SOME APPLICATIONS OF QUADRATURE RULES FOR MAPPINGS ON L p [ u, v ] SPACE VIA OSTROWSKI-TYPE INEQUALITY

. Some Ostrowski-type inequalities are stated for L p [ u, v ] space and for mappings of bounded variations. Applications are also given for obtaining error bounds of some composite quadrature formulae. MSC 2010. 26D15, 26D20.


INTRODUCTION
In 1938, Ostrowski introduced a bound for the absolute value of the difference of a function to its average over a finite interval. His well known result named as Ostrowski's inequality [10].
where the constant 1 4 is sharp. Let g, h : [u, v] → R be two absolutely continuous functions such that functions and their product are integrable, theČebyšev functional [2] is defined by In 1934, the following result proved by Grüss [5] (see also [6]): where m 1 , m 2 , M 1 , M 2 ∈ R and satisfy the conditions By G.V. Milovanović in [8], an application of classical Ostrowski inequality in quadrature formula was given for the very first time, also its generalization to functions in several variables was given in this article.
A generalization of Ostrowski inequality developed by Milovanović and Pečarić [9], which is stated as: In 1997, Dragomir and Wang [3] proved the following proposition by using (2) which is known as Ostrowski-Grüss inequality.
for all z ∈ [u, v] and for real constants m and M , then In this paper, we first derive an integral identity for differentiable functions by using the kernel (5). Then, we apply this equality to get our results for functions whose first derivative is bounded. First section is based on introduction and preliminaries. In the second and in the third section we prove inequalities for absolutely continuous mappings in which g ∈ L p [u, v] for p ≥ 1 and mappings of bounded variation, respectively. In the last section, we will give some applications for composite quadrature rules.

THE CASE WHERE
In order to prove our main results, we need the following lemma from [7]: the following identity holds In this section, we are going to present Ostrowski-type integral inequality Theorem 6. Let g : I → R be an absolutely continuous mapping on Proof. Using the Hölder inequality in (6), for any z ∈ [u, v], we get Remark 7. If we substitute q = 1 (and p = ∞) in (7), then we get the following Corollary.
Corollary 8. Let g : I → R be an absolutely continuously mapping on I o , the interior of the interval I, where u, v ∈ I with u < v. If g is bounded on [u, v], then the following inequality holds for any z ∈ [u, v] Remark 9. The inequality (8) is the generalization of Ostrowski inequality which is presented in Proposition 5, i.e., by replacing g(u) = g(v) in (8), we get (3) and also by choosing g ∞ = M we get (1).
Remark 10. If we replace z = u+v 2 in (8), then we get the following midpoint inequality where the constant 1 4 is sharp.
Remark 11. By replacing z = u or z = v in (8), we get the trapezoidal inequality Corollary 12. Let g be a function as defined in Theorem 7. 1) If we replace z = u+v 2 in (7), then we get the midpoint inequality ∀ p ≥ 1: (7), we get the trapezoidal inequality ∀ p ≥ 1: Remark 13. By the inequality (9) we retrieve the result of Corollary 5 and the inequality (11) gives us the result of Corollary 8 of M. W. Alomari paper [1], respectively.

THE CASE WHERE g IS OF BOUNDED VARIATION
Theorem 14. Let g : [u, v] → R be a function of bounded variation. Then the following inequality holds for any z ∈ [u, v]: g is the total variation of g over [u, v] Now using Lemma 6 with the inequality (16) for p(s) = P (z, s), and ν(s) = g(s), s ∈ [u, v], we get We notice that which proves the inequality (15). To prove that the constant 1 2 in inequality (15) is sharp, we suppose that the inequality (15) is valid for a constant K > 0, i.e., (17) Consider the mapping g : [u, v] → {0, 1} is defined as For z = u, we have By using (17), we obtain, 1 ≤ 2K or 1 2 ≤ K, and thus it is proved that the constant 1 2 is sharp.
Corollary 15. Let g be a function as defined in Theorem 15. 1) If we replace z = u+v 2 in (15), then we get the midpoint inequality where the constant 1 2 is sharp. 2) If we replace z = u or z = v in (15), then we get the trapezoidal inequality where the constant 1 2 is sharp. Remark 16. The inequalities (18)

APPLICATIONS TO NUMERICAL QUADRATURE RULES
Now, we are going to discuss some applications in numerical quadrature rules, which can be used to get some sharp bounds.
Theorem 17. Let g be defined as in Theorem 7. Then (20) holds where Q n (I n , g) is given by formula (21) and the remainder R n (I n , g) satisfies the estimates Proof. Applying inequality (7) on the intervals, [z k , z k+1 ], we can state that we sum the inequalities presented above over k from 0 to n − 1. This gives If we choose, ξ k = z k +z k+1 2 in (21), then quadrature formula becomes: Remark 20. If (20) holds and Q n (I n , g) is given by formula (25).
1) Let g be as in Theorem 7 where the remainder R n (I n , g) from (11) and (9)  2) Let g be as in Theorem 15 where the remainder R n (I n , g) from (18) becomes If we choose, ξ k = z k or ξ k = z k+1 in (21), then quadrature formula becomes: 2) Let g be as in Theorem 15 where the remainder R n (I n , g) from (19) becomes |R n (I n , g)| ≤ 1 2 n−1 k=0 z k+1 z k g.

CONCLUSION
We have given some remarks on Ostrowski type inequalities for absolutely continuous functions in which g ∈ L p space. Using the results of L p space, we have also given some special results for L ∞ space. Our Corollary 9 of Theorem 7 is the generalization of Ostrowski inequality [10] which is presented in 1938 by A. M Ostrowski. Furthermore, by putting suitable substitutions we get midpoint and trapezoidal rules which are presented in [1,4]. At the end we have also given some applications for numerical integration.