An application of matrix inequalities to certain functional inequalities involving fractional powers
© Watanabe; licensee Springer 2012
Received: 9 May 2012
Accepted: 20 September 2012
Published: 3 October 2012
We will show certain functional inequalities involving fractional powers, making use of the Furuta inequality and Tanahashi’s argument.
MSC:26D07, 26A09, 39B62, 47A63.
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.
An elementary approach to proving the inequality (1) might be to consider the power series expansion.
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 
as far as all parameters are positive.
Theorem 1.4 
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  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 . Our matrix A is a little different from that in , we use a variable y instead of ε and δ. It simplifies the argument to an extent, though the improvement is not essential.
Then we have . The eigenvalues of A are , where .
Lemma 2.1 and .
If , then we would have or , which is contrary to the assumption. □
Lemma 2.2 Let p, q, r be positive real numbers. Then and .
hence we have . Thus .
It is obvious that and , and hence . □
The following lemma is one of the most important points in Tanahashi’s argument. Although the substance is presented in the whole proof of , Theorem], we should restate and prove it in our context for the readers’ convenience.
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.
and denotes a term such that ().
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.
The author was supported in part by Grants-in-Aid for Scientific Research, Japan Society for the Promotion of Science.
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