Recent Advances in Function Spaces and its Applications in Fractional Differential Equations 2020View this Special Issue
New Modified Conformable Fractional Integral Inequalities of Hermite–Hadamard Type with Applications
In this study, a few inequalities of Hermite–Hadamard type are constructed via the conformable fractional operators so that the normal version is recovered in its limit for the conformable fractional parameter. Finally, we present some examples to demonstrate the usefulness of conformable fractional inequalities in the context of special means of the positive numbers.
Recently, fractional differential equations have attracted more and more attention, which can be used to describe some biological, chemical, and physical phenomenon more accurately than the classical differential equations of integer order. Nowadays, these appear naturally in modeling long-term behaviors, particularly in the areas of viscous fluid dynamics, control systems, physics, engineering, and viscoelastic materials [1–3]. The fractional calculus has many interesting applications when it was applied on a new mathematical model in thermoelectricity theory. Also, in modern physical engineering, the fractional system order is used to modify the mathematical models that describe the governing equations of those models to be more economical than the classic uses [4, 5].
There are many possible ways of defining fractional operators: Riemann–Liouville, Caputo, tempered, Marchaud, Atangana-Baleanu, and Hilfer. To find all these definitions, we advice our readers to visit the references [6–10]. Furthermore, all of these operators are considered special cases of a single, unifying, model of fractional operator by Fernandez et al. in .
In 2014, Khalil et al.  defined a new well-behaved simple fractional derivative, namely, the conformable fractional derivative relying only on the basic limit definition of the classical first derivative. It was first introduced as a conformable fractional derivative, but it lacks some of the desired properties for fractional derivatives [13–16]. This operator and its properties and applications have been intensely studied in other works, of which we mention  in particular. In , the following conformable derivative was defined: where , , and is a function. This operator relates to the large and thriving theory of conformable fractional derivatives, and it arises naturally in control theory.
Now, if is -differentiable in some , exist, then we define
Moreover, if is differentiable, then we have
Briefly, we can write for or to denote the conformable fractional derivatives of of order at . In addition, if exists, then we say that is -differentiable.
Theorem 1 (). Let and be -differentiable at a point . Then, (i) for all (ii)(iii)(iv) for any constant (v)
Now, conformable fractional integral and its few basic properties are stated.
Definition 2 (). For any and , the function is -fractional integrable on , if exists and is finite.
Remark 3. (a)We indicate each -fractional integrable functions by on (b)The usual Riemann improper integral is defined byfor each .
Theorem 5 (). Let and be a continuous function. Then, for the function and for all , we have This property is often called the inverse property.
Definition 8 (). A function is said to be convex on the interval , if the inequality holds for all and .
In view of recent results in theory of differential, integral, and fractional differential equations, it is becoming extremely worthless to ignore the existence of integral inequalities which are useful in determining bounds of unknown functions (see, e.g., [17–23]). Also, there are various integral inequalities in the literature and their numbers increase sharply every year. But the common inequality is the Hermite–Hadamard (HH) integral inequality, which was first found in 1893 by Hadamard in . It has the following formula: where is supposed to be convex on with and .
The inequality of HH type (12) has been extended and applied in all models of fractional calculus, such as Riemann–Liouville [25–29], -Riemann–Liouville , conformable fractional [31, 32], generalized fractional [33–35], time-scale fractional [36, 37], tempered fractional , and Atangana-Baleanu  models. Since our paper is about conformable fractional integral inequality, here, we give some inequalities involving conformable fractional integrals that are obtained in :
Lemma 9 (, Lemma 1). Let be an -fractional differentiable mapping on with . If; then, the following identity for conformable fractional integral holds: where
Theorem 10 (, Theorem 2). Let be an -fractional differentiable mapping on with . If and is convex on , then the following inequality for conformable fractional integral holds: where
Theorem 11 (, Theorem 3). Let be an - . is convex on , then the following inequality for conformable fractional integral holds: where
Theorem 12 (, Theorem 4). Let be an -fractional differentiable mapping on with . If and is concave on , then the following inequality for conformable fractional integral holds: where , and are given in Theorem 11 and
In view of the above indices, we establish new integral inequalities of HH type for the convex functions via the conformable fractional operators. By doing this, we will try to demonstrate the usefulness of conformable fractional inequalities in the context of special means of the positive numbers.
2. Main Results
The next result is necessary in the rest of the work.
Lemma 13. Let be an -differentiable function on , . If , then we have where
Proof. Making use of Definition 8, we have On integrating by parts, one can have making the change of variable , to get Similarly, we get Considering , and , we get (21) and thus, our proof is completed.
Theorem 14. Let be an -differentiable function on , . If and is a convex function on , then we have where
Proof. Making use of Lemma 13 and the properties of , we have For and , the function is convex. So, we have Then, by the convexity of , we can deduce Analogously, we can deduce Substituting’s into, we obtain the desired result.
Corollary 15. Theorem 14 with becomes where
Theorem 16. Let be an -differentiable function on such that . If , is a convex function on , then we have