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
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
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
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
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
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