Fractional Reverse Coposn’s Inequalities via Conformable Calculus on Time Scales

: This paper provides novel generalizations by considering the generalized conformable fractional integrals for reverse Copson’s type inequalities on time scales. The main results will be proved using a general algebraic inequality, chain rule, Hölder’s inequality, and integration by parts on fractional time scales. Our investigations unify and extend some continuous inequalities and their corresponding discrete analogues. In addition, when α = 1, we obtain some well-known time scale inequalities due to Hardy, Copson, Bennett, and Leindler inequalities.


Introduction
The Hardy discrete inequality is known as (see [1]): where w(l) > 0 for all l ≥ 1.
In [2], Hardy exemplified the continuous version of (1) by utilizing the calculus of variations, which has the form: where g ≥ 0, which is integrable over (0, y), g h is a convergent and integrable function over (0, ∞) and (h/(h − 1)) h is a sharp constant in (1) and (2).
For example, in [25], Saker et al. exemplified the time scale version of a converse of the inequalities (7) and (8), respectively, as follows: Assume that T be a time scale with w ∈ (0, ∞) T . If m ≤ 0 < h < 1, ϑ(ζ) = ∞ ζ k(s)∆s and Ω(ζ) = ζ w k(s)η(s)∆s, then where In the same paper [25], Saker et al. proved the time scale transcript of the Bennet-Leindler inequalities (9) and (10), respectively, as follows: Assume that T is a time scale In recent years, a lot of work has been published on fractional inequalities and the subject has become an active field of research with several authors interested in proving the inequalities of fractional type by using the Riemann-Liouville and Caputo derivative (see [26][27][28]).
On the other hand, the authors in [29,30] introduced a new fractional calculus called the conformable calculus and gave a new definition of the derivative with the base properties of the calculus based on the new definition of derivative and integrals.
The main question that arises now is: Is it possible to prove new fractional inequalities on timescales and give a unified approach of such studies? This in fact needs a new fractional calculus on timescales. Very recently Torres and others, in [31,32], combined a time scale calculus and conformable calculus and obtained the new fractional calculus on timescales. Thus, it is natural to look on new fractional inequalities on timescales and give an affirmative answer to the above question.
In particular, in this paper, we will prove the fractional forms of the classical Hardy, Copson type and its reversed and Leindler inequalities with employing conformable calculus on time scales. The article is structured as follows: Section 2 is an introduction of the basics of fractional calculus on timescales and Section 3 contains the main results.

Results
Here, we will exemplify our main results in this article by utilizing Hölder's inequality, chain rule, and integration by parts for fractional on time scale.
Proof. By utilizing the formula of integration by parts (18) on where Using Ω(w) = 0 and v(∞) = 0 in (21), we see that By utilizing chain rule, we get: Next note D ∆ α ϑ(y) = −x(y) ≤ 0. By the chain rule, we have (note k ≤ 0) This leads to and then, we have Raises (25) to the factor h, we have: By applying Hölder's inequality (19) on the term and This means that by substitution (27) This means that ∞ w x(y) which the wanted inequality (20).

Corollary 1.
If we put α = 1 in Theorem 6, then we get which is (11) in the Introduction.
Theorem 4. Suppose that T be a time scale with w ∈ (0, ∞) T , 0 < h < 1 < k and α ∈ (0, 1]. Assume that ϑ(y) is defined as in Theorem 6 such that: and define Proof. Utilizing the formula of integration by parts (18) on ∞ w x(y) But utilizing chain rule, we obtain: Since D ∆ α Ω(y) = −x(y)η(y) ≤ 0 and d ≥ y, we find that Ω(y) ≥ Ω(d). By substituting (35) into (34) and using that D ∆ α ζ(y) ≥ 0, we get ∞ w x(y) Next note D ∆ α ϑ(y) = −x(y) ≤ 0. By the chain rule, we have (note k ≤ 0) This implies that By substituting (37) into (36) yields Raising (38) to the factor h, we get: The rest of the proof is identical to the proof of Theorem 6 and hence is deleted.

This implies that
Raises (49) to the factor h, we get: The rest of the proof is identical to the proof of Theorem 6 and hence is deleted.

Applications
The applications of quantum calculus play an important role in mathematics and the field of natural sciences, such as physics and chemistry. It has many applications in orthogonal polynomials, number theory, quantum theory, etc. In this section, some example for Reverse Coposn's Inequalities in fractional quantum calculus are selected to fulfil the applicability of the obtained results. Now, we give an example using the time scale T = q N 0 , q > 1 which is a time scale with interesting applications in quantum calculus. Example 1. (Quantum calculus case 1.): Let T = q N 0 = {t : t = q n , n ∈ N 0 , q > 1}. Then for all t ∈ q N 0 , we have Now, with the help of Theorem 3 and the above identities in (66), we can deduce where, For an application of Theorem 4, we give the following example.
Note that. By using theorems 10 and 12, we can apply the technique used in the above examples to obtain different applications. In addition, the above result is important not only for arbitrary time scales, but also for quantum calculus.

Conclusions and Future Work
The new fractional calculus on timescales is presented with applications in new fractional inequalities on timescales like Hardy, Bennett, Copson, and Leindler types. In-equalities are considered in rather general forms and contain several special integral and discrete inequalities. The technique is based on the applications of well-known inequalities and new tools from fractional calculus. In future research, we will continue to generalize more dynamic inequalities by using Specht's ratio, Kantorovich's ratio, functional generalization, and n-tuple fractional diamond-α integral. It will be interesting to find the inequalities in α,β-symmetric quantum and stochastic calculus.
Author Contributions: M.Z. and H.M.R. contributed by preparing the introduction, preliminaries, and formulating (theorem 6, theorem 8) its proof and corollaries (cor.7, cor. 9 and cor.10). M.A. contributed by preparing the introduction, preliminaries, and formulating (theorem 10) its proof and corollaries (cor.11). G.A. contributed by preparing the introduction, preliminaries, and formulating theorem 12, its proof and corollaries (cor.7). C.C. and H.A.A.E.-H. contributed by preparing the introduction, preliminaries, and formulating theorem 6, its proof and corollaries (cor. 13). All authors contributed equally to the writing of the manuscript. All authors have read and agreed to the published version of the manuscript.