Abstract
Let A, B, X, and Y be \(n\times n\) complex matrices such that A is self-adjoint, \(B\geq 0\), \(\pm A\leq B\), \(\max ( \Vert X \Vert ^{2}, \Vert Y \Vert ^{2} ) \leq 1\), and let f be a nonnegative increasing convex function on \([ 0,\infty ) \) satisfying \(f(0)=0\). Then
for \(j=1,2,\ldots,n\). This singular value inequality extends an inequality of Audeh and Kittaneh. Several generalizations for singular value and norm inequalities of matrices are also given.
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1 Introduction
Let \(\mathbb{M}_{n}(\mathbb{C})\) denote the algebra of all \(n\times n\) complex matrices. For \(A\in \mathbb{M}_{n}(\mathbb{C})\), the singular values of A are denoted by \(s_{{1}}(A)\geq s_{2}(A)\geq \cdots\geq s_{{n}}(A)\), they are precisely the eigenvalues of the positive operator \(\vert A \vert = ( A^{\ast }A ) ^{1/2}\). Singular values have several properties: Let \(A,B\in \mathbb{M}_{n}(\mathbb{C})\). Then
(a) \(s_{j}(A)=s_{j}(A^{\ast })=s_{j}( \vert A \vert )\) for \(j=1,2,\ldots,n\).
(b) \(s_{j}(AA^{\ast })=s_{j}(A^{\ast }A)\) for \(j=1,2,\ldots,n\).
(c) If \(A,B\in \mathbb{M}_{n}(\mathbb{C})\), then \(s_{j}(A)\leq s_{j}(B)\) if and only if \(s_{j}(A\oplus A)\leq s_{j}(B\oplus B)\) for \(j=1,2,\ldots,n\).
Bhatia and Kittaneh proved in [15] the following inequalities:
-
(i)
If \(A,B\in \mathbb{M}_{n}(\mathbb{C})\) such that A is self-adjoint, \(B\geq 0\), and \(\pm A\leq B\), then
$$ s_{j}(A)\leq s_{j}(B\oplus B) $$(1)for \(j=1,2,\ldots,n\).
-
(ii)
If \(A,B\in \mathbb{M}_{n}(\mathbb{C})\), then
$$ s_{j}\bigl(AB^{\ast }+BA^{\ast }\bigr)\leq s_{j}\bigl(\bigl(AA^{\ast }+BB^{\ast }\bigr)\oplus \bigl(AA^{ \ast }+BB^{\ast }\bigr)\bigr) $$(2)for \(j=1,2,\ldots n\). Audeh and Kittaneh pointed out in [8] that:
-
(i)
If \(A,B\in \mathbb{M}_{n}(\mathbb{C})\) such that A is self-adjoint, \(B\geq 0\), and \(\pm A\leq B\), then
$$ 2s_{j}(A)\leq s_{j}\bigl((B+A)\oplus (B-A)\bigr) $$(3)\(for\) \(j=1,2,\ldots,n\).
-
(ii)
If \(A,B,C\in \mathbb{M}_{n}(\mathbb{C})\) such that , then
$$ s_{j}(B)\leq s_{j}(A\oplus C) $$(4)for \(j=1,2,\ldots,n\).
-
(iii)
If \(A,B\in \mathbb{M}_{n}(\mathbb{C})\), then
$$ s_{j}(A+B)\leq s_{j}\bigl(\bigl( \vert A \vert + \vert B \vert \bigr)\oplus \bigl( \bigl\vert A^{\ast } \bigr\vert + \bigl\vert B^{ \ast } \bigr\vert \bigr)\bigr) $$(5)for \(j=1,2,\ldots,n\).
Tao proved in [24] that if \(A,B,C\in \mathbb{M}_{n}(\mathbb{C})\) such that , then
for \(j=1,2,\ldots,n\). In addition, Bhatia and Kittaneh showed in [14] that if \(A,B\in \mathbb{M}_{n}(\mathbb{C})\), then
for \(j=1,2,\ldots,n\). For more details and comprehensive results related to this topic, we refer to [1–7, 9, 10] and [17]. In this paper, we provide considerable generalizations of inequalities (1)–(6).
Unitarily invariant norms on \(\mathbb{M}_{n}\) are denoted by \(|\!|\!|.|\!|\!|\), recall that these norms satisfying \(|\!|\!|UAV|\!|\!|=|\!|\!|A|\!|\!|\) for all \(U,V,A\in \mathbb{M}_{n}\) such that U and V are unitary. Important classes of such norms are the Schatten p-norms defined by \(\Vert A \Vert _{p}= ( \sum_{j=1}^{n}s_{j}^{p}(A) ) ^{1/p}\) where \(p\geq 1\) and the spectral norm defined by \(\Vert A \Vert =s_{1}(A)\). For the general theory of unitarily invariant norms, we refer the reader to [13], [16], and [23]. It follows easily from the basic properties of unitarily invariant norms that
Bhatia and Davis proved in [13] that if \(A,X,B\in \mathbb{M}_{n}\), then
This is a generalization of the arithmetic–geometric mean inequality for unitarily invariant norms. In this paper, we provide a considerable generalization of inequality (9). Hou and Du proved in [19] that if \(A\in \mathbb{M}_{n}\), then
Popovici and Sebestyen showed in [22] the following inequalities:
-
1.
If \(A_{1},A_{2},\ldots,A_{n}\in \mathbb{M}_{n}\) are positive, then
$$ \Biggl\Vert \sum_{k=1}^{n}A_{k} \Biggr\Vert \leq \bigl\Vert \bigl( \bigl\Vert A_{i}^{1/2}A_{j}^{1/2} \bigr\Vert \bigr) _{1 \leq i,j\leq n} \bigr\Vert . $$(11) -
2.
If \(A_{1},A_{2},\ldots,A_{n}\in \mathbb{M}_{n}\) are positive, then
$$ \Biggl\Vert \sum_{k=1}^{n}A_{k}A_{k}^{\ast } \Biggr\Vert \leq \bigl\Vert \bigl( \bigl\Vert A_{i}^{\ast }A_{j} \bigr\Vert \bigr) _{1\leq i,j\leq n} \bigr\Vert . $$(12)
We provide inequalities that are more general and sharper than inequalities (11) and (12).
Zou in [26] demonstrated the following generalization of arithmetic–geometric mean inequality: Let \(A,X,B\in \mathbb{M}_{n}\) such that X is positive semidefinite Then
Among our results, we obtain a generalization of inequality (13).
2 Singular value inequalities
The following lemmas are essential for supporting our conclusions. The first lemma is an immediate consequence of the min-max principle (see, e.g., [2, p. 75]). The second and third lemmas were shown in [11].
Lemma 1
Let \(A,B,X\in \mathbb{M}_{n}(\mathbb{C})\). Then
for \(j=1,2,\ldots,n\).
Lemma 2
Let \(A\in \mathbb{M}_{n}(\mathbb{C})\) and let f be a nonnegative increasing function on an interval I. Then
for \(j=1,2,\ldots,n\). If A is Hermitian and f is increasing on an interval I, then
for \(j=1,2,\ldots,n\).
Lemma 3
Let f be a monotone convex function on an interval I such that \(0\in I\) and \(f(0)\leq 0\), and let \(A,X\in \mathbb{M}_{n}(\mathbb{C})\) such that A is Hermitian and X is a contraction. Then
for \(j=1,2,\ldots,n\).
The first result in this paper is now ready to be presented.
Theorem 1
Let \(A,B,X,Y\in \mathbb{M}_{n}(\mathbb{C})\) such that \(\max \{ \Vert X \Vert , \Vert Y \Vert \}\leq 1\), and let f be a nonnegative increasing convex function on \([ 0,\infty ) \) satisfying \(f(0)=0\). Then
for \(j=1,2,\ldots,n\).
Proof
Consider the operator matrix \(Q= [ ( XA ) ^{\ast }\ (YB)^{\ast } ] \in M_{n,2n}(\mathbb{C} )\).
for any \(A,B,X,Y\in M_{2n,2n}(\mathbb{C} )\).
By making use of inequality (4) and by letting
gives
□
Corollary 1
Let \(A,B,X,Y\in \mathbb{M}_{n}(\mathbb{C})\) such that \(\max \{ \Vert X \Vert , \Vert Y \Vert \}\leq 1\). Then, for \(r\geq 1\),
and
for \(j=1,2,\ldots,n\).
Proof
Letting \(f(t)=t^{r}\), \(r\geq 1\), and \(f(t)=e^{t}-1\) in Theorem 1 gives inequalities (16) and (17), respectively. □
By using Theorem 1, we here by present the following theorem.
Theorem 2
Let \(A,B,C,X,Y\in \mathbb{M}_{n}(\mathbb{C})\) such that , \(\max \{ \Vert X \Vert ^{2}, \Vert Y \Vert ^{2} \} \leq 1\), and let f be a nonnegative increasing convex function on \([ 0,\infty ) \) satisfying \(f(0)=0\). Then
for \(j=1,2,\ldots,n\).
Proof
Let . Then there exists a matrix \(Q= [ ( H ) ^{\ast } (L)^{\ast } ] \in M_{n,2n}(\mathbb{C} )\) such that \(P=Q^{\ast }Q\) for some \(H,L\in M_{n}(\mathbb{C} )\). Then \(A=HH^{\ast }\), \(C=LL^{\ast }\) and \(B=HL^{\ast }\). Applying inequality (4) gives
Inequality (18) has thus been proved. □
Remark 1
Letting \(X=Y=I\) and \(f(t)=t\) in inequality (18) gives inequality (4). In that sense, inequality (18) is certainly a generalization of inequality (4).
Corollary 2
Let \(A,B,C,X,Y\in \mathbb{M}_{n}(\mathbb{C})\) such that and \(\max \{ \Vert X \Vert ^{2}, \Vert Y \Vert ^{2} \} \leq 1\). Then, for \(r\geq 1\),
and
for \(j=1,2,\ldots,n\).
Proof
Letting \(f(t)=t^{r}\), \(r\geq 1\), and \(f(t)=e^{t}-1\) in Theorem 2 gives inequalities (19) and (20), respectively. □
Using the proper incites of Theorem 2 gives the following inequality, which is a generalization of inequality (1).
Theorem 3
Let \(A,B,X,Y\in \mathbb{M}_{n}(\mathbb{C})\) such that A is self-adjoint, \(B\geq 0\), \(\pm A\leq B\), \(\max \{ \Vert X \Vert ^{2}, \Vert Y \Vert ^{2} \} \leq 1\), and let f be a nonnegative increasing convex function on \([ 0,\infty ) \) satisfying \(f(0)=0\). Then
for \(j=1,2,\ldots,n\).
Proof
Let . Since is unitarily equivalent to and since \(\pm A\leq B\), it follows that P is a positive matrix. Applying inequality (18) to the operator matrix P gives inequality (21). □
Remark 2
Letting \(f(t)=t\) and \(X=Y=I\) in inequality (21) gives inequality (1). In that sense, inequality (21) is a generalization of inequality (1).
Corollary 3
Let \(A,B,X,Y\in \mathbb{M}_{n}(\mathbb{C})\) such that A is self-adjoint, \(B\geq 0\), \(\pm A\leq B\), and \(\max \{ \Vert X \Vert ^{2}, \Vert Y \Vert ^{2} \} \leq 1\). Then
and
for \(j=1,2,\ldots,n\).
Proof
Letting \(f(t)=t^{r}\), \(r\geq 1\), and \(f(t)=e^{t}-1\) in Theorem 3 gives inequalities (22) and (23), respectively. □
The following lemma, which was proved in [12], is necessary to prove the next result.
Lemma 4
Let \(A\in \mathbb{M}_{n}(\mathbb{C})\). Then
Theorem 4
Let \(A,B,X,Y\in \mathbb{M}_{n}(\mathbb{C})\) such that \(\max \{ \Vert X \Vert ^{2}, \Vert Y \Vert ^{2} \} \leq 1\), and let f be a nonnegative increasing convex function on \([ 0,\infty ) \) satisfying \(f(0)=0\) and
Then
for \(j=1,2,\ldots,n\).
Proof
Let and . Then \(Y\geq 0\) and
this implies that
Applying the conclusion of inequality (21) to the operator matrix \(ZYZ^{\ast }\) gives
which is precisely (25). □
Remark 3
Substituting \(f(t)=t\), \(X,Y,C=I\), and \(R=(AA^{\ast }+BB^{\ast })\) in Theorem 4 gives the following inequality, which is a generalization of inequality (2):
for \(j=1,2,\ldots,n\).
Remark 4
Letting \(f(t)=t\) in inequality (27), we give inequality (2). In that sense, inequality (27) is certainly a generalization of inequality (2).
The following result is a direct consequence of Theorem 2.
Corollary 4
Let \(A,B,X,Y\in \mathbb{M}_{n}(\mathbb{C})\) where \(\max \{ \Vert X \Vert ^{2}, \Vert Y \Vert ^{2} \} \leq 1\), and let f be a nonnegative increasing convex function on \([ 0,\infty ) \) satisfying \(f(0)=0\) and \(K=M\oplus N\), where
Then
for \(j=1,2,\ldots,n\).
Proof
It was shown in [15] that the matrix
is positive semidefinite. Now, inequality (28) is a direct consequence of Theorem 2. □
Remark 5
Substituting \(X=Y=I\) in inequality (28) leads to inequality (5). In that sense, inequality (28) is certainly a generalization of inequality (5).
The following inequality is a generalization of inequality (6).
Theorem 5
Let \(A,B,C,X,Y\in \mathbb{M}_{n}(\mathbb{C})\) such that , \(\max \{ \Vert X \Vert ^{2}, \Vert Y \Vert ^{2} \} \leq 1\), and let f be a nonnegative increasing convex function on \([ 0,\infty ) \) satisfying \(f(0)=0\). Then
for \(j=1,2,\ldots,n\).
Proof
Let . Then there exist \(H,L\in \mathbb{M}_{n}(\mathbb{C})\) such that \(P=Q^{\ast }Q\), where \(Q= [ ( H ) ^{\ast } (L)^{\ast } ] \). This means that \(A=HH^{\ast }\), \(C=LL^{\ast }\), and \(B=HL^{\ast }\).
for \(j=1,2,\ldots,n\), where \(W=H^{\ast }X^{\ast }XH+L^{\ast }Y^{\ast }YL\). Now, letting gives
Inequality (29) has thus been substantiated. □
Remark 6
Letting \(f(t)=t\) and \(X=Y=I\) in Theorem 5 gives inequality (6). In that sense inequality (29) is a generalization of inequality (6).
At this stage of our discussion, we provide a considerable generalization of inequality (3).
Theorem 6
Let \(A,B,X,Y\in \mathbb{M}_{n}(\mathbb{C})\) such that A is self-adjoint, \(B\geq 0\), \(\pm A\leq B\), \(\max \{ \Vert X \Vert ^{2}, \Vert Y \Vert ^{2} \} \leq 1\), and let f be a nonnegative increasing convex function on \([ 0,\infty ) \) satisfying \(f(0)=0\). Then
for \(j=1,2,\ldots,n\).
Proof
Since \(\pm A\leq B\), it follows that
If , then
which is equivalent to stating that \(Q\geq 0\). Now applying Theorem 5 to the operator matrix Q leads to
Inequality (30) has thus been substantiated. □
Corollary 5
Let \(A,B,X,Y\in \mathbb{M}_{n}(\mathbb{C})\) such that A is self-adjoint, \(B\geq 0\), \(\pm A\leq B\), and \(\max \{ \Vert X \Vert ^{2}, \Vert Y \Vert ^{2} \} \leq 1\). Then
and
for \(j=1,2,\ldots,n\).
Proof
Letting \(f(t)=t^{r}\), \(r\geq 1\), and \(f(t)=e^{t}-1\) in Theorem 6 gives inequalities (31) and (32), respectively. □
3 Norm inequalities
We begin this section by the following lemmas that are essential in our study. The first lemma is folk lemma, the second lemma was introduced in [22], and the third lemma was obtained in [3].
Lemma 5
Let \(A,B\in \mathbb{M}_{n}\), then
Lemma 6
Let \(A,B\in \mathbb{M}_{n}\), where AB is Hermitian. Then
Let \(A,B,X\in B(H)\) such that X is positive. Then
The following theorem can be obtained from norm inequality (13).
Theorem 7
Let \(A_{i},B_{i},X_{i}\in \mathbb{M}_{n}\), \(i=1,2,\ldots,n\), such that every \(X_{i}\) is positive,
and
then
Proof
On \(\oplus \mathbb{M}_{n}\), define
Then
and
where
Applying inequality (35) gives
Thus, inequality (36) has been substantiated. □
Remark 7
Inequality (36) is a general norm inequality that involves inequality (13). To see this, substitute \(A_{i}=X_{i}=B_{i}=0\) for \(i=2,\ldots,n\) in inequality (36), which leads to inequality (13).
Theorem 8
Let \(A_{i},X_{i},B_{i}\in \mathbb{M}_{n}\), \(i=1,2,\ldots,n\). Then
where
and
Proof
Replacing the operator matrices A, B, and X in inequality (9) gives inequality (37). □
The next special case of Theorem 8 was proved by Bhatia and Davis in [13].
Corollary 6
Let \(A,X,B\in \mathbb{M}_{n}\). Then
Proof
Inequality (38) follows from inequality (37) by substituting \(A_{i}=B_{i}=X_{i}=0\) for \(i=2,3,\ldots,n\). □
Another conclusion of Theorem 8 is a generalization of arithmetic–geometric mean inequality for unitarily invariant norms.
Corollary 7
Let \(A_{i},B_{i}\in \mathbb{M}_{n}\), \(i=1,2,\ldots,n\), such that
Then
Proof
Substituting \(X=I\) in inequality (37) gives inequality (39). □
Remark 8
Letting \(A_{i}=B_{i}=0\) for \(i=2,3,\ldots,n\) in inequality (39) gives
which is the arithmetic–geometric mean inequality for unitarily invariant norms.
The following inequality is a generalization and more general than inequality (24).
Corollary 8
Let \(A_{i}\in \mathbb{M}_{n}\), \(i=1,2,\ldots,n\), where
Then
Proof
Substituting \(B_{i}=A_{i}\) in inequality (39) gives inequality (40). □
Remark 9
By making use of inequality (10), we note that when we specify inequality (40) to the usual spectral norm, it is sharper than inequality (12).
The following inequality is a generalization and more general than inequality (11).
Corollary 9
Let \(A_{i}\in \mathbb{M}_{n}\), \(i=1,2,\ldots,n\). Then
Proof
Replacing A by \(A^{1/2}\) in inequality (40) gives inequality (41). □
Remark 10
By using inequality (27), we note that when we specify inequality (41) to the spectral norm, it is sharper than inequality (21).
Because of the large number of papers that discuss unitarily invariant norms for \(2\times 2\) operator matrices, we specialize inequality (37) for \(n=2\). This special case contains several remarkable inequalities.
Corollary 10
Let \(A_{i},X_{i},B_{i}\in \mathbb{M}_{n}\), \(i=1,2\). Then
\(where\)
Proof
Substituting \(A_{i}=X_{i}=B_{i}=0\) for \(i=3,4,\ldots,n\) in inequality (37) gives inequality (42). □
By making use of inequality (42), we provide the following inequality.
Corollary 11
Let \(A_{i},B_{i}\in \mathbb{M}_{n}\), \(i=1,2\). Then
where
and
Proof
Throughout the proof of this theorem, let
Substituting \(X_{1}=X_{2}=I\) in inequality (42) gives
Inequality (43) has thus been substantiated. □
In turn, inequality (43) gives us the following finding.
Corollary 12
Let \(A,B\in \mathbb{M}_{n}\). Then
where
Proof
Substituting \(A_{1}=B_{2}=A\) and \(A_{2}=B_{1}=B\) in inequality (43) gives inequality (44). □
Another attractive special case of inequality (43) is the following result, which was shown in [21].
Corollary 13
Let \(A,B\in \mathbb{M}_{n}\) be positive. Then
Proof
Substituting \(A_{1}=B_{1}=A^{1/2}\) and \(A_{2}=B_{2}=B^{1/2}\) in inequality (43) gives inequality (45). □
If we look at inequality (42) from another side, we obtain the following inequality, which was proven in [20].
Corollary 14
Let \(A,B\in \mathbb{M}_{n}\). Then
Proof
Throughout this proof, let
and
Substituting \(A_{1}=B_{2}=A\) and \(A_{2}=B_{1}=B\) in inequality (42) gives
which is precisely inequality (46). □
The next inequality is a special case of inequality (42), which was shown in [18].
Corollary 15
Let \(X_{1}, X_{2}\in \mathbb{M}_{n}\). Then
Proof
Substituting \(A_{1}=A_{2}=B_{1}=B_{2}=I\) in inequality (42) gives
This is equivalent to saying that
□
Another special case of inequality (42) has been established by using a completely different technique in [25].
Corollary 16
Let \(A, B \in \mathbb{M}_{n}\) be positive. Then
Proof
Letting \(A_{1}=B_{1}=A^{1/2}\), \(A_{2}=-B_{2}=B^{1/2}\) in inequality (42) gives
which is inequality (48). □
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References
Audeh, W.: Singular value inequalities and applications. Positivity 25, 843–852 (2020)
Audeh, W.: Generalizations for singular value and arithmetic-geometric mean inequalities of operators. J. Math. Anal. Appl. 489, 1–8 (2020)
Audeh, W.: Generalizations for singular value inequalities of operators. Adv. Oper. Theory 5, 371–381 (2020)
Audeh, W.: Singular value and norm inequalities of Davidson-power type. J. Math. Inequal. 15, 1311–1320 (2021)
Audeh, W.: Some generalizations for singular value inequalities of compact operators. Adv. Oper. Theory 6 (2021)
Audeh, W.: Singular value inequalities for operators and matrices. Ann. Funct. Anal. 13 (2022)
Audeh, W.: Singular value inequalities for accretive-dissipative normal operators. J. Math. Inequal. 16, 729–737 (2022)
Audeh, W., Kittaneh, F.: Singular value inequalities for compact operators. Linear Algebra Appl. 437, 2516–2522 (2012)
Audeh, W., Moradi, H., Sababheh, M.: Singular values inequalities via matrix monotone functions. Anal. Math. Phys. 13 (2023). https://doi.org/10.1007/s13324-023-00832-8
Audeh, W., Moradi, H., Sababheh, M.: Davidson-Power type and singular value inequalities. Hokkaido Math. J. To appear
Aujla, J.S., Silva, F.C.: Weak majorization inequalities and convex functions. Linear Algebra Appl. 369, 217–233 (2003)
Bhatia, R.: Matrix Analysis, GTM169. Springer, New York (1997)
Bhatia, R., Davis, C.: More matrix forms of the arithmetic-geometric mean inequality. SIAM J. Matrix Anal. Appl. 14, 132–136 (1993)
Bhatia, R., Kittaneh, F.: On the singular values of a product of operators. SIAM J. Matrix Anal. Appl. 11, 272–277 (1990)
Bhatia, R., Kittaneh, F.: The matrix arithmetic-geometric mean inequality revisited. Linear Algebra Appl. 428, 2177–2191 (2008)
Gohberg, I.C., Krein, M.G.: Introduction to the Theory of Linear Nonselfadjoint Operators. Am. Math. Soc., Providence (1969)
Hirzallah, O.: Inequalities for sums and products of operators. Linear Algebra Appl. 407, 32–42 (2005)
Hirzallah, O., Kittaneh, F.: Inequalities for sums and direct sums of Hilbert space operators. Linear Algebra Appl. 424, 71–82 (2007)
Hou, J.C., Du, H.K.: Norm inequalities of positive operator matrices. Integral Equ. Oper. Theory 22, 281–294 (1995)
Kittaneh, F.: Some norm inequalities for operators. Can. Math. Bull. 42, 87–96 (1999)
Kittaneh, F.: Norm inequalities for sums of positive operators. II. Positivity 10, 251–260 (2006)
Popovici, D., Sebestyen, Z.: Norm estimations for finite sums of positive operators. J. Oper. Theory 56, 3–15 (2006)
Simon, B.: Trace Ideals and Their Applications. Cambridge University Press, Cambridge (1979)
Tao, Y.: More results on singular value inequalities of matrices. Linear Algebra Appl. 416, 724–729 (2006)
Zhan, X.: Singular values of differences of positive semidefinite matrices. SIAM J. Matrix Anal. Appl. 22(3), 819–823 (2000)
Zou, L.: An arithmetic-geometric mean inequality for singular values and its applications. Linear Algebra Appl. 528, 25–32 (2017)
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The authors are grateful to the referees for their valuable suggestions. The authors are indebted to University of Petra for its support.
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Wasim Audeh and Manal Al-Labadi wrote section two, Raja’a Al-Naimi and Anwar Boustanji wrote section three, all authors reviewed the paper.
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Audeh, W., Al-Boustanji, A., Al-Labadi, M. et al. Singular value inequalities of matrices via increasing functions. J Inequal Appl 2024, 114 (2024). https://doi.org/10.1186/s13660-024-03193-3
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DOI: https://doi.org/10.1186/s13660-024-03193-3