A Note on Reverse Minkowski Inequality via Generalized Proportional Fractional Integral Operator with respect to Another Function

This study reveals new fractional behavior of Minkowski inequality and several other related generalizations in the frame of the newly proposed fractional operators. For this, an eﬃcient technique called generalized proportional fractional integral operator with respect to another function Φ is introduced. This strategy usually arises as a description of the exponential functions in their kernels in terms of another function Φ . The prime purpose of this study is to provide a new fractional technique, which need not use small parameters for ﬁnding the approximate solution of fractional coupled systems and eliminate linearization and un-realistic factors. Numerical results represent that the proposed technique is eﬃcient, reliable, and easy to use for a large variety of physical systems. This study shows that a more general proportional fractional operator is very accurate and eﬀective for analysis of the nonlinear behavior of boundary value problems. This study also states that our ﬁndings are more convenient and eﬃcient than other available results.


Introduction
Recently, the idea of nonlocal operators of differentiation has boarded out numerous analysts from practically all parts of sciences and engineering due to their abilities to include progressively complex characteristics into numerical conditions. Fractional calculus has also been comprehensively utilized in several instances, but the concept has been popularized and implemented in numerous disciplines of science, technology, and engineering as a mathematical model [1,2]. Numerous distinguished generalized fractional integral operators consist of the Hadamard operator, Erdélyi-Kober operators, the Saigo operator, the Gaussian hypergeometric operator, the Marichev-Saigo-Maeda fractional integral operators, and so on, out of the which, the Riemann-Liouville fractional integral operator has been extensively utilized by researchers in theory as well as applications. For added information related to fractional calculus operators and their usefulness, one may also communicate to the expositions by Miller and Ross [3], Samko et al. [4], Kiryakova [5], and Baleanu et al. [6]. Almeida [7] proposed a new fractional derivative called Caputo derivative with respect to another function Φ, and Kilbas et al. [8] explored the concept of Riemann-Liouville fractional integrals with respect to another function Φ.
Within the structure of applied science and mathematical modeling, there exists an outstanding kind of operator known as generalized proportional fractional integral operator with respect to another function Φ in which the variable is a scaled according to proportionality index σ. is diversified operator was introduced by Rashid et al. [9], to conceivably role those physical problems for which classical physical law, for example, the well-known Mellin transform, Fourier transform, and probability theory, is suitable; such physical issue is accepted to be founded on the fractional calculus and pertinent to the media of nonintegral fractional operators. Amongst others, we estimate real-world issues such as Porous media, aquifer, and turbulence; furthermore, progressively, other media regularly show fractional properties [10][11][12][13][14][15][16][17][18][19][20][21][22].
During the most recent decade, integral inequalities have been expanding enthusiasm to employ fractional techniques that have capacious significance to many fields, including neural networks, remote sensing, optimization of structures, optimization of electromagnetic systems, and many other applied sciences [23][24][25][26][27][28][29][30][31][32][33]. Lately, much consideration has been given to the fractional calculus of integral inequalities. We comment that fractional calculus is imperative for a few reasons. We contemplate the subjective conduct of the solution of the integral-differential and difference equations when the given operator and the feasible variations occur in a parameter. Several integral inequalities and their modifications have been derived via the classical fractional operators [34][35][36][37][38][39][40][41][42]. e first fractional technique was employed on reverse Minkowski inequality in [43]. Lately, Anber et al. [44] proposed some fractional integral inequalities within the scope of Riemann-Liouville fractional integral. In [45], the researchers explored some Minkowski inequalities and other variants by contemplating Katugampola's fractional techniques. In [46][47][48], many researchers have been focused on their attentions in order to find the distinguished version of the reverse Minkowski inequality for generalized k−fractional conformable integral, by generalized proportional fractional integral operator and Hadamard fractional integral operators. e aim to deal with new operators of integration has been introduced in this paper comprising exponential functions in their kernels in terms of another function Φ and generalized some well-known fractional operators as generalized proportional fractional integral operator, Riemann-Liouville fractional integral operator, Katugampola fractional integrals, and Hadamard fractional integral operators. e new operators will be referred to as the generalized proportional fractional integral operator with respect to another function Φ. e new operators are expected to fascinate the reverse Minkowski inequality and other associated integral inequalities in the light of a generalized proportional fractional integral operator. Moreover, the numerical approximation of these new operators are additionally given a few utilities to a real-world problem.

Preliminaries
is segment is dedicated to some recognized definitions and outcomes associated with the generalized conformable fractional integral operators and their generalization related to the generalized conformable fractional integral operators. Set et al.,in [49], launched the fractional version of the Hermite-Hadamard and reverse Minkowski inequality. Additionally, Hardy's type and reverse Minkowski inequalities were supplied by Bougoffa in [36]. e subsequent consequences concerning the reverse Minkowski inequalities are the inducement of labor finished to date, concerning the classical integrals.
In [43], Dahmani used the Riemann-Liouville fractional integral operators to prove the subsequent reverse Minkowski inequalities.
Now, we present a new nonlocal fractional operator which is known as the generalized proportional fractional integral operator of a function with respect to another function Φ introduced by Rashid et al. [9].

Reverse Minkowski Inequalities via Generalized Proportional Fractional Integral Operator with respect to Another Function
is segment will consist of several generalizations by using generalized nonlocal proportional fractional integral operator with respect to another function Φ to derive reverse Minkowski integral inequalities.

Some Estimates for the Generalized Proportional Fractional Integral Operator with Respect to Another Function
is section is consisted to establishing several associated variants concerning to the generalized proportional fractional integral operator with respect to another function Φ.

Mathematical Problems in Engineering
Multiplying both sides of (54) by (56) and integrating with respect to z 1 on (x, φ) lead to Inequality (57) can be rewritten as Riemann-Liouville, Hadamard, and Katugampola sense can be found by choosing appropriate and suitable substitutions in the proportionality index σ and Φ. e variants obtained in this research will lead to the inequalities which are established earlier by Rahman et al. [47] and numerous outcomes can be generalized for the application of these newly introduced fractional integral operators by utilizing Remark 1. Note that the outcomes in this paper are like hypothetically surely understood proliferation properties of fractional Schrödinger equation [55,56]. Besides, our outcomes are practically identical to equality-time evenness in a fractional Schrödinger equation [57]. Indeed, the work established in the given arrangement is new and contributes suggestively to the study of integrodifferential and difference equations.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.