Mathematics in Applied Sciences and Engineering https://ojs.lib.uwo.ca/mase
Volume 4, Number 1, March 2023, pp.1-11 https://doi.org/10.5206/mase/15397

ON A NEW GENERALIZED TSALLIS RELATIVE OPERATOR ENTROPY

LAHCEN TARIK, MOHAMED CHERGUI, AND BOUAZZA EL WAHBI

Abstract. In this paper, we present a generalization of Tsallis relative operator entropy defined

for positive operators and we investigate some related properties. Some inequalities involving the

generalized Tsallis relative operator entropy are pointed out as well.

1. Introduction

The relative entropy plays an important role in many areas. In the classical information theory, it

serves as a notion to measure the difference between two probability distributions. For two discrete

probability distributions P = (p1,p2, . . . ,pn) and Q = (q1,q2, . . . ,qn), the relative entropy H(P |Q) is
defined as follows [12]

H(P|Q) =
i=n∑
i=1

pi log
pi
qi
.

For Q = ( 1
n
, 1
n
, . . . , 1

n
), we get

H(P |Q) = log n−Ss(P),

where Ss(P) := −
i=n∑
i=1

pi log pi stands for the famous Shannon entropy. It represents a fundamental tool

that caused an enormous change in studying many fields like physical quantum systems and modern

communication.

In [11], the authors provided a generalization for the entropy concept redefined by Tsallis in [17].

Namely, for a discrete probability distribution P of a random variable the Tsallis entropy is defined as

follows

Tq(P) ≡−
i=n∑
i=1

p
q
i logq

(
pi
)
,

where logq refers to the q-logarithmic function defined by the following formula

logq(x) =
x1−q − 1

1 −q
,

for any nonnegative real numbers x and q 6= 1.
Given the growing diffusion of the use of entropy, many studies have been interested in generalizing

this notion to positive operators. To give an overview, let us start by recalling some notions and fixing

some notations that will be used in the rest of this article.

Let H be a complex Hilbert space endowed with an inner product 〈., .〉. B(H) will stand for the C∗-
algebra of all bounded linear operators acting on H. An operator A ∈ B(H) is called positive, in brief
A ≥ 0, if A is selfadjoint and 〈Ax,x〉 ≥ 0 for all x ∈ H. We denote by B+(H) the closed cone of all

Received by the editors 23 October 2022; accepted 10 February 2023; published online 20 March 202.

2010 Mathematics Subject Classification. Primary 54C70, 94A17, 47A63.

Key words and phrases. Tsallis relative operator entropy, generalized Tsallis relative operator entropy.

1

https://ojs.lib.uwo.ca/mase
https://dx.doi.org/https://doi.org/10.5206/mase/15397


2 L. TARIK, M. CHERGUI, AND B. EL WAHBI

positive operators in B(H) and B+∗(H) the open cone of all positive invertible operators in B(H). For
A,B ∈B(H) selfadjoint, we set A ≤ B to mean that B −A ∈B+(H).
The relative operator entropy S(A|B) was introduced by Fujii and Kamei in [3] by the following ex-
pression

S(A|B) = A
1
2

(
log A−

1
2 BA−

1
2

)
A

1
2 .

Yanagi et al. provided in [19] a parameterized generalization of the operator S(A|B). The authors
introduced the Tsallis relative operator entropy Tp(A | B) for two operators A,B ∈ B+∗(H) and
p ∈ (0, 1] as follows

Tp(A | B) =
A]p B −A

p
. (1.1)

The generalization is to be understood by the following result

lim
p→0

Tp(A|B) = S(A|B).

In (1.1), A]p B stands for the well known as p-power mean defined as follows

A]pB := A
1
2

(
A−

1
2 BA−

1
2

)p
A

1
2 .

We recall also the operator means p-weighted arithmetic and p-weighted harmonic, defined for any

A,B ∈B+∗(H) respectively by [2, 13, 16]

A∇p B := (1 −p)A + pB and A!p B :=
{

(1 −p)A−1 + pB−1
}−1

,

where p ∈ [0, 1]. When p = 1
2

the subscript p will be omitted from the above notations.

In quantum systems, Tp(A|B) is an operator variant of the Tsallis entropy [1, 9]. Many nice properties
of Tp(A|B) can be found for example in [4, 5, 6, 7].
In [16], Räıssouli et al provided an integral representation of Tp(A|B) for p ∈ (0, 1] as follows

Tp(A|B) =
sin pπ

pπ

∫ 1
0

(
t

1 − t

)p (
A!t B −A

t

)
dt. (1.2)

The definition of p-weighted geometric operator mean A\p B for any p ∈ R by the following formula
[10]

A\pB := A
1
2

(
A−

1
2 BA−

1
2

)p
A

1
2 ,

allows to generalize (1.1) for any p ∈ R∗ by setting,

Tp(A|B) =
A\p B −A

p
.

In [9], the authors proved that for any unitary operator U ∈B+∗(H),

Tp(U
∗AU|U∗BU) = U∗Tp(A|B)U. (1.3)

Fujii et Kamei [3] and [20] provided the following upper and lower bounds of Tsallis relative operator

entropy,

A−AB−1A ≤ T−p(A|B) ≤ S(A|B) ≤ Tp(A|B) ≤ B −A.



GENERALIZED TSALLIS RELATIVE OPERATOR ENTROPY 3

By using the Hermite-Hadamard’s inequality, Moradi et al. established in [15] for any p ∈ (0, 1] the
following results,

A
1
2

(
A−

1
2 BA−

1
2 + IH

2

)p−1 (
A−

1
2 BA−

1
2 − IH

)
A

1
2 6 Tp(A | B)

6
1

2
(A]pB −A\p−1B + B −A) . (1.4)

From (1.4), the following inequalities can be deduced [9, 20]

A−AB−1A ≤ Tp(A|B) ≤ B −A.

The present work is centered on generalizing the operator Tp(A|B) and on investigating some related
properties. Some facets of our generalization are also highlighted.

The rest of this paper is organized as follows. In Section 2, we present a generalization for the Tsallis

relative operator entropy and we determine some of its properties. In section 3, after establishing some

Hermite-Hadamard type inequalities, we provide some inequalities which involve the generalized Tsallis

relative operator.

2. Generalized Tsallis Relative Operator Entropy

We begin this section by providing the definition of a generalized Tsallis relative operator entropy.

Then, we will deal with establishing some related properties.

Definition 2.1. Let A,B ∈ B+∗(H), p ∈ R∗ and 0 ≤ ν,µ ≤ 1. We define the generalized Tsallis
relative operator entropy T(p,µ,ν)(A|B) by

T(p,µ,ν)(A|B) =
A\p(A∇µB) −A\p(A∇νB)

p
. (2.1)

For the particular case µ = 1 and ν = 0, we get T(p,1,0)(A | B) = Tp(A | B). This confirms that
T(p,µ,ν)(A|B) represents effectively a generalization of Tp(A|B).

Our first result concerning the generalized Tsallis relative operator entropy T(p,µ,ν)(A|B) reads as
follows.

Proposition 2.1. Let A,B ∈B+∗(H), p ∈ R∗ and 0 ≤ ν,µ ≤ 1. We have

lim
p→0

T(p,µ,ν)(A|B) = S (A|A∇µ B) −S (A|A∇ν B) .

Proof. Noticing that

T(p,µ,ν)(A|B) =
A\p(A∇µB) −A

p
−
A\p(A∇νB) −A

p
,

the desired result can be deduced. �

Proposition 2.2. Let A,B ∈B+∗(H), p ∈ R∗ and 0 ≤ ν,µ ≤ 1. We have
i) T(p,µ,ν)(U∗AU|U∗BU) = U∗T(p,µ,ν)(A|B)U, for any unitary operator U ∈B+∗(H).
ii) T(p,µ,ν) is homogenous, i.e.

T(p,µ,ν)(αA|αB) = αT(p,µ,ν)(A|B) for any α > 0.

Proof. Let us notice at first that

T(p,µ,ν)(A|B) = Tp(A|A∇µB) −Tp(A|A∇νB). (2.2)



4 L. TARIK, M. CHERGUI, AND B. EL WAHBI

So,

T(p,µ,ν)(U∗AU|U∗BU) = Tp(U∗AU|(U∗AU)∇µ(U∗BU)) −Tp(U∗AU|(U∗AU)∇ν(U∗BU)).

Combining the formulas (2.2) and (1.3), we get

T(p,µ,ν)(U∗AU|U∗BU) = Tp(U∗AU|U∗(A∇µB)U) −Tp(U∗AU|U∗(A∇νB)U)
= U∗ (Tp(A|A∇µB) −Tp(A|A∇νB)) U
= U∗T(p,µ,ν)(A|B)U.

Using (2.2) and the fact that the operator mean ∇ and Tp are homogenous, one can easily deduce the
second assertion in the proposition. �

The following result provides an integral representation for the generalized Tsallis relative operator

entropy.

Theorem 2.3. Let A,B ∈B+∗(H). For any p ∈ R\{−1} and a,b ∈ [0, 1] with a < b, we have

T(p+1,b,a)(A | B) =
∫ b
a

[A\p(A∇t B)]
(
A−1B − IH

)
dt. (2.3)

Proof. Let us note that for all x ∈ R+, the following formula holds∫ b
a

(1 − t + tx)p(x− 1)dt =
(1 − b + bx)p+1 − (1 −a + ax)p+1

p + 1
.

So, by theory of functional calculus and substituting x by A
−1
2 BA

−1
2 , we get∫ b

a

(
(1 − t)IH + tA−

1
2 BA−

1
2

)p (
A−

1
2 BA−

1
2 − IH

)
dt =(

(1 − b)IH + bA−
1
2 BA−

1
2

)p+1
−
(

(1 −a)IH + aA−
1
2 BA−

1
2

)p+1
p + 1

. (2.4)

Noticing that A−
1
2 BA−

1
2 − IH = A

1
2

(
A−1BA−

1
2 −A−

1
2

)
, the formula (2.4) is equivalent to

∫ b
a

(
A−

1
2 ((1 − t)A + tB)A−

1
2

)p
A

1
2

(
A−1BA−

1
2 −A−

1
2

)
dt =(

A−
1
2 ((1 − b)A + bB)A−

1
2

)p+1
−
(
A−

1
2 ((1 −a)A + aB)A−

1
2

)p+1
p + 1

,

which leads to (2.3) by multiplying on its both sides by A
1
2 . �

Remark 2.1. For a = 0,b = 1 and p ∈ R∗, we obtain

Tp(A | B) =
∫ 1
0

[A\p−1(A∇t B)]
(
A−1B − IH

)
dt. (2.5)

It is worth mentioning that the formula (2.5) provides an integral representation for Tp(A | B) more
general than the one given by (1.2) stated by the authors in [16, Definition 3.1] only for parameters

p ∈ (0, 1) .

The following results in the ongoing section deal with the monotonicity of T(p,µ,ν)(A|B) according
to each of the parameters µ,ν and p.

Proposition 2.4. Let A,B ∈B+∗(H), p 6= 0 and µ,ν ∈ [0, 1]. We have



GENERALIZED TSALLIS RELATIVE OPERATOR ENTROPY 5

i) If A ≤ B (A ≥ B) then T(p,µ1,ν)(A|B) ≤ (≥)T(p,µ2,ν)(A|B) for ν ≤ µ1 ≤ µ2 ≤ 1.
ii) If A ≥ B (A ≤ B) then T(p,µ,ν1)(A|B) ≤ (≥)T(p,µ,ν2)(A|B) for 0 ≤ ν1 ≤ ν2 ≤ µ.

Proof. Let µ,ν ∈ [0, 1] and x ≥ 0.

If x ≥ 1 (0 < x ≤ 1), the function t 7→
(1 − t + tx)p − (1 −ν + νx)p

p
is increasing (decreasing) on [ν, 1].

Thus, for ν ≤ µ1 ≤ µ2 ≤ 1 it holds
(1 −µ1 + µ1x)p − (1 −ν + νx)p

p
≤ (≥)

(1 −µ2 + µ2x)p − (1 −ν + νx)p

p
.

So, by theory of functional calculus, after replacing x by A−
1
2 BA−

1
2 and multiplying left and right by

A
1
2 , we get

T(p,µ1,ν)(A|B) ≤ (≥) T(p,µ2,ν)(A|B).

The proof of the second statement can be done in a similar way to that of i). �

To study he monotonicity of the map p 7−→ T(p,µ,ν)(A|B), we need the following lemma.

Lemma 2.5. Let 0 ≤ ν ≤ µ ≤ 1, p ∈ R and x > 0. We have

(p log(1 −µ + µx) − 1)(1 −µ + µx)p − (p log(1 −ν + νx) − 1)(1 −ν + νx)p ≥ 0.

Proof. We define on (0,∞) the real function by fp(t) = (p log(t)−1)tp and we put α = 1−µ + µx and
β = 1 −ν + νx.
If α = β, that is µ = ν or x = 1, the desired result is obvious. So, let us consider α 6= β.
If 0 < x < 1 then by noticing that α−β = (µ−ν)(x− 1), we get 0 < α < β < 1.
Using the fact that fp is continuous on [α,β] and differentiable on (α,β), we can deduce by virtue of

Lagrange’s mean value theorem that there exists c ∈ (α,β) such that

fp(α) −fp(β) = (α−β)f
′

p(c),

or equivalently

(p log(α) − 1)αp − (p log(β) − 1)βp = p2cp−1(α−β) log c.

Since c ∈ (α,β) then log c < 0 and consequently

p2cp−1(α−β) log c ≥ 0.

If x ≥ 1, one can follow similar steps used for the previous case. Whence, the lemma is proved. �

Theorem 2.6. Let A,B ∈B+∗(H), 0 ≤ ν ≤ µ ≤ 1 and p,q ∈ R∗ with p ≤ q. We have

T(p,µ,ν)(A|B) ≤ T(q,µ,ν)(A|B). (2.6)

Proof. Let us put for x > 0 and p ∈ R∗

α = 1 −µ + µx, β = 1 −ν + νx and φµ,ν,x(p) =
αp −βp

p
.

We have,
d

dp
φµ,ν,x(p) =

(p log(α) − 1)αp − (p log(β) − 1)βp

p2
.

By Lemma 2.5, we have
d

dp
φµ,ν,x(p) ≥ 0. That is p 7−→ φµ,ν,x(p) is increasing on R∗.

So, if p ≤ q then
φ(p,µ,ν,x) ≤ φ(q,µ,ν,x),



6 L. TARIK, M. CHERGUI, AND B. EL WAHBI

which implies, by virtue of theory of functional calculus and after substituting x by A−
1
2 BA−

1
2 , the

following inequality

(A−
1
2 (A∇µ B) A−

1
2 )p − (A−

1
2 (A∇ν B) A−

1
2 )p

p
≤

(A−
1
2 (A∇µ B) A−

1
2 )q − (A−

1
2 (A∇ν B) A−

1
2 )q

q
.

Multiplying this last inequality by A
1
2 , we get (2.6). �

Remark 2.2. For µ = 1,ν = 0 and p,q ∈ [−1, 0)∪(0, 1] with p ≤ q, we get the well known inequality [8]

Tp(A|B) ≤ Tq(A|B).

This confirms, once more again, the generalization character of T(p,µ,ν)(A|B).

3. Inequalities involving T(p,µ,ν)(A|B)

In the current section, we aim to determine some estimations for T(p,µ,ν)(A|B). Our first result is
recited in the following proposition.

Proposition 3.1. Let A,B ∈B(H)+∗, p > 0 and 0 ≤ ν,µ ≤ 1. We have[
A\p (A∇ν B)

]
A−1 T(−p,µ,ν)(A|B) ≤ S(A|A∇µB) −S(A|A∇νB) ≤[

A\−p (A∇ν B)
]
A−1 T(p,µ,ν)(A|B). (3.1)

If p < 0, the inequalities (3.1) are reversed.

Proof. For p > 0 and y > 0, one can easily check by routine tools of real analysis that

y−p − 1
−p

≤ log y ≤
yp − 1
p

.

So, by setting y =
1 −µ + µx
1 −ν + νx

> 0 for x > 0, we obtain

(1 −ν + νx)p
(1 −µ + µx)−p − (1 −ν + νx)−p

−p
≤

log(1 −µ + µx) − log(1 −ν + νx) ≤

(1 −ν + νx)−p
(1 −µ + µx)p − (1 −ν + νx)p

p
. (3.2)

Thus, by theory of functional calculus, substitution of x by A−
1
2 BA−

1
2 and the following formula(

1 −ν + νA−
1
2 BA−

1
2

)p
= A−

1
2 [A\p (A∇νB)] A−

1
2 ,

allow us to state

(
1 −ν + νA−

1
2 BA−

1
2

)p (1 −µ + µA−12 BA−12 )−p −(1 −ν + νA−12 BA−12 )−p
−p

≤

log
(

1 −µ + µA−
1
2 BA−

1
2

)
− log

(
1 −ν + νA−

1
2 BA−

1
2

)
≤

(
1 −ν + νA−

1
2 BA−

1
2

)−p (1 −µ + µA−12 BA−12 )p −(1 −ν + νA−12 BA−12 )p
p

. (3.3)

Multiplying both sides of inequalities (3.3) by A
1
2 , we deduce the inequalities (3.1).

If p < 0, we apply the inequalities (3.1) for −p > 0 to deduce the desired result. �



GENERALIZED TSALLIS RELATIVE OPERATOR ENTROPY 7

Remark 3.1. It is worth mentioning that by taking µ = 1,ν = 0 and p ∈ (0, 1] in (3.1), we get
particularly the inequalities

T−p(A|B) ≤ S(A|B) ≤ Tp(A|B),

established by Furuichi et al in [9, Proposition 3.1].

Proposition 3.2. Let A,B ∈B+∗(H), q > 0, p ∈ [−q,q] and 0 ≤ ν ≤ µ ≤ 1. We have

mp,q A−
[
A\q(A∇ν B)

]
A−1

[
A\−q(A∇µ B)

]
≤[

A\−p(A∇ν B)
]
A−1T(p,µ,ν)(A|B) ≤[

A\−q(A∇ν B)
]
A−1

[
A\q(A∇µ B)

]
+ np,q A, (3.4)

with, mp,q :=
q

p
p+q −1
p

+ q
−q
p+q and np,q :=

q
p

p−q − 1
p

−q
q

p−q .

Proof. For −q ≤ p ≤ q and y > 0, a simple study leads to the following inequalities

q
p

p+q − 1
p

+ q
−q
p+q −y−q ≤

yp − 1
p

≤ yq +
q

p
p−q − 1
p

−q
q

p−q .

Whence, choosing y =
1 −µ + µx
1 −ν + νx

> 0 for x > 0, we deduce

q
p

p+q − 1
p

+ q
−q
p+q − (1 −ν + νx)q(1 −µ + µx)−q

≤ (1 −ν + νx)−p
(1 −µ + µx)p − (1 −ν + νx)p

p

≤ (1 −ν + νx)−q(1 −µ + µx)q +
q

p
p−q − 1
p

−q
q

p−q . (3.5)

Changing x by A−
1
2 BA−

1
2 and multiplying both sides of inequalities (3.5) by A

1
2 , we deduce the

inequalities (3.4). �

Remark 3.2. Proposition 3.2 provides a generalization of the Proposition 3.4 stated in [9]. In fact,

taking q = µ = 1,ν = 0 and p ∈ (0, 1] in (3.4), we get for all A,B ∈B+∗(H)

A−AB−1A ≤ Tp(A|B) ≤ B −A.

For further results, the following Hermite-Hadamard type inequalities will be very useful.

Theorem 3.3. Let f be a convex function on an open interval I ⊆ R. For all [x,y] ⊆ I and for each
λ ∈ [0, 1], we have

2µf ((1 −λ)x + λy) ≤
∫ λ+µ
λ−µ

f((1 − t)x + ty)dt ≤ 2µ [(1 −λ)f(x) + λf(y)] , (3.6)

where µ = min{λ, 1 −λ}.
If f is a concave function on I then the inequalities in (3.6) are reversed.

Proof. If f is a convex function on I, we get

f ((1 −λ)x + λy) ≤
1

2

[
f ((1 −λ− t)x + (λ + t)y) + f ((1 −λ + t)x + (λ− t)y)

]
≤ (1 −λ)f(x) + λf(y).



8 L. TARIK, M. CHERGUI, AND B. EL WAHBI

Integrating this inequalities over t ∈ [0,µ], we obtain

µf ((1 −λ)x + λy) ≤
1

2

∫ µ
0

f ((1 −λ− t)x + (λ + t)y) dt+

1

2

∫ µ
0

f ((1 −λ + t)x + (λ− t)y) dt ≤ µ ((1 −λ)f(x) + λf(y)) .

Using appropriately the changes of the variables u = λ + t and u = λ− t, it yields

µf ((1 −λ)x + λy) ≤
1

2

∫ λ+µ
λ−µ

f ((1 − t)x + ty) dt ≤ µ ((1 −λ)f(x) + λf(y)) ,

which ends the proof. �

Remark 3.3. For µ = λ =
1

2
, we find the following well known Hermite-Hadamard inequalities

f

(
a + b

2

)
≤

1

b−a

∫ b
a

f(t)dt ≤
f(a) + f(b)

2
. (3.7)

It is important to note that the inequalities (3.7) could also be deduced from the generalization pointed

out separately in [14] and [18] which differs from the one stated in (3.6). An investigation on the

extension of (3.6) to positive operators and the establishment of some applications will be an interesting

topic for future work.

The following theorem will be very useful to establish some inequalities involving generalized Tsallis

relative operator entropy.

Theorem 3.4. Let A,B ∈B+∗(H) and λ ∈ [0, 1]. For all p < 0 with p 6= −1 or p > 1, we have

2µ (A\p(A∇λ B)) ≤
∫ λ+µ
λ−µ

A\p (A∇t B)dt ≤ 2µ [A∇λ(A\p B)] , (3.8)

where µ = min{λ, 1 −λ}.
If p ∈ [0, 1], the inequalities in (3.8) are reversed.

Proof. Consider on (0, +∞) the function defined by f(t) = tp, p ∈ (−∞, 0) ∪ (1, +∞).
Since f is a convex function on (0, +∞), taking x = 1 in (3.6) we get the following inequalities

2µ (1 −λ + λy)p ≤
∫ λ+µ
λ−µ

(1 − t + ty)p dt ≤ 2µ (1 −λ + λyp) . (3.9)

By theory of functional calculus and replacing y by A
−1
2 BA

−1
2 , we have

2µ
(

(1 −λ)IH + λA
−1
2 BA

−1
2

)p
≤
∫ λ+µ
λ−µ

(
(1 − t)IH + tA

−1
2 BA

−1
2

)p
dt

≤ 2µ
[
(1 −λ)IH + λ

(
A
−1
2 BA

−1
2

)p]
. (3.10)

Multiplying both sides of inequalities (3.10) by A
1
2 , we deduce the desired result. �

In the following theorem, we will state another main result.

Theorem 3.5. Let A,B ∈ B+∗(H) be two selfadjoint operators, with A ≤ B. For any λ ∈ (0, 1] and
µ = min{λ, 1 −λ}, we have

2µ

λ
[A\p+1(A∇λ B) −A\p(A∇λ B)] 6 T(p+1,λ+µ,λ−µ)(A | B)

6 2µ [B∇λ(A\p+1 B) −A∇λ(A\p B)] , (3.11)



GENERALIZED TSALLIS RELATIVE OPERATOR ENTROPY 9

for all p < 0 with p 6= −1 or p > 1. If p ∈ [0, 1], the inequalities (3.11) are reversed.

Proof. According to the condition A ≤ B, we can set C = A
−1
2

(
A
−1
2 BA

−1
2 − IH

)1
2

A
1
2 .

For p < 0 with p 6= −1 or p > 1 and by the use of the Theorem 3.4, we get

2µ [A\p(A∇λ B)] 6
∫ λ+µ
λ−µ

[A\p (A∇t B)] dt 6 2µ [A∇λ(A\p B)] .

Thus,

2µC∗ [A\p(A∇λ B)] C 6
∫ λ+µ
λ−µ

C∗ [A\p (A∇t B)] Cdt 6 2µC∗ [A∇λ(A\p B)] C.

By setting D = A
−1
2 BA

−1
2 , we have

C∗ [A\p(A∇λ B)] C = A
1
2 (D − IH)

1
2

(
A
−1
2 (A∇λ B)A

−1
2

)p
(D − IH)

1
2 A

1
2

= A
1
2 (D − IH)

1
2 ((1 −λ)IH + D)

p
(D − IH)

1
2 A

1
2

= A
1
2 ((1 −λ)IH + D)

p
(D − IH) A

1
2

= A
1
2

(
A
−1
2 (A∇λ B)A

−1
2

)p (
A
−1
2 BA

−1
2 − IH

)
A

1
2

= A
1
2

(
A
−1
2 (A∇λ B)A

−1
2

)p
A

1
2 A

−1
2

(
A
−1
2 B −A

1
2

)
= [A\p(A∇λ B)] (A−1B − IH).

Furthermore, noticing that for any λ ∈ (0, 1],

A−1B =
1

λ

[
A
−1
2

(
A
−1
2 (A∇λ B)A

−1
2

)
A

1
2 − (1 −λ)IH

]
,

it yields

C∗ [A\p(A∇λ B)] C =
1

λ
[A\p+1(A∇λ B) − (1 −λ)A\p(A∇λ B)] −A\p(A∇λ B)

=
1

λ
[A\p+1(A∇λ B) −A\p(A∇λ B)] .

On the other hand, we have

C∗ [A∇λ(A\p B)] C = A
1
2 (D − IH)

1
2 [(1 −λ)IH + λDp] (D − IH)

1
2 A

1
2

= A
1
2 (D − IH)

1
2 [(1 −λ)IH + λDp] (D − IH)

1
2 A

1
2

= A
1
2 [(1 −λ)IH + λDp] (D − IH)

1
2 (D − IH)

1
2 A

1
2

= A
1
2 [(1 −λ)IH + λDp] (D − IH) A

1
2

= [A∇λ(A\p B)] (A−1B − IH)

= (1 −λ)B + λ(A\p B)A−1B −A∇λ(A\p B).

Noticing that A−1B = A
−1
2

(
A
−1
2 BA

−1
2

)
A

1
2 , we obtain

C∗ [A∇λ(A\p B)] C = (1 −λ)B + λ(A\p+1 B) −A∇λ(A\p B)
= B∇λ(A\p+1 B) −A∇λ(A\p B).

Finally, by Theorem 2.3 we have∫ λ+µ
λ−µ

C∗ [A\p (A∇t B)] Cdt =
∫ λ+µ
λ−µ

[A\p (A∇t B)] (A−1B − IH)dt = T(p+1,λ+µ,λ−µ).



10 L. TARIK, M. CHERGUI, AND B. EL WAHBI

For p ∈ [0, 1], using the inverses of inequalities (3.8) and following the same steps used for the proof of
(3.11), one can deduce the result. �

Remark 3.4. Inequalities (3.11) provides a generalization for (1.4), in the sense that for p ∈ (0, 1] and
λ = 1

2
in (3.11) we find (1.4).

The following result provides also an extension for the inequalities (1.4).

Corollary 3.6. Let A,B ∈B(H)+∗ with A ≤ B. For any p ∈ R∗\[1, 2], we have

A
1
2


(IH + A−12 BA−12

2

)p
−

(
IH + A

−1
2 BA

−1
2

2

)p−1A12
6 2A

1
2


(IH + A−12 BA−12

2

)p
−

(
IH + A

−1
2 BA

−1
2

2

)p−1A12 6 Tp(A | B)
6

1

2
(A\p B −A\p−1 B + B −A) . (3.12)

Proof. Let p ∈ R∗\[1, 2]. By inequalities (3.11) when taking λ =
1

2
and replacing p by p− 1, we find

2 [A\p(A∇B) −A\p−1(A∇B)] ≤ T(p,1,0)(A | B) ≤ B∇(A\p B) −A∇(A\p−1 B),

or equivalently

2A
1
2

[(
A
−1
2

(
B + A

2

)
A
−1
2

)p
−
(
A
−1
2

(
B + A

2

)
A
−1
2

)p−1]
A

1
2 6 T(p,1,0)(A | B)

6
1

2
(B + A\p B −A−A\p−1 B) .

Using the relation T(p,1,0)(A | B) = Tp(A | B), we deduce the inequalities (3.12). �

Corollary 3.7. Let A,B ∈ B+∗(H) be two selfadjoint operators with A ≤ B. For any λ ∈
(
0, 1

2

]
and

p ∈ R∗ \ [1, 2], we have

2[A\p(A∇λ B) −A\p−1(A∇λ B)] −
2 λ

p
(B −A) 6 Tp(A | A∇2λB)

6 2λ
[
B∇λ(A\p B) −A∇λ(A\p−1 B)

]
−

2 λ

p
(B −A). (3.13)

For p ∈ [1, 2], the inequalities reverse (3.13) are reversed.

Proof. As λ ∈
(
0, 1

2

]
then µ = λ. So, applying the inequalities (3.11) combined with the following

formula

T(p,2λ,0) = Tp (A|A∇2λB) +
2 λ

p
(B −A),

we get (3.13). �

Acknowledgments

The authors would like to thank the anonymous referee(s) for valuable comments and suggestions

that have been implemented in the final version of the paper.



GENERALIZED TSALLIS RELATIVE OPERATOR ENTROPY 11

References

[1] S. Abe, Monotonic decrease of the quantum non-additive divergence by projective measurements, Phys. Lett. A.,

312(5-6) (2003), 336-338.

[2] I. A. Al-Subaihi, M. Räıssouli, Further inequalities involving the weighted geometric operator mean and the Heinz

operator mean, Linear and Multilinear Algebra, (2021), 1-23.

[3] J. I. Fujii, E. Kamei, Relative operator entropy in non-commutative information theory, Math. Japon., 34 (1989),

341-348.

[4] J. I. Fujii and E. Kamei, Uhlmann’s interpolational method for operator means, Math.Japon, 34(1989), 541-547.

[5] J. I. Fujii, M. Fujii and Y. Seo, An extension of the Kubo-Ando theory: Solidarities, Math.Japonica, 35 (1990),

387-396.

[6] J. I. Fujii, Operator means and the relative operator entropy, Operator Theory, Advances and Applications, 59

(1992), 161-172.

[7] S. Furuichi, K. Yanagi and K. Kuriyama, Fundamental properties for Tsallis relative entropy, J.Math.Phys.,

45(2004),4868-4877.

[8] S. Furuichi, N. Minculete, Inequalities for Relative Operator Entropies and Operator Means, Acta Math Vietnam 43

(2018), 607-618.

[9] S. Furuichi, K. Yanagi and K. Kuriyama, A note on operator inequalities of Tsallis relative operator entropy, Linear

Algebra Appl., 407 (2005), 19-31.

[10] T. Furuta, Parametric extensions of Shannon inequality and its reverse one in Hilbert space operators, Linear Algebra

Appl., 381 (2004), 219-235.

[11] J. Havrda and F. Charvat, Quantification method of classification processes: concept of structural a-entropy, Kyber-

netika, 3(1)(1967), 30-35.

[12] S. Kullback, Information Theory and Statistics, Wiley, New York, 1959.

[13] W. Liao, J. Wu, Reverse arithmetic-harmonic mean and mixed mean operator inequalities, J. Inequal. Appl., 215

(2015).

[14] A. Lupaş, A generalisation of Hadamard’s inequality for convex functions, Univ. Beograd. Publ. Elek. Fak. Ser. Mat.

Fiz. 544/576 (1976), 115-121.

[15] H. R. Moradi, S. Furuichi and N. Minculete, Estimates for Tsallis relative operator entropy, Math Inequal Appl.,

20/4 (2017), 1079-1088.

[16] M. Räıssouli, M. S. Moslehian and S. Furuichi, Relative entropy and Tsallis entropy of two accretive operators, C.

R. Acad. Sci. Paris, Ser. I 355 (2017) 687-693.

[17] C.Tsallis, Possible generalization of Boltzman-Gibbs statistics, J.Stat. Phys., 52(1988),479-487.

[18] P. M. Vasić and I. B. Lacković, Some complements to the paper:On an inequality for convex functions, Univ. Beograd

Publ. Elek. Fak., Ser. Mat. Fiz. 544/576 (1976), 59-62.

[19] K. Yanagi, K. Kuriyamaa and S. Furuichi, Generalized Shannon inequalities based on Tsallis relative operator entropy,

Linear Algebra and its Applications, 394 (2005), 109-118.

[20] L. Zou, Operator inequalities associated with Tsallis relative operator entropy, Math. Inequal. Appl., 18/2 (2015),

401-406.

Corresponding author, Department of Mathematics, Science Faculty, LAGA-Lab Ibn Tofail University,

Kenitra, Morocco

Email address: lahcen.tarik@uit.ac.ma

Department of Mathematics, CRMEF-RSK, EREAM Team, LaREAMI-Lab, Kenitra, Morocco

Email address: chergui m@yahoo.fr

Department of Mathematics, Science Faculty, LAGA-Lab Ibn Tofail University, Kenitra, Morocco

Email address: bouazza.elwahbi@uit.ac.ma


	1. Introduction
	2. Generalized Tsallis Relative Operator Entropy
	3. Inequalities involving T(p, , )(A|B)
	Acknowledgments
	References