New generalized reverse Minkowski and related integral inequalities involving generalized fractional conformable integrals

This paper gives some novel generalizations by considering the generalized conformable fractional integrals operator for reverse Minkowski type and reverse Hölder type inequalities. Furthermore, novel consequences connected with this inequality, together with statements and confirmation of various variants for the advocated generalized conformable fractional integral operator, are elaborated. Moreover, our derived results are provided to show comparisons of convergence between old and modified operators towards a function under different parameters and conditions. The numerical approximations of our consequence have several utilities in applied sciences and fractional integro-differential equations.

conformable derivative. The exponential and Mittag-Leffler functions are used as kernels by several researchers for developing new fractional techniques. In [14], Khan  Conformable derivatives are nonlocal fractional derivatives. They can be called fractional since we can take derivatives up to arbitrary order. However, since in the community of fractional calculus, nonlocal fractional derivatives only are to be called fractional, we prefer to replace conformable fractional by conformable (as a type of local fractional).
The article is composed thus: in Sect. 2 we demonstrate the notations and primary definitions of our newly introduced operator generalized conformable fractional integrals.
Also, we present the results concerning the reverse Minkowski inequality. In Sect. 3, we advocate essential consequences such as the reverse Minkowski inequality via the generalized conformable fractional integral operators. In Sect. 4, we show the associated variants using this fractional integral.

Preliminaries
This section is dedicated to some recognized definitions and results associated with the generalized conformable fractional integral operators and their generalization related to the generalized conformable fractional integral operators. Set et al. in [60] proved the Hermite-Hadamard, and reverse Minkowski inequalities for Riemann-Liouville fractional integrals. Additionally, Hardy's type and reverse Minkowski inequalities were supplied by Bougoffa in [38]. The subsequent consequences concerning the reverse Minkowski inequalities are of significance for the classical integrals.
In [44], Dahmani used the Riemann-Liouville fractional integral operators to prove the subsequent reverse Minkowski inequalities.  1 , y], then the following inequality holds:  1 , y], then the following inequality holds: Recall the definition of the generalized conformable fractional integral which is mainly due to [14].

Reverse Minkowski inequalities via generalized conformable fractional integral operators
This section comprises our principal involvement of establishing the proof of the reverse Minkowski inequalities via generalized conformable fractional integral operators defined in (2.1) and (2.2) and an associated theorem insinuated as the reverse Minkowski inequalities.
Proof By the suppositions mentioned in Theorem 3.1, If we multiply both sides of (3.2) with 1 Γ (ς )ς 1-τ -( y τ + -η τ + τ + ) τ -1 and then integrate the subsequent inequality with respect to η from r 1 to y, we obtain Similarly, In contrast, as mf 2 (η) ≤ f 1 (η), it follows that Again, if we multiply both sides of (3.5) with ) τ -1 and then integrate the subsequent inequality with respect to η from r 1 to y, we obtain Thus adding (3.4) and (3.6) yields the desired inequality.

Certain associated inequalities via generalized conformable fractional integral operators (GCFI)
This section is dedicated to deriving certain associated variants regarding GCFI operator.