Abstract
In this paper, we establish a generalized OstrowskiGrüss 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. 2430, 2001), S. S. Dragomir and S. Wang (Comput. Math. Appl. 33:1522, 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 integration1 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 [35]. 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 [35]. Moreover, results investigated [6] were in weighted form instead of previous results [35] which were in nonweighted 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 OstrowskiGrüss type inequality using the weighted Grüss inequality for bounded differentiable mappings which generalizes the previous inequalities developed and discussed in [35]. 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,
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 wellknown midpoint 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
During the past few years, many researchers [710] 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
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
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 nonnegative 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
Proof The following weighted integral inequality for all is proved in [13].
where the weighted Peano kernel, , is given by
We observe that the mapping satisfies the estimation
Consider, and . Applying the weighted Grüss inequality to and , we get
Now, from (7), it can be easily seen that . Thus, (13) gives
Using (6), the inequality (10) gives
Further, we observe that
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
Proof This follows by inequality (5). □
Corollary 7Under the assumptions of Theorem 5, we have the perturbed trapezoidal inequality
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
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
The optimal point is closer to the origin than the midpoint , reflecting the strength of the log singularity.
Jacobi Substituting , , , into the moment 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
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 OstrowskiGrüss type inequality using the weighted Grüss inequality for bounded differentiable mappings which generalizes the previous inequalities developed and discussed in [35]. 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.
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