Abstract and Applied Analysis
Volume 2017 (2017), Article ID 5234181, 7 pages
https://doi.org/10.1155/2017/5234181

Research Article

On Weighted Montgomery Identity for Points and Its Associates on Time Scales

Eze R. Nwaeze and Ana M. Tameru

Department of Mathematics, Tuskegee University, Tuskegee, AL 36088, USA

Correspondence should be addressed to Eze R. Nwaeze; enwaeze@mytu.tuskegee.edu

Received 21 March 2017; Accepted 6 June 2017; Published 30 July 2017

Academic Editor: Patricia J. Y. Wong

Copyright © 2017 Eze R. Nwaeze and Ana M. Tameru. 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.

The purpose of this paper is to establish a weighted Montgomery identity for points and then use this identity to prove a new weighted Ostrowski type inequality. Our results boil down to the results of Liu and Ngô if we take the weight function to be the identity map. In addition, we also generalize an inequality of Ostrowski-Grüss type on time scales for points. For we recapture a result of Tuna and Daghan. Finally, we apply our results to the continuous, discrete, and quantum calculus to obtain more results in this direction.

1. Introduction

In 1938, Ostrowski [1] proved the following inequality which approximates a function by its integral average.

Theorem 1. Let be a differentiable mapping on with the property that for all Then

for all The constant is the best possible in the sense that it cannot be replaced by a smaller constant.

In 1997, Dragomir and Wang [2] obtained another inequality of this type.

Theorem 2. If is differentiable on and , for all for some constants , then

for all

In 1988, the German Mathematician Hilger [3] introduced the concept of time scales. The time scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering formalism for studying hybrid discrete-continuous dynamical system. Since the introduction of this theory, it became a point of research to extend known classical differential and integral results to time scales. Following this line of thought, Bohner and Matthews [4] extended Theorem 1 to time scales by proving the following result.

Theorem 3. Let , , and be differentiable. Then, for all , one has

where is given in Definition 12 and This inequality is sharp in the sense that the right-hand side of (3) cannot be replaced by a smaller one.

For more generalizations, extensions, and variants of Theorem 3, we refer the interested reader to papers [5–10] and the references therein. In 2008, Liu and Ngô [11] generalized Theorem 3 for points Specifically, they proved the following theorem.

Theorem 4. Suppose that

(1) is a division of the interval for ,(2) is points so that and ,(3) is differentiable function.

Then one has

where This inequality is sharp in the sense that the right-hand side of (4) cannot be replaced by a smaller one.

As a consequence (by taking ) of Theorem   in [12], Tuna and Daghan obtained the following time scale version of Theorem 2.

Theorem 5. Let , and be differentiable. If is rd-continuous and , for all and for some , then, for all , one has

The aim of this paper is twofold, namely,

(1)proving a generalized weighted Montgomery identity for points Using this identity, we then obtain a weighted version of Theorem 4. Our result boils down to Theorem 4, if the weight function is the identity map,(2)generalizing Theorem 5 for points For the case where , we recover Theorem 5.

This present paper is organized as follows. In Section 2, we provide some time scale essentials that will aid in better understanding of what follows. Our main results are then stated and proven in Section 3. Finally, we apply our results to the continuous, discrete, and quantum calculus to obtain more results in this direction. A brief conclusion follows thereafter in Section 4.

2. Preliminaries

Now, we briefly introduce the theory of time scales. For an in-depth study of the time scale calculus, we recommend the books of Bohner and Peterson [13, 14].

Definition 6. A time scale is an arbitrary nonempty closed subset of The forward jump operator and backward jump operator are defined by for and for , respectively. Clearly, we see that and for all If , then we say that is right-scattered, while if , then we say that is left-scattered. If , then is called right dense, and if , then is called left dense. Points that are both right dense and left dense are called dense. The set is defined as follows: if has a left-scattered maximum , then ; otherwise, For with , we define the interval in by Open intervals and half-open intervals are defined in the same manner.

Definition 7. The function , is called differentiable at , with delta derivative , if for any given there exists a neighborhood of such that

If , then , and if , then

Definition 8. The function is said to be -continuous if it is continuous at all right-dense points and its left-sided limits exist at all left-dense points

Definition 9. Let be -continuous function. Then is called an antiderivative of on if it is differentiable on and satisfies for any In this case, one has

Definition 10. The function is defined as

for any

Theorem 11. If with , and are -continuous, then one has the following:

(i)(ii).(iii).(iv)(v).(vi)

Definition 12. Let , be functions that are recursively defined as

3. Main Results

For the proof of our theorems, we will need the following lemma.

Lemma 13 (generalized weighted Montgomery identity for points). Suppose that

(1) is a division of the interval for ,(2) is points so that and ,(3) are differentiable functions.

Then one has the following equation:

where

Proof. Using items (i), (ii), (iv), and (vi) of Theorem 11, we have

Hence, the result follows.

Remark 14. The above lemma becomes Lemma in paper [11] if we take

Corollary 15. If , then and (10) boils down to

where

Corollary 16. If we take in Corollary 15, (13) becomes

Theorem 17 (weighted Ostrowski type inequality for points). Under the assumptions of Lemma 13, one has the following inequality:

where

Proof. The proof of inequality (16) follows by taking the absolute value of both sides of (10) and then applying item (v) of Theorem 11.

Remark 18. By choosing , we recapture Theorem 4.

Corollary 19 (continuous case). Let Then, one has from (16) the following inequality:

where and is given by (11).

Corollary 20 (discrete case). Let Suppose that

(1) is a division of for ;(2) is points so that and ;(3)

Then, for any differentiable function , and each , one has

where and

Next, we formulate and prove a generalization of Theorem 5.

Theorem 21 (generalized Ostrowski-Grüss type inequality for points). Suppose that

(1) is a division of the interval for ,(2) is points so that and ,(3) is differentiable, is rd-continuous, and there exist such that for all

Then one has the following inequality:

Inequality (20) is sharp in the sense that the constant 1/2 on the right-hand side cannot be replaced by a smaller one.

Proof. Using Lemma 13 with , we obtain (see also [11, Lemma ])

where

Also, from (22), we get

Similarly, one gets

Now, let From assumption (), for all implies that for all This further implies that , for all Hence,

Using (21) and (23), we obtain

The left-hand side of (26) is estimated as follows:

Using relation (25) in (27), we get

Hence, the desired result follows.

Remark 22. If , then Theorem 21 becomes Theorem 5 for the case where

We now apply Theorem 21 to different time scales.

Corollary 23 (continuous case). Taking in Theorem 21 amounts to the following inequality:

Proof. In this case, Using this in inequality (20) gives the desired result.

Remark 24. The above corollary is the same as Corollary   in [15]. In other words, Theorem 21 extends Corollary , in paper [15], to time scales.

Corollary 25 (discrete case). Let Suppose that

(1) is a division of for ;(2) is points so that and ;(3)

Then one has the following inequality:

Proof. The proof follows by using the inequality in Theorem 21 and observing that

Corollary 26 (quantum case). Let with Suppose that

(1) is a division of for ;(2) is points so that and ;(3) is differentiable.

Then one has

Proof. Using Theorem 21 and the fact that, for the quantum calculus, one has ,

Corollary 27. Suppose that Then one has the following inequality:

Proof. Inequality (34) follows by choosing , and , in Theorem 21.

Remark 28. () Taking in (34), we get

() Taking in (34), we get

() Now, for in (34), we obtain

4. Conclusion

The Ostrowski and Ostrowski-Grüss inequalities have received great deal of attention from the mathematical community dealing with inequalities. Giant steps have been made in extending some of the results to time scales. This work is tailored towards advancing this move. To be precise, we proved a generalization of the Montgomery identity and then used the resultant equation to obtain a weighted Ostrowski inequality for points, thus generalizing a result of Liu and Ngô [11]. Furthermore, we obtained an Ostrowski-Grüss type inequality which generalizes and extends results of Tuna and Daghan [12] and Feng and Meng [15].

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

References

S. S. Dragomir, “Grüss inequality in inner product spaces,” Gazette of the Australian Mathematical Society, vol. 26, no. 2, pp. 66–70, 1999.

S. S. Dragomir and S. Wang, “An inequality of Ostrowski-Grüss type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules,” Computers and Mathematics with Applications, vol. 33, no. 11, pp. 16–20, 1997.

S. Hilger, Ein Maβkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten [Ph.D. thesis], Universität Würzburg, Würzburg, Germany, 1988.

M. Bohner and T. Matthews, “Ostrowski inequalities on time scales,” Journal of Inequalities in Pure and Applied Mathematics, vol. 9, no. 1, article no. 6, 2008.

B. Karpuz and U. M. Özkan, “Ostrowski Inequality on time scales,” J. Inequal. Pure and Appl. Math, vol. 9, no. 4, article 112, 2008.

W. Liu, A. Tuna, and Y. Jiang, “On weighted Ostrowski type, Trapezoid type, Grüss type and Ostrowski-Grüss like inequalities on time scales,” Applicable Analysis, vol. 93, no. 3, pp. 551–571, 2014.

W. Liu, A. Tuna, and Y. Jiang, “New weighted Ostrowski and Ostrowski-Grüss type inequalities on time scales,” Analele Stiintifice ale Universitatii Al I Cuza din Iasi - Matematica, vol. 60, no. 1, pp. 57–76, 2014.

E. R. Nwaeze, “A new weighted Ostrowski type inequality on arbitrary time scale,” Journal of King Saud University - Science, vol. 29, no. 2, pp. 230–234, 2017.

E. R. Nwaeze, “Generalized weighted trapezoid and Grüss type inequalities on time scales,” Australian Journal of Mathematical Analysis and Applications, vol. 11, no. 1, article 4, pp. 1–13, 2017.

G. Xu and Z. B. Fang, “A Generalization of Ostrowski type inequality on time scales with k points,” Journal of Mathematical Inequalities, vol. 11, no. 1, pp. 41–48, 2017.

W. Liu and Q.-A. Ngô, “A generalization of Ostrowski inequality on time scales for k points,” Applied Mathematics and Computation, vol. 203, no. 2, pp. 754–760, 2008.

A. Tuna and D. Daghan, “Generalization of Ostrowski and Ostrowski-Grüss type inequalities on time scales,” Computers and Mathematics with Applications, vol. 60, no. 3, pp. 803–811, 2010.

M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhauser, Boston, Mass, USA, 2001.

M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Series, Birkhäuser, Boston, MA, USA, 2003.

Q. Feng and F. Meng, “New Ostrowski-Grüss type inequalities with the derivatives bounded by functions,” Journal of Inequalities and Applications, vol. 2013, article no. 456, 2013.