
dldEt ⊗ dFs

=

∫ 1

0

∫
I

∫
I

(s− t)2

16
l2f ′′

((
1− l

2

)
t+

l

2
s

)
dEt ⊗ dFsdl

=

∫ 1

0

(1⊗B −A⊗ 1)2

16
l2f ′′

((
1− l

2

)
A⊗ 1 +

l

2
1⊗B

)
dl,∫

I

∫
I

(
(s− t)2

16

∫ 1

0
l2f ′′

((
1− l

2

)
t+

(
1 + l

2

)
s

)
dldEt ⊗ dFs,

=

∫ 1

0

∫
I

∫
I

(s− t)2

16
l2f ′′

((
1− l

2

)
t+

(
1 + l

2

)
s

)
dEt ⊗ dFsdl

=

∫ 1

0

(1⊗B −A⊗ 1)2

16
l2f ′′

((
1− l

2

)
A⊗ 1 +

(
1 + l

2

)
1⊗B

)
dl.

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∥∥∥∥∫ 1

0
f((1− λ)A⊗ 1 + λ1⊗B)dλ− f

(
A⊗ 1 + 1⊗B

2

)∥∥∥∥
⩽ ∥1⊗B −A⊗ 1∥2

∥f ′′∥I,+∞
24

.

Proof. If we take the operator norm, we get∥∥∥∥∫ 1

0
f((1− λ)A⊗ 1 + λ1⊗B)dλ− f

(
A⊗ 1 + 1⊗B

2

)∥∥∥∥



V. Stojiljković / Eur. J. Pure Appl. Math, 16 (3) (2023), 1421-1433 1426

⩽
∥1⊗B −A⊗ 1∥2

16

∣∣∣∣∣∣∣∣ ∫ 1

0
l2f ′′

((
1− l

2

)
A⊗ 1 +

l

2
1⊗B

)
dl

+

∫ 1

0
(l − 1)2f ′′

((
1− l

2

)
A⊗ 1 +

(
1 + l

2

)
1⊗B

)
dl

∣∣∣∣∣∣∣∣.
Using the triangle inequality and the properties of the integral and the norm, we get

∥1⊗B −A⊗ 1∥2

16

∣∣∣∣∣∣∣∣ ∫ 1

0
l2f ′′

((
1− l

2

)
A⊗ 1 +

l

2
1⊗B

)
dl

+

∫ 1

0
(l − 1)2f ′′

((
1− l

2

)
A⊗ 1 +

(
1 + l

2

)
1⊗B

)
dl

∣∣∣∣∣∣∣∣
⩽

∥1⊗B −A⊗ 1∥2

16

(∫ 1

0
l2
∥∥∥∥f ′′((1− l

2

)
A⊗ 1 +

l

2
1⊗B

)∥∥∥∥ dl
+

∫ 1

0
(l − 1)2

∥∥∥∥f ′′((1− l

2

)
A⊗ 1 +

(
1 + l

2

)
1⊗B

)∥∥∥∥ dl).
Observe that by Lemma 1,∣∣∣∣f ′′((1− l

2

)
⊗ 1 +

l

2
1⊗B

) ∣∣∣∣ = ∫
I

∫
I

∣∣∣∣f ′′((1− l

2

)
t+

l

2
s

) ∣∣∣∣dEt ⊗ dFs.

Since ∣∣∣∣f ′′((1− l

2

)
t+

l

2
s

)
⩽
∥∥f ′′∥∥

I,+∞

for all l ∈ [0, 1] and t, s ∈ I. If we take the integral
∫
I

∫
I over dEt ⊗ dFs, then we get∣∣∣∣f ′′((1− l

2

)
⊗ 1 +

l

2
1⊗B

) ∣∣∣∣ = ∫
I

∫
I

∣∣∣∣f ′′((1− l

2

)
t+

l

2
s

) ∣∣∣∣dEt ⊗ dFs

⩽
∥∥f ′∥∥

I,+∞

∫
I

∫
I
dEt ⊗ dFs =

∥∥f ′∥∥
I,+∞ .

This implies that ∣∣∣∣f ′′((1− l

2

)
⊗ 1 +

l

2
1⊗B

) ∣∣∣∣ ⩽ ∥∥f ′′∥∥I,+∞

for l ∈ [0, 1], similarly we have∥∥∥∥f ′′((1− l

2

)
A⊗ 1 +

(
1 + l

2

)
1⊗B

)∥∥∥∥ ⩽
∥∥f ′′∥∥

I,+∞ .

Which combined gives us the following

∥1⊗B −A⊗ 1∥2

16

(∫ 1

0
l2
∥∥∥∥f ′′((1− l

2

)
A⊗ 1 +

l

2
1⊗B

)∥∥∥∥ dl



V. Stojiljković / Eur. J. Pure Appl. Math, 16 (3) (2023), 1421-1433 1427

+

∫ 1

0
(l − 1)2

∥∥∥∥f ′′((1− l

2

)
A⊗ 1 +

(
1 + l

2

)
1⊗B

)∥∥∥∥ dl)
⩽

∥1⊗B −A⊗ 1∥2

16

(∫ 1

0
l2
∥∥f ′′∥∥

I,+∞ dl +

∫ 1

0
(l − 1)2

∥∥f ′′∥∥
I,+∞ dl

)
.

Solving the resulting integrals and simplifying, we obtain the desired result.

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∥∥∥∥∫ 1

0
f((1− λ)A⊗ 1 + λ1⊗B)dλ− f

(
A⊗ 1 + 1⊗B

2

)∥∥∥∥
⩽

∥1⊗B −A⊗ 1∥2

48

(∥∥f ′′(A)∥∥+ ∥∥f ′′(B)
∥∥) .

Proof. Since |f ′′| is convex on I, then we get∣∣∣∣f ′′((1− l

2

)
t+

l

2
s

) ∣∣∣∣ ⩽ (1− l

2

)
|f ′′(t)|+ l

2
|f ′′(s)|

for all l ∈ [0, 1] and t, s ∈ I.
If we take the integral

∫
I

∫
I over dEt ⊗ dFs, then we get∣∣∣∣f ′′((1− l

2

)
A⊗ 1 +

l

2
1⊗B

) ∣∣∣∣ = ∫
I

∫
I

∣∣∣∣f ′′((1− l

2

)
t+

l

2
s

) ∣∣∣∣dEt ⊗ dFs

⩽
∫
I

∫
I

[(
1− l

2

)
|f ′′(t)|+ l

2
|f ′′(s)|

]
dEt ⊗ dFs

=

(
1− l

2

)
|f ′′(A)| ⊗ 1 +

l

2
1⊗ |f ′′(B)|

for all l ∈ [0, 1].
If we take the norm in the inequality, we get the following∥∥∥∥f ′′((1− l

2

)
A⊗ 1 +

l

2
1⊗B

)∥∥∥∥ ⩽

∥∥∥∥(1− l

2

)
|f ′′(A)| ⊗ 1 +

l

2
1⊗ |f ′′(B)|

∥∥∥∥
⩽

(
1− l

2

)∥∥|f ′′(A)| ⊗ 1
∥∥+ l

2

∥∥1⊗ |f ′′(B)|
∥∥

=

(
1− l

2

)∥∥f ′′(A)∥∥+ l

2

∥∥f ′′(B)
∥∥ .

Similarly, we get∥∥∥∥f ′′((1− l

2

)
A⊗ 1 +

(
1 + l

2

)
1⊗B

)∥∥∥∥ ⩽
1− l

2

∥∥f ′′(A)∥∥+ 1 + l

2

∥∥f ′′(B)
∥∥ .



V. Stojiljković / Eur. J. Pure Appl. Math, 16 (3) (2023), 1421-1433 1428

Which when applied to the inequality obtained in the previous Theorem, we obtain the
following

∥1⊗B −A⊗ 1∥2

16

(∫ 1

0
l2
∥∥∥∥f ′′((1− l

2

)
A⊗ 1 +

l

2
1⊗B

)∥∥∥∥ dl
+

∫ 1

0
(l − 1)2

∥∥∥∥f ′′((1− l

2

)
A⊗ 1 +

(
1 + l

2

)
1⊗B

)∥∥∥∥ dl)
⩽

∥1⊗B −A⊗ 1∥2

16

(∫ 1

0
l2
((

1− l

2

)∥∥f ′′(A)∥∥+ l

2

∥∥f ′′(B)
∥∥) dl

+

∫ 1

0
(l − 1)2

(
1− l

2

∥∥f ′′(A)∥∥+ 1 + l

2

∥∥f ′′(B)
∥∥) dl).

Which when simplified after integrating the terms, we obtain the original inequality.

We recall that the function f : I → R is quasi-convex, if f((1−λ)t+λs) ⩽ max(f(t), f(s)) =
1
2(f(t) + f(s) + |f(s)− f(t)|) for all t, s ∈ I and λ ∈ [0, 1].

Theorem 5. 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∥∥∥∥∫ 1

0
f((1− λ)A⊗ 1 + λ1⊗B)dλ− f

(
A⊗ 1 + 1⊗B

2

)∥∥∥∥
⩽

∥1⊗B −A⊗ 1∥2

48
(
∥∥|f ′′(A)| ⊗ 1 + 1⊗ |f ′′(B)|

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

Proof. Since |f ′′| is quasi-convex on I, then we get∣∣∣∣f ′′((1− l

2

)
t+

l

2
s

) ∣∣∣∣ ⩽ 1

2
(|f ′′(t)|+ |f ′′(s)|+ ||f ′′(t)| − |f ′′(s)||)

for all l ∈ [0, 1] and t, s ∈ I. If we take the integral
∫
I

∫
I over dEt ⊗ dFs, then we get∣∣∣∣f ′′((1− l

2

)
A⊗ 1 +

l

2
1⊗B

) ∣∣∣∣
⩽
∫
I

∫
I
|f ′′
((

1− l

2

)
t+

l

2
s

)
|dEt ⊗ dFs

⩽
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)||))



V. Stojiljković / Eur. J. Pure Appl. Math, 16 (3) (2023), 1421-1433 1429

for all l ∈ [0, 1].
If we take the norm, then we get∥∥∥∥f ′′((1− l

2

)
A⊗ 1 +

l

2
1⊗B

)∥∥∥∥
⩽

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

⩽
1

2

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

∥∥)
for all l ∈ [0, 1]. In a similar way, we obtain∥∥∥∥f ′′(1− l

2
A⊗ 1 +

1 + l

2
1⊗B

)∥∥∥∥
⩽

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

⩽
1

2

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

∥∥)
for all l ∈ [0, 1].
Using these inequalities in the inequality obtained during Theorem 4, we obtain the follow-
ing (∫ 1

0
l2
∥∥∥∥f ′′((1− l

2

)
A⊗ 1 +

l

2
1⊗B

)∥∥∥∥ dl
+

∫ 1

0
(l − 1)2

∥∥∥∥f ′′((1− l

2

)
A⊗ 1 +

(
1 + l

2

)
1⊗B

)∥∥∥∥ dl)

⩽
∫ 1

0
l2
(
1

2

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

∥∥)) dl
+

∫ 1

0
(l − 1)2

(
1

2

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

∥∥)) dl.
Which when simplified, we obtain the desired inequality.

3. Some comments

It is known that if U and V are commuting, that is UV = V U , then the exponential
function satisfies the property

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



V. Stojiljković / Eur. J. Pure Appl. Math, 16 (3) (2023), 1421-1433 1430

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

= (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)].

Corollary 1. If A,B are selfadjoint operators with Sp(A), Sp(B) ⊂ [m,M ] and 1⊗B −
A⊗ 1 is invertible, then by Theorem 3, we get∥∥∥∥(1⊗B −A⊗ 1)−1[exp(1⊗B)− exp(A⊗ 1)]− exp

(
A⊗ 1 + 1⊗B

2

)∥∥∥∥
⩽ ∥1⊗B −A⊗ 1∥2 exp(M)

24
.

Corollary 2. Since for f(t) = exp(t), t ∈ R, |f ′′| is convex, then by Theorem 4∥∥∥∥(1⊗B −A⊗ 1)−1[exp(1⊗B)− exp(A⊗ 1)]− exp

(
A⊗ 1 + 1⊗B

2

)∥∥∥∥
⩽

∥1⊗B −A⊗ 1∥2

48
(∥exp(A)∥+ ∥exp(B)∥)

4. Conclusion

Tensors have become important in various fields, for example in physics because they
provide a concise mathematical framework for formulating and solving physical problems
in fields such as mechanics, electromagnetism, quantum mechanics, and many others. As
such inequalities are crucial in numerical aspects. Reflected in this work is the tensorial
Ozdemir’s Lemma, which as a consequence enabled us to obtain Ostrowski type inequal-
ities in Hilbert space. New Ostrowski type inequalities are given, examples of specific
convex functions and their inequalities using our results are given in the section some ex-
amples. Plans for future research can be reflected in the fact that the obtained inequalities
in this work can be sharpened or generalized by using other methods. An interesting per-
spective can be seen in incorporating other techniques for Hilbert space inequalities with
the techniques shown in this paper. One direction is the technique of the Mond-Pecaric
inequality, on which we will work on.



REFERENCES 1431

References

[1] Afzal, W.; Abbas, M.; Maćıas-Dı́az, J.E.; Treanţă, S. Some H-Godunova–Levin Func-
tion Inequalities Using Center Radius (Cr) Order Relation. Fractal Fract. 2022, 6, 518.
https://doi.org/10.3390/fractalfract6090518

[2] Afzal, W.; Alb Lupa¸s, A.; Shabbir, K. Hermite–Hadamard and Jensen-Type Inequal-
ities for Harmonical (h1, h2)-Godunova–Levin Interval-Valued Functions. Mathemat-
ics 2022, 10, 2970. https://doi.org/10.3390/math10162970

[3] Afzal, W., Khurram Shabbir, Savin Treant¸˘a, Kamsing Nonlaopon. Jensen and
Hermite-Hadamard type inclusions for harmonical h-Godunova-Levin functions[J].
Aims Mathematics, 2023, 8(2): 3303-3321. doi: 10.3934/math.2023170

[4] Afzal, W., Khurram Shabbir, Thongchai Botmart. Generalized version of Jensen and
Hermite-Hadamard inequalities for interval-valued (h1, h2)- Godunova-Levin func-
tions[J]. AIMS Mathematics, 2022, 7(10): 19372-19387. doi: 10.3934/math.20221064

[5] Afzal,W., Waqas Nazeer, Thongchai Botmart, Savin Treant¸˘a. Some properties
and inequalities for generalized class of harmonical Godunova-Levin function via
center radius order relation[J]. AIMS Mathematics, 2023, 8(1): 1696-1712. doi:
10.3934/math.2023087

[6] H. Araki and F. Hansen, Jenseńıs operator inequality for functions of several variables,
Proc. Amer. Math. Soc. 128 (2000), No. 7, 20

[7] Butt, S.I.; Tariq, M.; Aslam, A.; Ahmad, H.; Nofal, T.A. Hermite–Hadamard type in-
equalities via generalized harmonic exponential convexity and applications. J. Funct.
Spaces 2021, 2021, 5533491.

[8] Chandola A., Agarwal R., Pandey M. R., Some New Hermite–Hadamard, Her-
mite–Hadamard Fejer and Weighted Hardy Type Inequalities Involving (k-p) Rie-
mann– Liouville Fractional Integral Operator, Appl. Math. Inf. Sci. 16, No. 2, 287–297
(2022).

[9] Chen, H., Katugampola, U.N. Hermite–Hadamard and Hermite–Hadamard–Fejr type
inequalities for generalized fractional integrals. J. Math. Anal. Appl. 2017, 446,
1274–1291

[10] S. S. Dragomir, An inequality improving the Örst Hermite-Hadamard inequality for
convex functions defined on linear spaces and applications for semi-inner products.
Journal of Inequalities in Pure Applied Mathematics, Volume: 3 (2002), Issue: 2,
Paper No. 31, 8 p.

[11] S. S. Dragomir, An inequality improving the second Hermite-Hadamard inequality for
convex functions defined on linear spaces and applications for semi-inner products.
Journal of Inequalities in Pure Applied Mathematics, Volume: 3 (2002), Issue: 3,
Paper No.



REFERENCES 1432

[12] S. S. Dragomir, Bounds for the normalized Jensen functional, Bull. Austral. Math.
Soc. 74(3)(2006), 417-478.

[13] S. S. Dragomir, A note on Younǵıs inequality, Revista de la Real Academia de Ciencias
Exactas, FÌsicas y Naturales. Serie A. Matematicas 111 (2017), no. 2, 349-354.

[14] S. S. Dragomir, P. Cerone and A. Sofo, Some remarks on the trapezoid rule in nu-
merical integration, Indian J. Pure Appl. Math. 31 (2000)

[15] S.S Dragomir, Tensorial Norm Inequalities For Taylor’s Expansions Of Functions Of
Selfadjoint Operators In Hilbert Spaces , ResearchGate, November 2022.

[16] S.S Dragomir, An Ostrowski Type Tensorial Norm Inequality For Continuous Func-
tions Of Selfadjoint Operators In Hilbert Spaces, Researchgate, November 2022.

[17] Guo. H., What Are Tensors Exactly?, World Scientific, June 2021,
https://doi.org/10.1142/12388

[18] Hezenci, F., Budak, H. Kara, H. New version of fractional Simpson type in-
equalities for twice differentiable functions. Adv Differ Equ 2021, 460 (2021).
https://doi.org/10.1186/s13662-021-03615-2

[19] Ozdemir, M. E., Ardic, A. A., Some companions of Ostrowski type inequality for
functions whose second derivatives are convex and concave with applications, Arab J
Math Sci 21(1) (2015), 53–66, https://doi.org/10.1016/j.ajmsc.2013.12.002

[20] A. Koranyi. On some classes of analytic functions of several variables. Trans. Amer.
Math. Soc., 101 (1961), 520-554.

[21] D. S. Mitrinović ,Analytic Inequalities, Springer-Verlag, Berlin, 1970.

[22] A. Ostrowski, Uber die Absolutabweichung einer differentienbaren Funktionen von
ihren Integralmittelwert, Comment. Math. Hel, 10 (1938), 226-227.

[23] Pečarić J., Proschan F., Tong Y., Convex Functions, Partial Orderings, and Statistical
Applications, Academic Press, INC, United States of America, 1992.

[24] Sarikaya,M.Z, E. Set, M.E. Özdemir, On new inequalities of Simpson’s type for convex
functions, RGMIA Res. Rep. Coll. 13 (2) (2010) Article2.

[25] Stojiljković, V.; Ramaswamy, R.; Abdelnaby, O.A.A.; Radenović, S. Some Novel In-
equalities for LR-(k,h-m)-p Convex Interval Valued Functions by Means of Pseudo Or-
der Relation. Fractal Fract. 2022, 6, 726. https://doi.org/10.3390/fractalfract6120726

[26] Stojiljković, V.; Ramaswamy, R.; Alshammari, F.; Ashour, O.A.; Alghazwani,
M.L.H.; Radenović, S. Hermite–Hadamard Type Inequalities Involving (k-p) Frac-
tional Operator for Various Types of Convex Functions. Fractal Fract. 2022, 6, 376.
https://doi.org/10.3390/fractalfract6070376



REFERENCES 1433

[27] Stojiljković, V.; Ramaswamy, R.; Ashour Abdelnaby, O.A.; Radenović, S. Riemann-
Liouville Fractional Inclusions for Convex Functions Using Interval Valued Setting.
Mathematics 2022, 10, 3491. https://doi.org/10.3390/math10193491

[28] Stojiljković, V.; Ramaswamy, R.; Abdelnaby, O.A.A.; Radenović, S. Some Refine-
ments of the Tensorial Inequalities in Hilbert Spaces. Symmetry 2023, 15, 925.
https://doi.org/10.3390/sym15040925

[29] Stojiljković, V.; Hermite–Hadamard–type fractional–integral inequalities for (p, h)-
convex fuzzy-interval-valued mappings, Electron. J. Math. 5 (2023) 18–28, DOI:
10.47443/ejm.2023.004