On a relation between Schur, Hardy-Littlewood-Pólya and Karamata’s theorem and an inequality of some products of derived from the Furuta inequality
© Watanabe; licensee Springer 2013
Received: 18 November 2012
Accepted: 13 March 2013
Published: 29 March 2013
We show a functional inequality of some products of as an application of an operator inequality. Furthermore, we will show it can be deduced from a classical theorem on majorization and convex functions.
MSC:26D07, 26A09, 26A51, 39B62, 47A63.
Keywordsinequalities fractional powers convex functions majorization matrix inequalities Furuta inequality
by simpler ones, how can we guess what forms and coefficients are possible?
In Section 2, we prove a certain functional inequality as mentioned above, although the efficiency and possible applications to other branches of mathematics are still to be clarified.
In Section 3, we show that the functional inequality derived in Section 2 can be easily deduced from Schur, Hardy-Littlewood-Pólya and Karamata’s theorem on majorization and convex functions. Although the proof presented in Section 2 looks like a detour, one should note that it naturally arises as a byproduct of the Furuta inequality, which is an epochmaking extension of the celebrated Löwner-Heinz inequality [1, 2]. It seems worthy to compare various ways to derive fundamental functional inequalities, for it might contribute to clarify relations between their background theories and to suggest further developments.
2 An inequality of some products of
The proof of the following theorem is based on an operator inequality by Furuta  and an argument related to the best possibility of that by Tanahashi . The main feature of the argument is applying an order-preserving operator inequality to matrices which contain variables as their entries. It might be a new method to obtain functional inequalities systematically.
Theorem 2.1 
Proof Put . Since , we have , and hence Proposition 2.2 immediately follows from Theorem 2.1. □
for arbitrary . By substituting to x in the above inequality, it is immediate to see the inequality (2). □
Definition 2.4 For a finite sequence of real numbers, we denote its decreasing rearrangement by .
If n is even, the inequality (3) holds for arbitrary . If n is odd, the reverse inequality of (3) holds for arbitrary .
by the assumption of the induction.
If , then the n-tuples and satisfy the assumption of the case n, so we may assume by using the inequality (4).
for arbitrary .
The last assertion of the theorem can be easily seen by substituting for and multiplying to both sides.
This completes the proof. □
3 A proof by Schur, Hardy-Littlewood-Pólya and Karamata’s theorem
Theorem 2.5 is a special case of a more general theorem on majorization and convex functions.
for every real-valued convex function f on .
is convex on the interval .
Although it is definitely elementary to prove this proposition, we will give it for the sake of completeness.
Therefore, is increasing on and so that (), and hence ().
Again, therefore, is increasing on and so that (), and hence ().
Once again, therefore, g is increasing on and so that (), and hence (), namely, f is convex on the interval . This completes the proof of Proposition 3.2. □
The completion of the proof of Theorem 2.5 by using Schur, Hardy-Littlewood-Pólya and Karamata’s theorem.
The rest is identical to the proof of Theorem 2.5. This completes the proof.
The author is grateful to the referee, for the careful reading of the paper and for the helpful suggestions and comments. The author was supported in part by Grants-in-Aid for Scientific Research, Japan Society for the Promotion of Science.
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