PERTURBATIONS OF AN OSTROWSKI TYPE INEQUALITY AND APPLICATIONS

Using (1.1), the authors obtained estimations of error for the mid-point, trapezoid, and Simpson quadrature formulae. They also gave applications of the mentioned results in numerical integration and for special means. In this paper, we establish two perturbations of (1.1). Using the perturbations, we derive some new error bounds for the mid-point, trapezoid, and Simpson quadrature formulae. Similar perturbed inequalities are also considered in [2, 3]. We give applications in numerical integration. It is shown that these new bounds can give much better results than the bounds obtained in [4].


Introduction.
In the recent years a number of authors have written about generalizations of Ostrowski's inequality.For example, this topic is considered in [1,4,5].In [4], Dragomir et al. proved the following generalization of Ostrowski's inequality.
Theorem 1.1 (see [4]).Using (1.1), the authors obtained estimations of error for the mid-point, trapezoid, and Simpson quadrature formulae.They also gave applications of the mentioned results in numerical integration and for special means.
In this paper, we establish two perturbations of (1.1).Using the perturbations, we derive some new error bounds for the mid-point, trapezoid, and Simpson quadrature formulae.Similar perturbed inequalities are also considered in [2,3].We give applications in numerical integration.It is shown that these new bounds can give much better results than the bounds obtained in [4].
Remark 2.3.In the above proof, we used 12) and (2.13), then we get corresponding mid-point inequalities.
If we now sum (3.23) over i from 0 to n − 1 and apply the triangle inequality and (3.22), then we get (3.17where (3.26) for S = 2γ/3, and where and S = 2Γ /3.
The results obtained in this paper can be much better than the results obtained in [4].We illustrate this fact for the composite trapezoid quadrature rule.
In [4], we can find the following result: Remark 3.9.In similar ways we can show that estimations for the mid-point and Simpson's composite rules (see Remarks 3.2 and 3.7) can be much better than corresponding estimations obtained in [4].

Theorem 2 . 1 .
Let I ⊂ R be an open interval and a, b ∈ I, a < b.If

3 .Theorem 3 . 1 .
Applications in numerical integration.The next approximations of the integral b a f (t)dt hold.Let all assumptions of Theorem 2.1 hold.If
Under the assumptions of Theorem 2.1, we have b 2.19) and (2.20) is now obvious.Remark 2.6.If we set x = (a + b)/2 in (2.19) and (2.20), then we get corresponding inequalities which do not depend on x.