# Refinements of the Heinz inequalities for matrices

## Abstract

This article aims to discuss Heinz inequalities involving unitarily invariant norms. We use a similar method to (Feng in J. Inequal. Appl. 2012:18, 2012; Wang in J. Inequal. Appl. 2013:424, 2013) and we get different refinements of the Heinz inequalities for matrices. Our results are better than some given in (Kittaneh in Integral Equ. Oper. Theory 68:519-527, 2010) and they are different from (Feng in J. Inequal. Appl. 2012:18, 2012; Wang in J. Inequal. Appl. 2013:424, 2013).

## 1 Introduction

If A, B, X are operators on a complex separable Hilbert space such that A and B are positive, then for every unitarily invariant norm $⦀\cdot ⦀$, the function $f\left(v\right)=⦀{A}^{v}X{B}^{1-v}+{A}^{1-v}X{B}^{v}⦀$ is convex on the interval $\left[0,1\right]$, attains its minimum at $v=\frac{1}{2}$, and attains its maximum at $v=0$ and $v=1$. Moreover, $f\left(v\right)=f\left(1-v\right)$ for $0\le v\le 1$. From [1] we know that for every unitarily invariant norm, we have the Heinz inequalities

$2⦀{A}^{\frac{1}{2}}X{B}^{\frac{1}{2}}⦀\le ⦀{A}^{v}X{B}^{1-v}+{A}^{1-v}X{B}^{v}⦀\le ⦀AX+XB⦀.$
(1)

In [2], Feng used the following inequalities to get refinements of (1):

$f\left(\frac{a+b}{2}\right)\le \frac{1}{b-a}{\int }_{a}^{b}f\left(t\right)\phantom{\rule{0.2em}{0ex}}dt\le \frac{1}{4}\left(f\left(a\right)+2f\left(\frac{a+b}{2}\right)+f\left(b\right)\right)\le \frac{f\left(a\right)+f\left(b\right)}{2},$

where f is a real-valued function which is convex on the interval $\left[a,b\right]$. With a similar method, Wang [3] got some new refinements of (1).

In this paper, we use a similar method to [2, 3] and we get different refinements of (1).

When we consider $⦀T⦀$, we are implicitly assuming that the operator T belongs to the norm ideal associated with $⦀\cdot ⦀$. Our results are better than those in [4] and different from [2, 3].

## 2 Main results

From page 122 of [5], we know the following Hermite-Hadamard integral inequality for convex functions.

Let f be a real-valued function which is convex on the interval $\left[a,b\right]$. Then

$f\left(\frac{a+b}{2}\right)\le \frac{1}{b-a}{\int }_{a}^{b}f\left(t\right)\phantom{\rule{0.2em}{0ex}}dt\le \frac{f\left(a\right)+f\left(b\right)}{2}.$

We will use the following lemma.

Lemma 2 Let f be a real-valued function which is convex on the interval $\left[a,b\right]$. Then

$f\left(\frac{a+b}{2}\right)\le \frac{1}{b-a}{\int }_{a}^{b}f\left(t\right)\phantom{\rule{0.2em}{0ex}}dt\le \frac{1}{8}\left(3f\left(a\right)+2f\left(\frac{a+b}{2}\right)+3f\left(b\right)\right)\le \frac{f\left(a\right)+f\left(b\right)}{2}.$

Proof Using the previous lemma, we can easily verify the inequality

$\frac{1}{8}\left(3f\left(a\right)+2f\left(\frac{a+b}{2}\right)+3f\left(b\right)\right)\le \frac{f\left(a\right)+f\left(b\right)}{2}.$

Next, we will prove the following inequality:

$\frac{1}{b-a}{\int }_{a}^{b}f\left(t\right)\phantom{\rule{0.2em}{0ex}}dt\le \frac{1}{8}\left(3f\left(a\right)+2f\left(\frac{a+b}{2}\right)+3f\left(b\right)\right).$

From the previous lemma, we have

$\begin{array}{rcl}\frac{1}{b-a}{\int }_{a}^{b}f\left(t\right)\phantom{\rule{0.2em}{0ex}}dt& =& \frac{1}{b-a}\left({\int }_{a}^{\frac{a+b}{2}}f\left(t\right)\phantom{\rule{0.2em}{0ex}}dt+{\int }_{\frac{a+b}{2}}^{b}f\left(t\right)\phantom{\rule{0.2em}{0ex}}dt\right)\\ \le & \frac{1}{b-a}\left(\frac{f\left(a\right)+f\left(\frac{a+b}{2}\right)}{2}\cdot \frac{b-a}{2}+\frac{f\left(\frac{a+b}{2}\right)+f\left(b\right)}{2}\cdot \frac{b-a}{2}\right)\\ =& \frac{1}{4}\left(f\left(a\right)+2f\left(\frac{a+b}{2}\right)+f\left(b\right)\right)\\ =& \frac{1}{8}\left(2f\left(a\right)+4f\left(\frac{a+b}{2}\right)+2f\left(b\right)\right)\\ \le & \frac{1}{8}\left(2f\left(a\right)+2f\left(\frac{a+b}{2}\right)+\left(f\left(a\right)+f\left(b\right)\right)+2f\left(b\right)\right)\\ =& \frac{1}{8}\left(3f\left(a\right)+2f\left(\frac{a+b}{2}\right)+3f\left(b\right)\right).\end{array}$

□

Applying the previous lemma to the function $f\left(v\right)=⦀{A}^{v}X{B}^{1-v}+{A}^{1-v}X{B}^{v}⦀$ on the interval $\left[\mu ,1-\mu \right]$ when $0\le \mu \le \frac{1}{2}$, and on the interval $\left[1-\mu ,\mu \right]$ when $\frac{1}{2}\le \mu \le 1$, we obtain a refinement of the first inequality in (1).

Theorem 1 Let A, B, X be operators such that A, B are positive. Then for $0\le \mu \le 1$ and for every unitarily invariant norm, we have

$\begin{array}{rl}2⦀{A}^{\frac{1}{2}}X{B}^{\frac{1}{2}}⦀& \le \frac{1}{|1-2\mu |}|{\int }_{\mu }^{1-\mu }⦀{A}^{v}X{B}^{1-v}+{A}^{1-v}X{B}^{v}⦀\phantom{\rule{0.2em}{0ex}}dv|\\ \le \frac{1}{4}\left(3⦀{A}^{\mu }X{B}^{1-\mu }+{A}^{1-\mu }X{B}^{\mu }⦀+2⦀{A}^{\frac{1}{2}}X{B}^{\frac{1}{2}}⦀\right)\\ \le ⦀{A}^{\mu }X{B}^{1-\mu }+{A}^{1-\mu }X{B}^{\mu }⦀.\end{array}$
(2)

Proof First assume that $0\le \mu \le \frac{1}{2}$. Then it follows by the previous lemma that

$\begin{array}{rcl}f\left(\frac{1-\mu +\mu }{2}\right)& \le & \frac{1}{1-2\mu }{\int }_{\mu }^{1-\mu }f\left(t\right)\phantom{\rule{0.2em}{0ex}}dt\\ \le & \frac{1}{8}\left(3f\left(\mu \right)+2f\left(\frac{1-\mu +\mu }{2}\right)+3f\left(1-\mu \right)\right)\\ \le & \frac{f\left(\mu \right)+f\left(1-\mu \right)}{2},\end{array}$

and so

$\begin{array}{rcl}f\left(\frac{1}{2}\right)& \le & \frac{1}{1-2\mu }{\int }_{\mu }^{1-\mu }f\left(t\right)\phantom{\rule{0.2em}{0ex}}dt\\ \le & \frac{1}{4}\left(3f\left(\mu \right)+f\left(\frac{1}{2}\right)\right)\\ \le & f\left(\mu \right).\end{array}$

Thus,

$\begin{array}{rcl}2⦀{A}^{\frac{1}{2}}X{B}^{\frac{1}{2}}⦀& \le & \frac{1}{1-2\mu }{\int }_{\mu }^{1-\mu }⦀{A}^{v}X{B}^{1-v}+{A}^{1-v}X{B}^{v}⦀\phantom{\rule{0.2em}{0ex}}dv\\ \le & \frac{1}{4}\left(3⦀{A}^{\mu }X{B}^{1-\mu }+{A}^{1-\mu }X{B}^{\mu }⦀+2⦀{A}^{\frac{1}{2}}X{B}^{\frac{1}{2}}⦀\right)\\ \le & ⦀{A}^{\mu }X{B}^{1-\mu }+{A}^{1-\mu }X{B}^{\mu }⦀.\end{array}$
(3)

Now, assume that $\frac{1}{2}\le \mu \le 1$. Then by applying (3) to $1-\mu$, it follows that

$\begin{array}{rcl}2⦀{A}^{\frac{1}{2}}X{B}^{\frac{1}{2}}⦀& \le & \frac{1}{2\mu -1}{\int }_{1-\mu }^{\mu }⦀{A}^{v}X{B}^{1-v}+{A}^{1-v}X{B}^{v}⦀\phantom{\rule{0.2em}{0ex}}dv\\ \le & \frac{1}{4}\left(3⦀{A}^{\mu }X{B}^{1-\mu }+{A}^{1-\mu }X{B}^{\mu }⦀+2⦀{A}^{\frac{1}{2}}X{B}^{\frac{1}{2}}⦀\right)\\ \le & ⦀{A}^{\mu }X{B}^{1-\mu }+{A}^{1-\mu }X{B}^{\mu }⦀.\end{array}$
(4)

Since

$\begin{array}{c}\underset{\mu \to \frac{1}{2}}{lim}\frac{1}{|1-2\mu |}|{\int }_{\mu }^{1-\mu }⦀{A}^{v}X{B}^{1-v}+{A}^{1-v}X{B}^{v}⦀\phantom{\rule{0.2em}{0ex}}dv|\hfill \\ \phantom{\rule{1em}{0ex}}=\underset{\mu \to \frac{1}{2}}{lim}\frac{1}{4}\left(3⦀{A}^{\mu }X{B}^{1-\mu }+{A}^{1-\mu }X{B}^{\mu }⦀+2⦀{A}^{\frac{1}{2}}X{B}^{\frac{1}{2}}⦀\right)\hfill \\ \phantom{\rule{1em}{0ex}}=2⦀{A}^{\frac{1}{2}}X{B}^{\frac{1}{2}}⦀,\hfill \end{array}$

the inequalities in (2) follow by combining (3) and (4). □

Applying the previous lemma to the function $f\left(v\right)=⦀{A}^{v}X{B}^{1-v}+{A}^{1-v}X{B}^{v}⦀$ on the interval $\left[\mu ,\frac{1}{2}\right]$ when $0\le \mu \le \frac{1}{2}$, and on the interval $\left[\frac{1}{2},\mu \right]$ when $\frac{1}{2}\le \mu \le 1$, we obtain the following.

Theorem 2 Let A, B, X be operators such that A, B are positive. Then for $0\le \mu \le 1$ and for every unitarily invariant norm, we have

$\begin{array}{c}⦀{A}^{\frac{2\mu +1}{4}}X{B}^{\frac{3-2\mu }{4}}+{A}^{\frac{3-2\mu }{4}}X{B}^{\frac{2\mu +1}{4}}⦀\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{2}{|1-2\mu |}|{\int }_{\mu }^{\frac{1}{2}}⦀{A}^{v}X{B}^{1-v}+{A}^{1-v}X{B}^{v}⦀\phantom{\rule{0.2em}{0ex}}dv|\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{8}\left(3⦀{A}^{\mu }X{B}^{1-\mu }+{A}^{1-\mu }X{B}^{\mu }⦀+2⦀{A}^{\frac{2\mu +1}{4}}X{B}^{\frac{3-2\mu }{4}}+{A}^{\frac{3-2\mu }{4}}X{B}^{\frac{2\mu +1}{4}}⦀+6⦀{A}^{\frac{1}{2}}X{B}^{\frac{1}{2}}⦀\right)\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{2}\left(⦀{A}^{\mu }X{B}^{1-\mu }+{A}^{1-\mu }X{B}^{\mu }⦀+2⦀{A}^{\frac{1}{2}}X{B}^{\frac{1}{2}}⦀\right).\hfill \end{array}$
(5)

The inequality (5) and the first inequality in (1) yield the following refinement of the first inequality in (1).

Corollary 1 Let A, B, X be operators such that A, B are positive. Then for $0\le \mu \le 1$ and for every unitarily invariant norm, we have

$\begin{array}{c}2⦀{A}^{\frac{1}{2}}X{B}^{\frac{1}{2}}⦀\hfill \\ \phantom{\rule{1em}{0ex}}\le ⦀{A}^{\frac{2\mu +1}{4}}X{B}^{\frac{3-2\mu }{4}}+{A}^{\frac{3-2\mu }{4}}X{B}^{\frac{2\mu +1}{4}}⦀\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{2}{|1-2\mu |}|{\int }_{\mu }^{\frac{1}{2}}⦀{A}^{v}X{B}^{1-v}+{A}^{1-v}X{B}^{v}⦀\phantom{\rule{0.2em}{0ex}}dv|\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{8}\left(3⦀{A}^{\mu }X{B}^{1-\mu }+{A}^{1-\mu }X{B}^{\mu }⦀+2⦀{A}^{\frac{2\mu +1}{4}}X{B}^{\frac{3-2\mu }{4}}+{A}^{\frac{3-2\mu }{4}}X{B}^{\frac{2\mu +1}{4}}⦀+6⦀{A}^{\frac{1}{2}}X{B}^{\frac{1}{2}}⦀\right)\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{2}\left(⦀{A}^{\mu }X{B}^{1-\mu }+{A}^{1-\mu }X{B}^{\mu }⦀+2⦀{A}^{\frac{1}{2}}X{B}^{\frac{1}{2}}⦀\right)\hfill \\ \phantom{\rule{1em}{0ex}}\le ⦀{A}^{\mu }X{B}^{1-\mu }+{A}^{1-\mu }X{B}^{\mu }⦀.\hfill \end{array}$
(6)

Applying the previous lemma to the function $f\left(v\right)=⦀{A}^{v}X{B}^{1-v}+{A}^{1-v}X{B}^{v}⦀$ on the interval $\left[0,\mu \right]$ when $0\le \mu \le \frac{1}{2}$, and on the interval $\left[\mu ,1\right]$ when $\frac{1}{2}\le \mu \le 1$, we obtain the following theorem.

Theorem 3 Let A, B, X be operators such that A, B are positive. Then

1. (1)

for $0\le \mu \le \frac{1}{2}$ and for every unitarily norm,

$\begin{array}{c}⦀{A}^{\frac{\mu }{2}}X{B}^{1-\frac{\mu }{2}}+{A}^{1-\frac{\mu }{2}}X{B}^{\frac{\mu }{2}}⦀\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{\mu }{\int }_{0}^{\mu }⦀{A}^{v}X{B}^{1-v}+{A}^{1-v}X{B}^{v}⦀\phantom{\rule{0.2em}{0ex}}dv\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{8}\left(3⦀AX+XB⦀+2⦀{A}^{\frac{\mu }{2}}X{B}^{1-\frac{\mu }{2}}+{A}^{1-\frac{\mu }{2}}X{B}^{\frac{\mu }{2}}⦀+3⦀{A}^{\mu }X{B}^{1-\mu }+{A}^{1-\mu }X{B}^{\mu }⦀\right)\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{2}\left(⦀AX+XB⦀+⦀{A}^{\mu }X{B}^{1-\mu }+{A}^{1-\mu }X{B}^{\mu }⦀\right);\hfill \end{array}$
(7)
2. (2)

for $\frac{1}{2}\le \mu \le 1$ and for every unitarily norm,

$\begin{array}{c}⦀{A}^{\frac{1+\mu }{2}}X{B}^{\frac{1-\mu }{2}}+{A}^{\frac{1-\mu }{2}}X{B}^{\frac{1+\mu }{2}}⦀\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{1-\mu }{\int }_{\mu }^{1}⦀{A}^{v}X{B}^{1-v}+{A}^{1-v}X{B}^{v}⦀\phantom{\rule{0.2em}{0ex}}dv\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{8}\left(3⦀AX+XB⦀+2⦀{A}^{\frac{1+\mu }{2}}X{B}^{\frac{1-\mu }{2}}+{A}^{\frac{1-\mu }{2}}X{B}^{\frac{1+\mu }{2}}⦀+3⦀{A}^{\mu }X{B}^{1-\mu }+{A}^{1-\mu }X{B}^{\mu }⦀\right)\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{2}\left(⦀AX+XB⦀+⦀{A}^{\mu }X{B}^{1-\mu }+{A}^{1-\mu }X{B}^{\mu }⦀\right).\hfill \end{array}$
(8)

Since the function $f\left(v\right)=⦀{A}^{v}X{B}^{1-v}+{A}^{1-v}X{B}^{v}⦀$ is decreasing on the interval $\left[0,\frac{1}{2}\right]$ and increasing on the interval $\left[\frac{1}{2},1\right]$, and using the inequalities (7) and (8), we obtain the refinement of the second inequality in (1).

Corollary 2 Let A, B, X be operators such that A, B are positive. Then for $0\le \mu \le 1$ and for every unitarily invariant norm, we have the following.

1. (1)

For $0\le \mu \le \frac{1}{2}$ and for every unitarily norm,

$\begin{array}{c}⦀{A}^{\mu }X{B}^{1-\mu }+{A}^{1-\mu }X{B}^{\mu }⦀\hfill \\ \phantom{\rule{1em}{0ex}}\le ⦀{A}^{\frac{\mu }{2}}X{B}^{1-\frac{\mu }{2}}+{A}^{1-\frac{\mu }{2}}X{B}^{\frac{\mu }{2}}⦀\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{\mu }{\int }_{0}^{\mu }⦀{A}^{v}X{B}^{1-v}+{A}^{1-v}X{B}^{v}⦀\phantom{\rule{0.2em}{0ex}}dv\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{8}\left(3⦀AX+XB⦀+2⦀{A}^{\frac{\mu }{2}}X{B}^{1-\frac{\mu }{2}}+{A}^{1-\frac{\mu }{2}}X{B}^{\frac{\mu }{2}}⦀+3⦀{A}^{\mu }X{B}^{1-\mu }+{A}^{1-\mu }X{B}^{\mu }⦀\right)\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{2}\left(⦀AX+XB⦀+⦀{A}^{\mu }X{B}^{1-\mu }+{A}^{1-\mu }X{B}^{\mu }⦀\right)\hfill \\ \phantom{\rule{1em}{0ex}}\le ⦀AX+XB⦀.\hfill \end{array}$
(9)
2. (2)

For $\frac{1}{2}\le \mu \le 1$ and for every unitarily norm,

$\begin{array}{c}⦀{A}^{\mu }X{B}^{1-\mu }+{A}^{1-\mu }X{B}^{\mu }⦀\hfill \\ \phantom{\rule{1em}{0ex}}\le ⦀{A}^{\frac{1+\mu }{2}}X{B}^{\frac{1-\mu }{2}}+{A}^{\frac{1-\mu }{2}}X{B}^{\frac{1+\mu }{2}}⦀\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{1-\mu }{\int }_{\mu }^{1}⦀{A}^{v}X{B}^{1-v}+{A}^{1-v}X{B}^{v}⦀\phantom{\rule{0.2em}{0ex}}dv\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{8}\left(3⦀AX+XB⦀+2⦀{A}^{\frac{1+\mu }{2}}X{B}^{\frac{1-\mu }{2}}+{A}^{\frac{1-\mu }{2}}X{B}^{\frac{1+\mu }{2}}⦀+3⦀{A}^{\mu }X{B}^{1-\mu }+{A}^{1-\mu }X{B}^{\mu }⦀\right)\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{2}\left(⦀AX+XB⦀+⦀{A}^{\mu }X{B}^{1-\mu }+{A}^{1-\mu }X{B}^{\mu }⦀\right)\hfill \\ \phantom{\rule{1em}{0ex}}\le ⦀AX+XB⦀.\hfill \end{array}$
(10)

It should be noticed that in the inequalities (7) to (10), we have

$\begin{array}{c}\underset{\mu \to 0}{lim}\frac{1}{\mu }{\int }_{0}^{\mu }⦀{A}^{v}X{B}^{1-v}+{A}^{1-v}X{B}^{v}⦀\phantom{\rule{0.2em}{0ex}}dv\hfill \\ \phantom{\rule{1em}{0ex}}\le \underset{\mu \to 1}{lim}\frac{1}{1-\mu }{\int }_{\mu }^{1}⦀{A}^{v}X{B}^{1-v}+{A}^{1-v}X{B}^{v}⦀\phantom{\rule{0.2em}{0ex}}dv\hfill \\ \phantom{\rule{1em}{0ex}}=⦀AX+XB⦀.\hfill \end{array}$

## References

1. Bhatia R, Davis C: More matrix forms of the arithmetic-geometric mean inequality. SIAM J. Matrix Anal. Appl. 1993, 14: 132-136. 10.1137/0614012

2. Feng Y: Refinements of the Heinz inequalities. J. Inequal. Appl. 2012., 2012: Article ID 18 10.1186/1029-242X-2012-18

3. Wang S: Some new refinements of Heinz inequalities of matrices. J. Inequal. Appl. 2013., 2013: Article ID 424 10.1186/1029-242X-2013-424

4. Kittaneh F: On the convexity of the Heinz means. Integral Equ. Oper. Theory 2010, 68: 519-527. 10.1007/s00020-010-1807-6

5. Bullen PS Pitman Monographs and Surveys in Pure and Applied Mathematics 97. In A Dictionary of Inequalities. Longman, Harlow; 1998.

## Acknowledgements

This work is supported by NSF of China (Grant Nos. 11171364 and 11271301).

## Author information

Authors

### Corresponding author

Correspondence to Guiyun Chen.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

YY carried out convex function. YF carried out unitarily invariant norm. GC carried out the calculation. All authors read and approved the final manuscript.

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Yan, Y., Feng, Y. & Chen, G. Refinements of the Heinz inequalities for matrices. J Inequal Appl 2014, 50 (2014). https://doi.org/10.1186/1029-242X-2014-50

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• DOI: https://doi.org/10.1186/1029-242X-2014-50

### Keywords

• convex function
• Heinz inequality