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Refinements of the Heinz inequalities for matrices
Journal of Inequalities and Applications volume 2014, Article number: 50 (2014)
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 is convex on the interval , attains its minimum at , and attains its maximum at and . Moreover, for . From [1] we know that for every unitarily invariant norm, we have the Heinz inequalities
In [2], Feng used the following inequalities to get refinements of (1):
where f is a real-valued function which is convex on the interval . With a similar method, Wang [3] 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 , we are implicitly assuming that the operator T belongs to the norm ideal associated with . Our results are better than those in [4] and different from [2, 3].
2 Main results
From page 122 of [5], we know the following Hermite-Hadamard integral inequality for convex functions.
Lemma 1 (Hermite-Hadamard integral inequality)
Let f be a real-valued function which is convex on the interval . Then
We will use the following lemma.
Lemma 2 Let f be a real-valued function which is convex on the interval . Then
Proof Using the previous lemma, we can easily verify the inequality
Next, we will prove the following inequality:
From the previous lemma, we have
 □
Applying the previous lemma to the function on the interval when , and on the interval when , 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 and for every unitarily invariant norm, we have
Proof First assume that . Then it follows by the previous lemma that
and so
Thus,
Now, assume that . Then by applying (3) to , it follows that
Since
the inequalities in (2) follow by combining (3) and (4). □
Applying the previous lemma to the function on the interval when , and on the interval when , we obtain the following.
Theorem 2 Let A, B, X be operators such that A, B are positive. Then for and for every unitarily invariant norm, we have
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 and for every unitarily invariant norm, we have
Applying the previous lemma to the function on the interval when , and on the interval when , we obtain the following theorem.
Theorem 3 Let A, B, X be operators such that A, B are positive. Then
-
(1)
for and for every unitarily norm,
(7) -
(2)
for and for every unitarily norm,
(8)
Since the function is decreasing on the interval and increasing on the interval , 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 and for every unitarily invariant norm, we have the following.
-
(1)
For and for every unitarily norm,
(9) -
(2)
For and for every unitarily norm,
(10)
It should be noticed that in the inequalities (7) to (10), we have
References
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
Feng Y: Refinements of the Heinz inequalities. J. Inequal. Appl. 2012., 2012: Article ID 18 10.1186/1029-242X-2012-18
Wang S: Some new refinements of Heinz inequalities of matrices. J. Inequal. Appl. 2013., 2013: Article ID 424 10.1186/1029-242X-2013-424
Kittaneh F: On the convexity of the Heinz means. Integral Equ. Oper. Theory 2010, 68: 519-527. 10.1007/s00020-010-1807-6
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).
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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
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DOI: https://doi.org/10.1186/1029-242X-2014-50