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1 Introduction
Let \(\mathbb{M}_{n}\) be the set of \(n\times n\) complex matrices. For \(A\in\mathbb{M}_{n}\), the singular values and eigenvalues of A are denoted by \(\sigma_{i}(A)\) and \(\lambda_{i}(A)\), respectively, \(i=1,\ldots,n\). The singular values \(\sigma_{1}(A), \sigma _{2}(A),\ldots, \sigma_{n}(A)\) of a matrix A are the eigenvalues of \(|A|=(A^{*}A)^{\frac{1}{2}}\) arranged in decreasing order and repeated according to multiplicity. The Ky Fan k-norm, a particular unitarily invariant norm, is defined as \(\|\cdot\|_{(k)}=\sum_{j=1}^{k}\sigma _{j}(A)\), \(1\leq k \leq n\). If A is Hermitian, then all eigenvalues of A are real and ordered as \(\lambda_{1}(A)\geq\cdots\geq\lambda_{n}(A)\).
Let \(A,B\in\mathbb{M}_{n}\). Bhatia and Kittaneh [8] proved an arithmetic–geometric mean inequality for unitarily invariant norms
As a generalization of (1), Bhatia and Davis [7] proved that
for \(A,X,B\in\mathbb{M}_{n}\).
Albadawi [3] obtained a stronger version of the Hölder inequality for unitarily invariant norms. Let \(A, X, B\in\mathbb {M}_{n}\) and \(\frac{1}{p}+\frac{1}{q}=1\), \(p,q>1\), \(r\geq0\). Then
which is a generalization of Horn and Zhan’s result [10] (also called the Hölder inequality)
Recently, Audenaert [5] proved that if \(A, B\in\mathbb{M}_{n}\) and \(\frac{1}{p}+\frac{1}{q}=1\), \(p, q>1\), \(r\geq0\), \(\alpha\in [0,1]\), then
which is a unification of inequalities (1) and (4). By setting \(r=1\) and \(p=p'=2\) in (5) we have
Lin [12] gave a new proof of inequality (6). Zou and Jiang [16] generalized it to the following inequality: Let \(A, B, X\in\mathbb{M}_{n}\) and \(q\in[0,1]\). Then
Al-khlyleh and Kittaneh [2, Theorem 2.5] presented an inequality that refines inequality (7) for the particular unitarily invariant norm, Hilbert–Schmidt norm. For more results on interpolation between the arithmetic–geometric mean inequality and the Cauchy–Schwarz inequality for matrices, see [1].
In this paper, we provide alternative proofs of inequalities (5) and (7), which provide new perspectives to the elegant results.
2 Main results
For presenting the new proofs, we need the following several lemmas.
Lemma 2.1
(see [6, Proposition IX.1.2])
Let\(A, B\in\mathbb{M}_{n}\)be any two matrices such that the productABis Hermitian. Then, for every unitarily invariant norm, we have
Lemma 2.2
(see [6, p. 41])
Let\(A, B\in\mathbb{M}_{n}\)and suppose thatfis convex and increasing on\([0, \infty)\). If
then
Lemma 2.3
(see [6, p. 35])
Let\(A, B\in\mathbb {M}_{n}\). Then
Lemma 2.4
(see [14, p. 63])
If\(A\in\mathbb {M}_{n}\), then
Lemma 2.5
Let\(A, B\in\mathbb{M}_{n}\)be positive semidefinite and\(0\leq q \leq1\). Then
Audenaert [5] proved the following theorem. We give a different proof of the result.
Theorem 2.6
Let\(A, B\in\mathbb{M}_{n}\)and\(\frac{1}{p}+\frac{1}{q}=1\), \(p,q>1\), \(r\geq0\), \(\alpha\in[0,1]\). Then
Proof
By Fan’s dominance theorem (see [11, Theorem 1.4]) (11) is equivalent to
for \(k=1,\ldots,n\).
First, let us show this inequality for the Ky Fan 1-norm, that is, the spectral norm:
which means that
Second, using a standard argument via the antisymmetric product (see [5, p. 18]), (13) yields
for \(k=1,\ldots,n\). Since weak log-majorization implies weak majorization (see, [9, p. 174]), by (10) we have
for \(k=1,\ldots,n\). The left-hand side is \(\||A^{*}B|^{r}\|_{(k)}\). By the Hölder inequality the right-hand side is bounded from above by
Thus (12) holds, and so does the conclusion. This completes the proof. □
In fact, by a similar technique used in the theorem, we may present a new proof of the following result due to Zou [10], which is a unified version of inequalities (2) and (3).
Theorem 2.7
Let\(A, B, X\in\mathbb{M}_{n}\)and\(\frac{1}{p}+\frac{1}{q}=1\), \(p,q>1\), \(r\geq\max \{\frac{1}{p},\frac{1}{q} \}\), \(\alpha\in[0,1]\). Then
Proof
There is a subtle difference between the proof of (14) and that of the previous theorem although most techniques are similar. For the readers’ convenience, we present the proof simply.
By Fan’s dominance theorem (14) is equivalent to
for all \(k=1,\ldots,n\).
If X is a positive semidefinite matrix, then for Ky Fan 1-norm, we have
which means that
Using a standard argument via the antisymmetric product (see [5, p. 18]), (15) yields
for \(k=1,\ldots,n\). Since weak log-majorization implies weak majorization (see, [9, p. 174]), we have
for \(k=1,\ldots,n\). The left-hand side is \(\||A^{*}XB|^{2r}\|_{(k)}\). By the Hölder inequality the right-hand side is bounded from above by
Thus
Since \((1-\alpha)X^{\frac{1}{2}}AA^{*}X^{\frac{1}{2}}+\alpha X^{\frac {1}{2}}BB^{*}X^{\frac{1}{2}}\) and \(\alpha X^{\frac{1}{2}}AA^{*}X^{\frac {1}{2}}+(1-\alpha) X^{\frac{1}{2}}BB^{*}X^{\frac{1}{2}}\) are Hermitian, since \(r\geq\max \{\frac{1}{p},\frac{1}{q} \} \), the previous inequalities become
Next, we consider the case where X is any matrix. By the singular value decomposition we know that there exist unitary matrices U and V such that \(X=UDV^{*}\), and then by (16) we have
where the last equality is due to the fact that \(\||U_{1}^{*}PU_{2}|^{r}\| =\||P|^{r}\|\) for any \(P\in\mathbb{M}_{n}\) and unitary matrices \(U_{1}, U_{2}\). This completes the proof. □
Finally, we give an alterative proof of (7) due to Zou and Jiang [16, Theorem 2.1].
Theorem 2.8
Let\(A, B, X\in\mathbb{M}_{n}\)and\(q\in[0,1]\). Then
Proof
First, consider the special case where A, B, X are Hermitian and \(A=B\). Then
Similarly,
which is just the desired inequality in this particular case.
Next, consider the more general situation where A and B are Hermitian and X is any matrix. Let
Then by the particular case considered before
Multiplying out the block-matrices, we have
Hence we obtain the following inequality from (19):
which means that
So by (17) we have
The following inequality can be proved in exactly the same way:
In this case, from (20) and (21) we have
Finally, Let \(A=UA_{1}\) and \(B=VB_{1}\) be polar decompositions of A and B. Then
whereas
So the theorem follows from inequality (22). This completes the proof. □
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The work is supported by Hainan Provincial Natural Science Foundation for High-level Talents grant no. 2019RC171, the Ministry of Education of Hainan grant no. Hnky2019ZD-13, China Scholarship Council grant no. 201908460006, the Ministry of Education of Hainan grant no. Hnky2019ZD-13, the Provincial Key Laboratory, Hainan Normal University grant no. JSKX201904, and National Natural Science Foundation of China grant 11671105.
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Yang, J., Lu, L. New proofs on two recent inequalities for unitarily invariant norms. J Inequal Appl 2020, 133 (2020). https://doi.org/10.1186/s13660-020-02402-z
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DOI: https://doi.org/10.1186/s13660-020-02402-z