EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5586
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

New Improvements to Heron and Heinz Inequality
Using Matrix Techniques

M.H.M Rashid1,∗, Wael Mahmoud Mohammad Salameh2

1 Department of Mathematics, Faculty of Science P.O.Box(7), Mutah University, Al-Karak,
Jordan
2 Faculty of Information Technology, Abu Dhabi University, Abu Dhabi 59911,
United Arab Emirates

Abstract. This paper presents a comprehensive study on matrix means interpolation and com-
parison, extending the parameter ϑ from the traditional closed interval [0, 1] to encompass the
entire positive real line, denoted as R+. The research delves into further results involving Heinz
means, proposing novel scalar adaptations of Heinz inequalities that integrate Kantorovich’s con-
stant. Additionally, the operator version of these inequalities is strengthened. A key contribution
of this work is the development of refined Young’s type inequalities tailored for the traces, deter-
minants, and norms of positive semi-definite matrices. These refinements offer deeper insights into
matrix analysis, especially in the context of operator theory and inequality theory. Through these
advancements, the paper enhances the mathematical framework for studying matrix means and
their associated inequalities, providing useful tools for both theoretical exploration and practical
applications in linear algebra and related fields.

2020 Mathematics Subject Classifications: 26D07, 26D15, 15A18, 47A63

Key Words and Phrases: Heinz mean inequalities, positive semi-definite matrices, Hilbert-
Schmidt norm, Young inequality

1. Introduction

Consider the algebra of complex matrices of size n× n, denoted as Mn(C). A matrix
T in Mn(C) is considered positive semi-definite, written as T ≥ 0, if it is Hermitian and
satisfies ⟨Tx, x⟩ ≥ 0 for all vectors x in Cn. If, for a Hermitian matrix T in Mn(C),
⟨Tx, x⟩ > 0 holds for all nonzero vectors x in Cn, it is termed a positive definite matrix,
denoted as T > 0. The set of all positive matrices is denoted as M+

n (C), and the subset
of definite matrices within M+

n (C) is represented as M++
n (C). The Schur product of two

matrices T = [tij ]i,j and S = [sij ]i,j in Mn(C) is defined as the matrix T ◦ S with entries

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5586

Email addresses: malik okasha@yahoo.com (M.H.M Rashid),
wael.salameh1@adu.ac.ae (W.M.M. Salameh)

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



M.H.M Rashid, W.M.M. Salameh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5586 2 of 21

tijsij . A norm |||.||| on the set of complex matrices of size n × n, denoted as Mn(C), is
termed unitarily invariant if |||UAV ||| = |||T ||| for any matrix T inMn(C) and for all unitary
matrices U and V in Mn(C).

For a matrix T = [tij ] ∈Mn(C), the Hilbert-Schmidt norm (also known as the Frobe-
nius norm) and the trace norm of T are defined as follows

∥T∥2 =

 n∑
j=1

s2j (T )

 1
2

, ∥T∥1 = tr(|T |) =
n∑

j=1

sj(T ) (1)

Here, s1(T ) ≥ s2(T ) ≥ · · · ≥ sn(T ) ≥ 0 represent the singular values of T , which are the
eigenvalues of the positive matrix |T | =

√
T ∗T arranged in decreasing order and repeated

according to multiplicity. The symbol tr(.) denotes the usual trace operation.
It’s important to note that the mathematical norms ∥·∥2 and ∥·∥1 are widely recognized

for being unitarily invariant.
The classic Young’s inequality for non-negative real numbers states that if ρ, σ ≥ 0

and 0 ≤ κ ≤ 1, then
ρκσ1−κ ≤ κρ+ (1− κ)σ (2)

Equality occurs if and only if ρ = σ. When κ is 1
2 , substituting into the inequality yields

the arithmetic-geometric mean inequality

√
ρσ ≤ ρ+ σ

2
. (3)

Manasarah and Kittaneh, as presented in [10], improved Young’s inequality with the
following refinement(

ρκσ1−κ
)m

+ rm0

(
ρ

m
2 − σ

m
2

)2
≤ (κρr + (1− κ)σr)

m
r , r ≥ 1 (4)

where m ∈ N and r0 = min{κ, 1− κ}.
The Kantorovich constant, denoted as K(t, 2), is defined as (t+1)2

4t . It possesses several
key properties: K(1, 2) = 1, K(t, 2) = K

(
1
t , 2
)
≥ 1 (t > 0) and K(t, 2) is monotone

increasing on [1,∞), and monotone decreasing on (0, 1]. For more detailed information
about the Kantorovich constant, interested readers can refer to [11, 15, 17, 22]..
The following multiplicative refinement and reversal of Young’s inequality, expressed in
terms of Kantorovich’s constant, can be stated as follows

K(h, 2)rρ♯κσ ≤ ρ∇κσ ≤ K(h, 2)Rρ♯κσ, (5)

where ρ and σ are both greater than 0, κ belongs to the interval [0, 1], r is the minimum
of κ and 1− κ, R is the maximum of κ and 1− κ, and h is defined as σ

ρ .
The second inequality in (5) is credited to Liao et al. [12], while the first one is

attributed to Zou et al. [11]. In [19], the authors obtained another improvement of the
Young inequality and its reverse as follows:

r(
√
ρ−

√
σ)2 +K(

√
h, 2)r

′
ρ♯κσ ≤ ρ∇κσ, (6)



M.H.M Rashid, W.M.M. Salameh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5586 3 of 21

and
ρ∇κσ ≤ K(

√
h, 2)−r′ρ♯κσ +R(

√
ρ−

√
σ)2 (7)

where h = σ
ρ , r = min{κ, 1 − κ}, R = max{κ, 1 − κ} and r′ = min{2r, 1 − 2r}. In

addition, another kind of the reversal of Young inequality utilizing Kantorovich’s constant
is described in [12] with the same notation as above.

ρ∇κσ −R(
√
ρ−

√
σ)2 ≤ K(

√
h, 2)R

′
ρ♯κσ, (8)

where R′ = max{2r, 1− 2r}.
For κ in the range of [0, 1] and two non-negative real numbers ρ and σ, the Heinz mean
serves as an interpolation between the κ-arithmetic mean and the κ-geometric mean.
These are defined by the expression

Hκ(ρ, σ) =
ρ♯κσ + ρ♯1−κσ

2
, (9)

where ρ♯κσ = ρκσ1−κ represents the κ-geometric mean.
The Heinz mean possesses certain properties, including convexity concerning κ within

the interval [0, 1]. Its minimum occurs at κ = 1
2 , and its maximum values are found at

κ = 0 and κ = 1. Additionally, the following inequalities are true

√
ρσ ≤ Hκ(ρ, σ) ≤

ρ+ σ

2
. (10)

It is worth noting that the function Hκ(ρ, σ) exhibits symmetry with respect to the point
κ = 1

2 , meaning that Hκ(ρ, σ) = H1−κ(ρ, σ).
The Heron mean is defined by the expression

Fϑ(ρ, σ) = (1− ϑ)
√
ρσ + ϑ

(
ρ+ σ

2

)
, ϑ ∈ [0, 1] and ρ, σ ∈ R+. (11)

where ϑ takes values in the interval [0, 1], and ρ and σ are positive real numbers.
Evidently, the Heron mean serves as a linear interpolation between the arithmetic and

geometric means. It adheres to the inequality Fϑ ≤ Fϱ whenever ϑ ≤ ϱ, with both ϑ and
ϱ belonging to the positive real numbers.

In a study by Bhatia published in [2], it was demonstrated that for ϑ(κ) = (2κ − 1)2

and κ within the range of [0, 1], the following relation holds

Hκ(ρ, σ) ≤ Fϑ(κ)(ρ, σ). (12)

Our paper is structured as follows: In the upcoming section, we will conduct an in-
depth investigation into matrix interpolation and mean comparisons. This analysis extends
the scope of ϑ beyond the closed interval [0, 1] to include all positive real numbers, repre-
sented as R+. Additionally, we will explore additional findings pertaining to Heinz means.
Section three is dedicated to exploring refinements in Heinz inequality, incorporating the
Kantorovich constant. Section 4 focuses on examining enhanced variations of Heinz-type
operator inequalities and their corresponding reversals. Finally, in section 5, we present
refined inequalities of Young’s type, specifically designed for traces, determinants, and
norms of positive semi-definite matrices.



M.H.M Rashid, W.M.M. Salameh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5586 4 of 21

2. Full Interpolation of Matrix Variants of Heron and Heinz MEANS

In the paper referenced as [2], R. Bhatia established a noteworthy result. In particular,
it was shown that for values of ϑ in the interval [0, 1/2], the function ψ(ϑ) adheres to the
inequality ψ(ϑ) ≤ ψ(1/2). Here, ψ(ϑ) denotes one of the potential matrix formulations of
Equation (11), and its definition is as follows

ψ(ϑ) =

∣∣∣∣∣∣∣∣∣∣∣∣(1− ϑ)T 1/2XS1/2 + ϑ

(
TX +XS

2

)∣∣∣∣∣∣∣∣∣∣∣∣, (13)

This definition involves matrices T , S, and X, subject to the conditions that T and S
belong to the set of positive definite matrices in C, denoted as M++

n (C), and X is a
member of the set of n× n matrices over C, denoted as Mn(C).

For further insights into the matrix formulations of Equation (9) and Equation (12),
as well as additional details, interested readers are encouraged to refer to the following
references: [2], [5], [6], [4], and [8].

Within the context of this article, the author endeavors to demonstrate that for ϑ
values within the interval [0, 1/2], the function ψ(ϑ, κ) adheres to the inequality ψ(ϑ, κ) ≤
ψ(1/2, κ). Additionally, it is asserted that ψ(ϑ, κ) displays an increasing trend as ϑ varies
within the range [1/2,∞).

This result serves as a generalization of the previously established monotonic property
associated with the matrix version of Equation (11). This generalization mirrors the
behavior of Fϑ(ρ, σ) for positive real numbers a and b when ϑ belongs to the set of positive
real numbers, R+.

As a consequential outcome of these findings, the author will introduce a potential
generalized matrix equivalent of Equation (12), which can be expressed as follows

1

2

∣∣∣∣∣∣TµXS1−µ + T 1−µXSµ
∣∣∣∣∣∣ ≤ ∣∣∣∣∣∣∣∣∣∣∣∣(1− ϑ)T κXS1−κ + ϑ

(
TX +XS

2

)∣∣∣∣∣∣∣∣∣∣∣∣ (14)

This inequality is valid for particular values of µ ∈ [1/4, 3/4], κ ∈ [0, 1], and ϑ ∈ [1/2,∞).
In this section, we will undertake a thorough investigation of matrix interpolation and

mean comparisons. This scrutiny broadens the range of ϑ from the closed interval [0, 1]
to encompass the entirety of positive real numbers, denoted as R+. Furthermore, we will
delve into additional findings associated with Heinz means.

Theorem 1. [6] Let T, S ∈Mn(C) such that T is a positive semi-definite. Then

|||T ◦ S||| ≤ max
1≤i≤n

tii|||S|||,

where tii for i = 1, 2, · · · , n are the diagonal entries of matrix T .

Lemma 1. [21] Let κ1, κ2, · · · , κn be positive numbers, r ∈ [−1, 1], and t ∈ (−2, 2]. Then
the n× n matrix matrix

Γ =

(
κri + κrj

κ2i + tκiκj + κ2j

)
is positive semi-definite.



M.H.M Rashid, W.M.M. Salameh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5586 5 of 21

Theorem 2. Let T, S,X ∈Mn(C) such that T and S are positive semi-definite, κ ∈ [0, 1]
and |||.||| any unitarily invariant norm, the function

ψ(ϑ, κ) =

∣∣∣∣∣∣∣∣∣∣∣∣(1− ϑ)T κXS1−κ + ϑ

(
TX +XS

2

)∣∣∣∣∣∣∣∣∣∣∣∣
is increasing for 1

2 ≤ ϑ <∞ and ψ(ϑ, κ) ≤ ψ
(
1
2 , κ
)
for all ϑ ∈

[
0, 12
]
.

Proof. We first prove the result for ϑ > 0 and T = S, that is,∣∣∣∣∣∣∣∣∣∣∣∣(1− ϑ)T κXS1−κ + ϑ

(
TX +XS

2

)∣∣∣∣∣∣∣∣∣∣∣∣ = ϑ

2
Z(ϑ),

where Z(ϑ) =
∣∣∣∣∣∣Q(ϑ)T κXT 1−κ + TX +XT

∣∣∣∣∣∣ and Q(ϑ) = 2
(
1
ϑ − 1

)
. We may assume

without loss of generality, T = diag (η1, · · · , ηn), ηj > 0. Then

Q(ϑ)T κXT 1−κ + TX +XT =
((
Q(ϑ)ηκi η

1−κ
j + ηi + ηj

)
xij

)
i,j

=

(
Q(ϑ)ηκi η

1−κ
j + ηi + ηj

Q(ϱ)ηκi η
1−κ
j + ηi + ηj

)
i,j

◦
(
Q(ϱ)T κXT 1−κ + TX +XT

)
= E ◦

(
Q(ϱ)T κXT 1−κ + TX +XT

)
,

where E =

(
Q(ϑ)ηκi η

1−κ
j +ηi+ηj

Q(ϱ)ηκi η
1−κ
j +ηi+ηj

)
i,j

. Now the matrix E can be written as

(
1 +

(Q(ϑ)−Q(ϱ))ηκi η
1−κ
j

Q(ϱ)ηκi η
1−κ
j + ηi + ηj

)
= (1)i,j +

(
ηκi

(
Q(ϑ)−Q(ϱ)

Q(ϱ)ηκi η
1−κ
j + ηi + ηj

)
η1−κ
j

)
which will be positive semidefinite if the matrix,

G =

(
Q(ϑ)−Q(ϱ)

Q(ϱ)ηκi η
1−κ
j + ηi + ηj

)
i,j

is positive semidefinite. According to Lemma 1, the latter matrix is positive semidefinite
if and only if Q(ϑ) ≥ Q(ϱ) and Q(ϱ) ∈ [−2, 2]. Since Q(ϑ) = 2

(
1
ϑ − 1

)
is a continuous and

decreasing function on the positive half-line, ranging from [12 ,∞) into [−2, 2], it follows
that Q(ϑ) ≥ Q(ϱ) for all ϱ ≥ ϑ. Consequently, using Theorem 1, we can deduce that

Z(ϑ) ≤
(
Q(ϑ)+2
Q(ϱ)+2

)
T (ϱ). Thus, the result holds for T = S and ϑ ≥ 1

2 .

For ϑ ∈ (0, 1/2], we have 2 ≤ Q(ϑ) <∞, and Q(ϑ) > Q
(
1
2

)
= 2. Therefore, the matrix

E with ϱ = 1
2 is positive semidefinite, as per Lemma 1. The case ϑ = 0 is straightforward

since, by Lemma 1, the matrix(
ηκi η

1−κ
j

ηκi η
1−κ
j + ηi + ηj

)
i,j

=

(
ηκi

(
1

ηκi η
1−κ
j + ηi + ηj

)
η1−κ
j

)
i,j



M.H.M Rashid, W.M.M. Salameh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5586 6 of 21

is positive semidefinite. Thus, we have established the desired result for this case, i.e.,
ϑZ(ϑ) ≤ 1

2Z
(
1
2

)
. In other words, ψ(ϑ, κ) ≤ ψ

(
1
2 , κ
)
for all ϑ ∈ [0, 1/2]. The general case

can be derived by substituting T with

(
T 0
0 S

)
and X by

(
X 0
0 0

)
.

Remark 1. By setting κ to be equal to half (i.e., κ = 1
2) in Theorem 2, we can deduce

that we arrive at Theorem 2.3 as presented in [8]. Consequently, our findings represent
an enhancement of the results established in that theorem.

As a consequence of Theorem 2, we have

Corollary 1. Let T, S,X ∈ Mn(C) with T, S positive definite. Then for any unitarily
invariant norm |||·||| and a matrix monotone increasing function ψ : (0,∞) −→ (0,∞) with
ψ∗(x) = x(ψ(x))−1,

1

2

∣∣∣∣∣∣∣∣∣T µ
2 (ψ(Tµ)Xψ∗(Sµ) + ψ∗(Tµ)Xψ(Sµ))S

µ
2

∣∣∣∣∣∣∣∣∣
≤
∣∣∣∣∣∣∣∣∣∣∣∣(1− ϑ)T κXS1−κ + ϑ

(
TX +XS

2

)∣∣∣∣∣∣∣∣∣∣∣∣.
Corollary 2. Let T, S,X ∈ Mn(C) with T, S positive definite. Then for any unitarily
invariant norm |||·|||, 1

4 ≤ µ ≤ 3
4 , κ ∈ [0, 1] and ϑ ∈ [1/2,∞),

1

2

∣∣∣∣∣∣TµXS1−µ + T 1−µXSµ
∣∣∣∣∣∣ ≤ ∣∣∣∣∣∣∣∣∣∣∣∣(1− ϑ)T κXS1−κ + ϑ

(
TX +XS

2

)∣∣∣∣∣∣∣∣∣∣∣∣.
Proof. Letting ψ(x) =

√
x in Corollary 1, we derived the result.

The following result is a consequence of Theorem 2.

Corollary 3. Let T, S,X ∈ Mn(C) with T, S positive definite, η = min{sp(T ), sp(S)},
µ ∈ [1/4, 3/4] and κ ∈ [0, 1]. Then for any unitarily invariant norm |||·||| and a matrix
monotone increasing function ψ : (0,∞) −→ (0,∞)

η

2f(η)

∣∣∣∣∣∣∣∣∣T µ
2 (ψ(Tµ)X +Xψ(Sµ))S

µ
2

∣∣∣∣∣∣∣∣∣
≤
∣∣∣∣∣∣∣∣∣∣∣∣(1− ϑ)T κXS1−κ + ϑ

(
TX +XS

2

)∣∣∣∣∣∣∣∣∣∣∣∣
holds for every ϑ ∈ [1/2,∞).

Choosing ψ(x) = log(1 + x) in Corollary 3, we have

Corollary 4. Let T, S,X ∈ Mn(C) with T, S positive definite, η = min{sp(T ), sp(S)},
µ ∈ [1/4, 3/4] and κ ∈ [0, 1]. Then for any unitarily invariant norm |||·|||

η

2 log(1 + η)

∣∣∣∣∣∣∣∣∣T µ
2 (log(1 + Tµ)X +X log(1 + Sµ))S

µ
2

∣∣∣∣∣∣∣∣∣
≤
∣∣∣∣∣∣∣∣∣∣∣∣(1− ϑ)T κXS1−κ + ϑ

(
TX +XS

2

)∣∣∣∣∣∣∣∣∣∣∣∣
holds for every ϑ ∈ [1/2,∞).



M.H.M Rashid, W.M.M. Salameh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5586 7 of 21

Theorem 3. Consider T, S,X ∈ Mn(C) with T and S being positive definite, κ ∈ [0, 1],
and |||.||| denoting any unitarily invariant norm. The function can be expressed as:

ϕ(ϑ, κ) =

∣∣∣∣∣∣∣∣∣∣∣∣(1− ϑ

2

)(
T κXS1−κ + T 1−κXSκ

)
+ ϑ

(
TX +XS

2

)∣∣∣∣∣∣∣∣∣∣∣∣
is increasing for 1

2 ≤ ϑ <∞ and ϕ(ϑ, κ) ≤ ϕ
(
1
2 , κ
)
for all ϑ ∈

[
0, 12
]
.

Proof. Once again following the same lines of the proof of Theorem (2), we shall prove
the result for ϑ > 0, T = S and T = diag(η1, · · · , ηn). Suppose

ϕ(ϑ, κ) =

∣∣∣∣∣∣∣∣∣∣∣∣(1− ϑ

2

)(
T κXT 1−κ + T 1−κXT κ

)
+ ϑ

(
TX +XT

2

)∣∣∣∣∣∣∣∣∣∣∣∣ = ϑ

2
Z(ϑ, κ),

where Z(ϑ, κ) =
∣∣∣∣∣∣W1(ϑ)

(
T κXT 1−κ + T 1−κXT κ

)
+ (TX +XT )

∣∣∣∣∣∣ and W1(ϑ) =
2
ϑ − 1.

W1(ϑ)
(
T κXT 1−κ + T 1−κXT κ

)
+ (TX +XT )

=
[(
W1(ϑ)

(
ηµi η

1−µ
j + η1−µ

i ηµj

)
+ ηi + ηj

)
xij

]
i,j

= Y ◦
(
W1(ϱ)

(
T κXT 1−κ + T 1−κXT κ

)
+ TX +XS

)
.

Now the matrix Y can be written asW1(ϑ)
(
ηµi η

1−µ
j + η1−µ

i ηµj

)
+ ηi + ηj

W1(ϱ)
(
ηµi η

1−µ
j + η1−µ

i ηµj

)
+ ηi + ηj


i,j

=

(
1 +

(W1(ϑ)−W1(ϱ))η
κ
i η

κ
j

(W1(ϱ)− 1)ηκi η
κ
j + η1−κ

i + η1−κ
j

)
i,j

= (1)i,j +

(
ηκi

(
W1(ϑ)−W1(ϱ)

(W1(ϱ)− 1)ηκi η
κ
j + η1−κ

i + η1−κ
j

)
ηµj

)
i,j

Once again, considering Lemma (1), we observe that the latter matrix is positive semidef-
inite if and only if W1(ϑ) > W1(ϱ) and 2 > W1(ϱ)− 1 > −2. Since W (ϑ) = W1(ϑ)− 1 =
(2ϑ−1 − 2), it is a continuously decreasing function in the positive half-line and maps to
the interval (2, 2] for ϑ in the range [1/2,∞). Therefore, as demonstrated in Theorem (2),
we can deduce that W (ϑ) > W (ϱ) and consequently, W1(ϑ) > W1(ϱ) for all ϑ ≤ ϱ.

Applying Theorem (1), we establish that T (ϑ, κ) ≤ W1(ϑ)+1
W1(ϱ)+1T (ϱ, κ). This verifies the

result for the case when T = S and ϑ ∈ [1/2,∞).
For ϑ ∈ (0, 1/2], we can observe that 3 = W1(1/2) ≤ W1(ϑ) < ∞, and according to

Lemma (1), the matrix Y with ϱ = 1/2 is positive semidefinite. Similarly, the case ϑ = 0
can be established through the positive semidefiniteness of the matrix(

ηκi

(
W1(ϑ)−W1(ϱ)

(W1(ϱ)−1)ηκi η
κ
j +η1−κ

i +η1−κ
j

)
ηµj

)
i,j

, which is confirmed by utilizing Lemma (1).

This leads us to the desired result for this case, i.e., ϑZ(ϑ, κ) ≤ 1
2Z(

1
2 , κ). In other

words, ϕ(ϑ, κ) ≤ ϕ(1/2, κ) holds for all ϑ ∈ [0, 1/2].

The general case can be obtained by substituting T with

(
T 0
0 S

)
and X by

(
X 0
0 0

)
.



M.H.M Rashid, W.M.M. Salameh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5586 8 of 21

The following outcome is an implication of Theorems 2, 3, and Corollary 2, resulting
in:

Corollary 5. Let T, S,X ∈ Mn(C) with T, S positive definite, κ ∈ [0, 1] and ψ(ϑ, κ) and
ϕ(ϑ, κ) are same as taken in Theorem (2) and (3) respectively. Then

ψ(0, κ) ≤ 1

2
ϕ(0, κ) ≤ ψ(ϑ, κ) (15)

for ϑ ∈ [1/2,∞), or equivalently, for any unitarily invariant norm |||·||| and 2 < t ≤ 2,∣∣∣∣∣∣T κXS1−κ
∣∣∣∣∣∣ ≤ 1

2

∣∣∣∣∣∣T κXS1−κ + T 1−κXSκ
∣∣∣∣∣∣ ≤ 1

t+ 2

∣∣∣∣∣∣TX +XS + tT κXS1−κ
∣∣∣∣∣∣.

Remark 2. (i) It’s worth noting that the corollary mentioned earlier (1) represents one
of the potential enhancements to an inequality introduced by Kaur and Singh in their work
(see [8, Corollary 2.4]).
(ii) Take note that when we set κ to the value of one-third (i.e., κ = 1

3), it is evident that
we arrive at the outcome outlined in Theorem 2.10 in [8]. This implies that our finding
constitutes a broader and more generalized version of their result.

3. Sharpening of the Heinz inequalities and its reverses with the
Kantorovich Constant

In this section, we make a refinement of Heinz inequality with the Kantorovich con-
stant.

Lemma 2. Let ρ, σ > 0 and 0 ≤ ν < κ ≤ 1. Then

r(
√
ρ♯κσ −

√
σ)2 +K(

√
h, 2)r

′
ρ♯νσ ≤ νρ+ (1− ν)σ −

(ν
κ

)
(ρ∇κσ − ρ♯κσ), (16)

where r = min{ ν
κ , 1−

ν
κ}, h = ρ

σ and r′ = min{2r, 1− 2r}.

Proof. An simple argument shows that

νρ+ (1− ν)σ − ν

κ
(ρ∇κσ − ρ♯κσ) = νρ+ (1− ν)σ − ν

κ

(
κρ+ (1− κ)σ − ρκσ1−κ

)
=
ν

κ
ρκσ1−κ +

(
1− ν

κ

)
σ = (ρ♯κσ)∇ ν

κ
σ. (17)

By applying the inequality (6) for the relation (17), it follows that

r(
√
ρ♯κσ −

√
σ)2 +K(

√
h, 2)r

′
ρ♯νσ ≤ (ρ♯κσ)∇ ν

κ
σ.

Hence, the inequality (16) follows.



M.H.M Rashid, W.M.M. Salameh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5586 9 of 21

Lemma 3. Let ρ, σ > 0 and 0 ≤ ν < κ ≤ 1. Then

νρ+ (1− ν)σ −
(ν
κ

)
(ρ∇κσ − ρ♯κσ) ≤ K(

√
h, 2)−r′ρ♯νσ +R(

√
ρ♯κσ −

√
σ)2 (18)

where R = max{ ν
κ , 1−

ν
κ}, h = ρ

σ and r′ = min{2r, 1− 2r}.

Proof. By applying the inequality (7) for the relation (17), it follows that

νρ+ (1− ν)σ −
(ν
κ

)
(ρ∇κσ − ρ♯κσ) ≤ K(

√
h, 2)−r′ρ♯νσ +R(

√
ρ♯κσ −

√
σ)2.

So, we get the inequality (18).

For two non-negative real numbers ρ and σ, we define the Heinz mean in the parameter
µ, 0 ≤ µ ≤ 1, as

Hµ =
ρµσ1−µ + ρ1−µσµ

2
. (19)

Note that H0(ρ, σ) = H1(ρ, σ) = ρ+σ
2 and H 1

2
(ρ, σ) =

√
ρσ. It is easy to see that as a

function of µ, Hµ(ρ, σ) is convex, attains its minimum at µ = 1
2 , and attains its maximum

at µ = 0 and µ = 1. Moreover, Hµ(ρ, σ) = H1−µ(ρ, σ) for 0 ≤ µ ≤ 1. Thus, the Heinz
mean interpolates between the geometric mean and the arithmetic mean:

√
ρσ ≤ Hµ(ρ, σ) ≤

ρ+ σ

2
for 0 ≤ µ ≤ 1. (20)

Theorem 4. Let ρ, σ > 0 and 0 ≤ ν < κ ≤ 1. Then

r
[
Hκ(ρ, σ) +H0(ρ, σ)−Hκ

2
(ρ, σ)

]
+K

[√
h, 2
]r′

Hν(ρ, σ)

≤ H0(ρ, σ)−
(
ν
κ

)
[H0(ρ, σ)−Hκ(ρ, σ)] ,

(21)

where r = min{ ν
κ , 1−

ν
κ}, h = ρ

σ and r′ = min{2r, 1− 2r}.

Proof. Interchanging ρ with σ and σ with ρ in inequality (16), we get

r(
√
σ♯κρ−

√
ρ)2 +K(

√
h, 2)r

′
σ♯νρ ≤ νσ + (1− ν)ρ−

(ν
κ

)
(σ∇κρ− σ♯κρ). (22)

Adding (16) and (22), we have

r
[
(
√
ρ♯κσ −

√
σ)2 + (

√
σ♯κρ−

√
ρ)2
]
+K

[√
h, 2
]r′

(2Hν(ρ, σ))

≤ 2H0(ρ, σ)−
(
ν
κ

)
[2H0(ρ, σ)− 2Hκ(ρ, σ)]

and so

r
[
Hκ(ρ, σ) +H0(ρ, σ)−Hκ

2
(ρ, σ)

]
+K

[√
h, 2
]r′

Hν(ρ, σ)

≤ H0(ρ, σ)−
(
ν
κ

)
[H0(ρ, σ)−Hκ(ρ, σ)] .

In similar of proof of Theorem 4, we can prove the following result.



M.H.M Rashid, W.M.M. Salameh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5586 10 of 21

Theorem 5. Let ρ, σ > 0 and 0 ≤ ν < κ ≤ 1. Then

H0(ρ, σ)−
(
ν
κ

)
[H0(ρ, σ)−Hκ(ρ, σ)] ≤ K

[√
h, 2
]−r′

Hν(ρ, σ)

+R
[
Hκ(ρ, σ) +H0(ρ, σ)−Hκ

2
(ρ, σ)

]
,

(23)

where R = min{ ν
κ , 1−

ν
κ}, h = ρ

σ and r′ = min{2r, 1− 2r}.

4. New operator versions of Heinz-type inequalities

Let H represent a complex Hilbert space, and B(H) denote the C∗-algebra comprising
all bounded linear operators on H. An operator T ∈ B(H) is considered positive if
⟨Tx, x⟩ ≥ 0 holds true for every x ∈ H. We express this as T ≥ 0. Now, let T and S be
two positive operators in B(H), and κ take on values in the interval [0, 1]. The κ-weighted
arithmetic mean of T and S, denoted as T∇κS, is defined as:

T∇κS = (1− κ)T + κS.

When T is invertible, the κ-geometric mean of T and S, represented as T♯κS, is defined
as:

T♯κS = T
1
2

(
T− 1

2ST− 1
2

)κ
T

1
2 .

In the case where κ = 1
2 , we can simplify the notation to T∇S and T♯S to refer to the

κ-weighted arithmetic mean and the κ-geometric mean, respectively. It is well-known that
for positive invertible operators T and S, the following inequality holds:

T♯κS ≤ T∇κS, κ ∈ [0, 1].

Additionally, we define the operator version of the Heinz mean as Hκ(T, S):

Hκ(T, S) =
T♯κS + T♯1−κS

2

for the case where T and S are positive invertible operators and κ ∈ [0, 1].
In this section, we will present improved variants of Heinz-type operator inequalities

and their converses, exploiting the monotonicity of operator functions as the foundational
concept for the ensuing discussion.

Lemma 4. [7] Suppose T ∈ B(H) is self-adjoint. If f and g are continuous functions
such that f(t) ≥ g(t) for t ∈ sp(T ) (where sp(T ) represents the spectrum of the operator
T ), then it follows that f(T ) ≥ g(T ).

Next, we present our main results on the basis of inequality (21). By Lemma 4, we
have the following.



M.H.M Rashid, W.M.M. Salameh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5586 11 of 21

Theorem 6. Let T, S ∈ B(H) be positive invertible operators, I is the identity operator
and 0 ≤ ν < κ ≤ 1. If all positive numbers m,m′ andM,M ′ satisfy either of the conditions
0 < mI ≤ T ≤ m′I < M ′I ≤ S ≤MI or 0 < mI ≤ S ≤ m′I ≤ T ≤MI, then:

r
[
Hκ(T, S) +H0(T, S)−Hκ

2
(T, S)

]
+K

[√
h, 2
]r′

Hν(T, S)

≤ H0(T, S)−
(
ν
κ

)
[H0(T, S)−Hκ(T, S)] ,

(24)

where r = min{ ν
κ , 1−

ν
κ}, h = M

m and r′ = min{2r, 1− 2r}.

Proof. Assuming that 0 ≤ ν < κ ≤ 1, according to inequality (21), for any positive
value of x, we can conclude:

r
[
Hκ(1, x) +H0(1, x)−Hκ

2
(1, x)

]
+K

[√
h, 2
]r′

Hν(1, x)

≤ H0(1, x)−
(
ν
κ

)
[H0(1, x)−Hκ(1, x)] ,

Regarding the operator X = T−1/2ST−1/2, within the framework of the first condition,
we establish the following range: I ≤ hI = M

m I ≤ X ≤ h′I = M ′

m′ I. Consequently, we infer
that σ(X) ⊆ [h, h′] ⊆ (1,∞). Applying Lemma 4, we obtain:

r
[
Hκ(I,X) +H0(I,X)−Hκ

2
(I,X)

]
+ min

h≤x≤h′
K
[√
x, 2
]r′
Hν(I,X)

≤ H0(I,X)−
(ν
κ

)
[H0(I,X)−Hκ(I,X)] ,

As the Kantorovich constant K(t, 2) = (1+t)2

4t exhibits monotonicity within the interval
(0,∞), it follows that:

r
[
Hκ(I, T

−1/2ST−1/2) +H0(I, T
−1/2ST−1/2)−Hκ

2
(I, T−1/2ST−1/2)

]
+ min

h≤x≤h′
K
[√
x, 2
]r′
Hν(I, T

−1/2ST−1/2) ≤ H0(I, T
−1/2ST−1/2)

−
(ν
κ

) [
H0(I, T

−1/2ST−1/2)−Hκ(I, T
−1/2ST−1/2)

]
, (25)

Likewise, within the context of the second condition, we observe that I ≤ 1
hI = m

M h ≤
X ≤ 1

h′ I = m′

M ′ I. Utilizing Lemma 4, we obtain the following:

r
[
Hκ(I,X) +H0(I,X)−Hκ

2
(I,X)

]
+min 1

h′≤x≤ 1
h
K [

√
x, 2]

r′
Hν(I,X)

≤ H0(I,X)−
(
ν
κ

)
[H0(I,X)−Hκ(I,X)] ,

Since the Kantorovich constant K(t, 2) = (1+t)2

4t is an increasing function on (0,∞), then

r
[
Hκ(I, T

−1/2ST−1/2) +H0(I, T
−1/2ST−1/2)−Hκ

2
(I, T−1/2ST−1/2)

]
+ min

1
h′≤x≤ 1

h

K
[√
x, 2
]r′
Hν(I, T

−1/2ST−1/2) ≤ H0(I, T
−1/2ST−1/2)



M.H.M Rashid, W.M.M. Salameh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5586 12 of 21

−
(ν
κ

) [
H0(I, T

−1/2ST−1/2)−Hκ(I, T
−1/2ST−1/2)

]
,

By multiplying both inequalities (25) and (26) on both the left-hand and right-hand sides
by the operator T 1/2, we can infer the desired inequality (24).

Theorem 7. Consider positive invertible operators T and S in a Hilbert space H, where
I represents the identity operator. Additionally, let κ be a non-negative number such that
0 ≤ ν < κ ≤ 1. Assuming that there exist positive real numbers m,m′,M,M ′ that satisfy
either of the following conditions:

(a) 0 < mI ≤ T ≤ m′I < M ′I ≤ S ≤MI

(b) 0 < mI ≤ S ≤ m′I ≤ T ≤MI

Then, the following conclusions hold:

H0(T, S)−
(
ν
κ

)
[H0(T, S)−Hκ(T, S)] ≤ K

[√
h, 2
]−r′

Hν(T, S)

+R
[
Hκ(T, S) +H0(T, S)−Hκ

2
(T, S)

]
,

(26)

where R = min{ ν
κ , 1−

ν
κ}, h = M

m and r′ = min{2r, 1− 2r}.

Proof. The proof process is similar to that of Theorem 6, and thus, we will not provide
it here.

Remark 3. The nature of the Kantorovich constant’s characteristics makes it clear that
the inequalities outlined in Theorems 6 and 7 signify improved results compared to those
detailed in [13], [14], [18], [20], and [23].

5. Utilizations of the improved Young-type inequalities for traces,
determinants, and norms of positive definite matrices

In this section, we introduce a collection of improved Young-type inequalities designed
specifically for traces, determinants, and norms of positive semi-definite matrices.

A matrix version proved in [1] says that if T, S ∈ Mn(C) are positive semi-definite,
then

sj(TS) ≤ sj

(
1

p
T p +

1

q
Sq

)
(27)

for j = 1, · · · , n

Lemma 5. Let ρ, σ > 0, p, q > 1 such that 1
p + 1

q = 1. Then for m ∈ N, we have

(
ρ

1
pσ

1
q

)m
+ rm0

(
ρ

m
2 − σ

m
2

)2
≤
(
ρr

p
+
σr

q

)m
r

, r ≥ 1 (28)

where r0 = min{1
p ,

1
q}.



M.H.M Rashid, W.M.M. Salameh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5586 13 of 21

Lemma 6. Let Ti ∈Mn(C) (i = 1, · · · , n),. Then

n∑
j=1

sj(T1 · · ·Tn) ≤
n∑

j=1

sj(T1) · · · sj(Tk).

Theorem 8. Let T, S ∈ B(H) be positive definite, p, q > 1 such that 1
p+

1
q = 1 and m ∈ N.

Then (
tr(T r)

p
+
tr(Sr)

q

)m
r

≥
(
tr
∣∣∣T 1

pS
1
q

∣∣∣)m + rm0

(
(tr(T ))

m
2 − (tr(S))

m
2

)2
, (29)

where r0 = min{1
p ,

1
q}.

Proof. By inequality (29), we have

s
m
r
j

(
T r

p
+
Sr

q

)
=

sm
r
j (T r)

p
+
s

m
r
j (Sr)

q


≥ smj

(
T

1
p

)
smj

(
S

1
q

)
+ rm0

(
s

m
2
j (T )− s

m
2
j (S)

)2
= smj

(
T

1
p

)
smj

(
S

1
q

)
+ rm0

(
smj (T ) + smj (S)− 2s

m
2
j (T )s

m
2
j (S)

)
for j = 1, · · · , n. Thus, by Lemma 6 and the Cauchy-Schwarz inequality, we have

tr
m
r

(
T r

p
+
Sr

q

)
=

n∑
j=1

s
m
r
j

(
T r

p
+
Sr

q

)

≥
n∑

j=1

smj

(
T

1
p

)
smj

(
S

1
q

)

+ rm0

 n∑
j=1

smj (T ) +

n∑
j=1

smj (S)− 2

n∑
j=1

s
m
2
j (T )s

m
2
j (S)


Hence

tr
m
r

(
T r

p
+
Sr

q

)
≥

n∑
j=1

smj

(
T

1
pS

1
q

)

+ rm0

 n∑
j=1

smj (T ) +
n∑

j=1

smj (S)− 2
n∑

j=1

s
m
2
j (T )s

m
2
j (S)


≥

(
tr
∣∣∣(T 1

pS
1
q

)∣∣∣)m + rm0 [(tr(T ))m + (tr(S))m



M.H.M Rashid, W.M.M. Salameh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5586 14 of 21

− 2

 n∑
j=1

sj(T )

m
2
 n∑

j=1

sj(S)

m
2


=

(
tr
∣∣∣(T 1

pS
1
q

)∣∣∣)m + rm0

(
(tr(T ))

m
2 − (tr(S))

m
2

)2

Remark 4. Ando’s singular value inequality (27) entails the norm inequality∣∣∣∣∣∣T κS1−κ
∣∣∣∣∣∣ ≤ |||κT + (1− κ)S|||. (30)

So, our Theorem 8 improves this inequality for the trace norm:∥∥∥T 1
pS

1
q

∥∥∥m
1
+ rm0

(
∥T∥

m
2
1 − ∥S∥

m
2
1

)2
≤
∥∥∥∥1pT r +

1

q
Sr

∥∥∥∥m
r

1

(31)

Theorem 9. Let T, S ∈ B(H) be positive definite, p, q > 1 such that 1
p+

1
q = 1 and m ∈ N.

Then for all r ≥ 1

det

(
T r

p
+
Sr

q

)m
r

≥ det
(
T

1
pS

1
q

)m
+ rmn

0 det
(
Tm + Sm − 2S

m
2

(
S− 1

2TS− 1
2

)m
S

m
2

)2
,

(32)
where r0 = min{1

p ,
1
q}.

Proof. By inequality (28), we have

s
m
r
j

(
1

p

(
S− r

2T rS− r
2

)
+

1

q
I

)
≥ s

m
p

j

(
S− 1

2TS− 1
2

)
+ rm0

(
s

m
2
j

(
S− 1

2TS− 1
2

)
− 1
)2

for all j = 1, · · · , n.

det

(
1

p
S− r

2TS− r
2 +

1

q

)m
r

=
n∏

j=1

(
1

p
s

m
r
j

(
S− r

2T rS− r
2 +

1

q

))

≥
n∏

j=1

[
s

m
p

j

(
S− 1

2TS− 1
2

)
+ rm0

(
s

m
2
j

(
S− 1

2TS− 1
2

)
− 1
)2]

≥
n∏

j=1

[
s

1
p

j

(
S− 1

2TS− 1
2

)m]

+ rmn
0

n∏
j=1

[
s

m
2
j

(
S− 1

2TS− 1
2

)
− 1
]2

= det
(
S− 1

2TS− 1
2

)m
p
+ rmn

0

[(
S− 1

2TS− 1
2

)m
2 − I

]2
.



M.H.M Rashid, W.M.M. Salameh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5586 15 of 21

Consequently

det

(
T r

p
+
Sr

q

)m
r

≥ det
(
T

1
pS

1
q

)m
+ rmn

0 det
(
Tm + Sm − 2S

m
2

(
S− 1

2TS− 1
2

)m
S

m
2

)2
.

Theorem 10. Let T, S,X ∈ Mn(C) such that T and S are positive semi-definite and
p, q > 1 with 1

p + 1
q = 1 and m ∈ N. Then for all r ≥ 1, we have

∣∣∣∣∣∣∣∣∣T 1
pXS

1
q

∣∣∣∣∣∣∣∣∣m + rm0

(
|||TX|||

m
2 − |||SX|||

m
2

)2
≤
(
1

p
|||TX|||r + 1

q
|||XB|||r

)m
r

, (33)

where r0 = min{1
p ,

1
q}.

To prove Theorem 10, we need the following lemma which is known as the Heinz-Kato
type for unitarily invariant norm.

Lemma 7 ([9]). Let T, S ∈ Mn(C) be positive definite matrices and 0 ≤ ϑ ≤ 1. Then we
have ∣∣∣∣∣∣∣∣∣T ϑXS1−ϑ

∣∣∣∣∣∣∣∣∣ ≤ |||TX|||ϑ|||XB|||1−ϑ. (34)

In particular

tr
∣∣∣T ϑXS1−ϑ

∣∣∣ ≤ (tr(T ))ϑ (tr(S))1−ϑ . (35)

Proof. [Proof of Theorem 10] We have∣∣∣∣∣∣∣∣∣T 1
pXS

1
q

∣∣∣∣∣∣∣∣∣m + rm0

(
|||TX|||

m
2 − |||SX|||

m
2

)2
≤

[
|||TX|||

1
p |||XB|||

1
q

]m
+ rm0

(
|||TX|||

m
2 − |||SX|||

m
2

)2
(by Lemma 7)

≤
(
1

p
|||TX|||r + 1

q
|||XB|||r

)m
r

(by inequality 28).

Lemma 8 ([3]). Let ω1, · · · , ωn be non-negative real numbers and ϑ1, · · · , ϑn be positive
real numbers with

∑n
i=1 ϑi = 1. Then we have

n∏
k=1

ωϑk
k + r

 n∑
k=1

ωk − n n

√√√√ n∏
k=1

ωk

 ≤
n∑

i=1

ϑkωk, (36)

where r = min {ϑk : k = 1, · · · , n}.



M.H.M Rashid, W.M.M. Salameh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5586 16 of 21

Theorem 11. Let Ti ∈ Mn(C) (i = 1, · · · , n) be positive semi-definite. If 0 ≤ ϑi ≤ 1
(i = 1, · · · , n) with

∑n
i=1 ϑi = 1, then

n∑
k=1

tr(ϑkTk) ≥ tr

∣∣∣∣∣
n∏

k=1

T ϑk
k

∣∣∣∣∣+ r

 n∑
k=1

tr(Tk)− n n

√√√√ n∏
k=1

tr(Tk)

 , (37)

where r = min {ϑk : k = 1, · · · , n}.

Proof. By inequality (36), we have

n∑
k=1

ϑksj(Tk) ≥
n∏

k=1

sj(Tk)
ϑk + r

 n∑
k=1

sj(Tk)− n n

√√√√ n∏
k=1

sj(Tk)


for j = 1, · · · , n. Thus, by Lemma 6 and the generalized Cauchy-Schwarz inequality, we
have

tr

(
n∑

k=1

ϑkTk

)
=

n∑
k=1

ϑktr(Tk) =

n∑
k=1

ϑk

n∑
j=1

sj(Tk) =

n∑
j=1

n∑
k=1

ϑksj(Tk)

≥
n∑

j=1

sj(T
ϑ1
1 ) · · · sj(T ϑn

k )

+ r

 n∑
j=1

n∑
k=1

sj(Tk)− n
n∑

j=1

n

√√√√ n∏
k=1

sj(Tn)


≥

n∑
j=1

sj(T
ϑ1
1 · · ·T ϑn

n )

+ r

 n∑
j=1

n∑
k=1

sj(Tk)− n n

√√√√ n∏
k=1

n∑
j=1

sj(Tk)


≥ tr

∣∣∣T ϑ1
1 · · ·T ϑn

n

∣∣∣+ r

 n∑
k=1

tr(Tk)− n n

√√√√ n∏
k=1

tr(Tk)


where r = min {ϑk : k = 1, · · · , n}.

Our Theorem 11 entils the following trace norm∥∥∥∥∥
n∑

k=1

ϑkTk

∥∥∥∥∥
1

≥

∥∥∥∥∥
n∏

k=1

T ϑk
k

∥∥∥∥∥
1

+ r

∥∥∥∥∥
n∑

k=1

Tk

∥∥∥∥∥
1

− n n

√√√√ n∏
k=1

∥Tk∥1

 , (38)

where r = min {ϑk : k = 1, · · · , n}.



M.H.M Rashid, W.M.M. Salameh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5586 17 of 21

Theorem 12. Let Ti ∈ Mn(C) (i = 1, · · · , n) be positive definite. If 0 ≤ ϑi ≤ 1 (i =
1, · · · , n) with

∑n
i=1 ϑi = 1, then

det

(
n∑

k=1

ϑkTk

)
≥

n∏
k=1

det
(
T ϑk
k

)
+ r

det

(
n∑

k=1

Tk

)
− n n

√√√√ n∏
k=1

det(Tk)

 , (39)

where r = min {ϑk : k = 1, · · · , n}.

To prove Theorem 12, we need the following lemma.

Lemma 9 ([5]). Let T, S ∈Mn(C) be positive definite. Then we have

det(T + S)
1
n ≥ det(T )

1
n + det(S)

1
n . (40)

Proof. [Proof of Theorem 12] We have

det

(
n∑

k=1

ϑkTk

)
=

det( n∑
k=1

ϑkTk

) 1
n

n

≥

[
n∑

k=1

det (ϑkTk)
1
n

]n
(by Lemma 9)

≥

[
n∑

k=1

ϑk det (Tk)
1
n

]n

≥

[
n∏

k=1

(
(Tk)

1
n

)ϑk

]n
+ rn

 n∑
k=1

det (Tk)
1
n − n n

√√√√det

n∏
k=1

Tk


=

n∏
k=1

det
(
T ϑk
k

)
+ rn

 n∑
k=1

det (Tk)
1
n − n n

√√√√det
n∏

k=1

Tk



Lemma 10 ([16]). Let γ1, γ2, · · · , γn be a set of non-negative real numbers constrained by
j∑

k=1

γk = Γj. If ω1, ω2, · · · , ωn are positive real numbers, then

1

Γn

n∑
k=1

γkωk +

√√√√1 +

(
1

Γn

n∑
k=1

γkωk

)2

≥

[
n∏

k=1

(
ωk +

√
1 + ω2

k

)γk
] 1

Γn

(41)

holds.



M.H.M Rashid, W.M.M. Salameh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5586 18 of 21

Theorem 13. Let T1, · · · , Tk ∈Mn(C) be positive define and let γ1, γ2, · · · , γn be a set of

non-negative real numbers such that

n∑
k=1

γk = Γn. Then

1

Γn

n∑
k=1

tr(Tk) +

√√√√1 +

(
1

Γn

n∑
k=1

tr(Tk)

)2

≥
n∏

k=1

[
tr(Tk) +

√
1 + tr(T 2

k )

] 1
Γn

. (42)

Proof. By inequality (41), we have

1

Γn

n∑
k=1

γksj(Tk)+

√√√√1 +

(
1

Γn

n∑
k=1

γksj(Tk)

)2

≥

[
n∏

k=1

(
sj(Tk) +

√
1 + s2j (Tk)

)γk] 1
Γn

(43)

for all j = 1, · · · , n. Hence we have

1

Γn

n∑
k=1

γk

n∑
j=1

sj(Tk) +

√√√√√1 +

 1

Γn

n∑
k=1

γk

n∑
j=1

sj(Tk)

2

≥

 n∏
k=1

 n∑
j=1

sj(Tk) +

√√√√1 +
n∑

j=1

s2j (Tk)

γk


1
Γn

Consequently,

tr

(
1

Γn

n∑
k=1

γkTk

)
+

√√√√1 +

(
1

Γn

n∑
k=1

γktr(Tk)

)2

=
1

Γn

n∑
k=1

γktr(Tk) +

√√√√1 +

(
1

Γn

n∑
k=1

γktr(Tk)

)2

≥

 n∏
k=1

tr(Tk) +
√√√√1 +

n∑
j=1

tr(T 2
k )

γk


1
Γn

6. Conclusion and Future Work

In conclusion, this paper has embarked on an extensive investigation into the domain
of matrix means interpolation and comparison. A key aspect of this research has been
the expansion of the parameter ϑ from the closed interval [0, 1] to encompass the entire
positive real line, represented as R+. This extension has allowed us to explore a broader
spectrum of mathematical relationships and properties within this framework.



M.H.M Rashid, W.M.M. Salameh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5586 19 of 21

Furthermore, our exploration has led to the development of various novel results related
to Heinz means. We have introduced scalar variants of Heinz inequalities, leveraging
Kantorovich’s constant, and have extended these inequalities to the operator realm. This
expansion not only deepens our understanding of Heinz means but also opens up new
avenues for applications in diverse mathematical contexts.

Lastly, we have presented refined Young’s type inequalities specifically tailored for
traces, determinants, and norms of positive semi-definite matrices. These refined inequal-
ities are expected to find utility in various matrix analysis and linear algebra problems,
enhancing our ability to derive meaningful conclusions and insights from the study of
positive semi-definite matrices.

As for future work, there are several intriguing directions to consider. Firstly, it may be
valuable to explore further extensions of the parameter space beyond R+ and investigate
the implications of such extensions on matrix means and related inequalities. Additionally,
the applicability of the developed results in practical fields such as physics, engineering,
and data science warrants investigation. Finally, refining and expanding upon the pre-
sented inequalities could lead to even more powerful tools for matrix analysis and opti-
mization, offering new insights and solutions to complex problems in mathematics and its
applications.

Declaration

• Author Contributions: The author have read and agreed to the published version
of the manuscript.

• Funding: No funding is applicable

• Conflicts of Interest: The authors declare no conflict of interest.

Acknowledgements

The authors sincerely thank the reviewers for their insightful feedback, valuable sug-
gestions, and expertise, which have greatly contributed to improving the clarity, quality,
and overall impact of this study.

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