Bounds of the logarithmic mean
© Furuichi and Yanagi; licensee Springer. 2013
Received: 4 August 2013
Accepted: 16 October 2013
Published: 12 November 2013
We give tight bounds for the logarithmic mean. We also give new Frobenius norm inequalities for two positive semidefinite matrices. In addition, we give some matrix inequalities on the matrix power mean.
MSC: 15A39, 15A45.
Keywordslogarithmic mean matrix mean Frobenius norm and inequality
We now have the following lemma.
Proof The second inequality of (3) can be proven easily. Indeed, we put . Then we have for and for . Thus we have for . □
In the paper , the following norm inequality was shown.
Theorem 1.2 ()
From Lemma 1.1, we have the following proposition.
To the first author’s best knowledge, the first inequality in Proposition 1.3 was suggested in .
This proposition can be proven in a similar way to the proof of Theorem 1.2 (or the proof of Theorem 2.2 which will be given in the next section), and this refines inequality (6) shown in .
2 Lower bound of the logarithmic mean
The following inequalities were given in . Hiai and Kosaki gave the norm inequalities for Hilbert space operators in . See also [6, 7]. Here we give them as a matrix setting to unify the description of this paper.
Theorem 2.1 ()
The Frobenius norm is one of unitarily invariant norms. We give the refinement of the lower bound of the first inequality above for the Frobenius norm. That is, we have the following inequalities.
To prove Theorem 2.2, we need a few lemmas.
where , and for . Here we have , whenever . By Lemma 2.3, if , then we have . Thus the proof of this lemma is completed. □
We then have the following lemma.
which implies the first inequality. The third inequality can be proven by the use of the arithmetic mean-geometric mean inequality. Thus the proof of this lemma is completed. □
We give some basic properties of the right-hand side of inequality (13) in the Appendix.
Applying inequality (11), we have the second inequality of (7) in a similar way. The third inequality holds due to Theorem 2.1 (or the third inequality of (12)). □
3 Upper bound of the logarithmic mean
In the paper , the following norm inequalities were also given for Hilbert space operators. Here we give them for matrices, as we mentioned in the beginning of Section 2.
Theorem 3.1 ()
We also give an improved upper bound of the logarithmic mean on Theorem 3.1 above, only for the Frobenius norm. Namely, we can prove the following inequalities in a similar way to the proof of Theorem 2.2, by the use of scalar inequalities which will be given in Lemma 3.4.
To prove Theorem 3.2, we need to prove the following lemmas.
where , and . The last inequality follows from Lemma 2.3, because we have whenever . □
Inequality (19) can be proven by putting in the famous inequality for .
To prove the second inequality of (18), it is sufficient to prove inequality (15) which follows from Lemma 3.3. We obtain actually the second inequality of (18) by putting in inequality (15), and then putting .
We put . Then we can prove by elementary calculations. Thus inequality (20) holds for . □
We give some basic properties of the right-hand side of inequality (19) in the Appendix.
4 Matrix inequalities on the geometric mean
Using Lemma 1.1, Lemma 2.5 and Lemma 3.4, we have following Proposition 4.1, Proposition 4.2 and Proposition 4.3, respectively.
where () is ν-weighted geometric mean introduced in .
We give the proof of Proposition 4.3. Proposition 4.1 and Proposition 4.2 are also proven in a similar way, using Lemma 1.1 and Lemma 2.5. In addition, by using the notion of the representing function for the operator mean m, it is well known  that holds for if and only if holds for all positive operators A and B. However, we give an elementary proof for the convenience of the readers.
Inserting and then multiplying two to all sides from both sides, we obtain the result. □
Inequalities (22) imply the following result in a similar way to the proof of Proposition 4.3.
Proposition 6.2 given in the Appendix shows that our upper bound is tighter than the standard upper bound for the case and our lower bound is tighter than the standard lower bound for any . In addition, our lower bound of the logarithmic mean is tighter than the lower bound given by Lin in  for . However, it may be a difficult problem to find the minimum such that for any . The right-hand side of the above inequality is the upper bound given by Lin in . (See inequalities (2).)
and converges to as . In addition, we have .
- (iii)Since we have
The arithmetic-geometric mean inequality proves for and . □
Then we have the following relations.
For , we have and .
For , we have and .
- (i)For the case , the following calculations show assertion:
- (ii)For the case , the following calculations show assertion:
The author (SF) was partially supported by JSPS KAKENHI Grant No. 24540146. The author (KY) was also partially supported by JSPS KAKENHI Grant No. 23540208.
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