New Modified Conformable Fractional Integral Inequalities of Hermite–Hadamard Type with Applications

Department of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia Department of Medical Research, China Medical University, Taichung 40402, Taiwan Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan Department of Mathematics, College of Education, University of Sulaimani, Sulaimani, Kurdistan Region, Iraq Department of Mathematics, Faculty of Technical Science, University Ismail Qemali, Vlora, Albania


Introduction
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 longterm behaviors, particularly in the areas of viscous fluid dynamics, control systems, physics, engineering, and viscoelastic materials [1][2][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][7][8][9][10]. Furthermore, all of these operators are considered special cases of a single, unifying, model of fractional operator by Fernandez et al. in [11].
In 2014, Khalil et al. [12] 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][14][15][16]. This operator and its properties and applications have been intensely studied in other works, of which we mention [13] in particular. In [13], the following conformable derivative was defined: where ρ ∈ ð0, 1Þ, t > 0, and f : ½0,∞Þ → R 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 f is ρ-differentiable in some ð0, ρÞ, ρ > 0, lim t→0 + f ðρÞ ðtÞ exist, then we define Moreover, if f is differentiable, then we have where Briefly, we can write f ðρÞ ðtÞ for D ρ ðf ÞðtÞ or ðd ρ /d ρ tÞð f ðtÞÞ to denote the conformable fractional derivatives of f of order ρ at t. In addition, if f ðρÞ ðtÞ exists, then we say that f is ρ-differentiable.
(a) We indicate each ρ -fractional integrable functions by The usual Riemann improper integral is defined by for each ρ ∈ ð0, 1.
Theorem 5 ([13]). Let ρ ∈ ðn, n + 1 and f ðnÞ be a continuous function. Then, for the function f : ½x 1 ,∞Þ → R and for all t > x 1 , we have This property is often called the inverse property.
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][18][19][20][21][22][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 [24]. It has the following formula: where f : E ⊆ R ⟶ R is supposed to be convex on E with x 1 , x 2 ∈ E and x 1 < x 2 .

Journal of Function Spaces
where Theorem 12 ([32], Theorem 4). Let h : , then the following inequality for conformable fractional integral holds: 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.

Main Results
The next result is necessary in the rest of the work. where Proof. Making use of Definition 8, we have
ρ ð½x 1 , x 2 Þ and jf ′ j is a convex function on ½x 1 , x 2 , then we have where Proof. Making use of Lemma 13 and the properties of D ρ , we have Journal of Function Spaces For ρ ∈ ð0, 1 and x > 0, the function x 1+ρ is convex. So, we have Then, by the convexity of j f ′ ðtÞj, we can deduce Analogously, we can deduce Substitutingκ i 's intoΔ, we obtain the desired result.

Journal of Function Spaces
where Proof. From Lemma 13, we have where Journal of Function Spaces making use of the power-mean inequality to get Since j f ′ j q is convex on ½x 1 , x 2 , then for any t ∈ ½0, 1, we So we have Analogously, we can deduce whereA i ðρÞ, B i ðρÞ, andC i ðρÞare as before. Then, by substituting the above inequalities into (37), we get the desired result.
making use of Hölder's inequality to get Since jf ′j q is convex on ½x 1 , x 2 , then for any t ∈ ½0, 1, we have It follows that 10 Journal of Function Spaces