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On the range of the parameters for the grand Furuta inequality to be valid
Journal of Inequalities and Applications volume 2014, Article number: 297 (2014)
In order to investigate the precise range D of the parameters p, s, t, and r for which the grand Furuta inequality is valid. We use the method of reductio ad absurdum. We find the following results: The area , , , and is contained in the complement of the range D. The condition seems essential for the grand Furuta inequality.
A bounded linear operator T on a Hilbert space is said to be positive semidefinite (denoted by ) if for all vectors h. We write if T is positive semidefinite and invertible.
Theorem 1.1 
Let , , and with . If holds, then
It is well known that Theorem 1.1 is equivalent to the next theorem, which is often called the essential case of the Furuta inequality.
Theorem 1.2 Let and . If holds, then
The following result by Tanahashi is a full description of the best possibility of the range
as far as all parameters are positive. We would like to emphasize that the theorem can be divided into two cases.
Theorem 1.3 
Let p, q, r be positive real numbers. If or , then there exist matrices A, B with that do not satisfy the inequality
The next proposition is corresponding to the case of the previous theorem by putting .
Proposition 1.4 Let , . If , then there exist matrices A, B with that do not satisfy the inequality
On the other hand, the following ‘α-free’ proposition corresponds to the case of Theorem 1.3 by putting .
Proposition 1.5 Let and . Then there exist matrices A, B with that do not satisfy the inequality
Since the condition is the essential case for the Furuta inequality, our interest in Proposition 1.5 is not at all less than Proposition 1.4.
Furuta gave a unifying extension of both Theorem 1.1 and the Ando-Hiai inequality , which is often called the grand Furuta inequality.
Theorem 1.6 
Let , , , and . If with , then the following inequality holds:
Again, Tanahashi showed that the outside powers in this theorem are the best possible.
Theorem 1.7 
Let , , , and . If , then there exist matrices A, B with that do not satisfy the inequality
Remark 1.8 In , Theorem 1.7 is originally stated as follows:
Let p, r, s, t be real numbers satisfying , , , . If
then there exist invertible matrices A, B with which do not satisfy
These are just a matter of rephrasing, although their α differs each other. Theorem 1.7 can be naturally considered as an extension of Proposition 1.4. Indeed, if we put , in Theorem 1.7, then we obtain Proposition 1.4 restricted to . On the other hand, being different from Theorem 1.3, even if all parameters are positive, Theorem 1.7 does not show that the range
cannot be expanded anymore for the grand Furuta inequality to be valid. Thus the clarification of the best possibility of the grand Furuta inequality is less satisfactory than that of the Furuta inequality. So our problem is to determine the range:
Although it would be a nice theorem if one could precisely determine the range (2) all at once, it seems difficult to the author. Therefore, we should treat several main cases of the problem.
It is quite natural to expect an ‘α-free’ version which can be regarded as corresponding to Proposition 1.5. The following result obtained by Koizumi and the author is such an attempt.
Theorem 1.9 
Let , , , and . Suppose that
Then there exist matrices A, B with that do not satisfy the inequality (1).
Remark 1.10 The quantity in the above assumption has an essential meaning. It also appears in a certain functional inequality (cf. ).
If , , , and , then .
If , , , and , then .
Remark 1.11 The case (ii) of [, Theorem 2.1] by Koizumi and the author treats the case , , . However, we have by the notations in  and the proof for (i) is not applicable to (ii). It seems still open.
The main purpose of this article is to show that the area , , , and is contained in the complement of the range (2).
Let A, B be matrices with and , and let U be a unitary which diagonalizes A as . Assume A and B satisfy the grand Furuta inequality (1). Put and . Then
hence we have
where k is a positive scalar to be specified later.
Lemma 2.1 Suppose that and . Let
Then , V is unitary and
The formula (3) implies
Write the left-hand matrix as
Lemma 2.2 Keep the situation as above. Assume that . Then the following inequality holds:
We begin with an elementary inequality for real numbers.
Lemma 3.1 If , then
As Theorem 1.7 is regarded as an extension of Proposition 1.4, our main theorem can be considered as an extension of Proposition 1.5, which shows one of the largest pieces of the complement of the range (2). The advantage is that the assumptions on the parameters other than are very mild. Note that, if we change to , then the inequality holds for .
Theorem 3.2 Let , , , and . Then there exist matrices A and B with that do not satisfy the inequality
Yamazaki’s simplified proof of Theorem 1.7 in  is not applicable in our context. The method of our proof is the same as Tanahashi’s argument, whose outline is explained in the previous section. We would like to emphasize there are several branching points such that the conditions about parameters in the assumption are to be reflected to powers or coefficients in calculations. Moreover, we have to estimate all terms up to the order of in the sequel. On the other hand, in the existing literature, such as [7, 8] and , it is sufficient to estimate only main and second terms.
Proof As in the preliminaries, we set and . Note that and . We will consider matrices
Then we have . The eigenvalues of A are . Let
Then U is unitary and , where
Assume A and B satisfy the grand Furuta inequality (1). Then we have
by the notation of the previous section. It is easy to see that , , , and as . Therefore, for sufficiently large x, we have . Further, we shall see later , as . Since , we have for sufficiently large x.
Now we estimate each term of the inequality (4) with respect to . The estimation of the factor in the right-hand side is a little delicate so that we need some calculation. Terms in other factors can be roughly estimated. As is a usual notation, means that
One can establish the following formulas:
Since is small, the estimations of and (resp. ) in these details are the same as (resp. ), and the main term of is the same as that of . Hence
Further, since , , and ,
The main term of is the same as , hence
so the estimation of is the same as in its detail, where the assumption is crucial. Hence
where k is the nonnegative integer determined by .
The main term of is obviously the same as that of and
For the following four factors in the formula (4), it is sufficient to estimate their main terms only,
Now we have the estimation of the most delicate factor in the formula (4), whose main terms are canceled by subtraction. We have
Applying these estimations to the inequality (4), we can obtain
The power of x in the right-hand may be positive, 0 or negative. However, by Lemma 3.1, the assumption implies that . By letting in (5), we have
This is a contradiction and completes the proof of Theorem 3.2. □
Corollary 3.3 Let , , , and . Then there exist matrices A, B with that do not satisfy the inequality
Proof It is obvious that and . □
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The author was supported in part by Grants-in-Aid for Scientific Research, Japan Society for the Promotion of Science.
The author declares that he has no competing interests.
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Watanabe, K. On the range of the parameters for the grand Furuta inequality to be valid. J Inequal Appl 2014, 297 (2014). https://doi.org/10.1186/1029-242X-2014-297
- Löwner-Heinz inequality
- Furuta inequality
- matrix inequality