OSTROWSKI TYPE INEQUALITIES PERTAINING STRONGLY CONVEX FUNCTIONS VIA CONFORMABLE FRACTIONAL INTEGRALS AND THEIR APPLICATIONS

In the article, by applied the concept of strongly convex function and one known identity, we establish several Ostrowski type inequalities involving conformable fractional integrals. As applications, some new error estimations for the midpoint formula are provided as well.


Introduction
The subsequent inequality is known as Ostrowski inequality.
Definition 3. A function h : I −→ R is called strongly convex with modulus c > 0, if holds for all a 1 , a 2 ∈ I and t ∈ [0, 1].
exists and is finite.All α-fractional integrable functions on [a 1 , a 2 ] is indicated by where the integral is the usual Riemann improper integral, α ∈ (0, 1].
The main purpose of the article is to find several Ostrowski type inequalities involving conformable fractional integrals using the concept of strongly convex functions and one known identity.At the end of the paper we give some error estimations for the midpoint formula.

Main Results
In order to prove our main results we need the following lemma. where Proof.By Lemma 1, the fact that x α−1 and −x α are both convex for x > 0, properties of the modulus and since the function |h (x)| is strongly convex with modulus c > 0, we have Hence, we have the result in (9).
Corollary 2. If we take x = (a 1 + a 2 )/2 in Theorem 3, we get ) and |h (x)| q is strongly convex function with modulus c > 0 for q > 1 and p where Proof.Using Lemma 1, properties of the modulus, Hölder's inequality and since the function |h (x)| q is strongly convex with modulus c > 0, we have .
Hence, we have the result in (10).
Corollary 3. If we take c −→ 0 + in Theorem 4, we get the following inequality . Corollary 4. If we take x = (a 1 + a 2 )/2 in Theorem 4, we get where Proof.Using Lemma 1, properties of the modulus, the well-known power mean inequality, |h (x)| ≤ M, ∀x ∈ [a 1 , a 2 ] and since the function |h (x)| q is strongly convex with modulus c > 0, we have Hence, we have the result in (11).
Corollary 6.If we take x = (a 1 + a 2 )/2 in Theorem 5, we get  where  where ) and |h (x)| q is strongly convex function with modulus c > 0 and q ≥ 1, then and ∆ 3 , ∆ 4 are defined as in Theorem 3. Proof.Using Lemma 1, properties of the modulus, the well-known power mean inequality and since the function |h (x)| q is strongly convex with modulus c > 0, we have Hence, we have the result in (12).
Corollary 8.If we take c −→ 0 + in Theorem 6, we get the following inequality Corollary 9.If we take x = (a 1 + a 2 )/2 in Theorem 6, we get , where L 1 (α) = + (L 2 (1)) and consider the quadrature formula where is the midpoint version and E α (h, P) denotes the associated approximation error.Here, we are going to derive some new estimates for the midpoint formula.
Proposition 2. Let 0 ≤ x 0 < x n and h : [x 0 , x n ] −→ R be an α-fractional differentiable function for α ∈ (0, 1]. ) and |h (x)| q is strongly convex function with modulus c > 0 for q > 1 and p −1 + q −1 = 1, then x α i+1 − t α p dt. Proof.The proof is analogous to that of Proposition 1 only by using Corollary 4 of Theorem 4. where C i,1 (α) = Proof.The proof is analogous to that of Proposition 1 only by using Corollary 6 of Theorem 5.

Conclusion
In this paper, using the concept of strongly convex functions and one known identity, we found several Ostrowski type inequalities pertaining conformable fractional integrals.Also, we give some error estimations for the midpoint formula.