Research

A generalized Ostrowski-Grüss type inequality for bounded differentiable mappings and its applications

Sabir Hussain1* and Ather Qayyum2

Author Affiliations

1 Department of Mathematics, Yanbu University, P.O. Box 31387, Yanbu Al Sinaiyah, Saudi Arabia

2 Department of Mathematical Sciences, University of Hail, P.O. Box 2440, Hail, Saudi Arabia

For all author emails, please log on.

Journal of Inequalities and Applications 2013, 2013:1  doi:10.1186/1029-242X-2013-1

 Received: 11 June 2012 Accepted: 28 November 2012 Published: 3 January 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we establish a generalized Ostrowski-Grüss type inequality for differentiable mappings using the weighted Grüss inequality which is another generalization of inequalities established and discussed by Barnett et al. (Inequality theory and applications, pp. 24-30, 2001), S. S. Dragomir and S. Wang (Comput. Math. Appl. 33:15-22, 1997) and A. Rafiq et al. (JIPAM. J. Inequal. Pure Appl. Math. 7(4):124, 2006). Perturbed midpoint and trapezoid inequalities are obtained. Some applications in different weights are given. This inequality is extended to account for applications in numerical integration.

Keywords:
Ostrowski inequality; Grüss inequality; weight function; numerical integration

1 Introduction

Integration with weight functions is used in countless mathematical problems such as approximation theory and spectral analysis, statistical analysis and the theory of distributions. Grüss developed an integral inequality [1] in 1935. In 1938, Ostrowski [2] established an interesting integral inequality associated with differentiable mappings which has powerful applications in numerical integration, probability and optimization theory, stochastic, statistics, information and integral operator theory. During the last few years, many researchers focused their attention on the study and generalizations of the above two inequalities [3-5]. Recently, Qayyum and Hussain [6] established a new inequality using the weighted Peano kernel, which is more generalized as compared to previous inequalities developed and discussed in [3-5]. Moreover, results investigated [6] were in weighted form instead of previous results [3-5] which were in non-weighted form. This approach not only generalized the results of [3], but also gave some other interesting inequalities as special cases. In this paper, we establish another generalization of the Ostrowski-Grüss type inequality using the weighted Grüss inequality for bounded differentiable mappings which generalizes the previous inequalities developed and discussed in [3-5]. Perturbed midpoint and trapezoid inequalities are also obtained. In Section 4, we give some applications in different weights. This inequality is extended to account for applications in numerical integration in Section 5.

2 Preliminaries

The classical Ostrowski integral inequality ([2] see also [[1], p.468]) in one dimension stipulates a bound between a function evaluated at an interior point x and the average of the function f over an interval. That is,

(1)

for all , where and is a differentiable mapping on .

The constant is sharp in the sense that it cannot be replaced by a smaller one. We also observe that the tightest bound is obtained at , resulting in the well-known mid-point inequality.

The integral inequality that establishes a connection between the integral of the product of two functions and the product of the integrals is known in the literature as the Grüss inequality. The inequality is as follows.

Theorem 1Letbe integrable functions such thatandfor all, whereφ, Φ, γ, Γ are constants. Then

(2)

where the constantis sharp.

During the past few years, many researchers [7-10] have given considerable attention to the inequality (1).

In [4], Dragomir and Wang improved the above inequality and proved the following Ostrowski type inequality in terms of the lower and upper bounds of the first derivative.

Theorem 2Letbe continuous onand differentiable on, and its derivative satisfy the conditionfor all. Then we have the inequality

(3)

for all.

In [3], Barnett et al. pointed out a similar result to the above for twice differentiable mappings in terms of the upper and lower bounds of the second derivative.

Theorem 3Letbe continuous onand twice differentiable on, and assume that the second derivativesatisfies the condition: for all.

Then, for all, we have the inequality

(4)

In the recent years, some authors (see, for example, [5,6,11]) also generalized the above inequality.

3 Some new results

We assume the weight function (or density) to be non-negative and integrable over its entire domain and consider . We denote the moments to be m, M and σ and define them as follows: , and . We start with the following weighted Grüss inequality [12].

Theorem 4Letbe two integrable functions such thatandfor all, and letϕ, θ, Γ, γbe constants. Then we have, the constantis sharp.

Now, we give our main result.

Theorem 5Letbe continuous onand differentiable onand. Then, for all, we have the inequality

(5)

Proof The following weighted integral inequality for all is proved in [13].

(6)

where the weighted Peano kernel, , is given by

(7)

We observe that the mapping satisfies the estimation

(8)

Consider, and . Applying the weighted Grüss inequality to and , we get

(9)

Now, from (7), it can be easily seen that . Thus, (13) gives

(10)

Using (6), the inequality (10) gives

(11)

Further, we observe that

(12)

Using (12) in (11), we get our main result (5). □

Corollary 6Under the assumptions of Theorem 5 and choosing, we have the perturbed midpoint inequality

(13)

Proof This follows by inequality (5). □

Corollary 7Under the assumptions of Theorem 5, we have the perturbed trapezoidal inequality

(14)

Proof Put and in (5) and sum up the obtained inequalities. Using the triangle inequality and dividing by two, we get the required inequality. □

4 Some weighted integral inequalities

Integration with weight functions is used in countless mathematical problems. Two main areas are: (i) approximation theory and spectral analysis and (ii) statistical analysis and the theory of distributions. In this section, inequality (5) is evaluated for the more popular weight functions.

Uniform (Legender) Substituting into the moment gives . Substituting it into (5) gives

(15)

Note that the interval mean is simply the midpoint.

Logarithm This weight is present in many physical problems, the main body of which exhibits some axial symmetry.

Putting , , , the moment and (5) imply

(16)

(17)

The optimal point is closer to the origin than the midpoint , reflecting the strength of the log singularity.

Jacobi Substituting , , , into the moment gives

(18)

Inequality (5) gives .

The optimal point is again shifted to the left of the midpoint due to the singularity at the origin.

Chebyshev Substituting , , , into the moment gives .

Hence, the inequality corresponding to the Chebyshev weight is .

The optimal point is at the midpoint of the interval reflecting the symmetry of the Chebyshev weight over its interval.

Laguerre The Laguerre weight , is defined for positive values, . From the moment , we have .

The appropriate inequality is , from which the optimal sample point of may be deduced.

Hermite Finally, the Hermite weight is defined over the entire real line . The inequality (5) with the Hermite weight function is thus , which results in an optimal sampling point of .

5 Application in numerical integration

Let be a division of the interval (). We have the following quadrature formula.

Theorem 8Letbe continuous onand differentiable on, andsatisfy the conditionfor all. Then we have the following perturbed Riemann type quadrature formula: , whereand the remainder satisfies the estimation

(19)

for all, where ().

Proof Apply Theorem 5 to the interval , , where (), to get .

Summing over i from 0 to and using the generalized triangular inequality, we deduce the desired estimation (19). □

Corollary 9Under the assumption of Theorem 5, by choosingin the above theorem, we recapture the midpoint like quadrature formula: , where, and the remainder term satisfies the estimation.

6 Conclusion

We established another generalization of the Ostrowski-Grüss type inequality using the weighted Grüss inequality for bounded differentiable mappings which generalizes the previous inequalities developed and discussed in [3-5]. Perturbed midpoint and trapezoid inequalities are also obtained. This inequality is extended to account for applications in different weights and numerical integration. This generalized inequality will be useful for the researchers working in the field of the numerical analysis to solve their problems in engineering and in practical life.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgements

The first author acknowledge the financial support from the Research and Development Center of Colleges and Institute of Royal Commission at Yanbu for this research.

References

1. Mitrinović, DS, Pěcarić, JE, Fink, AM: Classical and New Inequalities in Analysis, Kluwer Academic, Dordrecht (1993)

2. Ostrowski, AM: Über die absolutabweichung einer differentiebaren funktion von ihrem integralmittelwert. Comment. Math. Helv.. 10, 226–227 (1938)

3. Barnett, NS, Cerone, P, Dragomir, SS, Roumeliotis, J, Sofo, A: A survey on Ostrowski type inequalities for twice differentiable mappings and applications. Inequality Theory and Applications, pp. 24–30. Nova Publ., Huntington (2001)

4. Dragomir, SS, Wang, S: 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. Comput. Math. Appl.. 33, 15–22 (1997)

5. Rafiq, A, Mir, NA, Zafar, F: A generalized Ostrowski-Grüss type inequality for twice differentiable mappings and applications. JIPAM. J. Inequal. Pure Appl. Math.. 7(4), Article ID 124 (2006)

6. Qayyum, A, Hussain, S: A new generalized Ostrowski Grüss type inequality and applications. Appl. Math. Lett.. 25, 1875–1880 doi:10.1016/j.aml.2012.02.052 (2012)

doi:10.1016/j.aml.2012.02.052

Publisher Full Text

7. Milovanovic, GV, Pĕcairć, JE: On a generalization of the inequality of Ostrowski and some related applications. Publ. Elektroteh. Fak. Univ. Beogr., Ser. Mat. Fiz.. 544-576, 155–158 (1976)

8. Barnett, NS, Cerone, P, Dragomir, SS, Roumeliotis, J, Sofo, A: A survey on Ostrowski type inequalities for twice differentiable mappings and applications. Inequality Theory and Applications, pp. 24–30. Nova Publ., Huntington (2001)

9. Dragomir, SS, Sofo, A: Trapezoidal type inequalities for n-type differentiable functions. RGMIA, Research Report Collection 8(3), (2005)

10. Fink, AM: Bounds on the deviation of a function from its averages. Czechoslov. Math. J.. 42(117), 280–310 (1992)

11. Rafiq, A, Mir, NA, Ahmad, F: Weighted Ostrowski type inequality for differentiable mappings whose first derivatives belong to . Gen. Math.. 14(3), 91–102 (2006)

12. Dragomir, SS: Some integral inequalities of Grüss type. Indian J. Pure Appl. Math.. 31(4), 397–415 (2000)

13. Qayyum, A: A weighted Ostrowski-Grüss type inequality for twice differentiable mappings and applications. Int. J. Math. Comput. Sci. (Print). 1(8), 63–71 (2008)