Abstract

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 is introduced. This strategy usually arises as a description of the exponential functions in their kernels in terms of another function . 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 unrealistic 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.

1. 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 , and Kilbas et al. [8] explored the concept of Riemann–Liouville fractional integrals with respect to another function .

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 in which the variable is a scaled according to proportionality index . This 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 [10–22].

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 [23–33]. 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 [34–42].

The 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 [46–48], many researchers have been focused on their attentions in order to find the distinguished version of the reverse Minkowski inequality for generalized fractional conformable integral, by generalized proportional fractional integral operator and Hadamard fractional integral operators.

The 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 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. The new operators will be referred to as the generalized proportional fractional integral operator with respect to another function . The 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.

2. Preliminaries

This 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]. The subsequent consequences concerning the reverse Minkowski inequalities are the inducement of labor finished to date, concerning the classical integrals.

Theorem 1. (see [49]). Let , , , and and be two positive functions defined on such that , for all . Then, one has

Theorem 2. (see [49]). Let , , , and and be two positive functions defined on such that , for all . Then, the inequalityholds.
In [43], Dahmani used the Riemann–Liouville fractional integral operators to prove the subsequent reverse Minkowski inequalities.

Theorem 3. (see [43]). Let , , , , and and be two positive functions defined on such that and , for all . Then, the inequalityholds if , for all .

Theorem 4. (see [43]). Let , , , , and and be two positive functions defined on such that and , for all . Then, the inequalitytakes place if , for all .
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 introduced by Rashid et al. [9].

Definition 1. (see [9]). Let , , with , and be an increasing and positive monotone function on such that is continuous on and . Then, the left and right generalized proportional fractional integral operators and of the function with respect to the function of order are defined byrespectively, where is the Gamma function [50–52].

Remark 1. Many fractional integral operators are the special cases of (5) and (6). For example,(1)Let . Then, (5) and (6) lead to the left and right generalized proportional fractional integral operators proposed by Jarad et al. [53] as follows:(2)If , then (5) and (6) reduce to the left and right generalized Riemann–Liouville fractional integral operators introduced by Kilbas et al. [8] as follows:(3)Let . Then, (5) and (6) become the left and right generalized proportional Hadamard fractional integral operators [54]:(4)If and . Then, (5) and (6) lead to the left and right Hadamard fractional integral operators [8]:(5)Let and . Then, (5) and (6) become the left and right Riemann–Liouville fractional integral operators:

3. Reverse Minkowski Inequalities via Generalized Proportional Fractional Integral Operator with respect to Another Function

This segment will consist of several generalizations by using generalized nonlocal proportional fractional integral operator with respect to another function to derive reverse Minkowski integral inequalities.

Theorem 5. Let , , , , and be two positive functions defined on such that and , for all , and be an increasing and positive function defined on such that is continuous on and . Then,if , for all .

Proof. It follows from for thatMultiplying both sides of (13) byand integrating with respect to on , we obtainwhich can be written asthat is,On the contrary, from , one haswhich leads toMultiplying both sides of (19) byand integrating with respect to on , we obtainAdding inequalities (17) and (21) yields the desired inequality (12).

Remark 2. If , then Theorem 5 leads to Theorem 3.1 in [47]. If and , then Theorem 5 reduces to inequality (3). If and , then Theorem 5 becomes inequality (1).

Theorem 6. Let , , , , and be two positive functions defined on such that and , for all , and let be an increasing and positive function defined on such that is continuous on and . Then,if , for all .

Proof. Carrying out product between (17) and (21) yieldsApplying the Minkowski inequality to the right-hand side of (23), we obtainIt follows from (23) and (24) thatInequality (25) leads to the conclusion thatwhich complete the proof of Theorem 6.

Remark 3. If , then Theorem 6 leads to Theorem 3.2 of [47]; if and , then Theorem 6 reduces to inequality (4); if and , then Theorem 6 becomes inequality (2).

4. Some Estimates for the Generalized Proportional Fractional Integral Operator with Respect to Another Function

This section is consisted to establishing several associated variants concerning to the generalized proportional fractional integral operator with respect to another function .

Theorem 7. Let , , , with , and be two positive functions defined on such that and for , and be an increasing and positive function defined on such that is continuous on and . Then, one hasif , for all .

Proof. It follows from for thatMultiplying both sides of (28) by leads toMultiplying on both sides of (28) byand integrating with respect to on , we obtainInequality (31) can be written asOn the contrary, leads toMultiplying on both sides of (33) by and using the identity , we haveMultiplying on both sides of (34) byand integrating with respect to on , we obtainInequality (36) leads toFrom (32) and (37), together with , we clearly see thatwhich completes the proof of inequality (27).