Abstract
We will show certain functional inequalities involving fractional powers, making use of the Furuta inequality and Tanahashi’s argument.
MSC:26D07, 26A09, 39B62, 47A63.
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1 Introduction
Let x be an arbitrary positive real number. One can easily see the inequality
for instance, is reduced to a simple polynomial inequality by putting . However, at least to the author, it seems not easy to give an elementary proof of the inequality
which has a very similar form to the preceding one although their corresponding numerical parts are different.
The purpose of this article is to show the following theorem.
Theorem 1.1 Let , and with . If , then
An elementary approach to proving the inequality (1) might be to consider the power series expansion.
Put , and
Then we can expand around as
Thus, the constant term and the coefficients of t, and are 0. Further, one can obtain
and
Thus, if the assumption for the parameters p, q and r in Theorem 1.1 is satisfied, then we have . However, the signature of and depends on parameters, and one cannot see any signs of a simple rule among the coefficients of higher order terms. Although is non-negative on a sufficiently small neighborhood of , it seems difficult to show that is non-negative entirely on by such an argument as above.
Let us recall some fundamental concepts on related matrix inequalities. A capital letter means a matrix whose entries are complex numbers. A square matrix T is said to be positive semidefinite (denoted by ) if for all vectors x. We write if T is positive semidefinite and invertible. For two selfadjoint matrices and of the same size, a matrix inequality is defined by .
The celebrated Löwner-Heinz theorem includes:
Let . If , then .
For , does not always ensure . Furuta obtained an epoch-making extension of the Löwner-Heinz inequality by using the Löwner-Heinz inequality itself.
Theorem 1.3 [3]
Let , and with . If , then
The following result by Tanahashi is a full description of the best possibility of the range
as far as all parameters are positive.
Theorem 1.4 [4]
Let p, q, r be positive real numbers. If or , then there exist matrices A, B with that do not satisfy the inequality
One notices the coincidence between the assumption on parameters in Theorem 1.1 and Theorem 1.3. As a matter of fact, the inequality (1) is a particular conclusion of the Furuta inequality. We should point out that Tanahashi’s argument in [4] is almost sufficient to deduce the former from the latter. In the next section, we will prove Theorem 1.1 using Theorem 1.3 and Tanahashi’s argument.
2 Proof of Theorem 1.1
As we mentioned above, our proof of Theorem 1.1 has a major part which is parallel to [4]. Our matrix A is a little different from that in [4], we use a variable y instead of ε and δ. It simplifies the argument to an extent, though the improvement is not essential.
Throughout this paper, we assume that and . We will consider matrices
and
Then we have . The eigenvalues of A are , where .
Lemma 2.1 and .
Proof Obviously,
If , then we would have or , which is contrary to the assumption. □
Let
and
Then U is unitary and
where
By the assumption and Theorem 1.3, A and B satisfy the inequality (2). Then
hence we have
Denote
where
Lemma 2.2 Let p, q, r be positive real numbers. Then and .
Proof Since and , we have . Moreover,
and
hence we have . Thus .
It is obvious that and , and hence . □
Let
where
Then it is easy to see that , V is unitary and
The following lemma is one of the most important points in Tanahashi’s argument. Although the substance is presented in the whole proof of [4], Theorem], we should restate and prove it in our context for the readers’ convenience.
Lemma 2.3
where .
Proof The formula (3) implies
Write the left-hand matrix as
where
Then, by the formula (5), we have
So, its determinant is also non-negative. We expand it to obtain
Now,
Hence, the formula (6) implies
Cancel the common positive factor and substitute the definitions for and . Then a simple calculation shows that
By factorizing, we have
This completes the proof of Lemma 2.3. □
Now, we estimate each term of the inequality (4) with respect to . A key point in making use of the inequality (4) is that both estimations of the factor on the left-hand side and the factor on the right-hand side contain a common subfactor y. After the cancellation of this y, we will derive the desired functional inequality by letting , and applying l’Hopital’s rule. Terms in other factors can be roughly estimated.
In the following, o means , that is
and denotes a term such that ().
One can establish the following formulae:
Now, we have the estimation of the most delicate factor in the formula (4), whose constant term is canceled by subtraction.
Substitute these estimations for the inequality (4), cancel the positive factor y, and let , then we have
and hence
Letting and applying l’Hopital’s rule, we have
This implies that, for arbitrary ,
For arbitrary , substitute for b in (7) and multiply by x, , , both sides. It is easy to see that x itself satisfies (7). This completes the proof of Theorem 1.1.
References
Löwner K: Über monotone Matrixfunktionen. Math. Z. 1934, 38: 177–216. 10.1007/BF01170633
Heinz E: Beiträge zur Störungstheorie der Spektralzerlegung. Math. Ann. 1951, 123: 415–438. 10.1007/BF02054965
Furuta T: assures for , , with . Proc. Am. Math. Soc. 1987, 101(1):85–88.
Tanahashi K: Best possibility of the Furuta inequality. Proc. Am. Math. Soc. 1996, 124: 141–146. 10.1090/S0002-9939-96-03055-9
Acknowledgements
The author was supported in part by Grants-in-Aid for Scientific Research, Japan Society for the Promotion of Science.
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Watanabe, K. An application of matrix inequalities to certain functional inequalities involving fractional powers. J Inequal Appl 2012, 221 (2012). https://doi.org/10.1186/1029-242X-2012-221
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DOI: https://doi.org/10.1186/1029-242X-2012-221