GENERALIZED OSTROWSKI INEQUALITY WITH APPLICATIONS IN NUMERICAL INTEGRATION AND SPECIAL MEANS

Generalization of Ostrowski inequality with applications in numerical integration and special means are recognized for differentiable functions up to second order whose second order derivatives are bounded and first derivatives are absolutely continuous.


Introduction
In 1938, A. Ostrowski proved an inequality concerning function with bounded derivative which is known as Ostrowski inequality [10].The inequality is stated as follows: proposition 1.1.Let ψ : I ⊆ R → R be a differentiable function on (ρ, ) and let on (ρ, ), |ψ (v)| ≤ M for some positive real number M. Then for all v ∈ [ρ, ], (1.1) The constant 1 4 is the better possible in the sense that it cannot be replaced by the smaller one.
The explanation of Ostrowski inequality can be demonstrate in two ways as follows: (1) Estimate the deviation from its average value to functional value.
(2) Estimate the approximating area under the curve of a rectangle.
In last few decades, some work has been done on the generalizations of Ostrowski's inequality.
Some examples are mentioned in [4,7,11,2].In [3], an integral inequality has established of Ostrowski type by S. S. Dragomir et al. for mappings with bounded second derivatives.S. S.
Dragomir et al. in [5], established a similar inequality.In [5], S. S. Dragomir and N. S. Barnett, an indicated Ostrowski type integral inequality give a same sense to that of [3] or [5].
In order to recall some results we need here some definitions which can be found in [15, pp. 125, 128].
Let L p [ρ, ](1 ≤ p < 1) denotes the space of p-power integrable functions on the interval Then the inequality (1.2) According to [13] and [14] A 1 (ψ) : 1 In [16], Fiza Zafar and Nazir Ahmad Mir proved the following generalization of Ostrowski in equality: → R be a mapping whose first derivative is absolutely continuous on [ρ, ] and suppose that the second derivative φ ∈ L ∞ (ρ, ).Then, we have the inequality for . (1. 3) The objective of this paper is to further generalized Ostrowski inequality than inequality given in Proposition (1.3).

Main results
Theorem 2.1.Let all the assumptions of Propositions (1.3) are valid.Then, we have the inequality.
or equivalently, Proof.Consider the kernel K : [ρ, ] 2 → R, as defined in [4], which can be written as, this implies that, ) Now, we substitute (2.5) Integrating by parts, we have Using (2.3) and (2.6) in (2.5), we get: which can be written as and we obtain, ) where now, we let (2.9) we get: After some simple calculations,we get Using (2.8), (2.9), and (2.10) in (2.7), we get This completes the proof.
Remark 2.4.Choosing λ = 1 3 in the inequality (2.12) gives the better approximation: which has a new and a better approximation than the three-point quadrature inequalities. .Remark 2.5.If we select λ = 1 in the inequality (2.12), we get a perturbed trapezoid inequality as follows (2.13) which has a better approximation than the perturbed trapezoid quadrature inequalities mentioned in [5] and [12] for • ∞ norm.
Remark 2.6.If we select λ = 1 4 in the inequality (2.12), we get a perturbed trapezoid inequality as follows which has a new and a better approximation than the perturbed trapezoid quadrature inequalities.. Remark 2.7.If we select λ = 3 4 in the inequality (2.12), we get a perturbed trapezoid inequality as follows which has a new and a better approximation than the perturbed trapezoid quadrature inequalities. .
Remark 2.8.If we select λ = 3 10 in the inequality (2.12), we get a perturbed trapezoid inequality as follows which has a better approximation than the perturbed trapezoid quadrature inequalities presented in [2] and [5] .∞ norm.

Applications in numerical Integration
Using the inequality (2.1), we get the approximation of composite quadrature rules with smaller error found by the classical results. and Proof.Applying inequality (2.10) and summing over i from 0 to n − 1 and using triangular inequality, we get (3.2).Remark 3.2.By choosing λ = 0 gives as a special case [10], the modified version of approximations of composite quadrature rules.
, then we have the following quadrature rule: Which is a perturbed composite trapezoid inequality.

Applications for Probability Density Function
Let X be a continuous random variable having the probability density function φ : [ρ, ] → R + and the cumulative distribution function is the expectation of the random variable x on the interval is the expected of the random variable X on the interval [ρ, ].Then we may have the following theorem.
Theorem 4.1.Let the assumptions of Theorem (2.1) valid if probability density function belongs to L 2 [ρ, ] space, then we get the following inequality for all ξ ∈ [ρ, ].

Applications for some Special Mean
The inequality (2.1) may some written as where A(ρ, ) = ρ+ 2 .Choosing λ = 0 gives a special case, the modified version of the inequality in [10] as follows, 1 2 applying (4.1), to infer some inequalities for special means using some particular mappings.
The results of the special means are therefore as follows Example no 1: Consider From (4.1), we have: from which we obtain the approximation at the centre v = ρ+ 2 = A(ρ, ), so that from which we obtain the better approximation if we select λ = 3 10 , that is ln G 1 For λ = 0, we have For λ = 1 4 , we have For λ = 3 4 , we have ln G 1 For λ = 1 3 , we have Example no 2: Consider From (4.1), we have: and the approximation at the centre v = ρ+ 2 = A(ρ, ), so that from which we obtain the better approximation if we select λ = 3 10 , that is For λ = 0, we have For λ = 1 4 , we have For λ = 1 3 , we have For λ = 3 4 , we have Example no 3: Consider From (4.1), we obtain where or which gives the better approximation at λ = 3

Conclusion
In this paper, modified Peano Kernels are used for some remarkable Ostrowski type inequalities depending on the second derivatives are mentioned.Ostrowski type inequalities for twice differentiable functions have been lengthily mentioned.in the research paper [6] and [7].We have mentioned a generalization and extension of the inequalities mentioned in [6] and [7].we have mentioned a generalization (4) of the inequality (2) acquired in [1] for twice differentiable functions whose first derivatives are absolutely continuous and second derivative belong to L ∞ (ρ, ) by introducing a parameter λ ∈ [0, 1].This generalization also results in attaining a three-point inequality for a specific value of λ as mentioned in remarks (2.4) to (2.8).The three-point inequality thus acquired has a useful bound than the three-point inequalities mentioned in [6] and [7] for .∞ -norm.Remarks (2.4) to (2.8) also shows that the perturbed trapezoid inequality that can be acquired from (2.1)is useful than the perturbed inequalities [ρ, ] with the standard normψ p = ρ |ψ( †)| p d † 1 pand L 1 [ρ, ] denotes the space of all essentially bounded functions on [ρ, ] with the normψ p = ess sup v∈[ρ, ] |ψ (v)| < ∞In [12], S. S. Dragomir et al. proved the following generalization of Ostrowski in-equality: proposition 1.2.Let ψ : [ρ, ] → R be a function continuous on [ρ, ] and differentiable on (ρ, ).Assume that |ψ (v)| ≤ M for v ∈ (ρ, ) and M is positive real constant and denote