Abstract
For any unitarily invariant norm , the Heinz inequalities for operators assert that , for A, B, and X any operators on a complex separable Hilbert space such that A, B are positive and . In this paper, we obtain a family of refinements of these norm inequalities by using the convexity of the function and the Hermite-Hadamard inequality.
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1 Introduction
Let be the algebra of complex matrices. We denote by the set of all Hermitian matrices in . The set of all positive semi-definite matrices in shall be denoted by . A norm on is called unitarily invariant or symmetric if
for all and for all unitaries .
The arithmetic-geometric mean inequality for two nonnegative real numbers a and b is
which has been generalized to the context of matrices as follows:
where , , and is a unitarily invariant norm on .
For and two nonnegative numbers a and b, the Heinz mean is defined as
Clearly the Heinz mean interpolates between the geometric mean and the arithmetic mean:
The function has the following properties: it is convex, attains its minimum at , its maximum at and , and for . The generalization of the above inequalities to matrices is due to Bhatia and Davis [1] as follows:
where , , and . For a historical background and proofs of these norm inequalities as well as their refinements, and diverse applications, we refer the reader to the [2–8], and the references therein. Indeed, it has been proved, in [1], that is a convex function of ν on with symmetry about , and attains its minimum there and it has a maximum at and . Moreover, it increases on and decreases on .
In [4, 5], (1.1) is refined by using the so-called Hermite-Hadamard inequality:
where g is a convex function on .
Recently, in [3] and [7], respectively, the following inequalities were used to get new refinements of (1.1):
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 . Then for any positive integer n, we have
Proof Since g is convex on , we have
Thus
whence
To prove the middle inequality, we start by
□
Applying the previous lemma on the convex function defined earlier
on the interval when and on the interval when , 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 , and . Let n be any positive integer. Then for any , and for every unitarily invariant norm on , we have
Proof First assume that . Then it follows from Lemma 1 that
Since , we have
Thus,
Now, assume that . Then, by applying (2.2) to , it follows that
Since
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 in the interval on , and in the interval for , 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 , and . Then, for any positive integer n, any , and for every unitarily invariant norm on , we have
Inequalities (2.4) and the first inequality in (1.1) yield the following refinements of the first inequality in (1.1).
Corollary 1 Let , and . Then, for any positive integer n, any , and for every unitarily invariant norm on , we have
Applying the Lemma 1 to the function on the interval when , and on the interval when , 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 , and and let n be a positive integer. Then:
-
(1)
for any and for every unitarily invariant norm , we have
(2.6) -
(2)
for any and for every unitarily invariant norm , we have
(2.7)
Since the function is decreasing on the interval and increasing on the interval , and using the inequalities (2.6) and (2.7), we obtain a family of refinements of second inequality in (1.1).
Corollary 2 Let , and and let n be a positive integer. Then:
-
(1)
for any and for every unitarily invariant norm , we have
(2.8) -
(2)
for any and for every unitarily invariant norm , we have
(2.9)
It should be noted that in inequalities (2.8) and (2.9), we have
Remark 1 The two special values and give the refinements of Heinz inequalities obtained in [3] and [7], respectively.
References
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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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DOI: https://doi.org/10.1186/1029-242X-2014-267