Open Access

A family of refinements of Heinz inequalities of matrices

Journal of Inequalities and Applications20142014:267

https://doi.org/10.1186/1029-242X-2014-267

Received: 17 February 2014

Accepted: 30 June 2014

Published: 22 July 2014

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 A X + X B , 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.

Keywords

Heinz inequalityconvex functionHermite-Hadamard inequalityunitarily invariant norm

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
U A V = 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 A X + X B ,

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 0 ν 1 . 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 A X + X B ,
(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 ) d t 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
( 2 n 1 ) g ( a ) + 2 g ( a + b 2 ) + ( 2 n 1 ) g ( b ) 2 n g ( a ) + 2 n g ( 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 , 1 μ ] when 0 μ 1 2 and on the interval [ 1 μ , μ ] when 1 2 μ 1 , 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 0 μ 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 ( 1 μ ) , 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 μ 1 . Then, by applying (2.2) to 1 μ , 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 0 μ 1 2 , and in the interval [ 1 2 , μ ] for 1 2 μ 1 , 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 0 μ 1 2 , and on the interval [ 1 2 , μ ] when 1 2 μ 1 , 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 0 μ 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 μ 1 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 0 μ 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 μ 1 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.

Declarations

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.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Sciences, Lebanese University

References

  1. Bhatia R: Matrix Analysis. Springer, New York; 1997.View ArticleGoogle Scholar
  2. Bhatia R, Davis C: More matrix forms of the arithmetic-geometric mean inequality. SIAM J. Matrix Anal. Appl. 1993, 14: 132-136. 10.1137/0614012MathSciNetView ArticleMATHGoogle Scholar
  3. Feng Y: Refinements of Heinz inequalities. J. Inequal. Appl. 2012. Article ID 18, 2012: Article ID 18Google Scholar
  4. Kittaneh F, Manasrah Y: Improved Young and Heinz inequalities for matrices. J. Math. Anal. Appl. 2010, 361: 262-269. 10.1016/j.jmaa.2009.08.059MathSciNetView ArticleMATHGoogle Scholar
  5. Kittaneh F: On the convexity of the Heinz mean. Integral Equ. Oper. Theory 2010, 68: 519-527. 10.1007/s00020-010-1807-6MathSciNetView ArticleMATHGoogle Scholar
  6. Kaur, R, Moslehian, MS, Singh, M, Conde, C: Further refinements of the Heinz. J. Linear algebra and its applications (Available online 27 February 2013)Google Scholar
  7. Wang S: Some new refinements of Heinz inequalities for matrices. J. Inequal. Appl. 2013, 1: 132-136.Google Scholar
  8. Zou L, He C: On some inequalities for unitarily invariant norms and singular values. Linear Algebra Appl. 2012, 436: 3354-3361. 10.1016/j.laa.2011.11.030MathSciNetView ArticleMATHGoogle Scholar

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© Abbas and Mourad; licensee Springer. 2014

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