1 Introduction
We denote by \(\mathbb{M}_{n}(\mathbb{C})\) the set of \(n\times n\) complex matrices. For \(A\in \mathbb{M}_{n}(\mathbb{C})\), the conjugate transpose of A is denoted by \(A^{*}\), and the matrices \(\mathfrak{R}A= \frac{1}{2}(A+A^{*})\) and \(\mathfrak{I}A=\frac{1}{2i}(A-A^{*})\) are called the real part and imaginary part of A, respectively (e.g., [1, p. 6] and [6, p. 7]). Recall that a norm \(\|\cdot \|\) on \(\mathbb{M}_{n}(\mathbb{C})\) is unitarily invariant if \(\|UAV\|=\|A\|\) for any \(A\in \mathbb{M}_{n}(\mathbb{C})\) and unitarily matrices \(U, V \in \mathbb{M}_{n}(\mathbb{C})\). For \(p \geq 1\), the Schatten p-norm of \(A\in \mathbb{M}_{n}(\mathbb{C})\) is defined as \(\|A\|_{p}= (\sum_{j=1}^{n}\sigma _{j}^{p}(A) )^{\frac{1}{p}}\). If the eigenvalues of a square matrix \(A\in \mathbb{M}_{n}(\mathbb{C})\) are all real, then we denote \(\lambda _{j}(A)\) the jth largest eigenvalue of A. The singular values of a complex matrix \(A\in \mathbb{M}_{n}\) are the eigenvalues of \(|A|:=(A^{*}A)^{\frac{1}{2}}\), and we denote \(\sigma _{j}(A):=\lambda _{j}(|A|)\) the jth largest singular value of A. A positive semidefinite matrix A will be expressed as \(A\geq 0\). Likewise, we write \(A>0\) to refer that A is a positive definite matrix.
\(A\in \mathbb{M}_{n}(\mathbb{C})\) is an accretive-dissipative matrix if \(\mathfrak{R} A\) and \(\mathfrak{I} A\) are both positive semidefinite. This class of matrices has recently been considered by Lin [9] and Lin and Zhou [11].
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Let \(H\in \mathbb{M}_{n}(\mathbb{C})\) be a Hermitian matrix and let f be a real-valued function defined on an interval containing all the eigenvalues of H. Then \(f(H)\) is well defined through spectral decomposition. f is called matrix concave if \(f(\alpha A+(1-\alpha )B)\geq \alpha f(A)+(1-\alpha )f(B)\) for any two Hermitian matrices \(A,B\in \mathbb{M}_{n}(\mathbb{C})\) and all \(\alpha \in [0,1]\).
The numerical range of \(A\in \mathbb{M}_{n}(\mathbb{C})\) is defined by
For \(\alpha \in [0, \pi /2)\), let
be a sector region on the complex plane. A matrix whose numerical range is contained in a sector region \(S_{\alpha }\) is called a sector matrix [10]. Recent research interest in this class of matrices starts with a resolution of a problem from numerical analysis [3]. Some research results on sector matrices can be found in [3, 7, 8, 10, 13].
$$\begin{aligned} W(A)=\bigl\{ x^{*}Ax|x\in \mathbb{C}^{n}, x^{*}x=1 \bigr\} . \end{aligned}$$
$$\begin{aligned} S_{\alpha }=\bigl\{ z\in \mathbb{C}|\mathfrak{R} z \geq 0, \vert \mathfrak{I} z \vert \leq (\mathfrak{R} z)\tan (\alpha )\bigr\} \end{aligned}$$
Kittaneh and Sakkijha [7] proved the following Schatten-p norm inequalities.
Theorem 1.1
(see [7, Theorem 2.7])
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Let
\(S, T \in \mathbb{M}_{n}(\mathbb{C})\)
be accretive-dissipative. Then
$$\begin{aligned} 2^{\frac{-p}{2}}\bigl( \Vert S \Vert _{p}^{p}+ \Vert T \Vert _{p}^{p}\bigr)\leq \Vert S+T \Vert _{p}^{p} \leq 2^{\frac{3p}{2}-1}\bigl( \Vert S \Vert _{p}^{p}+ \Vert T \Vert _{p}^{p} \bigr) \quad\textit{for } p \geq 1. \end{aligned}$$
Theorem 1.2
(see [12, Theorem 2.3])
Let
\(A_{1},\ldots, A_{n} \in \mathbb{M}_{n}(\mathbb{C})\)
be accretive-dissipative. Then
$$\begin{aligned} 2^{\frac{-p}{2}}\sum_{j=1}^{n} \Vert A_{j} \Vert _{p}^{p}\leq \Biggl\Vert \sum_{j=1} ^{n}A_{j} \Biggr\Vert _{p}^{p} \leq \frac{(2n^{2})^{\frac{p}{2}}}{n}\sum _{j=1}^{n} \Vert A_{j} \Vert _{p}^{p}\quad \textit{for }p \geq 1. \end{aligned}$$
In [5], Garg and Aujla presented the following inequalities:
where \(A, B \in \mathbb{M}_{n}(\mathbb{C})\) and \(f: [0,\infty )\rightarrow [0,\infty )\) is a matrix concave function.
$$\begin{aligned} &\prod_{j=1}^{k}\sigma _{j}\bigl( \vert A+B \vert ^{r}\bigr) \\ &\quad\leq \prod _{j=1}^{k}\sigma _{j} \bigl(I_{n}+ \vert A \vert ^{r}\bigr)\prod _{j=1}^{k}\sigma _{j}\bigl(I_{n}+ \vert B \vert ^{r}\bigr) \quad\text{for } 1 \leq k\leq n, 1\leq r \leq 2; \end{aligned}$$
(1)
$$\begin{aligned} &\prod_{j=1}^{k}\sigma _{j} \bigl(I_{n}+f\bigl( \vert A+B \vert \bigr)\bigr) \\ &\quad \leq \prod_{j=1}^{k} \sigma _{j}\bigl(I_{n}+f\bigl( \vert A \vert \bigr)\bigr)\prod _{j=1}^{k}\sigma _{j} \bigl(I_{n}+f\bigl( \vert B \vert \bigr)\bigr) \quad\text{for } 1 \leq k \leq n, \end{aligned}$$
(2)
If \(A, B \geq 0, r=1\) and \(f(X)=X\) for any \(X \in \mathbb{M}_{n}( \mathbb{C})\) in (1) and (2), then
and
$$\begin{aligned} \prod_{j=1}^{k}\sigma _{j}(A+B)\leq \prod_{j=1}^{k} \sigma _{j}(I_{n}+A) \prod_{j=1}^{k} \sigma _{j}(I_{n}+B)\quad \text{for } 1 \leq k\leq n \end{aligned}$$
$$\begin{aligned} \prod_{j=1}^{k}\sigma _{j}(I_{n}+A+B)\leq \prod_{j=1}^{k} \sigma _{j}(I _{n}+A)\prod_{j=1}^{k} \sigma _{j}(I_{n}+B) \quad\text{for } 1 \leq k\leq n. \end{aligned}$$
(3)
Based on the above inequalities, Yang and Lu (see [12, Theorem 2.7]) gave the inequalities for sector matrices which removed the absolute values in (1) and (2) from the right sides as follows.
Theorem 1.3
(see [12, Theorem 2.7])
Let
\(A, B\in \mathbb{M}_{n}(\mathbb{C})\)
be such that
\(W(A), W(B)\subseteq S_{\alpha }\). Then, for
\(1 \leq k \leq n\),
and
$$\begin{aligned} \prod_{j=1}^{k}\sigma _{j}(A+B)\leq \sec ^{2k}(\alpha )\prod _{j=1}^{k} \sigma _{j}(I_{n}+A) \prod_{j=1}^{k}\sigma _{j}(I_{n}+B) \end{aligned}$$
(4)
$$\begin{aligned} \prod_{j=1}^{k}\sigma _{j}(I_{n}+A+B)\leq \sec ^{2k}(\alpha )\prod _{j=1} ^{k}\sigma _{j}(I_{n}+A) \prod_{j=1}^{k}\sigma _{j}(I_{n}+B). \end{aligned}$$
(5)
Theorem 1.4
(see [12, Corollary 2.8])
Let
\(A, B\in \mathbb{M}_{n}(\mathbb{C})\)
be such that
\(W(A), W(B) \subseteq S_{\alpha }\). Then
and
$$\begin{aligned} \Vert A+B \Vert \leq \sec ^{2}(\alpha ) \Vert I_{n}+A \Vert \Vert I_{n}+B \Vert \end{aligned}$$
(6)
$$\begin{aligned} \Vert I_{n}+A+B \Vert \leq \sec ^{2}( \alpha ) \Vert I_{n}+A \Vert \Vert I_{n}+B \Vert . \end{aligned}$$
(7)
2 Main results
We begin this section with some lemmas which are useful to establish our main results.
Lemma 2.1
(see [1, p. 73])
Let
\(A \in \mathbb{M}_{n}(\mathbb{C})\). Then
Consequently,
$$\begin{aligned} \lambda _{j}(\mathfrak{R} A)\leq \sigma _{j}( A),\quad j=1,2,\ldots,n. \end{aligned}$$
(8)
$$\begin{aligned} \Vert \mathfrak{R} A \Vert \leq \Vert A \Vert . \end{aligned}$$
(9)
Lemma 2.2
(see [13, Lemma 3.1])
Let
\(A \in \mathbb{M}_{n}(\mathbb{C})\)
be such that
\(W(A)\subseteq S_{\alpha }\)
for some
\(\alpha \in [0, \pi /2)\). Then
$$\begin{aligned} \Vert A \Vert \leq \sec (\alpha ) \Vert \mathfrak{R} A \Vert . \end{aligned}$$
(10)
Lemma 2.3
(see [2, (4)])
Let
\(A_{1},\ldots,A_{n} \in \mathbb{M} _{n}(\mathbb{C})\)
be positive semidefinite. Then, for
\(p \geq 1\),
$$\begin{aligned} \sum_{j=1}^{n} \Vert A_{j} \Vert _{p}^{p}\leq \Biggl\Vert \sum _{j=1}^{n}A_{j} \Biggr\Vert _{p}^{p}\leq n^{p-1}\sum _{j=1}^{n} \Vert A_{j} \Vert _{p}^{p}. \end{aligned}$$
(11)
Lemma 2.4
(see [4, Theorem 4.1])
Let
\(A \in \mathbb{M}_{n}( \mathbb{C})\)
be such that
\(W(A)\subseteq S_{\alpha }\). Then
$$\begin{aligned} \sigma _{j}(A)\leq \sec ^{2}(\alpha )\lambda _{j}(\mathfrak{R}A),\quad j=1,2, \ldots, n. \end{aligned}$$
The above inequality implies that there exists a unitary matrix \(U\in \mathbb{M}_{n}(\mathbb{C})\) such that
$$\begin{aligned} \vert A \vert \leq \sec ^{2}(\alpha )U \mathfrak{R}AU^{*}. \end{aligned}$$
(12)
Lemma 2.5
(see [1, Theorem III.5.6])
Let
\(A,B \in \mathbb{M} _{n}(\mathbb{C})\). Then there exist unitary matrices
\(U,V \in \mathbb{M}_{n}(\mathbb{C})\)
such that
$$\begin{aligned} \vert A+B \vert \leq U \vert A \vert U^{*}+V \vert B \vert V^{*}. \end{aligned}$$
(13)
For the above preparation, we present the first main result which is an extension of Theorem 1.1.
Theorem 2.6
Let
\(A_{1}, \ldots, A_{n}\in \mathbb{M}_{n}(\mathbb{C})\)
be sector matrices. Then
$$\begin{aligned} \cos ^{p}(\alpha )\sum_{j=1}^{n} \Vert A_{j} \Vert _{p}^{p}\leq \Biggl\Vert \sum_{j=1} ^{n}A_{j} \Biggr\Vert _{p}^{p}\leq \sec ^{p}(\alpha )n^{p-1}\sum _{j=1}^{n} \Vert A_{j} \Vert _{p}^{p}\quad \textit{for } p \geq 1. \end{aligned}$$
(14)
Proof
Let \(A_{j}=B_{j}+iC_{j}\) be the Cartesian decompositions of \(A_{j}\), \(j=1, \ldots,n\). Then we have
which proves the first inequality.
$$\begin{aligned} \Biggl\Vert \sum_{j=1}^{n}A_{j} \Biggr\Vert _{p}^{p} &= \Biggl\Vert \sum _{j=1}^{n}(B _{j}+iC_{j}) \Biggr\Vert _{p}^{p} \\ &= \Biggl\Vert \sum_{j=1}^{n}B_{j}+i \sum_{j=1}^{n}C_{j} \Biggr\Vert _{p}^{p} \\ &\geq \Biggl\Vert \sum_{j=1}^{n}B_{j} \Biggr\Vert _{p}^{p}\quad (\text{by }\text{(9)}) \\ &\geq \sum_{j=1}^{n} \Vert B_{j} \Vert _{p}^{p}\quad (\text{by }\text{(11)}) \\ &\geq \cos ^{p}(\alpha )\sum_{j=1}^{n} \Vert A_{j} \Vert _{p}^{p}\quad (\text{by }\text{(10)}), \end{aligned}$$
To prove the second inequality, compute
which completes the proof. □
$$\begin{aligned} \Biggl\Vert \sum_{j=1}^{n}A_{j} \Biggr\Vert _{p}^{p} &= \Biggl\Vert \sum _{j=1}^{n}(B _{j}+iC_{j}) \Biggr\Vert _{p}^{p} \\ &= \Biggl\Vert \sum_{j=1}^{n}B_{j}+i \sum_{j=1}^{n}C_{j} \Biggr\Vert _{p}^{p} \\ &\leq \sec ^{p}(\alpha ) \Biggl\Vert \sum _{j=1}^{n}B_{j} \Biggr\Vert _{p}^{p}\quad (\text{by }\text{(10)}) \\ &\leq \sec ^{p}(\alpha )n^{p-1}\sum _{j=1}^{n} \Vert B_{j} \Vert _{p}^{p} \quad(\text{by }\text{(11)}) \\ &\leq \sec ^{p}(\alpha )n^{p-1}\sum _{j=1}^{n} \Vert A_{j} \Vert _{p}^{p} \quad(\text{by }\text{(9)}), \end{aligned}$$
Remark 2.7
Next, we end this section with a generalization of singular value inequality for two positive semidefinite matrices to sector matrices.
Theorem 2.8
Let
\(A, B\in \mathbb{M}_{n}(\mathbb{C})\)
be such that
\(W(A), W(B) \subseteq S_{\alpha }\). Then, for
\(1 \leq k\leq n\), the following assertions hold:
and
$$\begin{aligned} \prod_{j=1}^{k}\sigma _{j}(A+B)\leq \prod_{j=1}^{k} \sigma _{j}\bigl(I_{n}+ \sec ^{2}(\alpha )A\bigr) \prod_{j=1}^{k}\sigma _{j} \bigl(I_{n}+\sec ^{2}(\alpha )B\bigr) \end{aligned}$$
(15)
$$\begin{aligned} \prod_{j=1}^{k}\sigma _{j}(I_{n}+A+B)\leq \prod_{j=1}^{k} \sigma _{j}\bigl(I _{n}+\sec ^{2}(\alpha )A\bigr) \prod_{j=1}^{k}\sigma _{j} \bigl(I_{n}+\sec ^{2}( \alpha )B\bigr). \end{aligned}$$
(16)
Proof
Compute
where \(U,V\) are unitary matrices.
$$\begin{aligned} \prod_{j=1}^{k}\sigma _{j}(A+B)\leq{}& \prod_{j=1}^{k} \sigma _{j}\bigl(I_{n}+ \vert A \vert \bigr) \prod _{j=1}^{k}\sigma _{j} \bigl(I_{n}+ \vert B \vert \bigr)\quad (\text{by}~\text{(1)}) \\ \leq{} &\prod_{j=1}^{k}\sigma _{j}\bigl(I _{n}+\sec ^{2}(\alpha )U \mathfrak{R}AU^{*}\bigr)\prod_{j=1}^{k} \sigma _{j}\bigl(I _{n}+\sec ^{2}(\alpha )V \mathfrak{R}BV^{*}\bigr) \\ & (\text{by}~\text{(12)}) \\ \leq{}& \prod_{j=1}^{k}\sigma _{j}\bigl(I _{n}+\sec ^{2}(\alpha )A\bigr)\prod _{j=1}^{k}\sigma _{j} \bigl(I_{n}+\sec ^{2}( \alpha )B\bigr) \quad(\text{by}~\text{ (8)}), \end{aligned}$$
To prove (16), compute
where \(U_{j}\) and \(V_{j}\), \(j=1,2,3\), are unitary matrices.
$$\begin{aligned} \prod_{j=1}^{k}\sigma _{j}\bigl( \vert I_{n}+A+B \vert \bigr)\leq {}&\prod _{j=1}^{k}\sigma _{j}\bigl(U _{1} \vert I_{n} \vert U_{1}^{*}+V_{1} \vert A+B \vert V_{1}^{*}\bigr)\quad (\text{by}~ \text{ (13)}) \\ ={}&\prod_{j=1}^{k}\sigma _{j} \bigl(I_{n}+ \vert A+B \vert \bigr) \\ \leq{}& \prod_{j=1}^{k}\sigma _{j}\bigl(I_{n}+U_{2} \vert A \vert U_{2}^{*}+V_{2} \vert B \vert V_{2} ^{*}\bigr)\quad (\text{by}~\text{(13)}) \\ \leq{} &\prod_{j=1}^{k} \sigma _{j}\bigl(I_{n}+\sec ^{2}(\alpha )U_{3} \mathfrak{R}AU_{3}^{*}+\sec ^{2}(\alpha )V_{3}\mathfrak{R}BV_{3}^{*}\bigr) \\ & (\text{by}~\text{(12)}) \\ \leq{}& \prod_{j=1}^{k}\sigma _{j}\bigl(I _{n}+\sec ^{2}(\alpha )\mathfrak{R}A \bigr)\prod_{j=1}^{k}s_{j} \bigl(I_{n}+\sec ^{2}(\alpha )\mathfrak{R}B\bigr) \\ & (\text{by}~\text{(3)}) \\ \leq{} &\prod_{j=1}^{k}\sigma _{j}\bigl(I _{n}+\sec ^{2}(\alpha )A\bigr)\prod _{j=1}^{k}\sigma _{j} \bigl(I_{n}+\sec ^{2}( \alpha )B\bigr), \\ & (\text{by}~\text{(8)}), \end{aligned}$$
Thus
This completes the proof. □
$$\begin{aligned} \prod_{j=1}^{k}\sigma _{j}(I_{n}+A+B) \leq \prod_{j=1}^{k}\sigma _{j} \bigl(I _{n}+\sec ^{2}(\alpha )A\bigr)\prod _{j=1}^{k}\sigma _{j}\bigl(I_{n}+ \sec ^{2}( \alpha )B\bigr). \end{aligned}$$
Example 2.9
Let
and
be such that \(W(A), W(B)\subseteq S_{\frac{\pi }{4}}\).
For the right side of the inequality (4),
When \(k=2\), for the right side of the inequality (15),
This shows that (15) is stronger than (4).
$$ \sec ^{4}(\alpha )\prod_{j=1}^{2} \sigma _{j}(I_{2}+A)\prod_{j=1}^{2} \sigma _{j}(I_{2}+B)=5.2098. $$
$$ \prod_{j=1}^{2}\sigma _{j} \bigl(I_{2}+\sec ^{2}(\alpha )A\bigr)\prod _{j=1}^{2} \sigma _{j}\bigl(I_{2}+ \sec ^{2}(\alpha )B\bigr)=1.6886. $$
Example 2.10
If
and
are such that \(W(A), W(B)\subseteq S_{\frac{\pi }{4}}\), we also suppose \(k=2\).
For the right side of the inequality (4),
For the right side of the inequality (15),
The example implies that the bound in (15) is weaker than that in (4).
$$ \sec ^{4}(\alpha )\prod_{j=1}^{2} \sigma _{j}(I_{2}+A)\prod_{j=1}^{2} \sigma _{j}(I_{2}+B)=69.8872. $$
$$ \prod_{j=1}^{2}\sigma _{j} \bigl(I_{2}+\sec ^{2}(\alpha )A\bigr)\prod _{j=1}^{2} \sigma _{j}\bigl(I_{2}+ \sec ^{2}(\alpha )B\bigr)=94.3901. $$
Remark 2.11
Corollary 2.12
Let
\(A, B\in \mathbb{M}_{n}(\mathbb{C})\)
be such that
\(W(A), W(B) \subseteq S_{\alpha }\). Then, for all unitarily invariant norms
\(\|\cdot \|\)
on
\(\mathbb{M}_{n}(\mathbb{C})\),
and
$$\begin{aligned} \Vert A+B \Vert \leq \bigl\Vert I_{n}+\sec ^{2}(\alpha )A \bigr\Vert \bigl\Vert I_{n}+\sec ^{2}(\alpha )B \bigr\Vert \end{aligned}$$
(17)
$$\begin{aligned} \Vert I_{n}+A+B \Vert \leq \bigl\Vert I_{n}+\sec ^{2}(\alpha )A \bigr\Vert \bigl\Vert I_{n}+\sec ^{2}(\alpha )B \bigr\Vert . \end{aligned}$$
(18)
Proof
From (15) and (16), we obtain for \(1 \leq k\leq n\)
and
By the property that weak log-majorization implies weak majorization, we have for \(1 \leq k\leq n\)
and
Then, by the Cauchy–Schwarz inequality, for \(1 \leq k\leq n\)
and
By Fan’s dominance principle [1, p. 93], we have
and
Let \(A+B=U|A+B|, I_{n}+A+B=V|I_{n}+A+B|\) be the polar decomposition of \(A+B\) and \(I_{n}+A+B\), respectively, where U and V are unitary matrices. Thus, by (19), we have
Similarly, by (20) we have
which completes the proof. □
$$\begin{aligned} \prod_{j=1}^{k}\sigma _{j}^{\frac{1}{2}}(A+B) \leq \prod_{j=1}^{k}\sigma _{j}^{\frac{1}{2}}\bigl(I_{n}+\sec ^{2}(\alpha )A \bigr)\sigma _{j}^{\frac{1}{2}}\bigl(I _{n}+\sec ^{2}(\alpha )B\bigr) \end{aligned}$$
$$\begin{aligned} \prod_{j=1}^{k}\sigma _{j}^{\frac{1}{2}}(I_{n}+A+B) \leq \prod_{j=1} ^{k}\sigma _{j}^{\frac{1}{2}}\bigl(I_{n}+\sec ^{2}(\alpha )A \bigr)\sigma _{j}^{ \frac{1}{2}}\bigl(I_{n}+\sec ^{2}(\alpha )B\bigr). \end{aligned}$$
$$\begin{aligned} \sum_{j=1}^{k}\sigma _{j}^{\frac{1}{2}}(A+B) \leq \sum_{j=1}^{k}\sigma _{j}^{\frac{1}{2}}\bigl(I_{n}+\sec ^{2}(\alpha )A \bigr)\sigma _{j}^{\frac{1}{2}}\bigl(I _{n}+\sec ^{2}(\alpha )B\bigr) \end{aligned}$$
$$\begin{aligned} \sum_{j=1}^{k}\sigma _{j}^{\frac{1}{2}}(I_{n}+A+B) \leq \sum_{j=1}^{k} \sigma _{j}^{\frac{1}{2}}\bigl(I_{n}+\sec ^{2}(\alpha )A \bigr)\sigma _{j}^{ \frac{1}{2}}\bigl(I_{n}+\sec ^{2}(\alpha )B\bigr). \end{aligned}$$
$$\begin{aligned} \sum_{j=1}^{k}\sigma _{j}^{\frac{1}{2}}(A+B) \leq \Biggl(\sum_{j=1}^{k} \sigma _{j}\bigl(I_{n}+\sec ^{2}(\alpha )A\bigr) \Biggr)^{\frac{1}{2}} \Biggl(\sum_{j=1} ^{k} \sigma _{j}\bigl(I_{n}+\sec ^{2}(\alpha )B\bigr) \Biggr)^{\frac{1}{2}} \end{aligned}$$
$$\begin{aligned} \sum_{j=1}^{k}\sigma _{j}^{\frac{1}{2}}(I_{n}+A+B) \leq \Biggl(\sum_{j=1} ^{k}\sigma _{j}\bigl(I_{n}+\sec ^{2}(\alpha )A\bigr) \Biggr)^{\frac{1}{2}} \Biggl(\sum_{j=1}^{k} \sigma _{j}\bigl(I_{n}+\sec ^{2}(\alpha )B\bigr) \Biggr)^{\frac{1}{2}}. \end{aligned}$$
$$\begin{aligned} \bigl\Vert \vert A+B \vert ^{\frac{1}{2}} \bigr\Vert ^{2}\leq \bigl\Vert I_{n}+\sec ^{2}(\alpha )A \bigr\Vert \bigl\Vert I_{n}+ \sec ^{2}(\alpha )B \bigr\Vert \end{aligned}$$
(19)
$$\begin{aligned} \bigl\Vert \vert I_{n}+A+B \vert ^{\frac{1}{2}} \bigr\Vert ^{2}\leq \bigl\Vert I_{n}+\sec ^{2}( \alpha )A \bigr\Vert \bigl\Vert I _{n}+\sec ^{2}(\alpha )B \bigr\Vert . \end{aligned}$$
(20)
$$\begin{aligned} \bigl\Vert \vert A+B \vert \bigr\Vert &= \bigl\Vert U \vert A+B \vert \bigr\Vert \\ &= \bigl\Vert \bigl( \vert A+B \vert ^{\frac{1}{2}}\bigr)^{2} \bigr\Vert \\ &\leq \bigl\Vert \vert A+B \vert ^{\frac{1}{2}} \bigr\Vert ^{2} \\ &\leq \bigl\Vert I_{n}+\sec ^{2}(\alpha )A \bigr\Vert \bigl\Vert I_{n}+\sec ^{2}(\alpha )B \bigr\Vert . \end{aligned}$$
$$\begin{aligned} \Vert I_{n}+A+B \Vert \leq \bigl\Vert I_{n}+\sec ^{2}(\alpha )A \bigr\Vert \bigl\Vert I_{n}+\sec ^{2}(\alpha )B \bigr\Vert , \end{aligned}$$
Remark 2.13
Taking \(k=n\) in Theorem 2.8, we get the following corollary.
Corollary 2.14
Let
\(A, B\in \mathbb{M}_{n}(\mathbb{C})\)
be such that
\(W(A), W(B) \subseteq S_{\alpha }\). Then
and
$$\begin{aligned} \bigl\vert \det (A+B) \bigr\vert \leq \bigl\vert \det \bigl(I_{n}+\sec ^{2}(\alpha )A\bigr) \bigr\vert \bigl\vert \det \bigl(I_{n}+\sec ^{2}(\alpha )B\bigr) \bigr\vert \end{aligned}$$
$$\begin{aligned} \bigl\vert \det (I_{n}+A+B) \bigr\vert \leq \bigl\vert \det \bigl(I_{n}+\sec ^{2}(\alpha )A\bigr) \bigr\vert \bigl\vert \det \bigl(I_{n}+ \sec ^{2}(\alpha )B\bigr) \bigr\vert . \end{aligned}$$
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