Ostrowski type inequalities involving conformable fractional integrals

In the article, we establish several Ostrowski type inequalities involving the conformable fractional integrals. As applications, we find new inequalities for the arithmetic and generalized logarithmic means.


Introduction
Let I ⊆ R be an interval and I • the interior of I. Then the classical Ostrowski inequality [1] states that a real-valued function f : I → R satisfies the inequality 2 2 ) 2 (a 2a 1 ) 2 (a 2a 1 ) f ∞ with the best possible constant 1/4 if a 1 , a 2 ∈ I • with a 1 < a 2 and |f (x)| ≤ M for all x ∈ [a 1 , a 2 ].
Recently, the Ostrowski inequality has attracted the attention of many researchers, many remarkable generalizations, extensions, variants and applications can be found in the literature .
g is said to be α-differentiable if the conformable fractional derivative of order α of g exists. In what follows, we write g α (t) or d α d α t (g) for D α (g)(t) to denote the conformable fractional derivative of order α of g. The conformable fractional derivative at 0 is defined as g α (0) = lim t→0 + g α (t).
Let α ∈ (0, 1] and 0 ≤ a < b. Then the function h : [a, b] → R is said to be α-fractional integrable on [a, b] if the integral Remark 1.1 Note that the relation between the Riemann integral and the conformable fractional integral is given by Let α ∈ (0, 1] and f , g be α-differentiable at t > 0. Then it is well known that for all a, b ∈ R; The main purpose of the article is to find the Ostrowski type inequalities involving the conformable fractional integrals and give their applications in certain bivariate means.

Corollary 2.3 Let x = (a 1 + a 2 )/2. Then Theorem 2.2 leads to
h (a 1 ) + a 1 12 Remark 2.4 If α = 1, then Corollary 2.3 becomes where the second inequality is obtained by using the convexity of |h |. , Proof From Lemma 2.1, power-mean inequality and the convexity of |h | q together with the identities we clearly see that (2.5) Therefore, Theorem 2.5 follows easily from (2.1)-(2.5).
Remark 2.6 Let α = 1. Then Theorem 2.5 leads to 2 , 1 , a 2 ]). Then the inequality Proof It follows from the proof of Theorem 2.2 that (2.6) From the power-mean inequality and convexity of |h | q together with the identities (2.10) Therefore, Theorem 2.7 follows easily from (2.6)-(2.10). 1 , a 2 ]). Then the inequality holds if |h | q is concave on [a 1 , a 2 ], where Proof It is well known that |h | is concave due to |h | q being concave. It follows from Making use of Jensen's integral inequality, we have where we have used the identities Remark 2.9 If α = 1, then Theorem 2.8 becomes Theorem 2.10 Let q > 1, 0 < α ≤ 1, 0 ≤ a 1 < a 2 , h : [a 1 , a 2 ] → R be an α-fractional differentiable function and D α (h) ∈ L 1 α ([a 1 , a 2 ]). Then the inequality , Proof From the concavity of |h | q we know that |h | is also concave, then from Lemma 2.1 we have It follows from the Jensen integral inequality that Remark 2.11 If α = 1, then Theorem 2.10 leads to Remark 2.12 If α = 1 and x = (a 1 + a 2 )/2, then Theorem 2.10 becomes

Applications to means
Let a, b > 0 with a = b. Then the arithmetic mean A(a, b), logarithmic mean L(a, b) and generalized logarithmic mean L (α,r) (a, b) of a and b are defined by Recently, the bivariate means have been the subject of intensive research, many remarkable inequalities for the bivariate means can be found in the literature .
Let h(x) = x r and h( holds for all a 1 , a 2 > 0.

Results and discussion
There are many results devoted to the well-known Ostrowski inequality. This inequality has many applications in the area of numerical analysis. In this paper, we give results for Ostrowski inequality containing conformable fractional integrals and their applications for means. First, we prove an identity associated with the Ostrowski inequality for conformable fractional integrals. By using this identity and convexity of different classes of functions and some well-known inequalities, we obtain several results for the inequality. The inequalities derived here are also pointed out to correspond to some known results, being special cases. At the end, we also present applications for means. The presented idea may stimulate further research in the theory of conformable fractional integrals.

Conclusion
In this paper, we prove an identity associated with the Ostrowski inequality for conformable fractional integral, present several Ostrowski type inequalities involving the conformable fractional integrals, and provide the applications in bivariate means theory. The idea and results presented are novel and interesting.