# Inequalities for partial determinants of accretive block matrices

## Abstract

Let $$A=[A_{i,j}]^{m}_{i,j=1}\in \mathbf{M}_{m}(\mathbf{M}_{n})$$ be an accretive block matrix. We write det1 and det2 for the first and second partial determinants, respectively. In this paper, we show that

\begin{aligned} \bigl\Vert \operatorname{det}_{1} (\operatorname{Re}A) \bigr\Vert \leq \biggl\Vert \biggl(\frac{\operatorname{tr}( \vert A \vert )}{m} \biggr)^{m}I_{n} \biggr\Vert \end{aligned}

and

\begin{aligned} \bigl\Vert \operatorname{det}_{2} (\operatorname{Re}A) \bigr\Vert \leq \biggl\Vert \biggl( \frac {\operatorname{tr}( \vert A \vert )}{n} \biggr)^{n}I_{m} \biggr\Vert \end{aligned}

hold for any unitarily invariant norm $$\|\cdot \|$$. The two inequalities generalize some known results related to partial determinants of positive-semidefinite block matrices.

## 1 Introduction

The set of $$n\times n$$ complex matrices is denoted by $$\mathbf{M}_{n}$$. $$I_{n}$$ is $$n\times n$$ identity matrix. Let $$\mathbf{M}_{m}(\mathbf{M}_{n})$$ be the set of all $$m\times m$$ block matrices with each block in $$\mathbf{M}_{n}$$. If $$A\in \mathbf{M}_{n}$$ is positive-semidefinite (definite), then we write $$A \geq 0$$ ($$A>0$$). For two Hermitian matrices A, B of the same size, $$A\geq B$$ ($$A> B$$) means that $$A-B\geq 0$$ ($$A-B>0$$). For $$A\in \mathbf{M}_{n}$$, the singular values of A, denoted by $$s_{1}(A), s_{2}(A), \ldots, s_{n}(A)$$, are the eigenvalues of the positive-semidefinite matrix $$\vert A \vert =(A^{*}A)^{1/2}$$, arranged in nonincreasing order and repeated according to multiplicity as $$s_{1}(A)\geq s_{2}(A)\geq \cdots \geq s_{n}(A)$$. If A is Hermitian, we enumerate eigenvalues of A in nonincreasing order $$\lambda _{1}(A)\geq \lambda _{2}(A)\geq \cdots \geq \lambda _{n}(A)$$. We denote by $$A^{T}$$ and $$A^{*}$$ the transpose and conjugate transpose of A, respectively. Recall that a norm $$\Vert \cdot \Vert$$ is unitarily invariant if $$\Vert UAV \Vert = \Vert A \Vert$$ for any unitary matrices $$U, V\in \mathbf{M}_{n}$$ and any $$A\in \mathbf{M}_{n}$$. The Ky Fan k-norms, a special class of unitarily invariant norms, are defined as $$\Vert \cdot \Vert _{(k)}=\sum_{j=1}^{k} s_{j}(A)$$, $$1\leq k\leq n$$. The Schatten p-norms ($$p\geq 1$$) are defined as

\begin{aligned} \Vert A \Vert _{p} = \bigl(\operatorname{tr}\bigl( \vert A \vert ^{p} \bigr) \bigr)^{\frac{1}{p}}= \Biggl[\sum _{j=1}^{n}s_{j}^{p}(A) \Biggr]^{\frac{1}{p}}. \end{aligned}

The Schatten p-norms ($$p\geq 1$$) are also typical examples of unitarily invariant norms. We say that $$A\in \mathbf{M}_{m}(\mathbf{M}_{n})$$ is an accretive block matrix if its real part $$\operatorname{Re}A:=\frac{A+A^{*}}{2}$$ is positive-semidefinite.

In the following, two partial traces [6, p.Â 12] of $$A=[A_{i, j}]_{i, j=1}^{m}\in \mathbf{M}_{m}(\mathbf{M}_{n})$$ are defined by

$$\operatorname{tr}_{1} A= \sum_{i=1}^{m} A_{i, i}\in \mathbb{M}_{n}\quad \text{and} \quad \operatorname{tr}_{2} A= [\operatorname{tr}A_{i, j} ]_{i,j=1}^{m} \in \mathbb{M}_{m}.$$

Assume that $$A=[A_{i, j}]_{i, j=1}^{m}\in \mathbf{M}_{m}(\mathbf{M}_{n})$$, where $$A_{i,j}=[a_{l,k}^{i,j}]_{l,k=1}^{n}$$. We introduce two partial determinants $$\operatorname{det}_{1} A\in \mathbf{M}_{n}$$ and $$\operatorname{det}_{2} A\in \mathbf{M}_{m}$$ analogous to the two partial traces as follows [2]:

$$\operatorname{det}_{1} A=[\det G_{l,k}]_{l,k=1}^{n},$$

where $$G_{l,k}=[a_{l,k}^{i,j}]_{i,j=1}^{m}$$, and

$$\operatorname{det}_{2} A=[\det A_{i,j}]_{i,j=1}^{m}.$$

For $$A=[[a_{l,k}^{i,j}]_{l,k=1}^{n}]_{i,j=1}^{m}\in \mathbf{M}_{m}( \mathbf{M}_{n})$$, we will denote by $$\tilde{A}\in \mathbf{M}_{n}(\mathbf{M}_{m})$$ and $$A^{\tau}\in \mathbf{M}_{m}(\mathbf{M}_{n})$$ the matrices

$$\widetilde{A}=[G_{l,k}]_{l,k=1}^{n}= \bigl[ \bigl[a_{l,k}^{i,j} \bigr]_{i,j=1}^{m} \bigr]_{l,k=1}^{n} \quad \text{and} \quad A^{\tau} =[A_{j,i}]^{m}_{i,j=1}= \bigl[ \bigl[a_{l,k}^{j,i} \bigr]^{n}_{l,k=1} \bigr]^{m}_{i,j=1}.$$

Note that $$\tilde{\tilde{A}}=A$$ and $$\operatorname{det}_{1} A=\operatorname{det}_{2} \tilde{A}$$ and therefore also $$\operatorname{det}_{2} A=\operatorname{det}_{1} \tilde{A}$$.

Recently, Xu et al. [8] presented the following unitarily invariant norm inequalities for two partial determinants of positive-semidefinite block matrices.

### Theorem 1.1

Let $$A=[A_{i,j}]^{m}_{i,j=1}\in \mathbf{M}_{m}(\mathbf{M}_{n})$$ be positive-semidefinite. Then, the inequalities

\begin{aligned} \Vert \operatorname{det}_{1} A \Vert \leq \biggl\Vert \biggl(\frac{\operatorname{tr}A}{m} \biggr)^{m}I_{n} \biggr\Vert \end{aligned}

and

\begin{aligned} \Vert \operatorname{det}_{2} A \Vert \leq \biggl\Vert \biggl(\frac {\operatorname{tr}A}{n} \biggr)^{n}I_{m} \biggr\Vert \end{aligned}

hold for any unitarily invariant norm $$\Vert \cdot \Vert$$.

This theorem is inspired by a determinantal inequality for partial traces given by Lin [5, TheoremÂ 1.2]. Actually, the two unitarily invariant norm inequalities for partial determinants of $$A^{\tau}$$ in TheoremÂ 1.1 also hold; see [8, TheoremÂ 2.12].

The main goal of this paper is to extend the above two inequalities to accretive block matrices that is a larger class of matrices than the class of positive-semidefinite block matrices. At the same time, some related results are obtained.

## 2 Partial determinant inequalities

We begin this section with some lemmas that are useful to present our main results. The following two results will be used in TheoremÂ 2.6.

### Lemma 2.1

[2, TheoremÂ 7 and RemarkÂ 9] For $$A=[A_{i, j}]_{i, j=1}^{m}\in \mathbf{M}_{m}(\mathbf{M}_{n})$$,

1. 1.

A and Ãƒ are unitarily similar;

2. 2.

if A is positive-semidefinite, so is Ãƒ.

The next lemma is standard in matrix analysis.

### Lemma 2.2

[4, p.Â 511] Let $$A, B\in \mathbf{M}_{n}$$ be positive-semidefinite. Then,

\begin{aligned} \det (A)+\det (B) \leq \det (A+B) . \end{aligned}

For the convenience of proofs, we also need to list some recent results as lemmas.

### Lemma 2.3

[7] Let $$A\in \mathbf{M}_{m}(\mathbf{M}_{n})$$ be positive-semidefinite. Then,

\begin{aligned} \operatorname{det}_{2} A \geq 0. \end{aligned}

### Lemma 2.4

[2, TheoremÂ 6] Let $$A\in \mathbf{M}_{m}(\mathbf{M}_{n})$$ be positive-semidefinite. Then,

1. 1.

$$\operatorname{det}_{1} A \geq 0$$,

2. 2.

$$\det (\operatorname{tr}_{2} A)\geq \operatorname{tr}(\operatorname{det}_{1} A)$$.

### Lemma 2.5

[3, PropositionÂ 2.1] Let $$A\in \mathbf{M}_{m}(\mathbf{M}_{n})$$ be positive-semidefinite. Then,

$$\det (\operatorname{tr}_{1} A) \leq \biggl(\frac{\operatorname{tr}A}{n} \biggr)^{n}\quad \textit{and} \quad \det (\operatorname{tr}_{2} A) \leq \biggl(\frac{\operatorname{tr}A}{m} \biggr)^{m}.$$

As an analog of TheoremÂ 1.1, we prove the following inequalities for unitarily invariant norms.

### Theorem 2.6

Let $$A=[A_{i,j}]^{m}_{i,j=1}\in \mathbf{M}_{m}(\mathbf{M}_{n})$$ be a sector block matrix. Then, the inequalities

\begin{aligned} \bigl\Vert \operatorname{det}_{1} (\operatorname{Re}A) \bigr\Vert \leq \biggl\Vert \biggl(\frac{\operatorname{tr}( \vert A \vert )}{m} \biggr)^{m}I_{n} \biggr\Vert \end{aligned}

and

\begin{aligned} \bigl\Vert \operatorname{det}_{2} (\operatorname{Re}A) \bigr\Vert \leq \biggl\Vert \biggl( \frac {\operatorname{tr}( \vert A \vert )}{n} \biggr)^{n}I_{m} \biggr\Vert \end{aligned}

hold for any unitarily invariant norm $$\Vert \cdot \Vert$$.

### Proof

To prove the desired results, by Ky Fanâ€™s dominance theorem [1, p.Â 93], we just need to show that for all $$k=1,\ldots,n$$,

$$\bigl\Vert \operatorname{det}_{1} (\operatorname{Re}A) \bigr\Vert _{(k)}\leq \biggl\Vert \biggl( \frac{\operatorname{tr}( \vert A \vert )}{m} \biggr)^{m}I_{n} \biggr\Vert _{(k)}$$

and for all $$k=1,\ldots,m$$,

$$\bigl\Vert \operatorname{det}_{2} (\operatorname{Re}A) \bigr\Vert _{(k)}\leq \biggl\Vert \biggl( \frac {\operatorname{tr}( \vert A \vert )}{n} \biggr)^{n}I_{m} \biggr\Vert _{(k)}.$$

Compute

\begin{aligned} \bigl\Vert \operatorname{det}_{1} (\operatorname{Re}A) \bigr\Vert _{(k)} =& \sum_{j=1}^{k} s_{j} \bigl( \operatorname{det}_{1} (\operatorname{Re}A) \bigr) \\ =& \sum_{j=1}^{k} \lambda _{j} \bigl(\operatorname{det}_{1} (\operatorname{Re}A) \bigr) \quad (\text{by Lemma 2.4}) \\ \leq & \operatorname{tr}\bigl(\operatorname{det}_{1} (\operatorname{Re}A) \bigr) \\ \leq & \det \bigl(\operatorname{tr}_{2} (\operatorname{Re}A) \bigr)\quad (\text{by Lemma 2.4}) \\ \leq & \biggl(\frac {\operatorname{tr}(\operatorname{Re}A)}{m} \biggr)^{m} \quad (\text{by Lemma 2.5}) \\ =& \frac{1}{k}\sum_{j=1}^{k} s_{j} \biggl( \biggl( \frac {\operatorname{tr}(\operatorname{Re}A)}{m} \biggr)^{m}I_{n} \biggr) \\ =& \frac{1}{k} \biggl\Vert \biggl(\frac {\operatorname{tr}(\operatorname{Re}A)}{m} \biggr)^{m}I_{n} \biggr\Vert _{(k)} \\ \leq & \biggl\Vert \biggl(\frac {\operatorname{tr}( \vert A \vert )}{m} \biggr)^{m}I_{n} \biggr\Vert _{(k)}, \quad k=1,\ldots, n, \end{aligned}

which means that

$$\bigl\Vert \operatorname{det}_{1} (\operatorname{Re}A) \bigr\Vert \leq \biggl\Vert \biggl(\frac {\operatorname{tr}( \vert A \vert )}{m} \biggr)^{m}I_{n} \biggr\Vert .$$

By LemmaÂ 2.1, we have $$\operatorname{tr}(\operatorname{Re}A)=\operatorname{tr}(\widetilde{\operatorname{Re}A})=\operatorname{tr}(\operatorname{Re}\widetilde{A})$$. Therefore, by $$\widetilde{\operatorname{Re}A}=\operatorname{Re}\widetilde{A}$$,

$$\bigl\Vert \operatorname{det}_{2} (\operatorname{Re}A) \bigr\Vert = \bigl\Vert \operatorname{det}_{1} (\widetilde{\operatorname{Re}A}) \bigr\Vert = \bigl\Vert \operatorname{det}_{1} (\operatorname{Re}\widetilde{A}) \bigr\Vert \leq \biggl\Vert \biggl( \frac{\operatorname{tr}( \vert \widetilde{A} \vert )}{n} \biggr)^{n}I_{m} \biggr\Vert = \biggl\Vert \biggl(\frac{\operatorname{tr}( \vert A \vert )}{n} \biggr)^{n}I_{m} \biggr\Vert .$$

â€ƒâ–¡

### Remark 1

When $$A=[A_{i,j}]^{m}_{i,j=1}\in \mathbf{M}_{m}(\mathbf{M}_{n})$$ is positive-semidefinite in TheoremÂ 2.6, our result is TheoremÂ 1.1. Thus, our result is a generalization of TheoremÂ 1.1.

Next, we will prove two determinantal inequalities for accretive block matrices involving partial determinants.

### Theorem 2.7

Let $$A=[A_{i,j}]^{m}_{i,j=1}\in \mathbf{M}_{m}(\mathbf{M}_{n})$$ be an accretive block matrix. Then,

\begin{aligned} \det \bigl(\operatorname{det}_{1} ( \operatorname{Re}A) \bigr) \leq \frac{(\operatorname{tr}( \vert A \vert ))^{mn}}{m^{mn}n^{n}} \end{aligned}

and

\begin{aligned} \det \bigl(\operatorname{det}_{2} (\operatorname{Re}A) \bigr) \leq \frac{(\operatorname{tr}( \vert A \vert ))^{mn}}{n^{mn}m^{m}}. \end{aligned}

### Proof

Let $$\lambda _{j}$$, $$j=1,\ldots, m$$, be the eigenvalues of $$\operatorname{det}_{2} (\operatorname{Re}A)$$. Then, by the AM-GM inequality and LemmaÂ 2.5, we have the following result:

\begin{aligned} \det \bigl(\operatorname{det}_{2} (\operatorname{Re}A) \bigr) =&\lambda _{1}\cdots \lambda _{m} \\ \leq & \biggl( \frac{\lambda _{1}+\lambda _{2}+\cdots + \lambda _{m}}{m} \biggr)^{m} \\ =& \biggl(\frac{\operatorname{tr}(\operatorname{det}_{2} (\operatorname{Re}A))}{m} \biggr)^{m} \\ =& \biggl(\frac{\sum_{i=1}^{m}\det (\operatorname{Re}A)_{ii}}{m} \biggr)^{m} \\ \leq & \biggl(\frac{\det (\sum_{i=1}^{m} (\operatorname{Re}A)_{ii})}{m} \biggr)^{m} \quad (\text{by Lemma 2.2}) \\ =& \biggl(\frac{\det (\operatorname{tr}_{1} (\operatorname{Re}A))}{m} \biggr)^{m} \\ \leq & \biggl(\frac{ (\frac{\operatorname{tr}(\operatorname{Re}A)}{n} )^{n}}{m} \biggr)^{m} \quad (\text{by Lemma 2.5}) \\ \leq & \frac{(\operatorname{tr}( \vert A \vert ))^{nm}}{n^{nm}m^{m}}, \end{aligned}

which means that

\begin{aligned} \det \bigl(\operatorname{det}_{2} (\operatorname{Re}A) \bigr) \leq \frac{(\operatorname{tr}( \vert A \vert ))^{nm}}{n^{nm}m^{m}}. \end{aligned}

On the other hand, by $$\operatorname{det}_{1} (\operatorname{Re}A)=\operatorname{det}_{2} (\widetilde{\operatorname{Re}A})$$ and LemmaÂ 2.1, we have

\begin{aligned} \det \bigl(\operatorname{det}_{1} (\operatorname{Re}A) \bigr) = \det (\operatorname{det}_{2} \widetilde{\operatorname{Re}A} )=\det \bigl( \operatorname{det}_{2} (\operatorname{Re}\widetilde{A}) \bigr)\leq \frac{(\operatorname{tr}( \vert \tilde{A} \vert ))^{mn}}{m^{mn}n^{n}}= \frac{(\operatorname{tr}( \vert A \vert ))^{mn}}{m^{mn}n^{n}}. \end{aligned}

â€ƒâ–¡

We would like to know whether or not the inequalities above hold in the case of replacing A with $$A^{\tau}$$. Now, we will present a result on sector block matrices that is the same as it was under the positive-semidefinite condition.

### Theorem 2.8

If $$A=[A_{i,j}]^{m}_{i,j=1}\in \mathbf{M}_{m}(\mathbf{M}_{n})$$ is a sector block matrix, then

$$\operatorname{det}_{1} \bigl(A^{\tau} \bigr)= \operatorname{det}_{1} A\quad \textit{and}\quad \operatorname{det}_{2} \bigl(A^{\tau} \bigr)=(\operatorname{det}_{2} A)^{T}= \operatorname{det}_{2} \bigl(A^{T} \bigr).$$

### Proof

Since $$A^{\tau}=[A_{j,i}]^{m}_{i,j=1}$$ and $$\widetilde{A}=[G_{l,k}]_{l,k=1}^{n}$$, we have $$\widetilde{A^{\tau}}=[G_{l,k}^{\;\;\;T}]^{n}_{l,k=1}$$.

Hence,

\begin{aligned} \operatorname{det}_{1} \bigl(A^{\tau} \bigr) =& \operatorname{det}_{2} \bigl( \widetilde{A^{\tau}} \bigr) \\ =& \bigl[\det G_{l,k}^{\;\;\; T} \bigr]_{l,k=1}^{n} \\ =& \bigl[\det \bigl[a_{l,k}^{j,i} \bigr]_{i,j=1}^{m} \bigr]_{l,k=1}^{n} \\ =& \bigl[\det \bigl[a_{l,k}^{i,j} \bigr]_{i,j=1}^{m} \bigr]_{l,k=1}^{n} \\ =&[\det G_{l,k}]_{l,k=1}^{n} \\ =& \operatorname{det}_{1} A. \end{aligned}

On the other hand,

\begin{aligned} \operatorname{det}_{2} \bigl(A^{T} \bigr) =& \bigl[\det A^{T}_{\;\;\; i,j} \bigr]^{m}_{i,j=1} \\ =& \bigl[\det A_{j,i}^{\;\;\; T} \bigr]^{m}_{i,j=1} \\ =& \bigl[\det \bigl[a_{k,l}^{j,i} \bigr]^{n}_{l,k=1} \bigr]_{i,j=1}^{m} \\ =& \bigl[\det \bigl[a_{l,k}^{j,i} \bigr]^{n}_{l,k=1} \bigr]_{i,j=1}^{m} \\ =&\operatorname{det}_{2} \bigl(A^{\tau} \bigr) \\ =&[\det A_{j,i}]_{i,j=1}^{m} \\ =& \bigl([\det A_{i,j}]_{i,j=1}^{m} \bigr)^{T} \\ =&(\operatorname{det}_{2} A)^{T}. \end{aligned}

â€ƒâ–¡

The following result for sector block matrices involving partial determinants of $$A^{\tau}$$ can be regarded as a complement of TheoremÂ 2.7.

### Theorem 2.9

Let $$A=[A_{i,j}]^{m}_{i,j=1}\in \mathbf{M}_{m}(\mathbf{M}_{n})$$ be a sector matrix. Then,

\begin{aligned} \det \bigl(\operatorname{det}_{1} (\operatorname{Re}A)^{\tau} \bigr) \leq & \frac{(\operatorname{tr}( \vert A \vert ))^{mn}}{m^{mn}n^{n}} \end{aligned}

and

\begin{aligned} \det \bigl(\operatorname{det}_{2} (\operatorname{Re}A)^{\tau} \bigr) \leq & \frac {(\operatorname{tr}( \vert A \vert ))^{mn}}{n^{nm}m^{m}}. \end{aligned}

### Proof

The proof is similar to that of TheoremÂ 2.7.â€ƒâ–¡

### Remark 2

In fact, the analogous inequalities below for partial traces are also valid using a similar idea to that of LemmaÂ 2.5:

$$\det \bigl(\operatorname{tr}_{1} \bigl(\operatorname{Re}A^{\tau} \bigr) \bigr) \leq \biggl(\frac{\operatorname{tr}(\operatorname{Re}A)}{n} \biggr)^{n}\leq \biggl(\frac{\operatorname{tr}( \vert A \vert )}{n} \biggr)^{n}$$

and

$$\det \bigl(\operatorname{tr}_{2} \bigl(\operatorname{Re}A^{\tau} \bigr) \bigr) \leq \biggl(\frac{\operatorname{tr}(\operatorname{Re}A)}{m} \biggr)^{m}\leq \biggl(\frac{\operatorname{tr}( \vert A \vert )}{m} \biggr)^{m}.$$

Next, we give inequalities for partial determinants of $$A^{\tau}$$ involving unitarily invariant norms.

### Theorem 2.10

Let $$A=[A_{i,j}]^{m}_{i,j=1}\in \mathbf{M}_{m}(\mathbf{M}_{n})$$ be a sector matrix. Then, the inequalities

$$\bigl\Vert \operatorname{det}_{1} \bigl(\operatorname{Re}A^{\tau} \bigr) \bigr\Vert \leq \biggl\Vert \biggl( \frac{\operatorname{tr} \vert A \vert }{m} \biggr)^{m}I_{n} \biggr\Vert$$

and

$$\bigl\Vert \operatorname{det}_{2} \bigl(\operatorname{Re}A^{\tau} \bigr) \bigr\Vert \leq \biggl\Vert \biggl( \frac{\operatorname{tr} \vert A \vert }{n} \biggr)^{n}I_{m} \biggr\Vert$$

hold for any unitarily invariant norm $$\Vert \cdot \Vert$$.

### Proof

Note that $$\operatorname{det}_{1} ((\operatorname{Re}A)^{\tau})=\operatorname{det}_{1} (\operatorname{Re}A)$$, $$\operatorname{det}_{2}( \operatorname{Re}A^{\tau})=\operatorname{det}_{2}( (\operatorname{Re}A)^{T})$$ by TheoremÂ 2.8 and $$\operatorname{tr}A=\operatorname{tr}(A^{T})$$, hence the proof is similar to that of TheoremÂ 2.6.â€ƒâ–¡

Not applicable.

## References

1. Bhatia, R.: Matrix Analysis. Springer, New York (1997)

2. Choi, D.: Inequalities related to trace and determinant of positive semidefinite block matrices. Linear Algebra Appl. 532, 1â€“7 (2017)

3. Fu, X., Lau, P.S., Tam, T.Y.: Inequalities on partial traces of positive semidefinite block matrices. Can. Math. Bull. 64, 964â€“969 (2021)

4. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (2013)

5. Lin, M.: AÂ determinantal inequality involving partial traces. Can. Math. Bull. 59, 585â€“591 (2016)

6. Petz, D.: Quantum Information Theory and Quantum Statistics, in Theoretical and Mathematical Physics. Springer, Berlin (2008)

7. Thompson, R.: AÂ determinantal inequality for positive definite matrices. Can. Math. Bull. 4, 57â€“62 (1961)

8. Xu, H., Fu, X.: Inequalities for partial determinants of positive semidefinite block matrices. Linear Multilinear Algebra. https://doi.org/10.1080/03081087.2022.2158299

## Authorsâ€™ information

The email addresses of the coauthors are hlh0922@163.com and haseeb2013.salarzay@gmail.com.

## Funding

The work is supported by the Hainan Provincial Natural Science Foundation of China (grant no. 120MS032), the National Natural Science Foundation (grant no. 12261030), the Hainan Provincial Natural Science Foundation for High-level Talents (grant no. 123RC474), the specific research fund of the Innovation Platform for Academicians of Hainan Province (grant no. YSPTZX202215), the Key Laboratory of Computational Science and Application of Hainan Province, and the National Natural Science Foundation (grant no. 11861031).

## Author information

Authors

### Contributions

Xiaohui Fu wrote the main manuscript text and Lihong Hu, Abdul Haseeb Salarzay checked the proofs. All authors reviewed the manuscript

### Corresponding author

Correspondence to Xiaohui Fu.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

### Publisherâ€™s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and permissions

Fu, X., Hu, L. & Salarzay, A.H. Inequalities for partial determinants of accretive block matrices. J Inequal Appl 2023, 101 (2023). https://doi.org/10.1186/s13660-023-03008-x