# Improved Heinz inequality and its application

## Abstract

We obtain an improved Heinz inequality for scalars and we use it to establish an inequality for the Hilbert-Schmidt norm of matrices, which is a refinement of a result due to Kittaneh.

Mathematical Subject Classification 2010: 26D07; 26D15; 15A18.

## 1. Introduction

Let M n be the space of n × n complex matrices and ||·|| stand for any unitarily invariant norm on M n . So, ||UAV|| = ||A|| for all A M n and for all unitary matrices U, V M n . If A = [a ij ] M n , then

$A 2 = ∑ i , j = 1 n | a i j | 2 1 / 2$

is the Hilbert-Schmidt norm of matrix A. It is known that the Hilbert-Schmidt norm is unitarily invariant.

The classical Young's inequality for nonnegative real numbers says that if a, b ≥ 0 and 0 ≤ v ≤ 1, then

$a v b 1 - v ≤ v a + ( 1 - v ) b$
(1.1)

with equality if and only if a = b. Young's inequality for scalars is not only interesting in itself but also very useful. If $v= 1 2$, by (1.1), we obtain the arithmetic-geometric mean inequality

$2 a b ≤ a + b .$
(1.2)

Kittaneh and Manasrah  obtained a refinement of Young's inequality as follows:

$a v b 1 - v + r 0 a - b 2 ≤ v a + ( 1 - v ) b ,$
(1.3)

where r0 = min {v, 1 − v}.

Let a, b ≥ 0 and 0 ≤ v ≤ 1. The Heinz means are defined as follows:

$H v ( a , b ) = a v b 1 - v + a 1 - v b v 2 .$

It follows from the inequalities (1.1) and (1.2) that the Heinz means interpolate between the geometric mean and the arithmetic mean:

$a b ≤ H v ( a , b ) ≤ a + b 2 .$
(1.4)

The second inequality of (1.4) is known as Heinz inequality for nonnegative real numbers.

As a direct consequence of the inequality (1.3), Kittaneh and Manasrah  obtained a refinement of the Heinz inequality as follows:

$H v ( a , b ) + r 0 a - b 2 ≤ a + b 2 ,$
(1.5)

where r0 = min {v, 1 − v}.

Bhatia and Davis  proved that if A, B, X M n such that A and B are positive semidefinite and if 0 ≤ v ≤ 1, then

$2 A 1 / 2 X B 1 / 2 ≤ A v X B 1 - v + A 1 - v X B v ≤ A X + X B .$
(1.6)

This is a matrix version of the inequality (1.4). Kittaneh  proved that if A, B, X M n such that A and B are positive semidefinite and if 0 ≤ v ≤ 1, then

$A v X B 1 - v + A 1 - v X B v ≤ 4 r 0 A 1 / 2 X B 1 / 2 + ( 1 - 2 r 0 ) A X + X B ,$
(1.7)

where r0 = min {v, 1 − v}. This is a refinement of the second inequality in (1.6).

In this article, we first present a refinement of the inequality (1.5). After that, we use it to establish a refinement of the inequality (1.7) for the Hilbert-Schmidt norm.

## 2. A refinement of the inequality (1.5)

In this section, we give a refinement of the inequality (1.5). To do this, we need the following lemma.

Lemma 2.1. [4, 5] Let f(x) be a real valued convex function on an interval [a, b]. For any x1, x2 [a, b], we have

$f ( x ) ≤ f ( x 2 ) - f ( x 1 ) x 2 - x 1 x - x 1 f ( x 2 ) - x 2 f ( x 1 ) x 2 - x 1 , x ∈ ( x 1 , x 2 ) .$

Theorem 2.1. Let a, b ≥ 0 and 0 ≤ v ≤ 1. If r0 = min {v, 1 − v}, then

$2 H v ( a , b ) ≤ ( 1 - 4 r 0 ) ( a + b ) + 4 r 0 ( a 1 / 4 b 3 / 4 + a 3 / 4 b 1 / 4 ) , v ∈ [ 0 , 1 4 ] ∪ [ 3 4 , 1 ] , 2 ( 4 r 0 - 1 ) a b + 2 ( 1 - 2 r 0 ) ( a 1 / 4 b 3 / 4 + a 3 / 4 b 1 / 4 ) , v ∈ [ 1 4 , 3 4 ] .$
(2.1)

Proof. It is known that as a function of v, H v (a, b) is convex and attains its minimum at $v= 1 2$. Let

$f ( v ) = 2 H v ( a , b ) = a v b 1 - v + a 1 - v b v , 0 ≤ v ≤ 1 .$

Obviously, f(v) is convex. For $0≤v≤ 1 4$, since f(v) is convex on 0, by Lemma 2.1, we have

$f v ≤ f 1 4 - f 0 1 4 - 0 v - 0 f 1 4 - 1 4 f 0 1 4 - 0 ,$

which is equivalent to

$f v ≤ 4 f 1 4 - f 0 v + f 0 .$

That is,

$f v ≤ 1 - 4 v f 0 + 4 v f 1 4 .$

So,

$a v b 1 - v + a 1 - v b v ≤ ( 1 - 4 r 0 ) a + b + 4 r 0 a 1 / 4 b 3 / 4 + a 3 / 4 b 1 / 4 .$

For $3 4 ≤v≤1$, similarly, we have

$f v ≤ f 1 - f 3 4 1 - 3 4 v - 3 4 f 1 - f 3 4 1 - 3 4 ,$

which is equivalent to

$f v ≤ 4 f 1 - f 3 4 v - 3 f 1 + 4 f 3 4 .$

That is,

$f v ≤ 4 v - 3 f 1 + 4 1 - v f 3 4 .$

So,

$a v b 1 - v + a 1 - v b v ≤ ( 1 - 4 r 0 ) a + b + 4 r 0 a 1 / 4 b 3 / 4 + a 3 / 4 b 1 / 4 .$

If $1 4 ≤v≤ 1 2$, then by Lemma 2.1, we have

$f v ≤ f 1 2 - f 1 4 1 2 - 1 4 v - 1 4 f 1 2 - 1 2 f 1 4 1 2 - 1 4 ,$

and so

$f v ≤ 4 v - 1 f 1 2 + 2 1 - 2 v f 1 4 ,$

which is equivalent to

$a v b 1 - v + a 1 - v b v ≤ 2 ( 4 r 0 - 1 ) a b + 2 ( 1 - 2 r 0 ) a 1 / 4 b 3 / 4 + a 3 / 4 b 1 / 4 .$

If $1 2 ≤v≤ 3 4$, similarly, we have

$f v ≤ f 3 4 - f 1 2 3 4 - 1 2 v - 1 2 f 3 4 - 3 4 f 1 2 3 4 - 1 2 ,$

and so

$f v ≤ 3 - 4 v f 1 2 + 2 2 v - 1 f 3 4 ,$

which is equivalent to

$f v ≤ 4 r 0 - 1 f 1 2 + 2 1 - 2 r 0 f 3 4 .$

That is,

$a v b 1 - v + a 1 - v b v ≤ 2 ( 4 r 0 - 1 ) a b + 2 ( 1 - 2 r 0 ) a 1 / 4 b 3 / 4 + a 3 / 4 b 1 / 4 .$

This completes the proof. □

Now, we give a simple comparison between the upper bound for avb1-v+ a1-vbv in (1.5) and (2.1). If $v∈ 0 , 1 4 ∪ 3 4 , 1$, then

$a + b - 2 r 0 a - b 2 - ( 1 - 4 r 0 ) ( a + b ) - 4 r 0 ( a 1 / 4 b 3 / 4 + a 3 / 4 b 1 / 4 ) = 2 r 0 a + b + 2 a b - 2 a 1 / 4 b 3 / 4 + a 3 / 4 b 1 / 4 ≥ 0 .$

If $v∈ 1 4 , 3 4$, then

$a + b - 2 r 0 a - b 2 - 2 ( 4 r 0 - 1 ) a b - 2 ( 1 - 2 r 0 ) ( a 1 / 4 b 3 / 4 + a 3 / 4 b 1 / 4 ) = 1 - 2 r 0 a + b + 2 a b - 2 a 1 / 4 b 3 / 4 + a 3 / 4 b 1 / 4 ≥ 0 .$

So, the inequality (2.1) is a refinement of the inequality (1.5).

## 3. An application

In this section, we give a refinement of the inequality (1.7) for the Hilbert-Schmidt norm based on the inequality (2.1).

Theorem 3.1. Let A, B, X M n such that A and B are positive semidefinite and suppose

that

$ϕ v = | | A v X B 1 - v + A 1 - v X B v | | 2 , 0 ≤ v ≤ 1 .$

Then

$ϕ ( v ) ≤ ( 1 - 4 r 0 ) ϕ ( 0 ) + 4 r 0 ϕ 1 4 , v ∈ [ 0 , 1 4 ] ∪ 3 4 , 1 ( 4 r 0 - 1 ) ϕ 1 2 + 2 ( 1 - 2 r 0 ) ϕ 1 4 , v ∈ 1 4 , 3 4 ,$
(3.1)

where r0 = min {v, 1 − v}.

Proof. Since every positive semidefinite matrix is unitarily diagonalizable, it follows that there exist unitary matrices U, V M n such that A = U Λ1U* and B = V Λ2V*, where Λ1 = diag (λ1,..., λ n ), Λ2 = diag(µ1,..., µ n ) and λ i , µ i 0, i = 1,..., n. Let

$Y = U * X V = [ y i j ] .$

If $v∈ 0 , 1 4 ∪ 3 4 , 1$, then by (2.1) and the Cauchy-Schwarz inequality, we have

$| | A v X B 1 - v + A 1 - v X B v | | 2 2 = ∑ i , j = 1 n ( λ i v μ j 1 - v + λ i 1 - v μ j v ) 2 | y i j | 2 ≤ ∑ i , j = 1 n ( 1 - 4 r 0 ) ( λ i + μ j ) + 4 r 0 ( λ i 1 / 4 μ j 3 / 4 + λ i 3 / 4 μ j 1 / 4 ) 2 | y i j | 2 = ( 1 - 4 r 0 ) 2 ∑ i , j = 1 n ( λ i + μ j ) 2 | y i j | 2 + 16 r 0 2 ∑ i , j = 1 n λ i 1 / 4 μ j 3 / 4 + λ i 3 / 4 μ j 1 / 4 2 | y i j | 2 + 8 r 0 ( 1 - 4 r 0 ) ∑ i , j = 1 n ( λ i + μ j ) λ i 1 / 4 μ j 3 / 4 + λ i 3 / 4 μ j 1 / 4 | y i j | 2 ≤ ( 1 - 4 r 0 ) 2 ϕ 2 ( 0 ) + 16 r 0 2 ϕ 2 1 4 + 8 r 0 ( 1 - 4 r 0 ) ϕ ( 0 ) ϕ 1 4 = ( 1 - 4 r 0 ) ϕ ( 0 ) + 4 r 0 ϕ 1 4 2 .$

If $v∈ 1 4 , 3 4$, the result follows from the inequality (2.1) and the same method above. This completes the proof. □

Remark. For the Hilbert-Schmidt norm, by the inequality (1.7), we have

$ϕ v ≤ 2 r 0 ϕ 1 2 + 1 - 2 r 0 ϕ 0 .$

So, for $v∈ 0 , 1 4 ∪ 3 4 , 1$, we have

$2 r 0 ϕ 1 2 + ( 1 - 2 r 0 ) ϕ 0 - ( 1 - 4 r 0 ) ϕ 0 - 4 r 0 ϕ 1 4 = 2 r 0 ϕ 1 2 + ϕ 0 - 2 ϕ 1 4 ≥ 0 .$

If $v∈ 1 4 , 3 4$, then

$2 r 0 ϕ 1 2 + ( 1 - 2 r 0 ) ϕ 0 - ( 4 r 0 - 1 ) ϕ 1 2 - 2 ( 1 - 2 r 0 ) ϕ 1 4 = ( 1 - 2 r 0 ) ϕ 1 2 + ϕ 0 - 2 ϕ 1 4 ≥ 0 .$

So, the inequality (3.1) is a refinement of the inequality (1.7) for the Hilbert-Schmidt norm.

## References

1. 1.

Kittaneh F, Manasrah Y: Improved Young and Heinz inequalities for matrices. J Math Anal Appl 2010, 361: 262–269. 10.1016/j.jmaa.2009.08.059

2. 2.

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

3. 3.

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

4. 4.

Bhatia R, Sharma R: Some inequalities for positive linear maps. Linear Algebra Appl 2012, 436: 1562–1571. 10.1016/j.laa.2010.09.038

5. 5.

Wang S, Zou L, Jiang Y: Some inequalities for unitarily invariant norms of matrices. J Inequal Appl 2011, 2011: 10. 10.1186/1029-242X-2011-10

## Acknowledgements

The authors wish to express their heartfelt thanks to the referees and Professor Gnana Bhaskar Tenali for their detailed and helpful suggestions for revising the manuscript. This research was supported by the Scientific Research Project of Chongqing Three Gorges University (No. 11QN-21).

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Correspondence to Limin Zou.

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

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Zou, L., Jiang, Y. Improved Heinz inequality and its application. J Inequal Appl 2012, 113 (2012). https://doi.org/10.1186/1029-242X-2012-113 