Abstract and Applied Analysis
Volume 2019 (2019), Article ID 5648095, 6 pages
https://doi.org/10.1155/2019/5648095

Research Article

Inequality of Ostrowski Type for Mappings with Bounded Fourth Order Partial Derivatives

Waseem Ghazi Alshanti

Department of General Studies, Jubail University College, Saudi Arabia

Correspondence should be addressed to Waseem Ghazi Alshanti; shantiw@ucj.edu.sa

Received 19 October 2018; Revised 7 January 2019; Accepted 15 January 2019; Published 3 March 2019

Academic Editor: Patricia J. Y. Wong

Copyright © 2019 Waseem Ghazi Alshanti. 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.

A general Ostrowski’s type inequality for double integrals is given. We utilize function whose partial derivative of order four exists and is bounded.

1. Introduction

In 1938, Ostrowski [1] introduced the following integral inequality.

Theorem 1. Let : be continuous mapping on and differentiable on , whose derivative is bounded on , i.e., , then for all

The constant is sharp in the sense that it cannot be replaced by a smaller one.

In 1975, Milovanović [2] proposed the following generalization of (1) for a function of several variables whose first order partial derivatives are bounded.

Theorem 2. Let : be a differentiable function defined on and let in . Then, for every ,

In 1998, Barnett and Dragomir [3] proved the following Ostrowski type inequality for mappings of two variables with bounded second order partial derivatives.

Theorem 3. Let : continuous on , exists on and is bounded, i.e.,

Then we have the inequality

for all .

In [4], Xue et al. derive the following inequality of Ostrowski type.

Theorem 4. Let : be an absolutely continuous function such that the partial derivatives of order two exist and suppose that there exist constants with for all . Then we have

for all and .

More recently, Sarikaya et al. [5] establish weighted Ostrowski type inequalities considering function whose second order partial derivatives are bounded as follows.

Theorem 5. Let : be an absolutely continuous function such that the partial derivatives of order two exist and are bounded, i.e.,

for all .Then we have

where

For other related work, we refer the reader to [6–15].

In this paper, motivated by the ideas in both [4, 5], we shall derive a new inequality of Ostrowski’s type similar to the inequalities (5) and (7), involving functions of two independent variables.

2. Main Results

In order to introduce our main results, we commence with the following lemma.

Lemma 6. Let : be an absolutely continuous function such that the partial derivative of order 4 exists for all and . Then for any two mappings and , where

and

the identity

holds.

Proof. By definitions of and in both (9) and (10), we have

For , integration by parts yields

Similarly, , , and can be obtained.
Thus, by adding , , , and , we easily deduce

By further algebraic manipulations and assuming result by [4], the proof of Lemma 6 is completed.

Theorem 7. Let : such that be an absolutely continuous function such that the partial derivative of order 4 exists and is bounded; i.e.,

for all . Then for all and , we have

where the functional is given by (11).

Proof. By considering (11), we have

But,

and

Now, substituting (18), (19) into (17) gives (16) and, hence, completes the proof.

Corollary 8. Under the assumption of Theorem 7 with , we have

Corollary 9. Under the assumption of Theorem 7 with , , and   we have

Corollary 10. Under the assumption of Theorem 7 with , , and   we have

Remark 11. In Corollaries 8, 9, and 10 we assume that the involved integrals can more easily be computed than the original double integral.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares no conflicts of interest.

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