GENERALIZATION OF WEIGHTED OSTROWSKI INEQUALITY WITH APPLICATIONS IN NUMERICAL INTEGRATION

An integral inequality of Ostrowski type with weights is established for differentiable functions up to second order, whose second derivatives are bounded and first derivatives are absolutely continuous. The inequality of weighted integral is then functional to weighted composite quadrature rules.


Introduction
It is a known that Ostrowski type inequalities can be used to estimate the absolute deviation from its integral mean.These inequalities can be used to provide explicit error bounds for numerical quadrature formulae.The weighted version of Ostrowski inequality was first presented in 1983 by J. E. Pečarić and B. Savić [10].Keeping in view the importance of this inequality, for last few decades, the researchers are in continuous effort to obtain sharp bounds of Ostrowski's inequality in terms of weight.
We have introduced a weighted inequality which has its applications in Numerical quadrature rules and Probability theory.The probability density function, defined by us, has produced the results of Ostrowski's inequality in a more organized way.
In 1938, a Ukrainian Mathematician Alexandar Markowich Ostrowski discovered an inequality called Ostrowski inequality, which states that.
Then the inequality is The number 1  4 is sharp which cannot be replaced by a smaller one.
[15] Let ϕ : I ⊆ R → R be a differentiable function on I 0 (I 0 is the interior of I) Then the inequality is 1  4 is sharp which cannot be replaced by a smaller one.
In 1999, the following inequality is proved by Cerone et.al. in [1].
In the same year, the following result is stated by Dragomir and Barnett in [2].
[15]Let the following assumption An Ostrowski type integral inequality for functions whose second derivatives are bounded is established by P. Cerone, S. S. Dragomir and J. Roumeliotis in [1].S. S. Dragomir and N.
S. Barnett in [2] have also established a similar inequality. in [4], an Ostrowski type integral inequality is similar in sense as that of [1] or [2] pointed out by S. S. Dragomir and A. Sofo.
The following integral inequality in [4] is proved by S. S. Dragomir and A. Sofo.
In present paper, we give weighted version of Ostrowski-type inequality (1) using different kernel and will also state its application in Numerical quadrature rules, probability theory and special means.This paper is arranged in the following approach.The first part is based on preliminaries whereas in second part we are using weights in inequality (1) from which we get main result, which is in fact probability density functions.In the last part we would state about the application in terms of composite quadrature rules.

Main results
We use here a weighted form of integral inequality which is proved by S. S. Dragomir and A. Sofo in [4], and also we are using it in numerical integration.
where w : Proof.Let us begin with the following integral identity [8] (see also [6]).
given that f is absolutely continuous on [a 0 , b 1 ] and the weighted kernel Using integration by parts, we get Now using equations (5) and (6) in (4), we get Also Now, we have a case: , we find After solving the equations,we find and finally Using equations ( 7), (8), and (9),we find which is our required result.remarks 3.2.If we put w(t) ≡ 1 b 1 −a 0 in (2), then we will get . remarks 3.4.If we put w(t) ≡ 1 b 1 −a 0 in (12), then we will get . remarks 3.5.If we examine an approximation for the end point v = a 0 in (2), then we will get . remarks 3.6.If we examine an approximation for the end point
Proof.Applying inequality (2) on ξ i ∈ [x i , x i+1 ] and summing over i from to n − 1 and using triangular inequality we get (14) remarks 3.6.If we put w(t) ≡ 1 b 1 −a 0 in (2) then we will get then the following quadrature rule is: . remarks 3.8.If we put w(t) ≡ 1 v i+1 −v i in (12) then we will get . remarks 3.9.If we examine an approximation for the end point v = v i in (2), we will get . remarks 3.10.If we examine an approximation for the end point v = b 1 in (2), we will get:

4.Application for Probability Density Function
Let X be a continuous random variable having the probability density function is the expectation of the random variable X on the interval [a 0 ; b 1 ] and weighted expectation would be is the expected of the random variable X on the interval [a 0 , b 1 ].so, we get the following theorem.

Conclusion
Weighted Peano Kernels are used for Ostrowski type inequalities, depending on the second derivatives, are stated in this paper.In the research paper [5] and [11], Weighted Ostrowski type inequalities for the second derivative of the functions are addressed.A generalization and extension of the inequalities is presented in [5] and [11].we have presented a generalization (2) of the inequality (1) found in [5] for twice differentiable function whose first derivatives are absolutely continuous and 2 nd derivative belong to L ∞ − (a 0 , b 1 ) by presenting a parameter λ ∈ [0, 1].This generalization also results in finding a inequality for a specific value of λ as stated in remarks .The inequality thus found has a better bound than the inequalities presented in [5] and [11].Remarks also shows that the perturbed trapezoid inequality that can be found from (2) is better than the perturbed inequalities presented in [5] and [11] of perturbed trapezoid type.
The inequality is then functional for a partition of the interval [a 0 , b 1 ] to find some composite quadrature rules and also functional to special means.

Theorem 4 .
Let the suppositions of Theorem 2.1 be valid if probability density function belongs to L 2 [a 0 , b 1 ] space, then we get the following inequality b