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

Authors: Saima Rashid; Fahd Jarad; Yu-Ming Chu
Journal: Mathematical Problems in Engineering
 (2020)

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## 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 functionΦ 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 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.


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## Body

## 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 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 functionΦ in which the variable is a scaled according to proportionality index σ. 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 k−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 functionΦ 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 Φ. 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 υ≥1, ϒ≥γ>0, y>x≥0, and G and H be two positive functions defined on 0,∞ such that γ≤Gz/Hz≤ϒ, for all z∈x,y. Then, one has(1)∫xyGυφdφ1/υ+∫xyHυφdφ1/υ≤1+ϒγ+2ϒ+1γ+1∫xyG+Hυφdφ1/υ.Theorem 2.
(see [49]). Let υ≥1, ϒ≥γ>0, y>x≥0, and G and H be two positive functions defined on 0,∞ such that γ≤Gz/Hz≤ϒ, for all z∈x,y. Then, the inequality(2)∫xyGυφdφ2/υ+∫xyHυφdφ2/υ≥1+γϒ+1ϒ−2∫xyGυφdφ1/υ∫xyHυφdφ1/υ,holds.
In [43], Dahmani used the Riemann–Liouville fractional integral operators to prove the subsequent reverse Minkowski inequalities.Theorem 3.
(see [43]). Let δ>0, υ≥1, y>x≥0, ϒ≥γ>0, and G and H be two positive functions defined on 0,∞ such that Tx+δGυφ<∞ and Tx+δHυφ<∞, for all φ>0. Then, the inequality(3)Tx+δGυφ1/υ+Tx+δHυφ1/υ≤1+ϒγ+2ϒ+1γ+1Tx+δG+Hυφ1/υ,holds if 0<γ≤Gz/Hz≤ϒ, for all z∈x,y.Theorem 4.
(see [43]). Let δ>0, υ≥1, y>x≥0, ϒ≥γ>0, and G and H be two positive functions defined on 0,∞ such that Tx+δGυφ<∞a and Tx+δHυφ<∞, for all φ>0. Then, the inequality(4)Tx+δGυφ2/υ+Tx+δHυφ2/υ≥1+γϒ+1ϒ−2Tx+δGυφ1/υTx+δHυφ1/υ,takes place if 0<γ≤Gz/Hz≤ϒ, for all z∈x,y.
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 functionΦ introduced by Rashid et al. [9].Definition 1.
(see [9]). Let δ>0, σ∈0,1, x,y∈ℝ with x<y, and Φ be an increasing and positive monotone function on x,y such that Φ′ is continuous on x,y and Φ0=0. Then, the left and right generalized proportional fractional integral operators Tx,Φδ,σℱφ and Ty,Φδ,σℱφ of the function ℱ with respect to the function Φ of order δ>0 are defined by(5)Tx,Φδ,σℱφ=1σδΓδ∫xφexpσ−1/σΦφ−Φz1Φ′z1Φφ−Φz11−δℱz1dz1,x<φ,(6)Ty,Φδ,σℱφ=1σδΓδ∫φyexpσ−1/σΦz1−ΦφΦ′z1Φz1−Φφ1−δℱz1dz1,φ<y,respectively, where Γx=∫0∞tx−1e−tdt is the Gamma function [50–52].Remark 1.
Many fractional integral operators are the special cases of (5) and (6). For example,(1)
LetΦφ=φ. Then, (5) and (6) lead to the left and right generalized proportional fractional integral operators proposed by Jarad et al. [53] as follows:(7)Txδ,σℱφ=1σδΓδ∫xφexpσ−1/σφ−z1φ−z11−δℱz1dz1,x<φ,Tyδ,σℱφ=1σδΓδ∫φyexpσ−1/σz1−φz1−φ1−δℱz1dz1,φ<y.(2)
Ifσ=1, then (5) and (6) reduce to the left and right generalized Riemann–Liouville fractional integral operators introduced by Kilbas et al. [8] as follows:(8)Tx,Φδℱφ=1Γδ∫xφΦ′z1ℱz1Φφ−Φz11−δdz1,x<φ,Ty,Φδℱφ=1Γδ∫φyℱz1Φ′z1Φz1−Φφ1−δdz1,φ<y.(3)
LetΦφ=lnφ. Then, (5) and (6) become the left and right generalized proportional Hadamard fractional integral operators [54]:(9)Txδ,σℱφ=1σδΓδ∫xφexpσ−1/σlnφ/z1lnφ/z11−δℱz1z1dz1,x<φ,Tyδ,σφ=1σδΓδ∫φyexpσ−1/σlnz1/φlnz1/φ1−δℱz1z1dz1,φ<y.(4)
IfΦφ=lnφ and σ=1. Then, (5) and (6) lead to the left and right Hadamard fractional integral operators [8]:(10)Txδℱφ=1Γδ∫xφℱz1z1lnφ/z11−δdz1,x<φ,Tyδℱφ=1Γδ∫φyℱz1z1lnz1/φ1−δdz1,φ<y.(5)
LetΦφ=φ and σ=1. Then, (5) and (6) become the left and right Riemann–Liouville fractional integral operators:(11)Txδℱφ=1Γδ∫xφℱz1φ−z11−δdz1,x<φ,Tyδℱφ=1Γδ∫φyℱz1z1−φ1−δdz1,φ<y.


## 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 functionΦ to derive reverse Minkowski integral inequalities.Theorem 5.
Letσ∈0,1, δ>0, υ≥1, ϒ≥γ>0, G and H be two positive functions defined on 0,∞ such that Tx,Φδ,σGυφ<∞ and Tx,Φδ,σHυφ<∞, for all φ>0, and Φ be an increasing and positive function defined on 0,∞ such that Φ′ is continuous on 0,∞ and Φ0=0. Then,(12)Tx,Φδ,σGυφ1/υ+Tx,Φδ,σHυφ1/υ≤1+ϒγ+2γ+1ϒ+1Tx,Φδ,σG+Hυφ1/υ,if 0<γ≤Gz/Hz≤ϒ, for all z∈x,φ⊆0,∞.Proof.
It follows fromGz/Hz≤ϒ for z1∈x,φ that(13)ϒ+1υGυz1≤ϒυG+Hυz1.
Multiplying both sides of (13) by(14)1σδΓδexpσ−1/σΦφ−Φz1Φ′z1Φφ−Φz11−δand integrating with respect to z1 on x,φ, we obtain(15)ϒ+1υσδΓδ∫xφexpσ−1/σΦφ−Φz1Φ′z1Φφ−Φz11−δGυz1dz1≤ϒυσδΓδ∫xφexpσ−1/σΦφ−Φz1Φ′z1Φφ−Φz11−δG+Hυz1dz1,which can be written as(16)Tx,Φδ,σGυφ≤ϒυϒ+1υTx,Φδ,σG+Hυφ,that is,(17)Tx,Φδ,σGυφ1/υ≤ϒΥ+1Tx,Φδ,σG+Hυφ1/υ.
On the contrary, fromγHz1≤Gz1, one has(18)1+1γHz1≤1γGz1+Hz1,which leads to(19)1+1γυHυz1≤1γυGz1+Hz1υ.
Multiplying both sides of (19) by(20)1σδΓδexpσ−1/σΦφ−Φz1Φ′z1Φφ−Φz11−δand integrating with respect to z1 on x,φ, we obtain(21)Tx,Φδ,σHυφ1/υ≤1γ+1Tx,Φδ,σG+Hυφ1/υ.
Adding inequalities (17) and (21) yields the desired inequality (12).Remark 2.
Ifσ=1, then Theorem 5 leads to Theorem 3.1 in [47]. If Φz1=z1 and σ=1, then Theorem 5 reduces to inequality (3). If Φz1=z1 and δ=σ=1, then Theorem 5 becomes inequality (1).Theorem 6.
Letσ∈0,1, δ>0, υ≥1, ϒ≥γ>0, G and H be two positive functions defined on 0,∞ such that Tx,Φδ,σGυφ<∞ and Tx,Φδ,σHυφ<∞, for all φ>0, and let Φ be an increasing and positive function defined on 0,∞ such that Φ′ is continuous on 0,∞ and Φ0=0. Then,(22)Tx,Φδ,σGυφ2/υ+Tx,Φδ,σHυφ2/υ≥ϒ+1γ+1ϒ−2Tx,Φδ,σGυφ1/υTx,Φδ,σHυφ1/υ,if 0<γ≤Gz/Hz≤ϒ, for all z∈x,φ⊆0,∞.Proof.
Carrying out product between (17) and (21) yields(23)ϒ+1γ+1ϒTx,Φδ,σGυφ1/υTx,Φδ,σHυφ1/υ≤Tx,Φδ,σGφ+Hφυ1/υ2.
Applying the Minkowski inequality to the right-hand side of (23), we obtain(24)Tx,Φδ,σGφ+Hφυ1/υ2≤Tx,Φδ,σGυφ1/υ+Tx,Φδ,σHφ1/υ2≤Tx,Φδ,σGυφ2/υ+Tx,Φδ,σHφ2/υ+2Tx,Φδ,σGυφ1/υTx,Φδ,σHφ1/υ.
It follows from (23) and (24) that(25)ϒ+1γ+1ϒ−2Tx,Φδ,σGυφ1/υTx,Φδ,σHυφ1/υ≤Tx,Φδ,σGυφ1/υ+Tx,Φδ,σHυφ1/υ2.
Inequality (25) leads to the conclusion that(26)Tx,Φδ,σGυφ2/υ+Tx,Φδ,σHυφ2/υ≥ϒ+1γ+1ϒ−2Tx,Φδ,σGυφ1/υTx,Φδ,σHυφ1/υ,which complete the proof of Theorem 6.Remark 3.
Ifσ=1, then Theorem 6 leads to Theorem 3.2 of [47]; if Φz1=z1 and σ=1, then Theorem 6 reduces to inequality (4); if Φz1=z1 and δ=σ=1, 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 functionΦ.Theorem 7.
Letσ∈0,1, δ>0, ϒ≥γ>0, υ1,υ2>1 with 1/υ1+1/υ2=1, G and H be two positive functions defined on 0,∞ such that Tx,Φδ,σGφ<∞ and Tx,Φδ,σHφ<∞ for φ>0, and Φ be an increasing and positive function defined on 0,∞ such that Φ′ is continuous on 0,∞ and Φ0=0. Then, one has(27)Tx,Φδ,σGφ1/υ1Tx,Φδ,σHφ1/υ2≤ϒγ1/υ1υ2Tx,Φδ,σG1/υ1φH1/υ2φ,if 0<γ≤Gz/Hz≤ϒ, for all z1∈x,φ⊆0,∞.Proof.
It follows fromGz1/Hz1≤ϒ for z1∈x,φ that(28)Hz11/υ2≥ϒ−1/υ2Gz11/υ2.
Multiplying both sides of (28) by G1/υ1z1 leads to(29)G1/υ1z1H1/υ2z1≥ϒ−1/υ2Gz1.
Multiplying on both sides of (28) by(30)1σδΓδexpσ−1/σΦφ−Φz1Φ′z1Φφ−Φz11−δand integrating with respect to z1 on x,φ, we obtain(31)ϒ−1/υ2σδΓδ∫xφexpσ−1/σΦφ−Φz1Φ′z1Φφ−Φz11−δGz1dz1≤1σδΓδ∫xφexpσ−1/σΦφ−Φz1Φ′z1Φφ−Φz11−δG1/υ1z1H1/υ2z1dz1.
Inequality (31) can be written as(32)ϒ−1/υ1υ2Tx,Φδ,σGφ1/υ1≤Tx,Φδ,σGφ1/υ1Hφ1/υ21/υ1.
On the contrary,γHz1≤Gz1 leads to(33)γ1/υ1H1/υ1z1≤G1/υ1z1.
Multiplying on both sides of (33) by H1/υ2z1 and using the identity υ1−1+υ2−1=1, we have(34)γ1/υ1Hz1≤H1/υ1z1G1/υ2z1.
Multiplying on both sides of (34) by(35)1σδΓδexpσ−1/σΦφ−Φz1Φ′z1Φφ−Φz11−δand integrating with respect to z1 on x,φ, we obtain(36)γ1/υ1σδΓδ∫xφexpσ−1/σΦφ−Φz1Φ′z1Φφ−Φz11−δHz1dz1≤1σδΓδ∫xφexpσ−1/σΦφ−Φz1Φ′z1Φφ−Φz11−δH1/υ1z1G1/υ2z1dz1.
Inequality (36) leads to(37)γ1/υ1υ2Tx,Φδ,σHφ1/υ2≤Tx,Φδ,σH1/υ1φG1/υ2φ1/υ2.
From (32) and (37), together with υ1−1+υ2−1=1, we clearly see that(38)Tx,Φδ,σGφ1/υ1Tx,Φδ,σHφ1/υ2≤ϒγ1/υ1υ2Tx,Φδ,σG1/υ1φH1/υ2φ,which completes the proof of inequality (27).Theorem 8.
Letσ∈0,1, δ>0, ϒ≥γ>0, υ1,υ2>1 with 1/υ1+1/υ2=1, G and H be two positive functions defined on 0,∞ such that Tx,Φδ,σGφ<∞ and Tx,Φδ,σHφ<∞ for φ>0, and Φ be an increasing and positive function defined on 0,∞ such that Φ′ is continuous on 0,∞ and Φ0=0. Then, one has(39)Tx,Φδ,σGφHφ≤2υ1−1ϒυ1υ1ϒ+1υ1Tx,Φδ,σGυ1+Hυ1φ+2υ2−1υ2γ+1υ2Tx,Φδ,σGυ2+Hυ2φ,if 0<γ≤Gz/Hz≤ϒ, for all z1∈x,φ⊆0,∞.Proof.
By the given conditions, we have the following inequality:(40)ϒ+1υ1Gυ1z1≤ϒυ1G+Hυ1z1.
Multiplying both sides of (40) by(41)1σδΓδexpσ−1/σΦφ−Φz1Φ′z1Φφ−Φz11−δand integrating with respect to z1 on x,φ lead to(42)ϒ+1υ1σδΓδ∫xφexpσ−1/σΦφ−Φz1Φ′z1Φ′z1Φφ−Φz11−δGυ1z1dz1≤ϒυ1σδΓδ∫xφexpσ−1/σΦφ−Φz1Φ′z1Φφ−Φz11−δG+Hυ1z1dz1.
Inequality (42) can be written as(43)Tx,Φδ,σGυ1φ≤ϒυ1ϒ+1υ1Tx,Φδ,σG+Hυ1φ.
On the contrary, it follows fromGz1/Hz1>ϒ that(44)γ+1υ2Hυ2z1≤G+Hυ2z1.
Multiplying on both sides of (44) by(45)1σδΓδexpσ−1/σΦφ−Φz1Φ′z1Φφ−Φz11−δand integrating with respect to z1 on x,φ, we obtain(46)Tx,Φδ,σHυ2φ≤1γ+1υ2Tx,Φδ,σG+Hυ2φ.
The well-known Young’s inequality states that(47)Gυ1z1υ1+Hυ2z1υ2≥Gz1Hz1.
Multiplying both sides of (47) with(48)1σδΓδexpσ−1/σΦφ−Φz1Φ′z1Φφ−Φz11−δand integrating with respect to z1 on x,φ give(49)1υ1Tx,Φδ,σGυ1φ+1υ2Tx,Φδ,σHυ2φ≥Tx,Φδ,σGφHφ.
From (43), (46), and (49), we clearly see that(50)Tx,Φδ,σGφHφ≤1υ1Tx,Φδ,σGυ1φ+1υ2Tx,Φδ,σHυ2φ≤ϒυ1υ1ϒ+1υ1Tx,Φδ,σG+Hυ1φ+1υ2γ+1υ2Tx,Φδ,σG+Hυ2φ.
Making use of the inequalitya1+a2q≤2q−1a1q+a2q, for a1,a2>0 and q>1, we can obtain(51)Tx,Φδ,σG+Hυ1φ≤2υ1−1Tx,Φδ,σGυ1+Hυ1φ,(52)Tx,Φδ,σG+Hυ2φ≤2υ2−1Tx,Φδ,σGυ2+Hυ2φ.
Therefore, inequality (39) follows easily from inequalities (50)–(52).Theorem 9.
Letσ∈0,1, δ>0, ϒ≥γ>0, υ≥1, G and H be two positive functions defined on 0,∞ such that Tx,Φδ,σGφ<∞ and Tx,Φδ,σHφ<∞ for φ>0, and Φ be an increasing and positive function defined on 0,∞ such that Φ′ is continuous on 0,∞ and Φ0=0. Then, one has(53)ϒ+1ϒ−ωTx,Φδ,σGφ−ωHφυ1/υ≤Tx,Φδ,σGυφ1/υ+Tx,Φδ,σHυφ1/υ≤γ+1γ−ωTx,Φδ,σGφ−ωHφυ1/υ,if 0<ω<γ≤Gz1/Hz1≤ϒ, for all z1∈x,φ⊆0,∞.Proof.
It follows from0<ω<γ≤Gz1/Hz1≤ϒ that(54)γω≤ϒω,γω+γ≤γω+ϒ≤ϒω+ϒ,ϒ+1γ−ω≤γ+1ϒ−ω,ϒ+1ϒ−ω≤γ+1γ−ω,γ−ω≤Gz1−ωHz1Hz1≤ϒ−ω,Gz1−ωHz1υΥ−ωυ≤Hυz1≤Gz1−ωHz1υγ−ωυ,1ϒ≤Hz1Gz1≤1γ,γ−ωγω≤Gz1−ωHz1ωGz1≤ϒ−ωωϒ,(55)ϒϒ−ωυGz1−ωHz1υ≤Gz1υ≤γγ−ωυGz1−ωHz1υ.
Multiplying both sides of (54) by(56)1σδΓδexpσ−1/σΦφ−Φz1Φ′z1Φφ−Φz11−δand integrating with respect to z1 on x,φ lead to(57)1ϒ−ωυσδΓδ∫xφexpσ−1/σΦφ−Φz1Φ′z1Φφ−Φz11−δGz1−ωHz1υdz1≤1σδΓδ∫xφ∫expσ−1/σΦφ−Φz1Φ′z1Φφ−Φz11−δHυz1dz1≤1γ−ωυσδΓδ∫xφexpσ−1/σΦφ−Φz1Φ′z1Φφ−Φz11−δGz1−ωHz1υdz1.
Inequality (57) can be rewritten as(58)1γ−ωTx,Φδ,σGφ−ωHφυ1/υ≤Tx,Φδ,σHυφ1/υ≤1γ−ωTx,Φδ,σGφ−ωHφυ1/υ.
Again, multiplying both sides of (55) with(59)1σδΓδexpσ−1/σΦφ−Φz1Φ′z1Φφ−Φz11−δand integrating with respect to z1 on x,φ give(60)ϒϒ−ωTx,Φδ,σGφ−ωHφυ1/υ≤Tx,Φδ,σGυφ1/υ≤γγ−ωTx,Φδ,σGφ−ωHφυ1/υ.
Therefore, inequality (53) follows from (58) and (60).Theorem 10.
Letσ∈0,1, δ>0, υ≥1, 0≤κ≤K, 0≤M≤ℳ, G and H be two positive functions defined on 0,∞ such that Tx,Φδ,σGυφ<∞ and Tx,Φδ,σHυφ<∞ for φ>0, and Φ be an increasing and positive function defined on 0,∞ such that Φ′ is continuous on 0,∞ and Φ0=0. Then, the inequality(61)Tx,Φδ,σGυφ1/υ+Tx,Φδ,σHυφ1/υ≤Kκ+ℳ+ℳM+KM+Kκ+ℳTx,Φδ,σGφ+Hφυφ1/υholds if κ≤Gz1≤K and M≤Hz1≤ℳ, for all z1∈x,φ.Proof.
It follows from the conditions given in Theorem10 that(62)1ℳ≤1Hz1≤1M.
Inequality (62) and 0≤κ≤Gz1≤K lead to the conclusion that(63)κℳ≤Gz1Hz1≤KM.
From (63), we clearly see that(64)Hυz1≤ℳκ+ℳυGz1+Hz1υ,(65)Gυz1≤KM+KυGz1+Hz1υ.
Multiplying both sides of (64) and (65) by(66)1σδΓδexpσ−1/σΦφ−Φz1Φ′z1Φφ−Φz11−δand integrating with respect to z1 on x,φ, we obtain(67)Tx,Φδ,σHυφ1/υ≤ℳκ+ℳTx,Φδ,σGz1+Hz1υ1/υ,(68)Tx,Φδ,σGυφ1/υ≤KM+KTx,Φδ,σGz1+Hz1υ1/υ.
Therefore, inequality (61) follows from (67) and (68).Theorem 11.
Letσ∈0,1, δ>0, υ≥1, 0<γ≤ϒ, G and H be two positive functions defined on 0,∞ such that Tx,Φδ,σGυφ<∞ and Tx,Φδ,σHυφ<∞ for all φ>0, and Φ be an increasing and positive function defined on 0,∞ such that Φ′ is continuous on 0,∞ and Φ0=0. Then, the double inequality(69)Tx,Φδ,σGφHφϒ≤Tx,Φδ,σG+H2φγ+1ϒ+1≤Tx,Φδ,σGφHφγholds if 0<γ≤Gz1/Hz1≤ϒ, for all z1∈x,φ.Proof.
It follows from0<γ≤Gz1/Hz1≤ϒ that(70)Hz1γ+1≤Hz1+Gz1≤Hz1ϒ+1,(71)Gz1ϒ+1ϒ≤Hz1+Gz1≤Gz1γ+1γ.
Inequality (70) and (71) lead to(72)Gz1Hz1ϒ≤Gz1+Hz12γ+1ϒ+1≤Gz1Hz1γ.
Multiplying both sides of (72) with(73)1σδΓδexpσ−1/σΦφ−Φz1Φ′z1Φφ−Φz11−δand integrating with respect to z1 on x,φ, we obtain(74)1ϒσδΓδ∫xφexpσ−1/σΦφ−Φz1Φ′z1Φφ−Φz11−δGz1Hz1dz1≤1γ+1ϒ+1σδΓδ∫xφexpσ−1/σΦφ−Φz1Φ′z1Φφ−Φz11−δGz1+Hz12dz1≤1γσδΓδ∫xφexpσ−1/σΦφ−Φz1Φ′z1Φz1−Φz11−δGz1Hz1dz1.
Inequality (74) can be rewritten as(75)Tx,Φδ,σGφHφϒ≤Tx,Φδ,σG+H2φγ+1ϒ+1≤Tx,Φδ,σGφHφγ.Theorem 12.
Letσ∈0,1, δ>0, υ≥1, 0<γ≤Υ, G and H be two positive functions defined on 0,∞ such that Tx,Φδ,σGυφ<∞ and Tx,Φδ,σHυφ<∞ for all φ>0, and Φ be an increasing and positive function defined on 0,∞ such that Φ′ is continuous on 0,∞ and Φ0=0. Then, the inequality(76)Tx,Φδ,σGυφ1/υ+Tx,Φδ,σHυφ1/υ≤2Tx,Φδ,σΔυGφ,Hφ1/υ,holds if γ≤Gz1/Hz1≤ϒ for all z1∈x,φ, where(77)ΔGφ,Hφ=maxϒϒγ+1Gz1−ϒHz1,γ+ϒHz1−Gz1γ.Proof.
It follows fromγ≤Gz/Hz≤ϒ that(78)γ≤ϒ+γ−Gz1Hz1,(79)ϒ+γ−Gz1Hz1≤ϒ.
From (78) and (79), we clearly see that(80)Hz1<ϒ+γHz1−Gz1γ≤ΔGφ,Hφ,where(81)ΔGφ,Hφ=maxϒϒγ+1Gz1−ϒHz1,γ+ϒHz1−Gz1γ.
Similarly, from0<1/ϒ≤Hz1/Gz1≤1/γ, we have(82)1ϒ≤1ϒ+1γ−Hz1Gz1,(83)1ϒ+1γ−Hz1Gz1≤1γ.
Inequalities (82) and (83) lead to(84)1ϒ≤1/ϒ+1/γGz1−Hz1Gz1≤1γ.
It follows that(85)Gz1=ϒ1ϒ+1γGz1−ϒHz1=ϒϒ+γGz1−ϒ2γHz1γϒ=ϒγ+1Gz1−ϒHz1≤ϒϒγ+1Gz1−ϒHz1≤ΔGφ,Hφ.
From (80) and (85), we clearly see that(86)Gυz1≤ΔυGφ,Hφ,(87)Hυz1≤ΔυGφ,Hφ.
Multiplying both sides of (86) with(88)1σδΓδexpσ−1/σΦφ−Φz1Φ′z1Φφ−Φz11−δand integrating with respect to z1 on x,φ, we obtain(89)1σδΓδ∫xφexpσ−1/σΦφ−Φz1Φ′z1Φφ−Φz11−δGυz1dz1≤1σδΓδ∫xφexpσ−1/σΦφ−Φz1Φ′z1Φφ−Φz11−δΔυGφ,Hφdz1.
Inequality (89) can be written as(90)Tx,Φδ,σGυφ1/υ≤Tx,Φδ,σΔυGφ,Hφ1/υ.
Similary, from (87), we obtain(91)Tx,Φδ,σHυφ1/υ≤Tx,Φδ,σΔυGφ,Hφ1/υ.
Therefore, inequality (76) follows easily from (90) and (91).


## 5. Conclusion

In this paper, we introduce a nonlocal generalized proportional fractional integral operator with respect to another functionΦ, and then we derived several variants concerning to the reverse Minkowski inequality by involving the generalized proportional fractional integral operator with respect to another function Φ; as a particular case, the inequality involving fractional integrals in the Riemann–Liouville, Hadamard, and Katugampola sense can be found by choosing appropriate and suitable substitutions in the proportionality index σ and Φ. The 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.



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