1. Introduction

In 1938, A.M. Ostrowski proved an interesting integral inequality, estimating the absolute value of the derivative of a differentiable function by its integral mean as follows

Theorem 1.1. [2] Let f: I , where I is an interval, be a mapping in the interior I° of I, and a,b Є I°, with a < b.

If |f'| ≤ M for all x Є [a, b], then

This is well-known as Ostrowski's inequality. In recent years, a number of authors have written about generalizations, extensions and variants of inequality (1).

In [¹], Cerone et al. proved the following identity

Lemma 1.2. [¹] Let f : [a,b] → be a differentiable mapping such that f ^(n-1) is absolutely continuous on [a, b]. Then for all x Є [a, b] we havethe identity

where the kernel Kn: [a, b]² → is given by

and n is natural number, n ≥ 1.

We also recall some definitions

Definition 1.3. [3] a function is said to be convex, if the following inequality

holds for all x, y Є I and t Є [0,1].

Definition 1.4. [2] A function is said to be is said to be φ-convex, if the following inequality

holds for all x, y Є I and t Є [0,1], where I is an interval of and is a bifunction.

Remark 1.5. Obviously if we choose y Definition 1.4 recaptures Definition 1.3.

In this paper we establish some new Ostrwoski's inequalities for n-times differentiable mappings which are φ-convex.

2. Main results

In what follows, we assume that n Є , and I ⊂ be an interval, where [a, b] C I, and y : be a bifunction

Theorem 2.1. Let he n-times differentiable on [a, b] such thatis φ-convex, then the following inequality

holds for all x Є [a,b].

Proof. From Lemma 1.2, and properties of modulus, we have

which is the desired result. The proof is completed.

Corollary 2.2. Let be n-times differentiable on [a, b] such that is convex, we have the following estimate

Proof. Taking in Theorem 2.1.

Theorem 2.3. Let be n-times differentiable on [a, b] such that, and let q > 1 with is φ-convex, then the following inequality

holds for all x Є [a,b].

Proof. From Lemma 1.2, properties of modulus, and Hölder's inequality, we have

Since is φ-convex, we deduce

Thus the proof is completed.

Corollary 2.4. Let be n-times differentiable on [a, b] such that, and let q > 1 with is convex,, then the following inequality holds

Proof. Taking in Theorem 2.3.

Corollary 2.5. Under the same assumptions of Corollary 2.4, we have

Proof. Taking in Theorem 2.3, we obtain (6). Then using the following algebraic inequality for all a, b ≥ 0, and 0 ≤ a ≤ 1 we have we get the desired result.

Theorem 2.6. Let be n-times differentiable on [a, b] such that and letis φ-convex, then the following inequality

holds for all x Є [a,b].

Proof. From Lemma 1.2, properties of modulus, and power mean inequality, we have

Since is φ-convex, we deduce

The proof is completed.

Corollary 2.7. Let be n-times differentiable on [a, b] such thatand letis convex, then the following inequality holds

Corollary 2.8. Let be n-times differentiable on [a,b] such thatand letis convex, then the following inequality holds

Theorem 2.9. Suppose that all the assumptions of Theorem 2.6 are satisfied, then the following inequality

holds for all x Є [a, b].

Proof. From Lemma 1.2, properties of modulus, and power mean inequality, we have

Since is φ-convex, we deduce

which is the desired result.

Corollary 2.10. be n-times differentiable on [a, b] such thatand letis convex, then the following inequality holds

Corollary 2.11. Let be n-times differentiable on [a, b] such thatand letis convex, then the following inequality holds

3. Applications for some particular mappings

In this section, we give some applications for the special case where the function

a) Consider with n ≥ 2. Then and

Using Corollary 2.2, we get

Moreover, if we choose we obtain

Particulary, if we choose α = 0, we obtain

b) Consider with . Then Corollary 2.2, we have

Choosing α = 0 and b = 1, we have for all x Є [0, 1]

Moreover, if we choose , we get