EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6461
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Some Refinements for Numerical Radius Inequalities of
Operators

Manal Al-Labadi, Wasim Audeh, Raja’a Al-Naimi, Jamal Oudetallah1,
Eman Almuhur2, Nazneen Khan3

1 Department of Mathematics, University of Petra, Amman, Jordan
2 Department of Mathematics, Applied Science Private University, Amman, Jordan
3 Department of Mathematics, Taibah University, Madina Munawwara, Chandigarh,
Saudi Arabia

Abstract. Several recent papers gave numerical radius inequalities for sums and products of
operators which are defined on a complex separable Hilbert space H. In this paper, we prove a
numerical radius inequality which generalizes and refines a recent inequality proved by Kittaneh.

2020 Mathematics Subject Classifications: 47A30, 15A18, 47A12, 15A60, 47B15

Key Words and Phrases: Inequality, numerical radius, Operator, Norm

1. Introduction

Let H be a Hilbert space over the field of complex numbers with inner product ⟨., .⟩.
The set of all operators in H is denoted by B(H). Upper case letters will denote elements
of B(H). If A ∈ B(H) such that the operator A is equal to its conjugate transpose,
then we say that the operator A is self-adjoint and in this case all its eigenvalues are real
numbers. The set of all eigenvalues of A ∈ B(H) is denoted by σ(A). The conjugate
transpose (adjoint) of A is denoted by A∗. A self-adjoint operator A ∈ B(H) is called
positive semi-definite if ⟨Ax, x⟩ ≥ 0 for all x ∈ H. For any operator A ∈ B(H), the
operator A∗A is positive semi-definite. The square root of A∗A, denoted by |A|, is defined
as |A| = (A∗A)

1
2 . The singular values of A ∈ B(H) are ordered descendingly as follows,

s1(A) ≥ s2(A) ≥ · · · and they are the eigenvalues of |A|. In fact sj(A) = λj(|A|) = sj(|A|)
for j = 1, 2, .... For recent studies about singular values, we advise the readers to read
[[1]-[4]], [[8]-[10]], [[13]] and [[14]-[18]]. We denote the identity operator on H by I ∈ B(H)
and we denote the zero operator on H by O ∈ B(H). Each operator A can be written as
a sum of its real and imaginary parts, as follows, A = B + iC, where B = Re(A) = A+A∗

2

DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6461

Email addresses: manal.allabadi@uop.edu.jo (M. Al-LAbadi), waudeh@uop.edu.jo (W. Audeh),
rajaa.alnaimi@uop.edu.jo (R. Al-Naimi), jamal.oudetallah@uop.edu.jo (J. Oudetallah),
e almuhur@asu.edu.jo (E. Almuhur), nkkhan@taibahu.edu.sa (N. Khan)

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



M. Al-Labadi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6461 2 of 13

and C = Im(A) = A−A∗

2i . Note that Re(A) and Im(A) are self-adjoint operators. For
A ∈ B(H), the spectral radius, numerical radius and the usual operator norm of A are
given, respectively, by

r(A) = max
λ∈σ(A)

{|λ|},

w(A) = sup
||x||=1

|⟨Ax, x⟩|

and
∥A∥ = sup

||x||=||y||=1
|⟨Ax, y⟩| .

The computation process to reach the exact value of w(A) is not always of that simplicity,
this gave the authors the area to obtain lower and upper bounds of w(A), these bounds
are usually in terms of ||A||. For recent studies of numerical radius inequalities of operaors,
we refer the reader to [5]-[6], [11]-[12] and [7]. The most known famous lower and upper
bounds for w(A) is the following equivalence between w(A) and ||A||, see [22],

∥A∥
2

≤ w(A) ≤ ∥A∥ . (1)

Several improvements and generalizations of inequality (1) has been given. For example,
in [26], it is shown that

w(A) ≤ 1

2
|| |A|+ |A∗| || (2)

and

w(A) ≤ 1

2

(
|| A ||+ || A2 ||

1
2

)
, (3)

as a refinement of the second inequality in (1).
In [25], another refinement of the second inequality in (1) has been given as follows:

w2(A) ≤ 1

2
|| A∗A+AA∗ ||. (4)

This bound for the numerical radius is given as one of the sharpest simple bounds in the
literature.
The author in [20] refines inequality (3) as follows,

w(A) ≤ 1

2

(
|| A ||+

√
r(|A||A∗|)

)
. (5)

Recently, in [23], it is shown that:

w(AB∗) ≤ 1

4
|| |A|+ |B| || || |A∗|+ |B∗| ||. (6)

In this paper, we give a remarkable refinement and a generalization of inequality (4). An
attractive generalization of inequality (5) is also given. Moreover, we give an inequality
that is equivalent, if A and B are self-adjoint, to inequality (6). A new proof of inequality
(2) is obtained. Several numerical radius inequalities are included.



M. Al-Labadi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6461 3 of 13

2. Numerical radius inequalities via inner product

The main result in this section is a numerical radius inequality for finite sums of
operators. To prove this result and to make some comparisons between some of its special
cases and recent inequalities proved by different authors, we need the following lemmas,
which are proved in [28], and they are essential in our analysis.

Lemma 1. Let A ∈ B(H) be positive semidefinite and x ∈ H such that ∥x∥ ≤ 1. Then

(i) ⟨Ax, x⟩r ≤ ⟨Arx, x⟩ for r ≥ 1.
(ii) ⟨Arx, x⟩ ≤ ⟨Ax, x⟩r for 0 < r ≤ 1.

Lemma 2. Let A ∈ B(H) be self-adjoint operator and x ∈ H. Then

|⟨Ax, x⟩| ≤ ⟨|A|x, x⟩ . (7)

Lemma 3. Let A ∈ B(H) and x, y ∈ H be any vectors. If f and g are nonnegative
continuous functions on [0,∞) satisfying the relation f(a)g(a) = a (a ∈ [0,∞)), then

|⟨Ax, y⟩|2 ≤ ⟨|A|x, x⟩ ⟨|A∗| y, y⟩ (8)

and more general
|⟨Ax, y⟩|2 ≤

〈
f2 (|A|)x, x

〉 〈
g2 (|A∗|) y, y

〉
. (9)

The following lemma follows by Weyl’s monotonocity principle (see, e.g., [5, p.63] or
[5, p. 20]).

Lemma 4. If A,B ∈ B(H) are positive semidefinite such that A ≤ B. Then

|| A || ≤ || B ||. (10)

Lemma 5. Let A,B ≥ O. Then

|| A+B ||2 ≤ 2|| A2 +B2 ||. (11)

Equality holds iff A = B.

Proof. It is well known that || (A + B)2 || = || A + B ||2 (since A,B ≥ 0). To reach
inequality (11), it is enough to prove that (A + B)2 ≤ 2(A2 + B2) and then applying
inequality (10).
Now,

2A2 + 2B2 − (A+B)2 = 2A2 + 2B2 − (A2 +B2 +AB +BA)

= A2 +B2 −AB −BA

= (A−B)2 ≥ 0 (since A-B is Hermitian).

This implies that (A+B)2 ≤ 2(A2 +B2), and so we reach our claim.



M. Al-Labadi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6461 4 of 13

Theorem 1. Let A1, A2, ..., An, B1, B2, ..., Bn be operators in B(H). If f and g are non-
negative continuous functions on [0,∞) satisfying the relation f(a)g(a) = a (a ∈ [0,∞)),
then

w

(
n∑

i=1

(Ai +Bi)

)
≤ 1

2

n∑
i=1

∣∣∣∣ f2(|Ai|) + g2(|A∗
i |) + f2(|Bi|) + g2(|B∗

i |)
∣∣∣∣ . (12)

Proof.∣∣∣∣∣
〈

n∑
i=1

(Ai +Bi)x, x

〉∣∣∣∣∣ = |⟨((A1 +B1) + (A2 +B2) + · · ·+ (An +Bn))x, x⟩|

= |⟨(A1 +B1)x, x⟩+ ⟨(A2 +B2)x, x⟩+ · · ·+ ⟨(An +Bn)x, x⟩|

≤ (|⟨(A1 +B1)x, x⟩|+ |⟨(A2 +B2)x, x⟩|+ · · ·+ |⟨(An +Bn)x, x⟩|)
( by using triangle inequality)

=

n∑
i=1

|⟨(Ai +Bi)x, x⟩|

=

n∑
i=1

|⟨Aix, x⟩+ ⟨Bix, x⟩|

≤
n∑

i=1

(|⟨Aix, x⟩|+ |⟨Bix, x⟩|) (by using triangle inequality)

≤
n∑

i=1

(〈
f2(|Ai|)x, x

〉 1
2
〈
g2(|A∗

i |)x, x
〉 1

2 +
〈
f2(|Bi|)x, x

〉 1
2
〈
g2(|B∗

i |)x, x
〉 1

2

)
(by using inequality (9))

≤ 1

2

n∑
i=1

(〈
f2(|Ai|)x, x

〉
+
〈
g2(|A∗

i |)x, x
〉
+
〈
f2(|Bi|)x, x

〉
+
〈
g2(|B∗

i |)x, x
〉)

(by using arithmetic geometric mean inequality)

=
1

2

n∑
i=1

〈(
f2(|Ai|) + g2(|A∗

i |) + f2(|Bi|) + g2(|B∗
i |)
)
x, x

〉
.

Taking the supremum over all unit vectors x ∈ H, we obtain the inequality (12).

Corollary 1. Let A1, A2, ..., An, B1, B2, ..., Bn be operators in B(H). Then

w

(
n∑

i=1

(Ai +Bi)

)
≤ 1

2

n∑
i=1

|| |Ai|+ |A∗
i |+ |Bi|+ |B∗

i | || . (13)



M. Al-Labadi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6461 5 of 13

Proof. Letting f(t) = g(t) =
√
t in inequality (12), we obtain the inequality (13).

Corollary 2. Let A,B be operators in B(H). Then

w(A+B) ≤ 1

2
|| |A|+ |A∗|+ |B|+ |B∗| || . (14)

Proof. Letting Ai = Bi = O for i = 2, 3, 4, ..., n in inequality (13), we obtain the
inequality (14).

Corollary 3. Let A,B be operators in B(H). Then

w2(A+B) ≤ 1

4
|| |A|+ |A∗|+ |B|+ |B∗| ||2 . (15)

Proof. By squaring both sides of inequality (14), we obtain the inequality (15).

Remark 1. Letting B = O in inequality (14), we derive inequality (2). This is considered
as a new proof of inequality (2).

Corollary 4. Let A ∈ B(H). Then

w2(A) ≤ 1

4
|| |A|+ |A∗| ||2. (16)

Proof. Letting B = O in inequality (15), we obtain the inequality (16).

Remark 2. Inequality (16) refines inequality (4). To show this, note that while the left
sides of inequalities (4) and (16) are the same. The right side of inequality (16) is

1

4
|| |A|+ |A∗| ||2 ≤ 1

2
|| |A|2 + |A∗|2 || (by using inequality (11))

=
1

2
|| A∗A+AA∗ ||,

which is the right side of inequality (4).

The following lemma [27] is essential to prove the following theorem which is a gener-
alization of inequality (5).

Lemma 6. Let A,B be positive semidefinite operators in B(H). Then

|| A+B || ≤ max{|| A ||, || B ||}+ || A
1
2B

1
2 || (17)

Theorem 2. Let A,B be operators in B(H). Then

w(A+B) ≤ 1

2

(
max{|| |A|+ |B| ||, || |A∗|+ |B∗| ||}+

√
r ((|A|+ |B|)(|A∗|+ |B∗|))

)
.

(18)



M. Al-Labadi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6461 6 of 13

Proof.

w(A+B) ≤ 1

2
|| |A|+ |A∗|+ |B|+ |B∗| || (by inequality (14))

=
1

2
|| (|A|+ |B|) + (|A∗|+ |B∗|) ||

≤ 1

2

(
max{|| |A|+ |B| ||, || |A∗|+ |B∗| ||}+ || (|A|+ |B|)

1
2 (|A∗|+ |B∗|)

1
2 ||
)

(by inequality (17)

=
1

2

(
max{|| |A|+ |B| ||, || |A∗|+ |B∗| ||}+

√
r ((|A|+ |B|)(|A∗|+ |B∗|))

)
(since || A

1
2B

1
2 ||2 = r(AB)).

Remark 3. Letting B = O in inequality (18), we obtain the inequality (5).

3. Numerical radius inequalities via block matrices

In this section, new numerical radius inequalities are proved by using block matrices.
We begin by proving the following inequality which is a direct application of inequality
(15).

Theorem 3. Let X,Y,W,Z be operators in B(H). Then

w2

[
X Z
W Y

]
≤ 1

4

∣∣∣∣∣∣∣∣[ |X|+ |X∗|+ |W |+ |Z∗| O
O |Y |+ |Y ∗|+ |W ∗|+ |Z|

]∣∣∣∣∣∣∣∣2 . (19)

Proof. Letting A =

[
X O
O Y

]
and B =

[
O Z
W O

]
in inequality (15), we obtain the

inequality (19).

Corollary 5. Let X,Y be operators in B(H). Then

w2

[
X O
O Y

]
≤ 1

4

∣∣∣∣∣∣∣∣[ |X|+ |X∗| O
O |Y |+ |Y ∗|

]∣∣∣∣∣∣∣∣2 . (20)

Proof. Letting Z = W = O in inequality (19), we obtain the inequality (20).

Remark 4. Letting Y = O in inequality (20), we obtain the inequality (16). In that sense,
inequality (20) is a generalization of inequality (16).

Corollary 6. Let W,Z be operators in B(H). Then

w2

[
O Z
W O

]
≤ 1

4

∣∣∣∣∣∣∣∣[ |W |+ |Z∗| O
O |W ∗|+ |Z|

]∣∣∣∣∣∣∣∣2 . (21)



M. Al-Labadi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6461 7 of 13

Proof. Letting X = Y = O in inequality (19), we obtain the inequality (21).

Remark 5. Replacing W by W ∗ in inequality (21), we obtain

w2

[
O Z
W ∗ O

]
≤ 1

4

∣∣∣∣∣∣∣∣[ |W ∗|+ |Z∗| O
O |W |+ |Z|

]∣∣∣∣∣∣∣∣2 = 1

4
max{ || |W ∗|+|Z∗| ||, || |W |+|Z| ||}2.

(22)

But max{w(ZW ∗), w(W ∗Z)} = w

[
ZW ∗ O
O W ∗Z

]
= w

[
O Z
W ∗ O

]2
≤ w2

[
O Z
W ∗ O

]
.

Thus

max{w(ZW ∗), w(W ∗Z)} ≤ w2

[
O Z
W ∗ O

]
. (23)

Combining inequalities (22) and (23), we obtain:

max{w(ZW ∗), w(W ∗Z)} ≤ 1

4
max{|| |W ∗|+ |Z∗| ||, || |W |+ |Z| ||}2. (24)

Note that inequality (24) is equivalent to inequality (6) if W and Z are self-adjoint oper-
ators.
In that sense, inequality (19) is a generalization of inequality (6) when W and Z are
self-adjoint operators.

Lemma 7. [7] Let A ∈ B(H) and r ≥ 2. Then

wr(A) ≤ 2r−3|| |A|r + |A∗|r ||. (25)

The following theorem is another generalization of inequality (4).

Theorem 4. Let X,Y be operators in B(H), r ≥ 2. Then

wr

[
O X
Y ∗ O

]
≤ 2r−3max{|| |X∗|r + |Y ∗|r ||, || |X|r + |Y |r ||}. (26)

In particular,

w2

[
O X
Y ∗ O

]
≤ 1

2
max{|| |X|2 + |Y |2 ||, || |X∗|2 + |Y ∗|2 ||} (27)

Proof. Letting A =

[
O X
Y ∗ O

]
in inequality (25), implies that

wr

[
O X
Y ∗ O

]
≤ 2r−3

∣∣∣∣∣∣∣∣[ |Y ∗|r O
O |X|r

]
+

[
|X∗|r O
O |Y |r

]∣∣∣∣∣∣∣∣
= 2r−3

∣∣∣∣∣∣∣∣[ |X∗|r + |Y ∗|r O
O |X|r + |Y |r

]∣∣∣∣∣∣∣∣
= 2r−3max{|| |X∗|r + |Y ∗|r ||, || |X|r + |Y |r ||}.



M. Al-Labadi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6461 8 of 13

Remark 6. Inequality (24) is sharper than inequality ( 29). To show this, note that

max{w(XY ∗), w(Y ∗X)} = w

[
XY ∗ O
O Y ∗X

]
= w

[
O X
Y ∗ O

]2
≤ w2

[
O X
Y ∗ O

]
. (28)

Combining inequalities (27) and (28), it follows that

max{ w(XY ∗), w(Y ∗X) } ≤ max{1
2
|| |X|2 + |Y |2 ||, 1

2
|| |X∗|2 + |Y ∗|2 ||}. (29)

By comparing inequality (24) and inequality (29), we note that while the left sides of both
inequalities are the same, the right side in inequality (24) is

1

4
max{|| |W |+ |Z| ||, || |W ∗|+ |Z∗| ||}2 ≤ 1

2
max{|| |W |2 + |Z|2 ||, || |W ∗|2 + |Z∗|2 ||}

(by inequality (11)), which is the right side of inequality (29). This implies that inequality
(24) is sharper than inequality (29).

Corollary 7. Let X be operators in B(H), r ≥ 2. Then

wr

[
O X
O O

]
≤ 2r−3max{|| |X∗|r ||, || |X|r ||}. (30)

Proof. Letting Y ∗ = O in inequality (26), we obtain the inequality (30).

Remark 7. Letting r = 2 in inequality (30), we obtain

w2

([
O X
O O

])
≤ 1

2
max{|| XX∗ ||, || X∗X ||} =

1

2
|| X∗X || = 1

2
|| X ||2. (31)

Theorem 5. Let X and Y be operators in B(H), r ≥ 2. Then

wr

[
X O
O Y ∗

]
≤ 1

2

∣∣∣∣∣∣∣∣[ |X|r + |X∗|r O
O |Y |r + |Y ∗|r

]∣∣∣∣∣∣∣∣ . (32)

In particular, if r = 2, we obtain

w2

[
X O
O Y ∗

]
≤ 1

2

∣∣∣∣∣∣∣∣[ |X|2 + |X∗|2 O
O |Y |2 + |Y ∗|2

]∣∣∣∣∣∣∣∣ . (33)

Proof. Let A =

[
X O
O Y ∗

]
in inequality (25), we obtain the inequality (32).

Remark 8. Letting Y = O in inequality (33), we obtain the inequality (4). In that sense,
inequality (33) is a generalization of inequality (4).



M. Al-Labadi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6461 9 of 13

Recall that the cartesian decomposition of the operator X is given by X = A + iB
where A = Re(X) and B = Im(X).
The author in [24] proved that if X = A+ iB and 0 < r ≤ 2, then

1

2
|| |A|r + |B|r || ≤ wr(X) ≤ || |A|r + |B|r ||. (34)

This implies , if r = 2,

1

2
|| |A|2 + |B|2 || ≤ w2(X) ≤ || |A|2 + |B|2 || = || A2 +B2 ||. (35)

In the following, we provide a new proof of the second inequality in (35).

Theorem 6. Let X ∈ B(H) with cartesian decomposition X = A+ iB. Then

w2(X) ≤ || A2 +B2 ||. (36)

Proof.

|⟨Xx, x⟩|2 = |⟨(A+ iB)x, x⟩|2

= |⟨Ax, x⟩+ i ⟨Bx, x⟩|2

= |⟨Ax, x⟩|2 + |⟨Bx, x⟩|2

= ⟨Ax, x⟩2 + ⟨Bx, x⟩2 (since < Ax, x >∈ R)
≤

〈
A2x, x

〉
+
〈
B2x, x

〉
(by Lemma (1) (i))

=
〈(
A2 +B2

)
x, x

〉
.

Taking the supremum over all unit vectors x ∈ H, we obtain the inequality (36).

4. Numerical radius inequalities via singular values and aluthge
transform

In this section, we prove numerical radius inequalities using recent singular values
inequalities and aluthge transform.
The author in [15] proves that if A,B ∈ B(H), then

2sj(AB
∗ +BA∗) ≤ s2j

[
A B
B A

]
. (37)

This inequality implies, since unitarily invariant norms and in particular the spectral norm,
are increasing functions of singular values, that

||AB∗ +BA∗|| ≤ 1

2

∣∣∣∣∣∣∣∣[ A B
B A

]∣∣∣∣∣∣∣∣2 . (38)



M. Al-Labadi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6461 10 of 13

Theorem 7. Let A and B be operators in B(H). Then

w(AB∗) ≤ 1

4
|| |A|+ |B| ||2 (39)

Proof. If A and B are operators in B(H), then

||Re(eiθAB∗)|| = 1

2
||eiθAB∗ + e−iθBA∗||

=
1

2
||AeiθB∗ + e−iθBA∗||

≤ 1

4

∣∣∣∣∣∣∣∣[ A e−iθB
e−iθB A

]∣∣∣∣∣∣∣∣2 (by inequality (38))

=
1

4

∣∣∣∣∣∣∣∣ ∣∣∣∣[ A e−iθB
e−iθB A

]∣∣∣∣ ∣∣∣∣∣∣∣∣2 (since ||A|| = || |A| ||)

=
1

4

∣∣∣∣∣∣∣∣ ∣∣∣∣[ A O
O A

]
+

[
O e−iθB

e−iθB O

]∣∣∣∣ ∣∣∣∣∣∣∣∣2

≤ 1

4

∣∣∣∣∣∣
∣∣∣∣∣∣
∣∣∣∣[ A O

O A

]∣∣∣∣+
∣∣∣∣∣∣
 O e−iθB

e−iθB O

∣∣∣∣∣∣
∣∣∣∣∣∣
∣∣∣∣∣∣
2

≤ 1

4

∣∣∣∣∣∣∣∣ ∣∣∣∣[ A O
O A

] ∣∣∣∣+ ∣∣∣∣[ |e−iθB| O
O |e−iθB|

]∣∣∣∣ ∣∣∣∣∣∣∣∣2 (by triangle inequality)

=
1

4

∣∣∣∣∣∣∣∣[ |A| O
O |A|

]
+

[
|B| O
O |B|

]∣∣∣∣∣∣∣∣2
=

1

4

∣∣∣∣∣∣∣∣[ |A|+ |B| O
O |A|+ |B|

]∣∣∣∣∣∣∣∣2
=

1

4
|| |A|+ |B| ||2.

Taking the supremum over all θ ∈ R, we obtain the inequality (39).

It can be shown easily that inequalities (6) and (39), if A and B are self-adjoint
operators, are the same.
It is obvious that in the general case if A and B are not self- adjoint operators, when
inequality (6) is sharper than inequality (39) then replacing A by A∗ and B by B∗ in the
same example will make inequality (39) is sharper than inequality (6).
The following example shows that inequality (39) is sharper than inequality (6).

Example 1. Let A =

[
0 4
2 0

]
, B =

[
9 0
0 1

]
. Then

|| |A|+ |B| || = 11 and || |A∗|+ |B∗| || = 13.



M. Al-Labadi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6461 11 of 13

This implies that inequality (39) is sharper than inequality (6), since

|| |A|+ |B| || < || |A∗|+ |B∗| ||.

This implies that

|| |A|+ |B| ||2 < || |A|+ |B| || || |A∗|+ |B∗| ||.

Remark 9. If we replace A by A∗ and B by B∗, in example 1, we obtain

|| |A∗|+ |B∗| || < || |A|+ |B| ||.

This implies that inequality (6) is sharper than inequality (39).

The Aluthge transform is used to study numerical radius bounds. Recall that if
A ∈ B(H), then the polar decomposition of A is given by A = U |A|, where U is a partial
isometry. For 0 ≤ t ≤ 1, the weighted Aluthge transform is defined by Ãt = |A|1−tU |A|t.
If t = 2, we write à instead of à 1

2
.

The following corollary is an application of Theorem 7, which inturns a generalization of
the second inequality of (1).

Corollary 8. Let T ∈ B(H). Then

w(T ) ≤ 1

4
|| |T |t + |T |1−t ||2, 0 ≤ t ≤ 1. (40)

In particular, if t = 1
2 , we obtain the second inequality of (1).

Proof. Letting A = |T |t and B∗ = |T |1−tU∗ in inequality (39), we obtain the inequality
(40).

5. Conclusions

Several numerical radius inequalities of operators are proved. We compare these new
inequalities with recent inequalities proved by Kittaneh. Our new inequalities refine and
generalize Kittaneh inequalities. We use several techniques to reach our new bounds for
numerical radius inequalities of operators. These techniques included inner products, block
matrices, singular values and aluthge transform.

Acknowledgements

The authors are grateful to the editor and referees for their comments and suggestions.
The authors are indebted to University of Petra for its support.



M. Al-Labadi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6461 12 of 13

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