Generalized 𝑛 Dimensional Ostrowski Type and Grüss Type Inequalities on Time Scales

Abstract

Some new generalized dimensional Ostrowski type and Grüss type integral inequalities on time scales are established in this paper. The present results unify continuous and discrete analysis, and extend some known results in the literature.

1. Introduction

In recent years, the research for the Ostrowski type and Grüss type inequalities has been an interesting topic in the literature. The Ostrowski type inequality can be used to estimate the absolute deviation of a function from its integral mean, and it was originally presented by Ostrowski in [1] as follows.

Theorem 1. Let be a differentiable mapping in the interior of , where is an interval, and let . . If , for all , then one has
The Grüss inequality, which can be used to estimate the absolute deviation of the integral of the product of two functions from the product of their respective integral, was originally presented by Grüss in [2] as follows.

Theorem 2. Let be integrable functions on and satisfy the condition , . Then one has
In the last few decades, various generalizations of the Ostrowski inequality and the Grüss inequality including continuous and discrete versions have been established (e.g., see [3–14] and the references therein). On the other hand, Hilger [15] initiated the theory of time scales as a theory capable of treating continuous and discrete analysis in a consistent way, based on which some authors have studied the Ostrowski type and Grüss type inequalities on time scales. For example, in [16], Bohner and Matthews established the following Ostrowski type inequality on time scales for the first time.

Theorem 3 (see [16, Theorem 3.5]). Let , , and let be differentiable, where is an arbitrary time scale. Then where . This inequality is sharp in the sense that the right-hand side of it cannot be replaced by a smaller one.

In [17], Özkan and Yildirim established the following Grüss type inequality for double integrals on time scales.

Theorem 4 (see [17, Theorem 2.2]). Let , and , where   and   are two arbitrary time scales. Then where  ,  , and  .
In [18], Liu and Ngô established an Ostrowski-Grüss type inequality in one independent variable as follows.

Theorem 5 (see [18, Theorem 4]). Suppose and is differentiable, where is an arbitrary time scale, is rd-continuous, and . Then one has .

Other results on the Ostrowski type and Grüss type inequalities on time scales can be found in [19–25]. These inequalities on time scales unify continuous and discrete analysis and can be used to provide explicit error bounds for some known and some new numerical quadrature formulae.

Motivated by the above works, in this paper, we establish some new generalized Ostrowski type and Grüss type inequalities on time scales involving functions of n independent variables, which extend some known results in the literature to dimensional case.

We first give the following definition for further use.

Definition 6. , are defined by where is an arbitrary time scale and .

Throughout this paper, denotes the set of real numbers and , while denotes the set of integers and denotes the set of nonnegative integers. For a function and two integers   and  , one has , provided that . denotes an arbitrary time scale, . For an interval , , .

2. Main Results

For the sake of convenience, we present the following notations:

Lemma 7 (generalized dimensional Montgomery identity). Let Then one has

Proof. Suppose that . Then we have That is, (9) holds for .

Suppose that (9) holds for . Then for we have which is the desired result.

Remark 8. Lemma 7 is the dimensional extension of [17, Lemma 2.3].
Based on Lemma 7, we present two generalized dimensional Ostrowski type inequalities on time scales.

Theorem 9. Let , , and such that the partial delta derivative of order exists and there exists a constant such that = K. Then one has The inequality (12) is sharp in the sense that the right-hand side of it cannot be replaced by a smaller one.

Proof. From the definition of we can obtain Then by Lemma 7 we have which is the desired inequality.

The proof of the sharpness of (12) can be referred to [16, Theorem 3.5], which is equivalent to the case in Theorem 9 with .

Theorem 10. Under the conditions of Theorem 9, if there exist constants   and   such that ≤ , then one has

Proof. We notice that     .
We also have − and Collecting the information above and Lemma 7 we can get the desired result.

Remark 11. Theorem 9 is the dimensional extension of [16, Theorem 3.5].
In Theorem 9, if we take , , for some special time scales, then we immediately obtain the following three corollaries.

Corollary 12 (continuous case). Let , , in Theorem 9. Then , and one obtains where   =   .

Corollary 13 (discrete case). Let , , in Theorem 9. Then for , and one has where denotes the maximum value of the absolute value of the difference over .

Corollary 14 (quantum calculus case). Let , , in Theorem 9, where , . Then one has where denotes the maximum value of the absolute value of the -difference over .

Proof. Since for , , then we have Substituting (20) into (12) we get the desired result.

Next we propose the dimensional Grüss inequalities in the following two theorems.

Theorem 15. Let , ,  and  such that the partial delta derivative of order exists and there exist constants and such that / =  ,/ = . Then one has

Proof. For a function , from Lemma 7, we have Then multiplying (9) by and (22) by , and adding the resulting identities, we obtain An integration for (23) on yields that
So we have
After simple computation we can get the desired result.

Remark 16. Theorem 15 is the dimensional extension of [17, Theorem 2.2].

Theorem 17. Let such that and for all , , where ,   and   are constants. Then one has The proof for Theorem 17 is similar to [26, pp: 295–296], which is omitted here.

Finally, we present one Ostrowski-Grüss type inequality as follows.

Theorem 18. Suppose that such that the partial delta derivative of order exists and there exist constants   and   such that . Then one has

Proof. From the definition of in (8) we obtain that . On the other hand, we have where , , . So by Theorem 17 we deduce that Then combining (9), (28), and (29) we get the desired result.

Remark 19. Theorem 18 is the dimensional extension of [18, Theorem 4] and is the dimensional extension on time scales of [12, Theorem 2.1].

Remark 20. For Theorems 10, 15, 17, and 18, we can also obtain similar results as shown in Corollaries 12–14, which are omitted here.

3. Conclusions

In this paper, we have presented some new generalized dimensional Ostrowski type and Grüss type integral inequalities on time scales. These inequalities are of new forms compared with the existing results in the literature and are dimensional extension on time scales of some known continuous and discrete inequalities. Based on these inequalities, some new bounds as well as some sharp bounds for unknown functions are derived. These inequalities can also be used in the estimate of explicit error bounds for some numerical quadrature formulae.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was partially supported by the Natural Science Foundation of Shandong Province,China (Grant no. ZR2013AQ009), and Doctoral Initializing Foundation of Shandong University of Technology, China (Grant no. 4041-413030). The authors would like to thank the anonymous reviewers very much for their valuable suggestions on improving this paper.