EXTENSION OF OSTROWKI TYPE INEQUALITY VIA MOMENT GENERATING FUNCTION

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, generalizations of weighted Ostrowski inequality are derived by using moment generating functions in bounded variation, L∞ and Lp spaces. Applications to composite quadrature formulae are developed in which 3 Simpson’s, 3 8 Simpson’s, trapezoidal and midpoint inequalities are derived.


INTRODUCTION
In 1938 Ostrowski developed an important inequality [11] which states that: Theorem 1.1. Let φ : I ⊂ R → R be differentiable mapping on I o the interior of interval I such that φ ∈ L [u, v], where u, v ∈ I with u < v. If |φ (y)| ≤ M, then the following inequality holds which holds ∀ y ∈ [u, v] and 1 4 is the best possible constant in a sense that it cannot be replaced by a smaller constant.

(1.3)
where y ∈ u + λ v−u 2 , v − λ v−u 2 . We should also know the definition of bounded variation.
Also according to [7, p.318] We denote the first two moments to be m and M, where In this paper our aim is to give generalization of Preposition 1.2 by using moment generating function and to study three different cases namely φ is bounded variation, φ ∈ L ∞ [u, v] and φ ∈ L p [u, v]. We use first two moments of weighted function.

IF
Applying the above inequality for p(t) = K(y,t) and Now we us the fact to get To proof our next theorem we need the following lemma.  and Proof. Use Integration-by-parts on kernal (2.3), we get By adding equations (2.4) and (2.5), we get (2.2).
, then the following weighted inequality holds (2.6) Proof. By using the same fact we used in the previous theorem Applying the above inequality for p(t) = P w (y,t) and By using the same fact Corollary 2.4. By replacing w(s) = 1 v−u , and the values of µ and ϑ in (2.6), we will get the inequality (2.1).

FOR THE
i.e., Then the following inequality holds we have Corollary 3.2. By replacing w(s) = 1 v−u , µ and ϑ in (3.1), we will get the inequality of Proposition 1.2. Corollary 3.3. By replacing λ = 0, we will get the following inequality Remark 3.4. By replacing w(s) = 1 v−u in (3.2), we will get the following inequality Corollary 3.5. By replacing y = u+v 2 in Corollary 3.2, we will get following inequality

FOR THE CASE φ ∈ L p [u, v]
Theorem 4.1. Let φ : I ⊂ R → R is absolutely continuous mapping on I o , the interior of the interval I, where u, v ∈ I with u < v. If φ ∈ L p [u, v], p > 1. Then the following inequality holds Proof. We have Applying Hölder's Inequality If φ belongs to L p [u, v], p > 1, then the following inequality holds Proof. By using the identity (2.2) and kernel (2.3) and applying Hölder's Inequality which completes the proof.

Corollary 4.3
By replacing w(s) = 1 v−u and the values of µ and ϑ in (4.1), we will get the following inequality Corollary 4.4 By replacing λ = 0 in (4.1), we will get the following inequality Remark 4.5 By replacing w(s) = 1 v−u in (4.4), we will get the following inequality The inequality (4.6) is the result of Corollary 7 in [1].

APPLICATION TO QUADRATURE FORMULA
If I n : u = y 0 < y 1 < y 2 < ... < y n = v be a partition of the interval [u, v] and let h i = y i+1 − y i for i ∈ {0, 1, 2, ..., n − 1} Consider a general Quadrature Formula which yields following theorems.
where Q n (I n , φ ) is given in (5.1), then the remainder satisfies the estimation Proof. Applying inequality (2.1) on the interval [y i , y i+1 ], we get Sum the equalities presented above over i from 0 to n, we get which implies where Q n (I n , φ ) is given in (5.1), then the remainder satisfies the estimation Proof. By using the similar technique use in Theorem 5.1, we get Theorem 5.3. Let φ be as defined in Theorem 5.1 and we have v u φ (t)w(t)dt = R n (I n , φ ) + Q n (I n , φ ) where Q n (I n , φ ) is given in (5.1), then the remainder satisfies the estimation |R n (I n , φ )| ≤ 1

(5.4)
Proof By using the similar technique use in Theorem 5.1, we get

CONCLUSION
We have derived three different versions of Ostrowski type inequality, namely for bounded variation, L p and L ∞ space involving weights in terms of moment generating functions and by using them we also discussed their few applications in numerical integration.

FUNDING
This research is financially supported by Dean Science Research Grant, University of Karachi.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.