# Norm inequalities for submultiplicative functions involving contraction sector $$2 \times 2$$ block matrices

## Abstract

In this article, we show unitarily invariant norm inequalities for sector $$2\times 2$$ block matrices which extend and refine some recent results of Bourahli, Hirzallah, and Kittaneh (Positivity, 2020, https://doi.org/10.1007/s11117-020-00770-w).

## Introduction

Let $$\mathbb{M}_{n}$$ be a set of all $$n\times n$$ complex matrices. A matrix $$A\in \mathbb{M}_{n}$$ is said to be positive semidefinite if $$x^{*}Ax\geq 0$$ for all $$x\in \mathbb{C}^{n}$$. If the eigenvalues $$\lambda _{1}(A), \dots , \lambda _{n}(A)$$ of A are all real, we arrange them in nonincreasing order $$\lambda _{1}(A)\ge \cdots \ge \lambda _{n}(A)$$. Singular values of A are the eigenvalues of $$|A|$$ and are arranged in nonincreasing order $$s_{1}(A)\ge \cdots \ge s_{n}(A)$$. For $$A \in \mathbb{M}_{n}$$, we denote by $$|A|=(A^{*}A)^{\frac{1}{2}}$$, $$A^{*}$$, $$\|A\|$$, and $$\|A\|_{\infty }=s_{1}(A)$$ the absolute value, the conjugate transpose, the unitarily invariant norm, and the operator norm, respectively. We say A is a contraction if $$\|A\|_{\infty }\leq 1$$. By convention, the $$n \times n$$ identity matrix is denoted by $$I_{n}$$. $$\|A\|$$ and $$\|A\|_{\infty }=s_{1}(A)$$ are unitarily invariant, i.e., $$\|UAV\|=\|A\|$$ for all unitary matrices U, V. For $$A, B\in \mathbb{M}_{n}$$, the weak majorization relation $$s(A)\prec _{w} s(B)$$ means

$$\sum_{j=1}^{k}s_{j}(A) \leq \sum _{j=1}^{k}s_{j}(B),\quad k=1,2, \ldots , n.$$

For $$A\in \mathbb{M}_{n}$$, recall the Cartesian (or Toeplitz) decomposition (see, e.g., [2, p. 6] and [3, p. 7])

$$A =\operatorname{Re}A+i \operatorname{Im}A,$$

where

$$\operatorname{Re}A:=\frac{1}{2}\bigl(A+A^{*}\bigr),\qquad \operatorname{Im}A:= \frac{1}{2i}\bigl(A-A^{*}\bigr).$$

The Cartesian decomposition of a matrix is unique. There are many interesting properties for such a decomposition. A celebrated result due to Fan and Hoffman (see, e.g., [2, p.73]) states that

\begin{aligned} \lambda _{j}(\operatorname{Re}A) \leq & s_{j}(A),\quad j=1, \ldots , n. \end{aligned}
(1)

The numerical range of $$A\in \mathbb{M}_{n}$$ is defined by

$$W(A) = \bigl\{ x^{*}Ax\mid x\in \mathbb{C}^{n}, x^{*}x=1\bigr\} ,$$

which is a compact convex set (see, e.g., [4, Chap. 1]). For $$\alpha \in [0, \pi /2)$$, a sector on the complex plane is

$$S_{\alpha } =\bigl\{ z\in \mathbb{C}\mid \operatorname{Re}z \geq 0, \vert \operatorname{Im}z \vert \leq (\operatorname{Re}z)\tan \alpha \bigr\} .$$

A sector matrix $$A\in \mathbb{M}_{n}$$ is a matrix whose numerical range is contained in $$S_{\alpha }$$ for some $$\alpha \in [0, \pi /2)$$. It is clear that if $$A\in \mathbb{M}_{n}$$ is a sector matrix, then ReA is positive semidefinite. The interested readers can refer to , and  for recent results on sector matrices. If $$W(A)$$ is contained in the first quadrant of the complex plane, then ReA and ImA are positive semidefinite. We call such a matrix A accretive-dissipative. Note that if A is accretive-dissipative, then $$W(e^{-\frac{i\pi }{4}}A)\subseteq S_{\frac{\pi }{4}}$$. Recently this class of matrices has been studied by researchers partly due to the fact that it contains the class of positive semidefinite matrices (see, e.g., [1, 1117]).

Next we introduce a special class of functions. Let $$\mathcal{C}$$ be the class of all nonnegative increasing functions f on $$[0, \infty )$$ preserving the weak-log majorization, i.e., for two nonincreasing sequences of nonnegative real numbers $$(x_{1}, x_{2},\ldots , x_{n})$$ and $$(y_{1}, y_{2}, \ldots , y_{n})$$, $$\prod_{j=1}^{k} x_{j} \leq \prod_{j=1}^{k} y_{j}$$ for $$k=1,\ldots , n$$ implies $$\prod_{j=1}^{k} f(x_{j}) \leq \prod_{j=1}^{k} f(y_{j})$$ for $$k=1,\ldots , n$$. There are many other properties on this class of functions; see [12, 18]. A nonnegative function $$f\in \mathcal{C}$$ on the interval $$[0, \infty )$$ is said to be submultiplicative if $$f(ab)\leq f(a)f(b)$$ whenever $$a, b\in [0, \infty )$$. Recently, some unitarily invariant norm inequalities for submultiplicative functions of accretive-dissipative matrices have been shown in  and .

Bourahli et al. [1, Lemma 3.4] showed that if $A= ( A 11 A 12 A 21 A 22 ) ∈ M 2 n$ is a positive semidefinite contraction and s, t are positive real numbers such that $$\frac{1}{s}+\frac{1}{t}=1$$, then

$$\bigl\Vert f\bigl( \vert A_{12} \vert ^{2}\bigr) \bigr\Vert \leq \bigl\Vert f^{s} \bigl(A_{11}^{\frac{1}{2}} \bigr) \bigr\Vert ^{\frac{1}{s}} \bigl\Vert f^{t} \bigl(A_{22}^{\frac{1}{2}} \bigr) \bigr\Vert ^{\frac{1}{t}},$$
(2)

where f is an increasing submultiplicative function on $$[0, \infty )$$ with $$f(0)=0$$. Moreover, if $$A\in \mathbb{M}_{2n}$$ is just positive semidefinite (not necessarily contraction matrices), then they presented a result related to (2) in [1, Remark 3.5] as follows:

$$\bigl\Vert f\bigl( \vert A_{12} \vert ^{2}\bigr) \bigr\Vert \leq f \bigl( \bigl\Vert A_{11}^{\frac{1}{2}} \bigr\Vert _{\infty } \bigl\Vert A_{22}^{\frac{1}{2}} \bigr\Vert _{\infty } \bigr) \bigl\Vert f^{s} \bigl(A_{11}^{\frac{1}{2}} \bigr) \bigr\Vert ^{ \frac{1}{s}} \bigl\Vert f^{t} \bigl(A_{22}^{\frac{1}{2}} \bigr) \bigr\Vert ^{ \frac{1}{t}}.$$
(3)

## Unitarily invariant norms for submultiplicative functions

In  and , some unitarily invariant norms for accretive-dissipative matrices involving a special class of functions have been shown. In this section, we present inequalities for sector block matrices involving the class of function.

### Lemma 2.1

([19, p. 280])

Let A, X, B be $$m\times p$$, $$p\times q$$, $$q\times n$$ matrices, respectively. Then

$$s_{i}(AXB)\leq s_{1}(A)s_{j}(X)s_{1}(B), \quad i\leq \min \{m, p, q, n \}.$$

### Lemma 2.2

([8, Theorem 2.1])

Let $$A\in \mathbb{M}_{n}$$ be $$n\times n$$ such that $$W(A)\subseteq S_{\alpha }$$ for some $$\alpha \in [0, \pi /2)$$. Then there exist an invertible matrix X and a unitary and diagonal matrix $$Z=\operatorname{diag}(e^{i\theta _{1}},\ldots , e^{i\theta _{n}})$$ with all $$|\theta _{j}|\leq \alpha$$ such that $$A=XZX^{*}$$. Moreover, such a matrix Z is unique up to permutation.

### Lemma 2.3

([8, Corollary 2.3 (ii)])

Let $$A\in \mathbb{M}_{n}$$ be such that $$W(A)\subseteq S_{\alpha }$$ for some $$\alpha \in [0, \pi /2)$$, and let $$A=XZX^{*}$$ be a sectoral decomposition of A, where X is invertible and Z is unitary and diagonal. Then

\begin{aligned} RR^{*}\leq \sec (\alpha ) \bigl(R(\operatorname{Re}Z)R^{*} \bigr)=\sec (\alpha ) \bigl( \operatorname{Re}\bigl(RZR^{*}\bigr)\bigr) \end{aligned}

for every matrix $$R\in \mathbb{M}_{n}$$.

We are ready to present our main result of this section.

### Theorem 2.4

Let $$f\in \mathcal{C}$$ be an increasing submultiplicative function on $$[0, \infty )$$ and $$A\in \mathbb{M}_{2n}$$ be a contraction matrix partitioned as

$$A= \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix},$$
(4)

with $$W(A)\subseteq S_{\alpha }$$ for some $$\alpha \in [0, \pi /2)$$. Then, for all $$r, s, t>0$$ with $$\frac{1}{s}+\frac{1}{t}=1$$ and all unitarily invariant norms,

$$\bigl\Vert f\bigl( \vert A_{12} \vert ^{2r}\bigr) \bigr\Vert \leq \bigl\Vert f^{s}\bigl(\bigl( \sec ^{2}(\alpha ) \vert A_{11} \vert \bigr)^{r/2}\bigr) \bigr\Vert ^{1/s} \bigl\Vert f^{t}\bigl(\bigl(\sec ^{2}(\alpha ) \vert A_{22} \vert \bigr)^{r/2}\bigr) \bigr\Vert ^{1/t}$$
(5)

and

$$\bigl\Vert f \bigl( \vert A_{21} \vert ^{2r} \bigr) \bigr\Vert \le \bigl\Vert f^{s}\bigl(\bigl( \sec ^{2}( \alpha ) \vert A_{11} \vert \bigr)^{r/2}\bigr) \bigr\Vert ^{1/s} \bigl\Vert f^{t}\bigl(\bigl(\sec ^{2}(\alpha ) \vert A_{22} \vert \bigr)^{r/2}\bigr) \bigr\Vert ^{1/t}.$$
(6)

### Proof

Note that A is a sector matrix with $$W(A)\subseteq S_{\alpha }$$. By Lemma 2.2, we have $$A=XZX^{*}$$, where X is invertible and Z is unitary and diagonal. We partition X as $( X 1 X 2 )$, $$X_{1},X_{2}\in \mathbb{M}_{n \times 2n}$$. Thus, $$\operatorname{Re}A_{11}=X_{1}(\operatorname{Re}Z)X_{1}^{*}$$, $$\operatorname{Re}A_{22}=X_{2}(\operatorname{Re}Z)X_{2}^{*}$$, and $$A_{12}=X_{1}ZX_{2}^{*}$$. Consider the Cartesian decomposition

\begin{aligned} A = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix}=R+iS= \begin{pmatrix} R_{11} & R_{12} \\ R_{21} & R_{22} \end{pmatrix}+i \begin{pmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{pmatrix}, \end{aligned}

where R is positive semidefinite and S is Hermitian. Since A is a contraction matrix,

\begin{aligned} AA^{*}=R^{2}+S^{2}+i(SR-RS)\leq I \end{aligned}
(7)

and

\begin{aligned} A^{*}A=R^{2}+S^{2}+i(RS-SR)\leq I. \end{aligned}
(8)

Adding (7) and (8), we get

\begin{aligned} 2\bigl(R^{2}+S^{2}\bigr) \leq 2I. \end{aligned}

Thus, R and S are also contraction matrices, which implies that both $$\operatorname{Re}A_{11}=R_{11}$$ and $$\operatorname{Re}A_{22}=R_{22}$$ are positive semidefinite contractions.

Now

\begin{aligned} s_{\ell }\bigl( \vert A_{12} \vert ^{r}\bigr) =&s_{\ell }^{r}\bigl( \vert A_{12} \vert \bigr) =s_{\ell }^{r}(A_{12})= s_{\ell }^{r}\bigl(X_{1}ZX_{2}^{*} \bigr) \\ \leq & s_{1}^{r}(X_{1})s_{\ell }^{r} \bigl(ZX_{2}^{*}\bigr) \quad \text{(by Lemma 2.1)} \\ =& \lambda _{1}^{\frac{r}{2}}\bigl(X_{1}^{*}X_{1} \bigr)\lambda _{\ell }^{ \frac{r}{2}}\bigl(X_{2}Z^{*}ZX_{2}^{*} \bigr) \\ = &\lambda _{1}^{\frac{r}{2}}\bigl(X_{1}X_{1}^{*} \bigr)\lambda _{\ell }^{ \frac{r}{2}}\bigl(X_{2}X_{2}^{*} \bigr) \\ \le &\lambda _{1}^{\frac{r}{2}}\bigl(\sec (\alpha )X_{1}( \operatorname{Re}Z)X_{1}^{*}\bigr)\lambda _{\ell }^{\frac{r}{2}}\bigl(\sec ( \alpha )X_{2}( \operatorname{Re}Z)X_{2}^{*}\bigr) \quad \text{(by Lemma 2.3)} \\ =&\lambda _{1}^{\frac{r}{2}}\bigl(\sec (\alpha ) \operatorname{Re}A_{11}\bigr) \lambda _{\ell }^{\frac{r}{2}} \bigl(\sec (\alpha )\operatorname{Re}A_{22}\bigr) \end{aligned}
(9)
\begin{aligned} \le & \sec ^{\frac{r}{2}}(\alpha ) \lambda _{\ell }^{\frac{r}{2}} \bigl( \sec (\alpha )\operatorname{Re}A_{22}\bigr)\quad ( \text{since } \operatorname{Re}A_{11}\text{ is a contraction}) \\ \le & \sec ^{r}(\alpha ) s_{\ell }^{\frac{r}{2}} \bigl( \vert A_{22} \vert \bigr)\quad ( \text{by (1)}) \end{aligned}
(10)

for $$l=1,2, \ldots , n$$.

Since $$\operatorname{Re}A_{22}$$ is also a contraction, it follows from (1) and (9) that

\begin{aligned} s_{\ell }\bigl( \vert A_{12} \vert ^{r}\bigr) \leq & \sec ^{r}(\alpha ) s_{\ell }^{\frac{r}{2}} \bigl( \vert A_{11} \vert \bigr) \end{aligned}
(11)

for $$l=1,2, \ldots , n$$.

Multiplying inequalities (10) and (11) by each other implies that

\begin{aligned} s_{\ell }\bigl( \vert A_{12} \vert ^{2r}\bigr) \leq & \sec ^{2r}(\alpha ) s_{\ell }^{ \frac{r}{2}} \bigl( \vert A_{11} \vert \bigr)s_{\ell }^{\frac{r}{2}} \bigl( \vert A_{22} \vert \bigr) \end{aligned}
(12)

for $$l=1,2, \ldots , n$$.

So,

\begin{aligned} s_{\ell }\bigl(f\bigl( \vert A_{12} \vert ^{2r}\bigr)\bigr) =&f\bigl(s_{\ell }\bigl( \vert A_{12} \vert ^{2r}\bigr)\bigr) \\ \leq &f\bigl(\sec ^{2r}(\alpha ) s_{\ell }^{\frac{r}{2}} \bigl( \vert A_{11} \vert \bigr)s_{\ell }^{ \frac{r}{2}} \bigl( \vert A_{22} \vert \bigr)\bigr) \quad \text{(by (12))} \\ = &f\bigl(\sec ^{r}(\alpha ) s_{\ell }^{\frac{r}{2}} \bigl( \vert A_{11} \vert \bigr)\bigr)f\bigl(\sec ^{r}( \alpha ) s_{\ell }^{\frac{r}{2}}\bigl( \vert A_{22} \vert \bigr)\bigr)\\ & (\text{since }f\text{ is submultiplicative}) \end{aligned}
(13)

for $$l=1,2, \ldots , n$$. Let $$\alpha =(\alpha _{1},\alpha _{2}, \ldots , \alpha _{n})$$ be a decreasing sequence of nonnegative real numbers. The α-norm of a matrix $$B\in \mathbb{M}_{n}$$ is defined by

$$\Vert B \Vert _{\alpha } = \sum_{\ell =1}^{n} \alpha _{\ell }s_{\ell }(B).$$

The α-norms are unitarily invariant [4, p. 204].

Actually, inequality (13) means that

$$\prod_{\ell =1}^{k}\alpha _{\ell }s_{\ell } \bigl(f\bigl( \vert A_{12} \vert ^{2r}\bigr)\bigr) \leq \prod_{\ell =1}^{k}\alpha _{\ell }s_{\ell } \bigl(f\bigl(\bigl(\sec ^{2}(\alpha ) \vert A_{11} \vert \bigr)^{ \frac{r}{2}}\bigr)\bigr)s_{\ell }\bigl(f\bigl(\bigl(\sec ^{2}(\alpha ) \vert A_{22} \vert \bigr)^{\frac{r}{2}} \bigr)\bigr)$$

for $$k=1,2, \ldots , n$$, which implies that

\begin{aligned} \begin{aligned} \sum _{\ell =1}^{k}\alpha _{\ell }s_{\ell } \bigl(f\bigl( \vert A_{12} \vert ^{2r}\bigr)\bigr) &\leq \sum_{\ell =1}^{k}\alpha _{\ell }s_{\ell } \bigl(f\bigl(\bigl(\sec ^{2}(\alpha ) \vert A_{11} \vert \bigr)^{ \frac{r}{2}}\bigr)\bigr)s_{\ell }\bigl(f\bigl(\bigl(\sec ^{2}(\alpha ) \vert A_{22} \vert \bigr)^{\frac{r}{2}} \bigr)\bigr) \end{aligned} \end{aligned}

for $$k=1,2, \ldots , n$$. Thus,

\begin{aligned}& \bigl\Vert f\bigl( \vert A_{12} \vert ^{2r}\bigr) \bigr\Vert _{\alpha } \\& \quad = \sum _{\ell =1}^{n}\alpha _{\ell }s_{\ell } \bigl(f\bigl( \vert A_{12} \vert ^{2r}\bigr)\bigr) \\& \quad \leq \sum_{\ell =1}^{n}\alpha _{\ell }s_{\ell }\bigl(f\bigl(\bigl(\sec ^{2}(\alpha ) \vert A_{11} \vert \bigr)^{r/2}\bigr)\bigr)s_{\ell } \bigl(f\bigl(\bigl(\sec ^{2}(\alpha ) \vert A_{22} \vert \bigr)^{r/2}\bigr)\bigr) \\& \quad = \sum_{\ell =1}^{n}\alpha _{\ell }^{1/s}s_{\ell }\bigl(f\bigl(\bigl(\sec ^{2}( \alpha ) \vert A_{11} \vert \bigr)^{r/2} \bigr)\bigr)\alpha _{\ell }^{1/t}s_{\ell }\bigl(f\bigl( \bigl(\sec ^{2}( \alpha ) \vert A_{22} \vert \bigr)^{r/2}\bigr)\bigr) \\& \quad \leq \Biggl(\sum_{\ell =1}^{n}\alpha _{\ell }s_{\ell }^{s}\bigl(f\bigl(\bigl(\sec ^{2}( \alpha ) \vert A_{11} \vert \bigr)^{r/2} \bigr)\bigr) \Biggr)^{1/s} \Biggl(\sum_{\ell =1}^{m} \alpha _{\ell }s_{\ell }^{t}\bigl(f\bigl(\bigl(\sec ^{2}(\alpha ) \vert A_{22} \vert \bigr)^{r/2} \bigr)\bigr) \Biggr)^{1/t} \\& \quad \bigl(\text{by H\"{o}lder's inequality} \bigr) \\& \quad = \Biggl(\sum_{\ell =1}^{n}\alpha _{\ell }s_{\ell }\bigl(f^{s}\bigl(\bigl(\sec ^{2}( \alpha ) \vert A_{11} \vert \bigr)^{r/2} \bigr)\bigr) \Biggr)^{1/s} \Biggl(\sum_{\ell =1}^{m} \alpha _{\ell }s_{\ell }\bigl(f^{t}\bigl(\bigl(\sec ^{2}(\alpha ) \vert A_{22} \vert \bigr)^{r/2} \bigr)\bigr) \Biggr)^{1/t} \\& \quad = \bigl\Vert f^{s}\bigl(\bigl(\sec ^{2}(\alpha ) \vert A_{11} \vert \bigr)^{r/2}\bigr) \bigr\Vert _{\alpha }^{1/s} \bigl\Vert f^{t}\bigl(\bigl(\sec ^{2}(\alpha ) \vert A_{22} \vert \bigr)^{r/2} \bigr) \bigr\Vert _{\alpha }^{1/t} \end{aligned}

for all decreasing sequences $$\alpha =(\alpha _{1},\ldots , \alpha _{n})$$ of nonnegative real numbers. It follows from the above inequality that

$$\bigl\Vert f\bigl( \vert A_{12} \vert ^{2r}\bigr) \bigr\Vert \leq \bigl\Vert f^{s}\bigl(\bigl(\sec ^{2}( \alpha ) \vert A_{11} \vert \bigr)^{r/2}\bigr) \bigr\Vert ^{1/s} \bigl\Vert f^{t}\bigl(\bigl(\sec ^{2}( \alpha ) \vert A_{22} \vert \bigr)^{r/2}\bigr) \bigr\Vert ^{1/t}.$$

The inequality for $$A_{21}$$ is similarly proved. □

### Remark 1

In particular, when A is a positive semidefinite contraction ($$\alpha =0$$) and $$r=1$$, Theorem 2.4 gives

$$\bigl\Vert f\bigl( \vert A_{12} \vert ^{2}\bigr) \bigr\Vert \le \bigl\Vert f^{s} \bigl(A_{11}^{1/2}\bigr) \bigr\Vert ^{1/s} \bigl\Vert f^{t}\bigl(A_{22}^{1/2}\bigr) \bigr\Vert ^{1/t},$$
(14)

which is due to Bourahli et al. [1, Lemma 3.4]. Thus, our result (5) is a generalization of (14).

### Remark 2

If A is just a general sector matrix with $$W(A)\subseteq S_{\alpha }$$ for $$\alpha \in [0, \frac{\pi }{2})$$ (not a contraction matrix), then we have the following result: Let $$f\in \mathcal{C}$$ be an increasing submultiplicative function on $$[0, \infty )$$ and $A= ( A 11 A 12 A 21 A 22 ) ∈ M 2 n$ be a sector matrix with $$W(A)\subseteq S_{\alpha }$$ for some $$\alpha \in [0, \pi /2)$$. Then, for all $$r, s, t>0$$ with $$\frac{1}{s}+\frac{1}{t}=1$$ and all unitarily invariant norms,

\begin{aligned} \bigl\Vert f\bigl( \vert A_{12} \vert ^{2r}\bigr) \bigr\Vert = &f \bigl(\sec ^{2r}(\alpha ) \Vert A_{11} \Vert _{ \infty }^{\frac{r}{2}} \Vert A_{22} \Vert _{\infty }^{\frac{r}{2}} \bigr) \bigl\Vert f^{s}\bigl( \vert A_{11} \vert ^{ \frac{r}{2}}\bigr) \bigr\Vert ^{\frac{1}{s}} \bigl\Vert f^{t}\bigl( \vert A_{22} \vert ^{\frac{r}{2}}\bigr) \bigr\Vert ^{ \frac{1}{t}}. \end{aligned}
(15)

By (9), (10), and Lemma 2.1, we have

\begin{aligned} s_{\ell }\bigl( \vert A_{12} \vert ^{r}\bigr) =&s_{\ell }^{r}\bigl( \vert A_{12} \vert \bigr) =s_{\ell }^{r}(A_{12})= s_{\ell }^{r}\bigl(X_{1}ZX_{2}^{*} \bigr) \\ \le & \sec ^{r}(\alpha ) \Vert A_{11} \Vert _{\infty }^{\frac{r}{2}}s_{\ell }^{ \frac{r}{2}}\bigl( \vert A_{22} \vert \bigr) \end{aligned}
(16)

and

\begin{aligned} s_{\ell }\bigl( \vert A_{12} \vert ^{r}\bigr) \leq & \sec ^{r}(\alpha ) \Vert A_{22} \Vert _{\infty }^{ \frac{r}{2}}s_{\ell }^{\frac{r}{2}} \bigl( \vert A_{11} \vert \bigr) \end{aligned}
(17)

for $$l=1,2, \ldots , n$$.

Multiplying inequalities (16) and (17) by each other implies that

\begin{aligned} s_{\ell }\bigl( \vert A_{12} \vert ^{2r}\bigr) \leq & \sec ^{2r}(\alpha ) \Vert A_{11} \Vert _{\infty }^{ \frac{r}{2}} \Vert A_{22} \Vert _{\infty }^{\frac{r}{2}}s_{\ell }^{\frac{r}{2}}\bigl( \vert A_{11} \vert \bigr)s_{\ell }^{\frac{r}{2}}\bigl( \vert A_{22} \vert \bigr) \end{aligned}
(18)

for $$l=1,2, \ldots , n$$.

So,

\begin{aligned} s_{\ell }\bigl(f\bigl( \vert A_{12} \vert ^{2r}\bigr)\bigr) =&f\bigl(s_{\ell }\bigl( \vert A_{12} \vert ^{2r}\bigr)\bigr) \\ \leq &f(\sec ^{2r}(\alpha ) \Vert A_{11} \Vert _{\infty }^{\frac{r}{2}} \Vert A_{22} \Vert _{\infty }^{\frac{r}{2}}s_{\ell }^{\frac{r}{2}}\bigl( \vert A_{11} \vert \bigr)s_{\ell }^{ \frac{r}{2}}\bigl( \vert A_{22} \vert \bigr) \quad \text{(by (18))} \\ = &f \bigl(\sec ^{2r}(\alpha ) \Vert A_{11} \Vert _{\infty }^{\frac{r}{2}} \Vert A_{22} \Vert _{\infty }^{\frac{r}{2}} \bigr)f \bigl(s_{\ell }^{\frac{r}{2}} \bigl( \vert A_{11} \vert \bigr)\bigr)f\bigl(s_{\ell }^{\frac{r}{2}} \bigl( \vert A_{22} \vert \bigr) \bigr) \end{aligned}
(19)

for $$l=1,2, \ldots , n$$. Based on (19), we can obtain the desired result by a proof similar to that given for inequality (13). Therefore, when $$\alpha =0$$ and $$r=1$$, our result (15) is (2).

### Theorem 2.5

Let $$f\in \mathcal{C}$$ be an increasing submultiplicative function on $$[0, \infty )$$ and $$A\in \mathbb{M}_{2n}$$ be a contraction matrix partitioned as

$$A= \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix},$$

with $$W(A)\subseteq S_{\alpha }$$ for some $$\alpha \in [0, \pi /2)$$. Then, for all $$r, s, t>0$$ with $$\frac{1}{s}+\frac{1}{t}=1$$ and all unitarily invariant norms,

\begin{aligned}[b] &\bigl\Vert f\bigl( \vert A_{12} \vert ^{2r}\bigr) \bigr\Vert + \bigl\Vert f \bigl( \vert A_{21} \vert ^{2r} \bigr) \bigr\Vert \\ &\quad \leq 2 \bigl\Vert f^{s}\bigl(\bigl( \sec ^{2}(\alpha ) \vert A_{11} \vert \bigr)^{r/2}\bigr) \bigr\Vert ^{1/s} \bigl\Vert f^{t}\bigl(\bigl(\sec ^{2}( \alpha ) \vert A_{22} \vert \bigr)^{r/2}\bigr) \bigr\Vert ^{1/t}. \end{aligned}
(20)

### Proof

By Theorem 2.4, we can have the desired result. □

### Remark 3

When $$f(t)=t$$, $$r=1$$, and $$\alpha =\frac{\pi }{4}$$, result (20) becomes

$$\bigl\Vert \vert A_{12} \vert ^{2} \bigr\Vert + \bigl\Vert \vert A_{21} \vert ^{2} \bigr\Vert \leq 2 \bigl\Vert f^{s}\bigl(\bigl(\sec ^{2}(\alpha ) \vert A_{11} \vert \bigr)^{r/2}\bigr) \bigr\Vert ^{1/s} \bigl\Vert f^{t}\bigl(\bigl( \sec ^{2}( \alpha ) \vert A_{22} \vert \bigr)^{r/2}\bigr) \bigr\Vert ^{1/t}.$$

Not applicable.

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## Acknowledgements

The author wants to express her sincere thanks to the referee for his or her valuable remarks and suggestions, which made this paper more readable.

## Funding

Zhou’s work was supported by the High-level Talents Program of Natural Science Foundation of Hainan Province, China (2019RC193) and the Scientific Research Project of Colleges and Universities of Hainan Province, China (Hnky2020-31).

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The author drafted and approved the final manuscript.

### Corresponding author

Correspondence to Xiaoying Zhou.

## Ethics declarations

### Competing interests

The author declares that they have no competing interests.

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