New Hermite–Hadamard Type Inequalities Involving Non-Conformable Integral Operators

: At present, inequalities have reached an outstanding theoretical and applied development and they are the methodological base of many mathematical processes. In particular, Hermite– Hadamard inequality has received considerable attention. In this paper, we prove some new results related to Hermite–Hadamard inequality via symmetric non-conformable integral operators.


Introduction
The significant role of inequalities in the development and evolution of Mathematics is well known. Some basic notions related to them were already in use by the ancient Greeks, such as triangle and isoperimetric inequalities. However, inequalities were not employed either in arithmetic or any other kind of number manipulation [1]. The formalization of the Mathematical Theory of Inequalities essentially begins in the 18th century with the studies carried out by Gauss. It was continued by Cauchy, and Chebyshov, who had the idea to apply some inequalities to Mathematical Analysis. Later, the Russian mathematician Bunyakovsky, proved in 1859 the well-known Cauchy-Schwarz inequality for the case of infinite dimensions.
Likewise, the research conducted by Hardy on this subject should be recognized as particularly significant, since it went beyond particular inequalities. Hardy succeeded in gathering together the best mathematicians of the moment to solve problems related to inequalities. Furthermore, he founded the Journal of the London Mathematical Society, a magazine especially suitable to publish papers on inequalities. Together with renowned mathematicians such as Littlewood and Polya, he developed the famous volume entitled "Inequalities" [2], which was the first monograph on this subject.
The book became a milestone in the field of inequalities, and it achieved the goal of giving structure, systematization and formalization to an apparently isolated set of results, and, by doing so, it changed them into a theory. At present, inequalities have reached an outstanding theoretical and applied development and they are the methodological base of processes of approximation, estimation, boundedness, interpolation, etc. In general, they are fundamental in every modeling problem.
As usual, a function f : I ⊆ R → R is said to be convex on the interval I, if the inequality holds for all x, y ∈ I and t ∈ [0, 1]. We say that f is concave if − f is convex. It is well known that every convex function is continuous and thus integrable on any compact interval. Among many important inequalities involving convex functions, we will focus here on the following ones. If f : I → R is a convex function on the interval I, then The converse inequalities hold if the function f is concave on the interval I. This seminal result was proved in [3] and it is known as Hermite-Hadamard inequality (see [4,5] for more details). Since its discovery, this inequality has received considerable attention.
The authors in [17] introduced a useful conformable derivative; in addition, a non-conformable derivative is introduced in [18]. These derivatives are interesting from a theoretical viewpoint and useful in many applications [19][20][21].
Next, we give the definition of the non-conformable derivative related to our results.

Definition 1.
Given an interval I ⊆ [0, ∞), a function f : I → R, α ∈ (0, 1) and t ∈ I, the non-conformable derivative of f of order α at t is defined by We say that f is α-differentiable at t if there exists N α 3 ( f )(t) and it is finite.
Note that if f is differentiable at t, then where f (t) denotes the usual derivative.
Following the ideas in [18], we can easily prove the next result.

Definition 2.
Let α ∈ R and a < b. We define the following linear spaces: Motivated by this non-conformable derivative, we define the non-conformable integrals that appear in the inequalities of this paper.
Definition 4. Let α ∈ R and a < b. For each function f ∈ L α,0 [a, b], let us define the fractional integrals The symmetry of these non-conformable integral operators will allow for obtaining new results related to Hermite-Hadamard inequality.

Main Results
We start with an equality that will be useful.
We can write I 0 as follows: Integration by parts gives that the first integral is equal to We obtain, in a similar way, These equalities give the desired result.
Lemma 1 allows for proving several inequalities.
Proof. We have since the integrand is the product of two non-negative functions. Thus, Lemma 1 gives the inequality.
The argument in the proof of Theorem 2 also allows for dealing with the case α ≤ 0.
We deal now with the case α = 0.
Proof. Let us define α = 0. First of all, note that In addition, we have (1 − t) 1−α − t 1−α = 1 − 2t for every t ∈ [0, 1]. Thus, the argument in the proof of Theorem 3 allows for concluding Proof. By Lemma 1, we have Since | f | is convex on [a, b], we have |I 0 | ≤ J 1 + J 2 , with A simple computation gives and we obtain the inequality by adding these expressions of J 1 and J 2 .
Let us state a result relating the three integral operators.
for every t ∈ (a, b), we obtain for every t ∈ (a, b).