New bounds for the Ostrowski-type inequalities via conformable fractional calculus

In this paper, we have introduced the new upper bounds for Ostrowski-type integral inequalities by using conformable fractional integral. In accordance with this purpose, we have beneﬁted from the Taylor expansion for conformable fractional derivatives which was introduced by Anderson.

Some authors have argued that conformable derivatives are not considered as fractional derivatives in the fractional calculus community; it is an interesting derivative that enables to derive with respect to arbitrary order but without memory effect. This question seems today to still be open, and perhaps, it is a philosophical issue. Such derivative makes it possible to generalize many mathematical concepts depending on ordinary derivatives. For instance, it contributes in generalizing certain mathematical inequalities [2]. It also contributed to of general form of Sturm-Liouville problems [21]. Conformable local-type derivatives also make it possible to obtain generalized-type fractional derivatives by iterating their corresponding integrals [22,23]. Conformable (fractional) derivatives have the drawback that the limiting case α → 0 does not give us the function itself. To improve this drawback, Anderson [24] made use of proportional calculus to define better well-behaved derivatives in the limiting case, and therefore, he improved conformable (fractional) derivatives.
In this study, we present new Ostrowski-type conformable fractional integral inequalities using the rules of conformable fractional calculus and Taylor formula for conformable fractional derivatives.
This work is organized as follows: in Sect. 2, the conformable fractional derivatives and integrals are summarised. In Sect. 3, the new upper bounds for Ostrowski-type inequalities with the help of conformable fractional calculus are introduced. Application to numerical integration is given in Sect. 4, while some conclusions and further directions of research are discussed in Sect. 5.

Definitions and properties of conformable fractional derivative and integral
The following definitions and theorems with respect to conformable fractional derivative and integral [1,6,11] are summarised. Definition 2.1 [11] (Conformable fractional derivative) "Given a function f : [0, ∞) → R. Then, the conformable fractional derivative" of f of order α is defined by We can write f (α) (t) for D α ( f ) (t) to denote the conformable fractional derivative of f of order α. In addition, if the conformable fractional derivative of f of order α exists, then we simply say f is α-differentiable.
We will also use the following important results, which can be derived from the results above.

Lemma 2.6 [1]
Let the conformable differential operator D α be given as in (1.1), where α ∈ (0, 1] and t ≥ 0, and assume the functions f and g are α-differentiable as needed. Then is continuous and α ∈ (n, n + 1]. Then, for all t > a we have We can give the Hölder's inequality in conformable integral as follows:

Remark 2.9
If we take p = q = 2 in Lemma 2.8 then, we have the Cauchy-Schwartz inequality for conformable integral.
The following lemma and theorems are given by Anderson in [3].
Using Taylor's Theorem, we define the remainder function by and for n > −1, This inequality is sharp in the sense that the right-hand side of (2.9) cannot be replaced by a smaller one. Now, we present the main results: ([a, b]) , p, q > 1 and 1 p + 1 q = 1. Then for all x ∈ [a, b], we have the following inequality:

Using the identities (3.1) and (3.2), we obtain
Using the change of variable Moreover, we have Thus, putting the identities (3.4) and (3.5) in (3.3), we deduce That is, Using Hölder's inequality, we have and similarly Thus, we obtain the inequality which completes the proof.

Corollary 3.4 Under the assumption of Theorem
Similarly, we get and Therefore, using (3.6)-(3.8), we obtain

Applications to numerical integration
We now deal with applications of the integral inequalities involving conformable fractional integral. Consider the partition of the interval [a, b] , given by Define the quadrature: where i = 0, . . . , n − 1. ([a, b]) , p, q > 1 and 1 where S α ( f, I n ) is as defined in (4.1) and the remainder satisfies the estimation: Proof Applying Corollary 3.4 on the interval x i , x i+1 , we obtain for all i = 0, . . . , n − 1. Summing over i from 0 to n − 1 and using the triangle inequality, we obtain |R( f, I n , ξ)| ≤ 1 α 1 α (q + 1)  where S α ( f, I n ) is as defined in (4.1) and the remainder satisfies the estimation: |R