Some New Integral Inequalities via Variant of Pompeiu ’ s Mean Value Theorem

The main of this paper is to establish an inequality pro- viding some better bounds for integral mean by using a mean value theorem. Our results generalize the results of Ahmad et. al in (8).


Introduction
The inequality of Ostrowski [7] gives us an estimate for the deviation of the values of a smooth function from its mean value.More precisely, if f : [a, b] → R is a differentiable function with bounded derivative, then for every x ∈ [a, b].Moreover the constant 1/4 is the best possible.
In this paper, we establish an general form with the p-norm, 1 ≤ p ≤ ∞, which will give the Ahmad et.al result for p = ∞.Our results generalize the results of Ahmad et.al in [8].

Main Results
Before stating the main results, we will give the following lemma proved by Pecaric and Ungar in [4]: A(x, q) := where for p = 1, i.e. q = ∞, the integrals are to be interpreted as the ∞norms, i.e. as maxima of the function (u, t) → 1 u 2 on the corresponding domains of integration.Then, , for 1 < p, q < ∞, p, q = 2; To prove our theorems, we need the following lemma: Using the change of the variable in last integrals with u = 1 t , we get Denote x 1 = 1 x and x 2 = 1 t .Then for all x, t ∈ [a, b] from (3), we have which gives (2) and completes the proof.Then for 1 p + 1 q = 1, with 1 ≤ p, q ≤ ∞, and all x ∈ [a, b], we have 92 Some New Integral Inequalities Proof.From Lemma 2.2, we have ( 5) Integrating with respect to t on [a, b] and dividing by 3x 2 , we get Firstly, we consider the case 1 < p, q < ∞.By using Hölder's inequality, the sum in the last line of ( 6) can be written The first factor in ( 7) is equal with and, by Lemma 2.1, the second factor equals A (x, q).Thus, putting ( 8) into ( 6) and dividing b − a gives the required inequality (4).
Proof.Multiplying (5) by w(t) x and integrating with respect to t on [a, b], we have and as in the proof of Theorem 2.1, we get

M.Z. Sarikaya 91 Lemma 2 . 2 .
Let f : [a, b]→ R be continuous function on [a, b] and twice order differentiable function on (a, b) with 0 < a < b.Then for any t, x ∈ [a, b], we have

Theorem 2 . 1 .
Let f : [a, b]→ R be continuous function on [a, b] and twice order differentiable function on (a, b) with 0 < a < b.