- Open Access
Refinements of the Heinz inequalities
© Feng; licensee Springer. 2012
- Received: 1 October 2011
- Accepted: 27 January 2012
- Published: 27 January 2012
This article aims to discuss Heinz inequalities involving unitarily invariant norms. We obtain refinements of the Heinz inequalities. In particular, our results refine some results given in Kittaneh.
- Heinz inequality
- convex function
- Hermite-Hadamard inequality
- unitarily invariant norm
on [0, 1] to obtain new refinements of the inequalities (1.1). Our analysis enables us to discuss the equality conditions in (1.1) for certain unitarily invariant norms. When we consider |||T|||, we are implicitly assuming that the operator T belongs to the norm ideal associated with ||| · |||. Our results are better than those in .
In , Kittaneh obtained several refinements of the Heinz inequalities by using the previous lemma. In the following, we will use the following lemma to obtain several better refinements of the Heinz inequalities.
The following lemma can be proved by using the previous lemma.
Applying the previous lemma to the function f(v) = |||A v XB1-v+ A1-vXB v ||| on the interval [μ, 1 - μ] when , and on the interval [1 - μ, μ] when , we obtain refinement of the first inequality in (1.1).
Now, assume that . Then by applying (2.2) to 1 - μ, it follows
the inequalities in (2.1) follow by combining (2.2) and (2.3).
Applying the previous lemma to the function f(v) = |||A v XB1-v+ A1-vXB v ||| on the interval when , and on the interval when , we obtain the following.
The inequality (2.4) and the first inequality in (1.1) yield the following refinement of the first inequality in (1.1).
Applying the previous lemma to the function f(v) = |||A v XB1-v+ A1-vXB v ||| on the interval [0, μ] when , and on the interval [μ, 1] when , we obtain the following theorem.
Theorem 3. Let A, B, X be operators such that A, B are positive. Then
Since the function f(v) = |||A v XB1-v+ A1-vXB v ||| is decreasing on the interval and increasing on the interval , and using the inequalities (2.6) and (2.7), we obtain the refinement of the second inequality in (1.1).
Corollary 2. Let A, B, X be operators such that A, B are positive. Then for 0 ≤ μ ≤ 1 and for every unitarily invariant norm, we have
This article is prepared before the author's visit to Udine University, he wishes to express his gratitude to Prof. Corsini, Dr. Paronuzzi and Prof. Russo for their hospitality. Also, he wishes to thank Mr. Baojie Zhang, from Qujing Normal University, for the discussion. This research is financed by CMEC (KJ091104, KJ111107), CSTC, CTGU (10QN-27) and QNU (2008QN-034).
- Bhatia R, Davis C: More matrix forms of the arithmeticgeometric mean inequality. SIAM J Matrix Anal Appl 1993, 14: 132–136. 10.1137/0614012MATHMathSciNetView ArticleGoogle Scholar
- Kittaneh F: On the convexity of the Heinz means. Integ Equ Oper Theory 2010, 68: 519–527. 10.1007/s00020-010-1807-6MATHMathSciNetView ArticleGoogle Scholar
- Bullen PS: A Dictionary of Inequalities. In Pitman Monographs and Surveys in Pure and Applied Mathematics. Volume 97. Addison Wesley Longman Ltd., U.K; 1998.Google Scholar
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