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 be an interval and the interior of I. Then the classical Ostrowski inequality [1] states that a real-valued function satisfies the inequality with the best possible constant if with and for all .

Recently, the Ostrowski inequality has attracted the attention of many researchers, many remarkable generalizations, extensions, variants and applications can be found in the literature [2-24].

Let and g be a real-valued function defined on . Then the (conformable) fractional derivative [23] of order α of g at is defined by

g is said to be α-differentiable if the conformable fractional derivative of order α of g exists. In what follows, we write or for to denote the conformable fractional derivative of order α of g. The conformable fractional derivative at 0 is defined as .

Let and . Then the function is said to be α-fractional integrable on if the integral exists and is finite. All α-fractional integrable functions on are denoted by .

Remark 1.1

Note that the relation between the Riemann integral and the conformable fractional integral is given by

Let and f, g be α-differentiable at . Then it is well known that for all ; for all constant ; for all ; if f is differentiable at .

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.

Main results

Lemma 2.1

Let , and be an α-fractional differentiable function. Then the identity holds if .

Proof

It follows from integration by parts that  □

Theorem 2.2

Let , , be an α-fractional differentiable function and . Then the inequality holds if is convex, where

Let , and . Then we clearly see that the functions and both are convex. It follows from Lemma 2.1 and the convexity of , and that  □

Corollary 2.3

Let . Then Theorem 2.2 leads to

Remark 2.4

If , then Corollary 2.3 becomes where the second inequality is obtained by using the convexity of .

Theorem 2.5

Let , , , , be an α-fractional differentiable function and . Then the inequality holds if is convex on and , where

From Lemma 2.1, power-mean inequality and the convexity of together with the identities and we clearly see that

Therefore, Theorem 2.5 follows easily from (2.1)–(2.5). □

Remark 2.6

Let . Then Theorem 2.5 leads to where

Theorem 2.7

Let , , , , be an α-fractional differentiable function and . Then the inequality holds if is convex on and , where

It follows from the proof of Theorem 2.2 that

From the power-mean inequality and convexity of together with the identities and we get

Therefore, Theorem 2.7 follows easily from (2.6)–(2.10). □

Theorem 2.8

Let , , , be an α-fractional differentiable function and . Then the inequality holds if is concave on , where

It is well known that is concave due to being concave. It follows from Lemma 2.1 that

Making use of Jensen’s integral inequality, we have where we have used the identities  □

Remark 2.9

If , then Theorem 2.8 becomes

Theorem 2.10

Let , , , be an α-fractional differentiable function and . Then the inequality holds if is concave on , where

From the concavity of we know that is also concave, then from Lemma 2.1 we have

It follows from the Jensen integral inequality that  □

Remark 2.11

If , then Theorem 2.10 leads to

Remark 2.12

If and , then Theorem 2.10 becomes

Applications to means

Let with . Then the arithmetic mean , logarithmic mean and generalized logarithmic mean of a and b are defined by respectively.

Recently, the bivariate means have been the subject of intensive research, many remarkable inequalities for the bivariate means can be found in the literature [25-60].

Let and . Then Corollary 2.3 immediately leads to Theorems 3.1 and 3.2.

Theorem 3.1

Let and . Then the inequality holds for all .

Theorem 3.2

Let and . Then the inequality holds for all .

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.