New Fractional Dynamic Inequalities via Conformable Delta Derivative on Arbitrary Time Scales

: Building on the work of Josip Peˇcari´c in 2013 and 1982 and on the work of Srivastava in 2017. We prove some new α -conformable dynamic inequalities of Steffensen-type on time scales. In the case when α = 1, we obtain some well-known time scale inequalities due to Steffensen inequalities. For some speciﬁc time scales, we further show some relevant inequalities as special cases: α -conformable integral inequalities and α -conformable discrete inequalities. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities.

Several inequalities such as Hardy's inequality [38,39], Hermite-Hadamard's inequality [40][41][42], Opial's inequality [43,44], and Steffensen's inequality [45] have been introduced. For example, in 2016, Anderson [46] gave an α-conformable version of Steffensen inequality as follows: In 2017, Sarikaya et al. [45] gave a generalization for Theorem 2 as follows: In this article, we explore new generalizations of the integral Steffensen inequality given in [37,47,48] via a conformable integral on a general time-scale measure space. We also retrieve some of the integral inequalities known in the literature as special cases of our tests.

Main Results
Next, we enroll the accompanying suppositions for the verifications of our primary outcomes: ( 3 ) ‫/ג‬ is nonincreasing, and is nonnegative.
is the solution of the equations listed below: Now, we are ready to state and prove our main results.
The proof is complete.
We will need the following lemma to prove the subsequent results.
Lemma 1. Let 1 , 2 , and 5 hold such that Then, and Proof. The suppositions of the Lemma imply that 1 ≤ 1 +ˆ ≤ 2 and 1 ≤ 2 −ˆ ≤ 2 . Now, we prove (5), and we see that Since A combination of (7) and (8) led to the required integral identity (5) asserted by the Lemma. The integral identity (10) can be proved similarly. The proof is completed.
Theorem 9. If 1 , 2 , 6 , 7 , and 14 hold, then Proof. Follow a similar to the proof of the right-hand side inequality in Theorem 7.
Corollary 9. T = R in Corollary 8, we have Proof. Carry out the same proof of the left-hand side inequality in Theorem 7.

Conclusions
In this important work, we discussed some new dynamic inequalities of Steffensentype using delta integral on time scales. By employing the conformable fractional α-integral on time scales, several α-conformable Steffensen-type inequalities on time scales are proved. Our proposed results show the potential for producing some original continuous, discrete, and quantum inequalities. We further presented some relevant inequalities as special cases: discrete inequalities and integral inequalities. These results may be used to obtain more generalized results of several obtained inequalities before. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities.