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A family of refinements of Heinz inequalities of matrices

Abstract

For any unitarily invariant norm , the Heinz inequalities for operators assert that 2 A 1 2 X B 1 2 A ν X B 1 ν + A 1 ν X B ν AX+XB, for A, B, and X any operators on a complex separable Hilbert space such that A, B are positive and ν[0,1]. In this paper, we obtain a family of refinements of these norm inequalities by using the convexity of the function f(ν)= A ν X B 1 ν + A 1 ν X B ν and the Hermite-Hadamard inequality.

1 Introduction

Let M n (C) be the algebra of n×n complex matrices. We denote by H n (C) the set of all Hermitian matrices in M n (C). The set of all positive semi-definite matrices in M n (C) shall be denoted by H n + (C). A norm on M n (C) is called unitarily invariant or symmetric if

UAV=A

for all A M n (C) and for all unitaries U,V M n (C).

The arithmetic-geometric mean inequality for two nonnegative real numbers a and b is

a b a + b 2 ,

which has been generalized to the context of matrices as follows:

2 A 1 2 X B 1 2 AX+XB,

where A,B H n + (C), X M n , and is a unitarily invariant norm on M n (C).

For ν[0,1] and two nonnegative numbers a and b, the Heinz mean is defined as

H ν (a,b)= a ν b 1 ν + a 1 ν b ν 2 .

Clearly the Heinz mean interpolates between the geometric mean and the arithmetic mean:

a b H ν (a,b) a + b 2 .

The function H ν (a,b) has the following properties: it is convex, attains its minimum at ν= 1 2 , its maximum at ν=0 and ν=1, and H ν (a,b)= H 1 ν (a,b) for 0ν1. The generalization of the above inequalities to matrices is due to Bhatia and Davis [1] as follows:

2 A 1 2 X B 1 2 A ν X B 1 ν + A 1 ν X B ν AX+XB,
(1.1)

where A,B H n + (C), X M n (C), and ν[0,1]. For a historical background and proofs of these norm inequalities as well as their refinements, and diverse applications, we refer the reader to the [28], and the references therein. Indeed, it has been proved, in [1], that f(ν)= A ν X B 1 ν + A 1 ν X B ν is a convex function of ν on [0,1] with symmetry about ν= 1 2 , and attains its minimum there and it has a maximum at ν=0 and ν=1. Moreover, it increases on [0, 1 2 ] and decreases on [ 1 2 ,1].

In [4, 5], (1.1) is refined by using the so-called Hermite-Hadamard inequality:

g ( a + b 2 ) 1 b a a b g(t)dt g ( a ) + g ( b ) 2 ,

where g is a convex function on [a,b].

Recently, in [3] and [7], respectively, the following inequalities were used to get new refinements of (1.1):

g ( a + b 2 ) 1 b a a b g ( t ) d t 1 4 ( g ( a ) + 2 g ( a + b 2 ) + g ( b ) ) g ( a ) + g ( b ) 2 , g ( a + b 2 ) 1 b a a b g ( t ) d t 1 32 ( 15 g ( a ) + 2 g ( a + b 2 ) + 15 g ( b ) ) g ( a ) + g ( b ) 2 .

The purpose of this note is to obtain a family of new refinements of Heinz inequalities for matrices. Also the two refinements, given in [3] and [7], are two special cases of this new family.

2 Main results

We start by the following key lemma which plays a central role in our investigation to obtain a further series of refinements of the Heinz inequalities.

Lemma 1 Let g be a convex function on the interval [a,b]. Then for any positive integer n, we have

g ( a + b 2 ) 1 b a a b g ( t ) d t 1 4 n [ ( 2 n 1 ) g ( a ) + 2 g ( a + b 2 ) + ( 2 n 1 ) g ( b ) ] g ( a ) + g ( b ) 2 .

Proof Since g is convex on [a,b], we have

g ( a + b 2 ) g ( a ) + g ( b ) 2 .

Thus

(2n1)g(a)+2g ( a + b 2 ) +(2n1)g(b)2ng(a)+2ng(b),

whence

1 4 n ( ( 2 n 1 ) g ( a ) + 2 g ( a + b 2 ) + ( 2 n 1 ) g ( b ) ) g ( a ) + g ( b ) 2 .

To prove the middle inequality, we start by

1 b a a b g ( t ) d t = 1 b a [ a a + b 2 g ( t ) d t + a + b 2 b g ( t ) d t ] 1 b a [ g ( a + b 2 ) + g ( a ) 2 b a 2 + g ( b ) + g ( a + b 2 ) 2 b a 2 ] = 1 4 [ g ( a ) + 2 g ( a + b 2 ) + g ( b ) ] = 1 4 n [ n g ( a ) + 2 n g ( a + b 2 ) + n g ( b ) ] = 1 4 n [ n g ( a ) + 2 g ( a + b 2 ) + ( 2 n 2 ) g ( a + b 2 ) + n g ( b ) ] 1 4 n [ n g ( a ) + 2 g ( a + b 2 ) + ( 2 n 2 ) [ g ( a ) + g ( b ) 2 ] + n g ( b ) ] = 1 4 n [ n g ( a ) + 2 g ( a + b 2 ) + ( n 1 ) g ( a ) + ( n 1 ) g ( b ) + n g ( b ) ] = 1 4 n [ ( 2 n 1 ) g ( a ) + 2 g ( a + b 2 ) + ( 2 n 1 ) g ( b ) ] .

 □

Applying the previous lemma on the convex function defined earlier

f(ν)= A ν X B 1 ν + A 1 ν X B ν

on the interval [μ,1μ] when 0μ 1 2 and on the interval [1μ,μ] when 1 2 μ1, we obtain the following refinement of the first inequality (1.1) which is a kind of refinements of Theorem 1 in a paper Kittaneh [5] and Theorem 1 in a paper of Feng [3].

Theorem 1 Let A,B H n + (C), and X M n (C). Let n be any positive integer. Then for any μ[0,1], and for every unitarily invariant norm on M n (C), we have

2 A 1 2 X B 1 2 1 | 1 2 μ | | μ 1 μ A ν X B 1 ν + A 1 ν X B ν d ν | 1 2 n [ ( 2 n 1 ) 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 .
(2.1)

Proof First assume that 0μ 1 2 . Then it follows from Lemma 1 that

f ( μ + 1 μ 2 ) 1 1 2 μ μ 1 μ f ( t ) d t 1 4 n [ ( 2 n 1 ) f ( μ ) + 2 f ( 1 μ + μ 2 ) + ( 2 n 1 ) f ( 1 μ ) ] f ( μ ) + f ( 1 μ ) 2 = f ( μ ) .

Since f(μ)=f(1μ), we have

f ( 1 2 ) 1 1 2 μ μ 1 μ f ( t ) d t 1 4 n [ ( 4 n 2 ) f ( μ ) + 2 f ( 1 2 ) ] 1 2 n [ ( 2 n 1 ) f ( μ ) + f ( 1 2 ) ] f ( μ ) .

Thus,

2 A 1 2 X B 1 2 1 1 2 μ μ 1 μ A ν X B 1 ν + A 1 ν X B ν d ν 1 2 n [ ( 2 n 1 ) 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 .
(2.2)

Now, assume that 1 2 μ1. Then, by applying (2.2) to 1μ, it follows that

2 A 1 2 X B 1 2 1 2 μ 1 1 μ μ A ν X B 1 ν + A 1 ν X B ν d ν 1 2 n [ ( 2 n 1 ) 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 .
(2.3)

Since

lim μ 1 2 1 | 1 2 μ | | μ 1 μ A ν X B 1 ν + A 1 ν X B ν d ν | = lim μ 1 2 1 4 n [ ( 4 n 2 ) f ( μ ) + f ( 1 2 ) ] = 2 A 1 2 X B 1 2 ,

the inequalities in (2.1) follow by combining (2.2) and (2.3) and so the required result is proved. □

Applying Lemma 1 to the function f(ν)= A ν X B 1 ν + A 1 ν X B ν in the interval [μ, 1 2 ] on 0μ 1 2 , and in the interval [ 1 2 ,μ] for 1 2 μ1, we obtain the following, which is a kind of refinements of Theorem 2 in a paper Kittaneh [5] and Theorem 2 in a paper of Feng [3].

Theorem 2 Let A,B H n + (C), and X M n (C). Then, for any positive integer n, any μ[0,1], and for every unitarily invariant norm on M n (C), we have

A 1 + 2 μ 4 X B 3 2 μ 4 + A 3 2 μ 4 X B 1 + 2 μ 4 2 | 1 2 μ | | μ 1 2 A ν X B 1 ν + A 1 ν X B ν d ν | 1 4 n [ ( 2 n 1 ) A μ X B 1 μ + A 1 μ X B μ + 2 A 1 + 2 μ 4 X B 3 2 μ 4 + A 3 2 μ 4 X B 1 + 2 μ 4 + 2 ( 2 n 1 ) 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 ] .
(2.4)

Inequalities (2.4) and the first inequality in (1.1) yield the following refinements of the first inequality in (1.1).

Corollary 1 Let A,B H n + (C), and X M n (C). Then, for any positive integer n, any μ[0,1], and for every unitarily invariant norm on X M n (C), we have

2 A 1 2 X B 1 2 A 1 + 2 μ 4 X B 3 2 μ 4 + A 3 2 μ 4 X B 1 + 2 μ 4 2 | 1 2 μ | | μ 1 2 A ν X B 1 ν + A 1 ν X B ν d ν | 1 4 n [ ( 2 n 1 ) A μ X B 1 μ + A 1 μ X B μ + 2 A 1 + 2 μ 4 X B 3 2 μ 4 + A 3 2 μ 4 X B 1 + 2 μ 4 + 2 ( 2 n 1 ) 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 .
(2.5)

Applying the Lemma 1 to the function f(ν)= A ν X B 1 ν + A 1 ν X B ν on the interval [μ, 1 2 ] when 0μ 1 2 , and on the interval [ 1 2 ,μ] when 1 2 μ1, we obtain the following theorem, which is a kind of refinements of Theorem 3 in a paper Kittaneh [5] and Theorem 3 in a paper of Feng [3].

Theorem 3 Let A,B H n + (C), and X M n (C) and let n be a positive integer. Then:

  1. (1)

    for any 0μ 1 2 and for every unitarily invariant norm , we have

    A μ 2 X B 1 μ 2 + A 1 μ 2 X B μ 2 1 μ 0 μ A ν X B 1 ν + A 1 ν X B ν d ν 1 4 n [ ( 2 n 1 ) A X + X B + 2 A μ 2 X B 1 μ 2 + A 1 μ 2 X B μ 2 + ( 2 n 1 ) 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 ] ;
    (2.6)
  2. (2)

    for any 1 2 μ1 and for every unitarily invariant norm , we have

    A 1 + μ 2 X B 1 μ 2 + A 1 μ 2 X B 1 + μ 2 1 1 μ μ 1 A ν X B 1 ν + A 1 ν X B ν d ν 1 4 n [ ( 2 n 1 ) A X + X B + 2 A 1 + μ 2 X B 1 μ 2 + A 1 μ 2 X B 1 + μ 2 ] 1 2 [ A X + X B + A μ X B 1 μ + A 1 μ X B μ ] .
    (2.7)

Since the function f(ν)= A ν X B 1 ν + A 1 ν X B ν is decreasing on the interval [0, 1 2 ] and increasing on the interval [ 1 2 ,1], and using the inequalities (2.6) and (2.7), we obtain a family of refinements of second inequality in (1.1).

Corollary 2 Let A,B H n + (C), and X M n (C) and let n be a positive integer. Then:

  1. (1)

    for any 0μ 1 2 and for every unitarily invariant norm , we have

    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 ν X B 1 ν + A 1 ν X B ν d ν 1 4 n [ ( 2 n 1 ) A X + X B + 2 A μ 2 X B 1 μ 2 + A 1 μ 2 X B μ 2 + ( 2 n 1 ) 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 ;
    (2.8)
  2. (2)

    for any 1 2 μ1 and for every unitarily invariant norm , we have

    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 ν X B 1 ν + A 1 ν X B ν d ν 1 4 n [ ( 2 n 1 ) A X + X B + 2 A 1 + μ 2 X B 1 μ 2 + A 1 μ 2 X B 1 + μ 2 + ( 2 n 1 ) 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 .
    (2.9)

It should be noted that in inequalities (2.8) and (2.9), we have

lim μ 0 1 μ 0 μ A ν X B 1 ν + A 1 ν X B ν d ν = lim μ 1 1 1 μ μ 1 A ν X B 1 ν + A 1 ν X B ν d ν = A X + X B .

Remark 1 The two special values n=1 and n=8 give the refinements of Heinz inequalities obtained in [3] and [7], respectively.

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Acknowledgements

Thanks for both reviewers for their helpful comments and suggestions. The authors wish also to express their thanks to professor Mohammad S Moslehian for helpful suggestions for revising the manuscript. This research is supported by the Lebanese University grants program for the Discrete Mathematics and Algebra group.

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Abbas, H., Mourad, B. A family of refinements of Heinz inequalities of matrices. J Inequal Appl 2014, 267 (2014). https://doi.org/10.1186/1029-242X-2014-267

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