Abstract
We prove new Hermite-Hadamard inequalities for conformable fractional integrals by using convex function, -convex, and coordinate convex functions. We prove new Montgomery identity and by using this identity we obtain generalized Hermite-Hadamard type inequalities.
1. Introduction
The class of convex functions is well known in the literature and is usually defined in the following way: let be an interval in ; then a function is said to be convex on if the inequalityholds for all and Also, we say that is concave, if the inequality in (1) holds in the reverse direction. There are several generalizations of the convex function. Here we mention basic definition of -convex function and coordinate convex function. In the paper [1], Hudzik and Maligranda considered a generalization of convex function, which is known as -convex function in the second sense. This class of function is defined in the following way: a function is said to be -convex in the second sense ifholds for all and and for some fixed . The class of -convex functions in the second sense is usually denoted by .
In [2], the concept of convex functions defined on the coordinates of the bidimensional interval of the plane of two variables was introduced.
Definition 1. Let us consider the bidimensional interval in with and A function is called convex on the coordinates if the partial mappings defined as and defined as are convex for all and .
Remark 2. Note that every convex function is convex on the coordinates, but the converse is not generally true [2].
Many important inequalities have been obtained for this class of functions but here we will present only one of them.
If is a convex function on the interval , then, for any with , we have the following double inequality:
Both inequalities hold in reverse direction if the function is concave on the interval . This remarkable result was given in ([3], 1893) and is well known in the literature as Hermite-Hadamard inequality. Since its discovery, this inequality has become the center of interest for many prolific researchers and received a considerable attention. Also, a number of extensions, generalizations, and variants of (3) have been provided in the theory of mathematical inequalities. For example, see [4–12] and the references cited therein.
Now we recall some definitions and important results in the theory of conformable fractional calculus. For detailed treatment of the results, we refer the interested readers to [13–20].
Definition 3 (see [20]). Given a function , the conformable fractional derivative of of order is defined byfor all and . If the conformable fractional derivative of of order exists, then we say that is -differentiable. Let be -differentiable in , and exists; then defineWe will, sometimes, write and for to denote the conformable fractional derivatives of of order .
Theorem 4 (see [20]). Let and be -differentiable at a point . Then we have the following: (1), for all .(2), for all constant functions (3), for all (4)(5)(6), for differentiable at If, in addition, the function is differentiable, then
Also, it is important to note the following:(1)(2)(3)(4)(5)(6)(7)(8)
Definition 5 (see [21] (conformable fractional integral)). Let and . A function is -fractional integrable on if the integralexists and is finite. All -fractional integrable functions on are indicated by
Remark 6. Note that the relation between the Riemann integral and conformable fractional integral is given byThe -fractional integrable functions are strongly related to fractional Lebesgue and Sobolev spaces. General definitions of fractional Lebesgue and Sobolev spaces can be found in the monograph [22]. Moreover, in recent years, they have been widely used in the theory of regularity for PDE. For interested readers, we recommend [23–25] and some of the references therein.
Theorem 7 (see [13]). Let be differentiable and Then, for all , one has
Theorem 8 (see [13] (integration by parts)). Let be two functions such that is differentiable. Then
Theorem 9 (see [13]). Assume that such that is continuous and Then, for all , one has
Very recently, Anderson [21] investigated the following conformable integral version of Hermite-Hadamard inequality.
Theorem 10 (see [21]). Let and let be an -differentiable function with , such that is increasing; then one has the following inequality:Moreover, if the function is decreasing on , then one has
Remark 11. It is obvious that if we choose , then inequalities (12) and (13) reduce to inequality (3).
Several important variants of Hermite-Hadamard inequality have been provided in the literature, such as the versions established by Anderson [21] and Sarikaya et al. [26] and so forth.
In this paper, we prove new Hermite-Hadamard inequalities for conformable fractional integrals by using convex function, -convex, and coordinate convex functions. We prove new Montgomery identity for conformable fractional integral. By using this identity, we obtain Hermite-Hadamard type inequalities. These results give us the generalizations of the earlier results.
2. Hermite-Hadamard Inequalities
Theorem 12. Let be a convex function defined on , where ; then the following double inequality holds:
Proof. Let us define a function on byObviously the function is increasing and continuous function on . Therefore, and henceNowBy changing of variable and convexity of , we get Hence,Now let us define a function on byClearly the function is decreasing and continuous on . Therefore, and henceNowHence,From (20) and (25), we deduce the right-hand side of (14).
Now we prove left inequality in (14).
It is well known thatAlso from the functions and as defined in (15) and (21), respectively, we have Therefore,By using (28) in (26), we obtainNow, by changing of variable and using the fact that for , we obtainSimilarly,NowCombining (29), (30), (31), and (32), we get which is equivalent to the left inequality in (14).
Corollary 13. Under the assumptions of Theorem 12, if we put , we get the following well-known Hermite-Hadamard inequality for convex function:
Now we prove Hermite-Hadamard inequality for conformable fractional integral by using -convex function.
Theorem 14. Let and let be an -convex function defined on , where ; then the following double inequality holds: where is Euler beta function defined for
Proof. By definition of -convex function, we haveLet be defined in (15). Then, as in the proof of Theorem 12, we have NowBy changing of variable and -convexity of , we get Hence,Let be defined in (21). Then similar proof leads toNowHence,From (41) and (44), we deduce the right-hand side of (35).
Now we prove left inequality in (35).
It is well known thatAlso from the functions and as defined in (15) and (21), respectively, we have Therefore,By using (47) in (45), we obtainSimilar to (30), we haveAlso,Now, using -convexity of , we haveCombining (48), (49), (50), and (51), we getwhich is equivalent to the left inequality in (35).
Corollary 15. Under the assumptions of Theorem 14, if we put , we get the following well-known Hermite-Hadamard inequality for -convex function [27]:
In the following theorem, we prove Hermite-Hadamard inequality for conformable fractional integral by using coordinate convex function.
Theorem 16. Let and let be a convex function on the coordinates on , where and ; then the following double inequality holds:
Proof. Since is convex on the coordinates, it follows that the mapping , , is convex on for all Then, by Theorem 12, we havethat is,Integrating (56) on , we haveSimilarly, using the mapping , , we getSumming inequalities (57) and (58), we deduce the left-hand side of (54).
Now we prove right inequality in (54).
Also, by Theorem 12, using inequality (20), we haveSimilarly, using inequality (25), we haveSumming inequalities (59) and (60), we deduce the right-hand side of (54).
Remark 17. Under the assumptions of Theorem 16, if we put , we get the Hermite-Hadamard inequality for coordinate convex function obtained by Dragomir in [2].
Remark 18. Under the assumptions of Theorem 16, using inequalities (57), (58), (59), and (60), the following double inequality holds:
3. Generalization of Hermite-Hadamard Type Inequalities
Now, we are in position to find some new estimations for the left-hand side of Hermite-Hadamard inequality for conformable fractional integral as given in Theorem 12 by using a new Montgomery identity.
Lemma 19 (new Montgomery identity). Let with , and let be -fractional differentiable function, where Thenwhere
Proof. Integrating by parts, we have Adding and solving for yields the required result.
Theorem 20. Let be an -fractional differentiable function such that is a convex function, where and Then, for and , one has where
Proof. Using Lemma 19 with defined in (63), convexity of , Hölder’s inequality, and property of the modulus, we have
Theorem 21. Let be -fractional differentiable and is a convex function, where and Then, for , one haswhereand are defined as in Theorem 20.
Proof. Using Lemma 19 with defined in (63), convexity of , the well-known power mean inequality, and property of the modulus, we have
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The research was supported by the National Natural Science Foundation of China (Grants nos. 61673169, 61374086, 11371125, 11401191, and 11701176), the Tianyuan Special Funds of the National Natural Science Foundation of China (Grant no. 11626101), and the Natural Science Foundation of the Department of Education of Zhejiang Province (Grant no. Y201635325).