A family of refinements of Heinz inequalities of matrices
© Abbas and Mourad; licensee Springer. 2014
Received: 17 February 2014
Accepted: 30 June 2014
Published: 22 July 2014
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.
for all and for all unitaries .
where , , and is a unitarily invariant norm on .
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 , 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 .
where g is a convex function on .
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.
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  and Theorem 1 in a paper of Feng .
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  and Theorem 2 in a paper of Feng .
Inequalities (2.4) and the first inequality in (1.1) yield the following refinements of the first inequality in (1.1).
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  and Theorem 3 in a paper of Feng .
- (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).
- (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)
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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