Refinements of the Heinz inequalities for matrices
© Yan et al.; licensee Springer. 2014
Received: 10 November 2013
Accepted: 8 January 2014
Published: 31 January 2014
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).
Keywordsconvex function Heinz inequality Hermite-Hadamard inequality unitarily invariant norm
where f is a real-valued function which is convex on the interval . With a similar method, Wang  got some new refinements of (1).
2 Main results
From page 122 of , we know the following Hermite-Hadamard integral inequality for convex functions.
Lemma 1 (Hermite-Hadamard integral inequality)
We will use the following lemma.
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).
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.
The inequality (5) and the first inequality in (1) yield the following refinement of the first inequality in (1).
Applying the previous lemma to the function on the interval when , and on the interval when , we obtain the following theorem.
- (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).
- (1)For and for every unitarily norm,(9)
- (2)For and for every unitarily norm,(10)
This work is supported by NSF of China (Grant Nos. 11171364 and 11271301).
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