A PERTURBED VERSION OF GENERAL WEIGHTED OSTROWSKI TYPE INEQUALITY AND APPLICATIONS

The main purpose of this paper is to derive some new generalizations of weighted Ostrowski type inequalities. The new established inequalities are carried out for a twice differentiable mapping in different Lp spaces. Applications throught considering Grüss type inequality and numerical integration are also provided.


Introduction
The Ostrowski's inequality [1] can be considered as a very powerful tool for enhancement of numerical integration rules.It provides convenient potintial window for establishing bounds for the well known Newton-Cotes rules.To illustrate this point, consider f : f exists and is bounded, the inequality of Ostrowski may be stated as follows where Consequently, over the past few decades, there have been many studies on obtaining sharp bounds of (1.1) by considering the mappings and their derivatives in various Lebesgue spaces.Further, the new bounds have been carried out by implementing weighted and non-weighted Peano kernel.Several weighted and nonweighted versions of (1.1) have been derived.Applications in both numerical integration and probability are also presented in this regards.For instance, Roumeliotis et.al [2] proved a weighted integral inequality of Ostrowski's type for mappings whose second derivatives are bounded.Cerone [3] obtained bounds for the deviation of a function from a combination of integral means over the end intervals covering the entire interval.Qayyum et.al [4] established a new Ostrowski's type inequality using weight function which generalizes the inequality in [3].Barnett [5] reported a companion of (1.1) and the generalized trapezoid inequalites for various classes of functions, including functions of bounded variation, Lipschitzian, convex and absolutely continuous functions.Recently, Budak et.al [6] presented a new generalization of weighted Ostrowski's type inequality for mappings of bounded variation.Several further generalizations of (1.1) are provided in [7] - [16].
In [12], Qayyum et.al proved the following non-weighted generalization of Ostrowski's type integral inequality.
Theorem 1.1.Let f : [a, b] → R be a twice differentiable mapping.Then where α and β are non-negative real numbers such that not both zero.
In this paper, motivated by the non-weighted case in [12], new general weighted Peano kernel has been defined.To obtain new general weighted inequality of Ostrowski's type that is more generalized and extended as compare to [12].We consider a twice differentiable mapping f where, respectively, f ∈ L ∞ , f ∈ L p and f ∈ L 1 .Moreover, we utilize Grüss type inequality to present the perturbed verion of our result.
Finally, we investigate the new general weighted inequality in numerical integration.
Before we introduce our main result for a general weighted inequality of Ostrowski's type, we commence with the following definition and lemma.
The domain of ω may be finite or infinite and may vanish at the boundary points.We denote the moments Furthermore, for a function f : [a, b] → R, we define the functional where α, β ≥ 0 and not both zero.Then the following identity where holds.
Proof: From (1.5), we have After further simplification, the identity (1.6) can be obtained.
Proof: Taking the modulus of the right hand side of (1.6) , yields b a Thus, by combining (2.3) and (2.4) , the first inequality of (2.1) results.
Further, from (2.2) and by using Hölder's integral inequality for f ∈ L p [a, b] , we have where 1 p + 1 q = 1 with p > 1.Now, by (1.5) and utilizing the weighted mean value theorem for integrals, we have and so, by considering (2.5) and (2.6) the second inequality of (2.1) is obtained. where, Therefore, combining (2.7) and (2.8) gives the third inequality of (2.1) , and so, the theorem is now completely proven.
For different weights, a variety of results can be obtained.
Now, by noting that The desired second inequality of (2.9) can be obtained.

Perturbed Results
The Grüss inequality is as follows [13].
Now, the perturbed verions of the results in the pervious section may be obtained by using Grüss type inequalities involving the Čebyŝev functional [14], where where .
Proof: Replacing f (t) by P ω (x, t) and g(t) by f (x) in (3.2) yields, Now, by using both (1.6) and (2.4) , we have where k is the secant slope of f over [a, b] as given in (3.4) .Moreover, by [3], we have But, where Now, for T 1 2 (P ω (x, t), P ω (x, t)) , we consider (1.5) as follows where N (x) is given in (3.4) .
[a, b] → R to be a bounded function such that b − a is small, then I = b a f (x) dx, can be, simply, approximated by sampling at one point as I * (x) = (b − a) f (x) for some x ∈ [a, b] .Now, if

. 1 ) 4 . 1 .
Theorem Let the conditions of Theorem (2.1) hold.The following weighted quadrature rule for weighted integral holds α δ x a