Abstract and Applied Analysis

Volume 2013 (2013), Article ID 124510, 6 pages

http://dx.doi.org/10.1155/2013/124510

## Some Operator Inequalities on Chaotic Order and Monotonicity of Related Operator Function

College of Mathematics and Information Science, Henan Normal University, Xinxiang 453002, China

Received 24 March 2013; Accepted 24 April 2013

Academic Editor: Yisheng Song

Copyright © 2013 Changsen Yang and Yanmin Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We will discuss some operator inequalities on chaotic order about several operators, which are generalization of Furuta inequality and show monotonicity of related Furuta type operator function.

#### 1. Introduction

An operator is said to be positive (denoted by ) if for all vectors in a Hilbert space, and is said to be strictly positive (denoted by ) if is positive and invertible.

*Theorem LH [Löwner-Heinz inequality, denoted by (LH) briefly]*

If holds, then for any .

This was originally proved in [1, 2] and then in [3]. Although (LH) asserts that ensures for any , unfortunately does not always hold for . The following result has been obtained from this point of view.

Theorem F (Furuta inequality). *If , then for each ,*(i)*,*(ii)*
hold for and with .*

The original proof of Theorem F is shown in [4], an elementary one-page proof is in [5], and alternative ones are in [6, 7]. We remark that the domain of the parameters , , and in Theorem F is the best possible for the inequalities (i) and (ii) under the assumption ; see [8].

We write if for , which is called the chaotic order.

Theorem A. *For , the following (i) and (ii) hold:*(i)* holds if and only if for ;*(ii)* holds if and only if for any fixed is a decreasing function of and .*

(i) in Theorem A is shown in [9, 10], an excellent proof in [11], a proof in the case in [12], (ii) in [9, 10], and so forth.

Lemma B (see [11]). *Let be a positive invertible operator, and let be an invertible operator. For any real number ,
*

*Definition 1. *Let , , and for a natural number .

Let be defined by
For example,
Let be defined by
For example,
For the sake of convenience, we define
and these definitions in (6) may be reasonable by (2) and (4).

Lemma 2. *For and any natural number , we have*(i)*,
*(ii)*. *

*Proof. *(i) and (ii) can be easily obtained by definitions (2) and (4).

#### 2. Basic Results Associated with and

We will give some operator inequalities on chaotic order, and Theorem 5 is further extension of Theorem 3.1 in [13].

Lemma 3. *If , for and , then .*

*Proof. *Since , we can obtain the following inequality.

holds for and by (i) of Theorem A.

Take the logarithm on both sides of the previous inequality; that is,
therefor we have

Theorem 4. *If and , for a natural number . Then the following inequality holds:
**
where and are defined in (2) and (4).*

*Proof. *We will show (9) by mathematical induction. In the case .

Since implies
holds for any and by Lemma 3, whence (9) for .

Assume that (9) holds for a natural number (). We will show that (9) holds and for .

Put , and , and (9) holds for implying
Equation (11) yields the following by Lemma 3, for and
that is,
Put in (13), then by (ii) of Lemma 2, the exponential power of the right hand side of (13) can be written as follows:
and we have the following desired (15) by (12) and (13):
so that (15) shows that (9) holds for .

Theorem 5. *If and for a natural number . For any fixed , let be satisfied by
**
The operator function for any natural number such that is defined by
**
Then the following inequality holds:
**
for every natural number such that , where and are defined in (2) and (4).*

*Proof. *Since , in (6), we may define for .

Because , then for any fixed ,
since holds by (ii) of Theorem A. And (19) can be expressed as
We can apply Theorem 4, and we have the following (21) for any natural number such that :
Since implies that holds for any , (21) ensures
Putting , and applying (19) for and , we have
holds for and .

Putting in (23), then (23) can be rewritten by
Putting , since in (16), then we have
and we have (18) for such that by (25) and (20) since (20) means (18) for .

Corollary 6. *If and for a natural number . For any fixed , let be satisfied by (16).**Then the following inequalities hold:
**
where , , and are defined in (2), (4), and (17).*

*Proof. *Applying (18) of Theorem 5 for such that , we have

#### 3. Monotonicity Property on Operator Functions

We would like to emphasize that the condition of Theorem 7 is stronger than Theorem 5, and moreover when we discuss monotonicity property on operator functions, we can only apply Theorem 7.

Theorem 7. *If and , for a natural number . Then the following inequality holds:
**
where and are defined in (2) and (4).*

*Proof. *We will show (28) by mathematical induction. In the case .

Since implies
holds for any, and by (i) of Theorem A, whence (28) for .

Assume that (28) holds for a natural number (). We will show (28) for and for .

We can obtain the following inequality from the hypothesis (28) for the case :
hence we have , and (i) of Theorem A ensures
Putting and , then we have the following inequality:
so that (32) shows (28) for .

Theorem 8. *If and for a natural number . For any fixed , let be satisfied by (16).**Then
**
is a decreasing function of both and which satisfies
**
where and are defined in (2) and (4).*

*Proof. *Since the condition (16) with suffices (28) in Theorem 7, we have the following inequality by Theorem 7; see (28).

We state the following important inequality (35) for the forthcoming discussion which is the inequality in (16):
because the inequality in (35) follows by (ii) of Lemma 2, and the inequality follows by
obtained by (34).

(a) Proof of the result that is a decreasing function of .

Without loss of generality, we can assume that . We can obtain the following inequality by (28) and by (i) of Lemma 2:
and (37) implies
Put for , then we raise each side of (38) to the power , then

Whence we have
and the last inequality holds by LH because (39) and which is ensured by (35) and by (4), so that is a decreasing function of .

(b) Proof of the result that is a decreasing function of .

Without loss of generality, we can assume that . Raise each side of (28) to the power for by LH, then
We state the following inequality by (ii) of Lemma 3 and (35):
Then we have
and the last inequality holds by LH because (41) and
so that is a decreasing function of .

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (1127112; 11201127), Technology and Pioneering project in Henan Province (122300410110).

#### References

- E. Heinz, “Beiträge zur Störungstheorie der Spektralzerlegung,”
*Mathematische Annalen*, vol. 123, pp. 415–438, 1951. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Löwner, “Über monotone Matrixfunktionen,”
*Mathematische Zeitschrift*, vol. 38, no. 1, pp. 177–216, 1934. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. K. Pedersen, “Some operator monotone functions,”
*Proceedings of the American Mathematical Society*, vol. 36, pp. 309–310, 1972. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Furuta, “$A\ge B$ assures ${{(B}^{r}{A}^{p}{B}^{r})}^{1/q}\ge {B}^{p+r/q}$ for $r\ge 0,p\ge 0,q\ge 1$ with $\left(1+2r\right)q\ge p+2r$,”
*Proceedings of the American Mathematical Society*, vol. 101, no. 1, pp. 85–88, 1987. View at Publisher · View at Google Scholar · View at MathSciNet - T. Furuta, “An elementary proof of an order preserving inequality,”
*Proceedings of the Japan Academy*, vol. 65, no. 5, p. 126, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Fujii, “Furuta's inequality and its mean theoretic approach,”
*Journal of Operator Theory*, vol. 23, no. 1, pp. 67–72, 1990. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Kamei, “A satellite to Furuta's inequality,”
*Mathematica Japonica*, vol. 33, no. 6, pp. 883–886, 1988. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Tanahashi, “Best possibility of the Furuta inequality,”
*Proceedings of the American Mathematical Society*, vol. 124, no. 1, pp. 141–146, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Furuta, “Applications of order preserving operator inequality,”
*Operator Theory*, vol. 59, pp. 180–190, 1992. View at Google Scholar - M. Uchiyama, “Some exponential operator inequalities,”
*Mathematical Inequalities and Applications*, vol. 2, no. 3, pp. 469–471, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Furuta, “Extension of the Furuta inequality and Ando-Hiai log-majorization,”
*Linear Algebra and Its Applications*, vol. 219, pp. 139–155, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Fujii, T. Furuta, and E. Kamei, “Furuta's inequality and its application to Ando's theorem,”
*Linear Algebra and Its Applications*, vol. 179, pp. 161–169, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Furuta, “Operator functions on chaotic order involving order preserving operator inequalities,”
*Journal of Mathematical Inequalities*, vol. 6, no. 1, pp. 15–31, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet