# 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 $⦀⋅⦀$, the function $f(v)=⦀ A v X B 1 − v + A 1 − v X B v ⦀$ is convex on the interval $[0,1]$, attains its minimum at $v= 1 2$, and attains its maximum at $v=0$ and $v=1$. Moreover, $f(v)=f(1−v)$ for $0≤v≤1$. From  we know that for every unitarily invariant norm, we have the Heinz inequalities

$2⦀ A 1 2 X B 1 2 ⦀≤⦀ A v X B 1 − v + A 1 − v X B v ⦀≤⦀AX+XB⦀.$
(1)

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

$f ( a + b 2 ) ≤ 1 b − a ∫ a b f(t)dt≤ 1 4 ( f ( a ) + 2 f ( a + b 2 ) + f ( b ) ) ≤ f ( a ) + f ( b ) 2 ,$

where f is a real-valued function which is convex on the interval $[a,b]$. With a similar method, Wang  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 $⦀⋅⦀$. Our results are better than those in  and different from [2, 3].

## 2 Main results

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

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

$f ( a + b 2 ) ≤ 1 b − a ∫ a b f(t)dt≤ f ( a ) + f ( b ) 2 .$

We will use the following lemma.

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

$f ( a + b 2 ) ≤ 1 b − a ∫ a b f(t)dt≤ 1 8 ( 3 f ( a ) + 2 f ( a + b 2 ) + 3 f ( b ) ) ≤ f ( a ) + f ( b ) 2 .$

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

$1 8 ( 3 f ( a ) + 2 f ( a + b 2 ) + 3 f ( b ) ) ≤ f ( a ) + f ( b ) 2 .$

Next, we will prove the following inequality:

$1 b − a ∫ a b f(t)dt≤ 1 8 ( 3 f ( a ) + 2 f ( a + b 2 ) + 3 f ( b ) ) .$

From the previous lemma, we have

$1 b − a ∫ a b f ( t ) d t = 1 b − a ( ∫ a a + b 2 f ( t ) d t + ∫ a + b 2 b f ( t ) d t ) ≤ 1 b − a ( f ( a ) + f ( a + b 2 ) 2 ⋅ b − a 2 + f ( a + b 2 ) + f ( b ) 2 ⋅ b − a 2 ) = 1 4 ( f ( a ) + 2 f ( a + b 2 ) + f ( b ) ) = 1 8 ( 2 f ( a ) + 4 f ( a + b 2 ) + 2 f ( b ) ) ≤ 1 8 ( 2 f ( a ) + 2 f ( a + b 2 ) + ( f ( a ) + f ( b ) ) + 2 f ( b ) ) = 1 8 ( 3 f ( a ) + 2 f ( a + b 2 ) + 3 f ( b ) ) .$

□

Applying the previous lemma to the function $f(v)=⦀ A v X B 1 − v + A 1 − v X B v ⦀$ on the interval $[μ,1−μ]$ when $0≤μ≤ 1 2$, and on the interval $[1−μ,μ]$ when $1 2 ≤μ≤1$, 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 $0≤μ≤1$ and for every unitarily invariant norm, we have

$2 ⦀ A 1 2 X B 1 2 ⦀ ≤ 1 | 1 − 2 μ | | ∫ μ 1 − μ ⦀ A v X B 1 − v + A 1 − v X B v ⦀ d v | ≤ 1 4 ( 3 ⦀ A μ X B 1 − μ + A 1 − μ X B μ ⦀ + 2 ⦀ A 1 2 X B 1 2 ⦀ ) ≤ ⦀ A μ X B 1 − μ + A 1 − μ X B μ ⦀ .$
(2)

Proof First assume that $0≤μ≤ 1 2$. Then it follows by the previous lemma that

$f ( 1 − μ + μ 2 ) ≤ 1 1 − 2 μ ∫ μ 1 − μ f ( t ) d t ≤ 1 8 ( 3 f ( μ ) + 2 f ( 1 − μ + μ 2 ) + 3 f ( 1 − μ ) ) ≤ f ( μ ) + f ( 1 − μ ) 2 ,$

and so

$f ( 1 2 ) ≤ 1 1 − 2 μ ∫ μ 1 − μ f ( t ) d t ≤ 1 4 ( 3 f ( μ ) + f ( 1 2 ) ) ≤ f ( μ ) .$

Thus,

$2 ⦀ A 1 2 X B 1 2 ⦀ ≤ 1 1 − 2 μ ∫ μ 1 − μ ⦀ A v X B 1 − v + A 1 − v X B v ⦀ d v ≤ 1 4 ( 3 ⦀ A μ X B 1 − μ + A 1 − μ X B μ ⦀ + 2 ⦀ A 1 2 X B 1 2 ⦀ ) ≤ ⦀ A μ X B 1 − μ + A 1 − μ X B μ ⦀ .$
(3)

Now, assume that $1 2 ≤μ≤1$. Then by applying (3) to $1−μ$, it follows that

$2 ⦀ A 1 2 X B 1 2 ⦀ ≤ 1 2 μ − 1 ∫ 1 − μ μ ⦀ A v X B 1 − v + A 1 − v X B v ⦀ d v ≤ 1 4 ( 3 ⦀ A μ X B 1 − μ + A 1 − μ X B μ ⦀ + 2 ⦀ A 1 2 X B 1 2 ⦀ ) ≤ ⦀ A μ X B 1 − μ + A 1 − μ X B μ ⦀ .$
(4)

Since

$lim μ → 1 2 1 | 1 − 2 μ | | ∫ μ 1 − μ ⦀ A v X B 1 − v + A 1 − v X B v ⦀ d v | = lim μ → 1 2 1 4 ( 3 ⦀ A μ X B 1 − μ + A 1 − μ X B μ ⦀ + 2 ⦀ A 1 2 X B 1 2 ⦀ ) = 2 ⦀ A 1 2 X B 1 2 ⦀ ,$

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

Applying the previous lemma to the function $f(v)=⦀ A v X B 1 − v + A 1 − v X B v ⦀$ on the interval $[μ, 1 2 ]$ when $0≤μ≤ 1 2$, and on the interval $[ 1 2 ,μ]$ when $1 2 ≤μ≤1$, we obtain the following.

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

$⦀ A 2 μ + 1 4 X B 3 − 2 μ 4 + A 3 − 2 μ 4 X B 2 μ + 1 4 ⦀ ≤ 2 | 1 − 2 μ | | ∫ μ 1 2 ⦀ A v X B 1 − v + A 1 − v X B v ⦀ d v | ≤ 1 8 ( 3 ⦀ A μ X B 1 − μ + A 1 − μ X B μ ⦀ + 2 ⦀ A 2 μ + 1 4 X B 3 − 2 μ 4 + A 3 − 2 μ 4 X B 2 μ + 1 4 ⦀ + 6 ⦀ A 1 2 X B 1 2 ⦀ ) ≤ 1 2 ( ⦀ A μ X B 1 − μ + A 1 − μ X B μ ⦀ + 2 ⦀ A 1 2 X B 1 2 ⦀ ) .$
(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 $0≤μ≤1$ and for every unitarily invariant norm, we have

$2 ⦀ A 1 2 X B 1 2 ⦀ ≤ ⦀ A 2 μ + 1 4 X B 3 − 2 μ 4 + A 3 − 2 μ 4 X B 2 μ + 1 4 ⦀ ≤ 2 | 1 − 2 μ | | ∫ μ 1 2 ⦀ A v X B 1 − v + A 1 − v X B v ⦀ d v | ≤ 1 8 ( 3 ⦀ A μ X B 1 − μ + A 1 − μ X B μ ⦀ + 2 ⦀ A 2 μ + 1 4 X B 3 − 2 μ 4 + A 3 − 2 μ 4 X B 2 μ + 1 4 ⦀ + 6 ⦀ A 1 2 X B 1 2 ⦀ ) ≤ 1 2 ( ⦀ A μ X B 1 − μ + A 1 − μ X B μ ⦀ + 2 ⦀ A 1 2 X B 1 2 ⦀ ) ≤ ⦀ A μ X B 1 − μ + A 1 − μ X B μ ⦀ .$
(6)

Applying the previous lemma to the function $f(v)=⦀ A v X B 1 − v + A 1 − v X B v ⦀$ on the interval $[0,μ]$ when $0≤μ≤ 1 2$, and on the interval $[μ,1]$ when $1 2 ≤μ≤1$, we obtain the following theorem.

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

1. (1)

for $0≤μ≤ 1 2$ and for every unitarily norm,

$⦀ A μ 2 X B 1 − μ 2 + A 1 − μ 2 X B μ 2 ⦀ ≤ 1 μ ∫ 0 μ ⦀ A v X B 1 − v + A 1 − v X B v ⦀ d v ≤ 1 8 ( 3 ⦀ A X + X B ⦀ + 2 ⦀ A μ 2 X B 1 − μ 2 + A 1 − μ 2 X B μ 2 ⦀ + 3 ⦀ A μ X B 1 − μ + A 1 − μ X B μ ⦀ ) ≤ 1 2 ( ⦀ A X + X B ⦀ + ⦀ A μ X B 1 − μ + A 1 − μ X B μ ⦀ ) ;$
(7)
2. (2)

for $1 2 ≤μ≤1$ and for every unitarily norm,

$⦀ A 1 + μ 2 X B 1 − μ 2 + A 1 − μ 2 X B 1 + μ 2 ⦀ ≤ 1 1 − μ ∫ μ 1 ⦀ A v X B 1 − v + A 1 − v X B v ⦀ d v ≤ 1 8 ( 3 ⦀ A X + X B ⦀ + 2 ⦀ A 1 + μ 2 X B 1 − μ 2 + A 1 − μ 2 X B 1 + μ 2 ⦀ + 3 ⦀ A μ X B 1 − μ + A 1 − μ X B μ ⦀ ) ≤ 1 2 ( ⦀ A X + X B ⦀ + ⦀ A μ X B 1 − μ + A 1 − μ X B μ ⦀ ) .$
(8)

Since the function $f(v)=⦀ A v X B 1 − v + A 1 − v X B v ⦀$ is decreasing on the interval $[0, 1 2 ]$ and increasing on the interval $[ 1 2 ,1]$, 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 $0≤μ≤1$ and for every unitarily invariant norm, we have the following.

1. (1)

For $0≤μ≤ 1 2$ and for every unitarily norm,

$⦀ A μ X B 1 − μ + A 1 − μ X B μ ⦀ ≤ ⦀ A μ 2 X B 1 − μ 2 + A 1 − μ 2 X B μ 2 ⦀ ≤ 1 μ ∫ 0 μ ⦀ A v X B 1 − v + A 1 − v X B v ⦀ d v ≤ 1 8 ( 3 ⦀ A X + X B ⦀ + 2 ⦀ A μ 2 X B 1 − μ 2 + A 1 − μ 2 X B μ 2 ⦀ + 3 ⦀ A μ X B 1 − μ + A 1 − μ X B μ ⦀ ) ≤ 1 2 ( ⦀ A X + X B ⦀ + ⦀ A μ X B 1 − μ + A 1 − μ X B μ ⦀ ) ≤ ⦀ A X + X B ⦀ .$
(9)
2. (2)

For $1 2 ≤μ≤1$ and for every unitarily norm,

$⦀ A μ X B 1 − μ + A 1 − μ X B μ ⦀ ≤ ⦀ A 1 + μ 2 X B 1 − μ 2 + A 1 − μ 2 X B 1 + μ 2 ⦀ ≤ 1 1 − μ ∫ μ 1 ⦀ A v X B 1 − v + A 1 − v X B v ⦀ d v ≤ 1 8 ( 3 ⦀ A X + X B ⦀ + 2 ⦀ A 1 + μ 2 X B 1 − μ 2 + A 1 − μ 2 X B 1 + μ 2 ⦀ + 3 ⦀ A μ X B 1 − μ + A 1 − μ X B μ ⦀ ) ≤ 1 2 ( ⦀ A X + X B ⦀ + ⦀ A μ X B 1 − μ + A 1 − μ X B μ ⦀ ) ≤ ⦀ A X + X B ⦀ .$
(10)

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

$lim μ → 0 1 μ ∫ 0 μ ⦀ A v X B 1 − v + A 1 − v X B v ⦀ d v ≤ lim μ → 1 1 1 − μ ∫ μ 1 ⦀ A v X B 1 − v + A 1 − v X B v ⦀ d v = ⦀ A X + X B ⦀ .$

## References

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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. 2.

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

3. 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. 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. 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 