Chinese Journal of Mathematics

Volume 2014 (2014), Article ID 363050, 10 pages

http://dx.doi.org/10.1155/2014/363050

Research Article

New Čebyšev Type Inequalities and Applications for Functions of Self-Adjoint Operators on Complex Hilbert Spaces

Mohammad W. Alomari

Department of Mathematics, Faculty of Science and Information Technology, Jadara University, Irbid 21110, Jordan

Received 28 January 2014; Accepted 20 May 2014; Published 5 June 2014

Academic Editor: Prasanna Kumar Sahoo

Copyright © 2014 Mohammad W. Alomari. 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.

Several new error bounds for the Čebyšev functional under various assumptions are proved. Applications for functions of self-adjoint operators on complex Hilbert spaces are provided as well.

1. Introduction

In recent years the approximation problem of the Riemann-Stieltjes integral via the famous Čebyšev functional

increasingly became essential. In 1882, Čebyšev [1] derived an interesting result involving two absolutely continuous functions whose first derivatives are continuous and bounded and is given by

and the constant is the best possible.

In 1935, Grüss [2] proved another result for two integrable mappings , such that and ; the inequality

holds, and the constant is the best possible.

In [3, p 302] Beesack et al. have proved the following Čebyšev inequality for absolutely continuous functions whose first derivatives belong to spaces:

where , , and .

For the constant

we have

for all , . Furthermore, we have the following particular cases in (4).

(1)If , we have

(2)If , we have

In 1970, Ostrowski [4] has proved the following combination of the Čebyšev and Grüss results:

where is absolutely continuous with and is Lebesgue integrable on and satisfying , for all . The constant is the best possible.

In 1973, Lupaş [5] has improved Beesack et al. inequality (7), as follows:

provided that , are two absolutely continuous functions on with , where . The constant is the best possible.

More recently, and using the identity ([3], page 246),

Dragomir [6] has proved the following inequality.

Theorem 1. Let be of bounded variation on and a Lebesgue integrable function on ; then

where denotes the total variation of on the interval . The constant is best possible in (12).

Another result when both functions are of bounded variation was considered in the same paper [6], as follows.

Theorem 2. If are of bounded variation on , then

The constant is best possible in (13).

Many authors have studied the functional (1) and, therefore, several bounds under various assumptions have been obtained; for more new results and generalizations the reader may refer to [6–21].

On other hand and in order to study the difference between two Riemann integral means, Barnett et al. [22] have proved the following estimates.

Theorem 3. Let be an absolutely continuous function with the property that ; that is,

Then for , we have the inequality

The constant in the first inequality and in the second inequality are the best possible.

After that, Cerone and Dragomir [23] have obtained the following three results as well.

Theorem 4. Let be an absolutely continuous mapping. Then for , we have the inequalities

where , and . Both inequalities in (16) are sharp.

Theorem 5. Assume that the mapping is of --Hölder type on . Then for , we have the inequality

Inequality (17) is best possible in the sense that we cannot put in the right-hand side a constant less than .

Theorem 6. Let be a mapping of bounded variation on . The following bounds hold:

where .

In this paper by utilising amongst others the inequalities from Theorems 3–6, several new bounds for the Čebyšev functional are provided.

The inequalities (15)–(18) are used in an essential way to obtain new error bounds for the , which gives a significant application for these inequalities. Applications for functions of self-adjoint operators on complex Hilbert Spaces are provided as well.

2. The Case When Is of Bounded Variation

We may start with the following result.

Theorem 7. Let be such that is of bounded variation on and is absolutely continuous on ; then

where are the usual Lebesgue norms; that is,

Proof. Using integration by parts, we have

It is known that for a continuous function and a function of bounded variation, the Riemann-Stieltjes integral exists and one has the inequality

As is of bounded variation on , by (22) we have

In the inequality (15), setting and , we get

Substituting (24) into (23), we get

since , occurs at , therefore, , which proves the first inequality in (19).

In the inequality (16), setting and , we get

Substituting (26) into (23), we get

where and , which proves the second and the third inequalities in (19).

Another result when is of --Hölder type is as follows.

Theorem 8. Let be such that is of bounded variation on and is of --Hölder type on ; then

Proof. As is of bounded variation and is of --Hölder type on , by (23) and using (17) we have

since , which completes the proof.

Theorem 9. Let be such that is of bounded variation on and is monotonic nondecreasing on ; then

Proof. As and is of bounded variation on and is monotonic nondecreasing on , by (22) and using (23) we have

In the third part of inequality (18), setting and , we get

Substituting (32) into (31), we get

and thus the proof is finished.

3. The Case When Is Lipschitzian

In this section, we give some new bounds when is -Lipschitzian.

Theorem 10. Let be such that is -Lipschitzian on and is an absolutely continuous on ; then

where and .

Proof. Using the fact that for a Riemann integrable function and -Lipschitzian function , one has the inequality

As is -Lipschitzian on , by (35) we have

where for the last inequality we used the inequality (15), with and , (see (24)).

In the inequality (16), setting and , we get

Substituting (37) into (36), we get

where , and

which proves the second and the third inequalities in (34).

Theorem 11. Let be such that is -Lipschitzian on and is of --Hölder type on ; then

Proof. As is -Lipschitzian and is of --Hölder type on , by (36) and using (17) we have

where, for the last inequality, a simple calculation yields that

which completes the proof.

Corollary 12. In Theorem 11, if is -Lipschitzian on , then we have

Theorem 13. Let be such that is of bounded variation on and is monotonic nondecreasing on ; then

Proof. As is -Lipschitzian on and is monotonic nondecreasing on , by (35) and using (36) we have

In the third part of inequality (18), setting and , we get

Substituting (46) into (45), we get

which completes the proof.

4. More Inequalities

In this section we give other related results.

Theorem 14. Let be such that and are of bounded variation on ; then

The constant is the best possible.

Proof. As and is of bounded variation on , by (22) and using (23), we have

In the first inequality of (18), setting and , we get

Substituting (50) into (49), we get

which proves the inequality. The sharpness case trivially holds by taking , which completes the proof.

When the integrator is of bounded variation we have the following.

Theorem 15. Let be such that is -Lipschitzian on and is of bounded variation on ; then

Proof. As is -Lipschitzian on and is of bounded variation on , by (22) and using (23), we have

In the second inequality of (18), setting and , we get

Substituting (54) into (53), we get

and the proof is completed.

When both functions are Lipschitzian we have the following.

Theorem 16. Let be, respectively, such that and are - and -Lipschitzian on ; then

The constant is the best possible.

Proof. As and are - and -Lipschitzian on , respectively, by (35) and using (36), we have

In the second inequality of (18), setting and , we get

Substituting (58) into (57), we get

which proves the inequality. The sharpness case trivially holds by taking , which completes the proof.

Remark 17. Let be as in Theorems 7–16. By applying the same techniques used in the corresponding proofs of each theorem, we may obtain several inequalities for monotonic nondecreasing integrator using the fact that for a monotonic nondecreasing function and continuous function , one has the inequality

We leave the details to the interested reader.

5. Applications for Self-Adjoint Operators

We denote by the Banach algebra of all bounded linear operators on a complex Hilbert space . Let be self-adjoint and let be defined for all as follows:

Then for every the operator

is a projection which reduces .

The properties of these projections are collected in the following fundamental result concerning the spectral representation of bounded self-adjoint operators in Hilbert spaces; see for instance [24, page 256].

Theorem 18 (Spectral Representation Theorem). Let be a bonded self-adjoint operator on the Hilbert space and let and . Then there exists a family of projections , called the spectral family of , with the following properties:

(a) for ;(b), and for all ;(c)we have the representation

More generally, for every continuous complex-valued function defined on and for every there exists a such that

whenever

this means that

where the integral is of Riemann-Stieltjes type.

Corollary 19. With the assumptions of Theorem 18 for , , and we have the representations

In particular,

Moreover, we have the equality

We recall the following result (see [25]) that provides an upper bound for the total variation of the function on an interval .

Theorem 20. Let be the spectral family of the bounded self-adjoint operator and let and . Then for any and we have the inequality

where denotes the total variation of the function on .

Remark 21. For with and we get from (70) the inequality

for any .

This implies, for any , that

where denotes the limit .

The inequality (72) was also proved in the recent monographs [26, 27] and will be utilized in the following.

After these preparations we can state and prove the following trapezoidal type inequality for functions of self-adjoint operators on Hilbert spaces.

Theorem 22. Let be a bonded self-adjoint operator on the Hilbert space and let and . If is such that its derivative is of bounded variation on , then we have the inequality

for any .

Proof. Utilising the inequality (48) for the function of bounded variation and the continuous function we have

for any .

By the Spectral Representation Theorem we have

for any .

Substituting these values into (74) we deduce after simple calculations the desired result (73).

The above inequality (73) can be utilized for different particular functions of interest in Operator Theory, such as the power, logarithmic, and exponential functions.

If we take with , then for any positive operator with we have the inequality

for any .

If we take the function , then for any positive definite operator with we have the inequality

for any .

Finally, if we take , then we have for any self-adjoint operator with the inequality

for any , .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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