Abstract
The main objective of this paper is to develop a novel framework to study a new fractional operator depending on a parameter > 0, known as the generalized -fractional integral operator. To ensure appropriate selection and with the discussion of special cases, it is shown that the generalized -fractional integral operator generates other operators. Meanwhile, we derived notable generalizations of the reverse Minkowski inequality and some associated variants by utilizing generalized -fractional integrals. Moreover, two novel results correlate with this inequality, and other variants associated with generalized -fractional integrals are established. Additionally, this newly defined integral operator has the ability to be utilized for the evaluation of many numerical problems.
1. Introduction
Fractional calculus is truly considered to be a real-world framework, for example, a correspondence framework that comprises extravagant interfacing, has reliant parts that are utilized to achieve a bound-together objective of transmitting and getting signals, and can be portrayed by utilizing complex system models (see [1–8]). This framework is considered to be a mind-boggling system, and the units that create the whole framework are viewed as the hubs of the intricate system. An attractive characteristic of this field is that there are numerous fractional operators, and this permits researchers to choose the most appropriate operator for the sake of modeling the problem under investigation (see [9–13]). Besides, because of its simplicity in application, researchers have been paying greater interest to recently introduced fractional operators without singular kernels [2, 14, 15], after which many articles considering these kinds of fractional operators have been presented. These techniques had been developed by numerous mathematicians with a barely specific formulation, for instance, the Riemann-Liouville (RL), the Weyl, Erdelyi-Kober, Hadamard integrals, and the Liouville and Katugampola fractional operators (see [16–18]). On the other hand, there are numerous approaches to acquiring a generalization of classical fractional integrals. Many authors have introduced new fractional operators generated from general classical local derivatives (see [9, 19, 20]) and the references therein. Other authors have introduced a parameter and enunciated a generalization for fractional integrals on a selected space. These are called generalized -fractional integrals. For such operators, we refer to Mubeen and Habibullah [21] and Singh et al. [22] and the works cited in them. Inspired by these developments, future research can bring revolutionary thinking to provide novelties and produce variants concerning such fractional operators. Fractional integral inequalities are an appropriate device for enhancing the qualitative and quantitative properties of differential equations. There has been a continuous growth of interest in several areas of science: mathematics, physics, engineering, amongst others, and particularly, initial value problems, linear transformation stability, integral-differential equations, and impulse equations [23–30].
The well-known integral inequality, as perceived in Dahmani [31], is referred to as the reverse Minkowski inequality. In Nisar et al. [32, 33], the authors investigated numerous variants of extended gamma and confluent hypergeometric -functions and also established Gronwall inequalities involving the generalized Riemann-Liouville and Hadamard -fractional derivatives with applications. In Dahmani [25], Dahmani explored variants on intervals that are known as generalized -fractional integral operators for positive continuously decreasing functions for a certain family of n(n ∈ ℕ). In Chinchane and Pachpatte [34], the authors obtained Minkowski variants and other associated inequalities by employing Katugampola fractional integral operators. Recently, some generalizations of the reverse Minkowski and associated inequalities have been established via generalized fractional conformable integrals by Mubeen et al. in [35]. Additionally, Hardy-type and reverse Minkowski inequalities are supplied by Bougoffa [36]. Aldhaifallah et al. [37], explored several variants by employing the -fractional integral operator.
In the present paper, the authors introduce a parameter and enunciate a generalization for fractional integrals on a selected space, which we name generalized -fractional integrals. Taking into account the novel ideas, we provide a new version for reverse Minkowski inequality in the frame of the generalized -fractional integral operators and also provide some of its consequences that are advantageous to current research. New outcomes are introduced, and new theorems relating to generalized -fractional integrals are derived that correlate with the earlier results.
The article is composed as follows. In the second section, we demonstrate the notations and primary definitions of our newly described generalized -fractional integrals. Also, we present the results concerning reverse Minkowski inequality. In the third section, we advocate essential consequences such as the reverse Minkowski inequality via the generalized -fractional integral. In the fourth section, we show the associated variants using this fractional integral.
2. Prelude
In this section, we demonstrate some important concepts from fractional calculus that play a major role in proving the results of the present paper. The essential points of interest are exhibited in the monograph by Kilbas et al. [20].
Definition 2.1. ([9, 20]) A function is said to be in Lp, u[0, ∞] space if
For r = 0,
Definition 2.2. ([38]) “Let and Ψ be an increasing and positive monotone function on [0, ∞) and also derivative Ψ′ be continuous on [0, ∞) and Ψ(0) = 0. The space of those real-valued Lebesgue measureable functions on [0, ∞) for which
and for the case p = ∞
In particular, when Ψ(λ) = λ (1 ≤ p < ∞), the space matches with the Lp[0, ∞)-space and, furthermore, if we take Ψ(λ) = ln λ (1 ≤ p < ∞), the space concurs with Lp, u[1, ∞)-space.
Now, we present a new fractional operator that is known as the generalized -fractional integral operator of a function in the sense of another function Ψ.
Definition 2.3. Let and let Ψ be an increasing positive monotone function defined on [0, ∞), containing continuous derivative Ψ′(λ) on [0, ∞) with Ψ(0) = 0. Then, the left- and right-sided generalized -fractional integral operators of a function in the sense of another function Ψ of order η > 0 are stated as:
and
where ρ ∈ ℂ, ℜ(ρ) > 0, and is the -Gamma function introduced by Daiz and Pariguan [39].
Remark 2.1. Several existing fractional operators are just special cases of (2.1) and (2.2).
(1) Choosing , it turns into the both sided generalized RL-fractional integral operator [20].
(2) Choosing Ψ(λ) = λ, it turns into the both-sided -fractional integral operator [21].
(3) Choosing Ψ(λ) = λ along with , it turns into the both-sided RL-fractional integral operators.
(4) Choosing Ψ(λ) = logλ along with , it turns into the both-sided Hadamard fractional integral operators [9, 20].
(5) Choosing along with , it turns into the both-sided Katugampola fractional integral operators [17].
(6) Choosing along with , it turns into the both-sided conformable fractional integral operators defined by Jarad et al. [2].
(7) Choosing along with , it turns into the both-sided generalized conformable fractional integrals defined by Khan et al. [40].
Definition 2.4. Let and let Ψ be an increasing positive monotone function defined on [0, ∞), containing continuous derivative Ψ′(λ) in [0, ∞) with Ψ(0) = 0. Then, the one-sided generalized -fractional integral operator of a function in the sense of another function Ψ of order η > 0 is stated as:
where is the -Gamma function.
In Set et al. [41] proved the Hermite-Hadamard and reverse Minkowski inequalities for an RL-fractional integral. The subsequent consequences concerning the reverse Minkowski inequalities are the motivation of work finished to date concerning the classical integrals.
Theorem 2.5. Set et al. [41] For s ≥ 1, let be two positive functions on [0, ∞). If , then
Theorem 2.6. Set et al. [41] For s ≥ 1, let be two positive functions on [0, ∞). If , then
In Dahmani [31], introduced the subsequent reverse Minkowski inequalities involving the RLFI operators.
Theorem 2.7. Dahmani [31] For ρ ∈ ℂ, ℜ(ρ) > 0, s ≥ 1, and let be two positive functions on [0, ∞) such that, for all If then
Theorem 2.8. Dahmani [31] For ρ ∈ ℂ, ℜ(ρ) > 0, s ≥ 1, and let be two positive functions on [0, ∞) such that, for all If then
3. Reverse Minkowski Inequality via Generalized -Fractional Integrals
Throughout the paper, it is supposed that all functions are integrable in the Riemann sense. Also, this segment incorporates the essential contribution for obtaining the proof of the reverse Minkowski inequality via the newly described generalized -fractional integrals defined in section (2.4).
Theorem 3.1. For and s ≥ 1, and let two positive functions be defined on [0, ∞). Assume that Ψ is an increasing positive monotone function on [0, ∞) having derivative Ψ′ and is continuous on [0, ∞) with Ψ(0) = 0 such that, for all and If for ς, Ω ∈ ℝ+ and for all η ∈ [0, λ], then
with
Proof: Under the given conditions it can written as
which implies that
If we multiply both sides of (3.2) by and integrate w.r.t η over [0, λ], one obtains
Accordingly, it can be written as
In contrast, as , it follows
Again, taking the product of both sides of (3.5) with and integrating w.r.t η over [0, λ], we obtain
The desired inequality (3.1) can be obtained from 3.4 and 3.6.
□
Inequality (3.1) is referred to as the reverse Minkowski inequality related to the generalized -fractional integral.
Theorem 3.2. For and s ≥ 1, let two positive functions be defined on [0, ∞). Assume that Ψ is an increasing positive monotone function on [0, ∞) having derivative Ψ′ and is continuous on [0, ∞) with Ψ(0) = 0 such that, for all and If for ς, Ω ∈ ℝ+ and for all η ∈ [0, λ], then
with
Proof: Multiplying 3.4 and 3.6 results in
Involving the Minkowski inequality, on the right side of (3.8), we get
From 3.9, we conclude that
□
4. Certain Associated Inequalities via the Generalized -Fractional Integral Operator
Theorem 4.1. For and let two positive functions be defined on [0, ∞). Assume that Ψ is an increasing, positive monotone function on [0, ∞) having derivative Ψ′ and is continuous on [0, ∞) with Ψ(0) = 0 such that, for all and If for ς, Ω ∈ ℝ+ and for all η ∈ [0, λ], then
Proof: Under the given condition it can be expressed as
which implies that
Taking the product of both sides of (4.2) by we are able to rewrite it as follows:
Multiplying both sides of (4.3) with and integrating w.r.t η over [0, λ], one obtains
As a consequence, we can rewrite as follows
Similarly, as it follows that
Again, taking the product of both sides of (4.6) by and using the relation gives
If we multiply both sides of (4.7) by and integrate w.r.t η over [0, λ], we obtain
Finding the product between (4.5) and (4.8) and using the relation we get the desired inequality (4.1).
□
Theorem 4.2. For , and let two positive functions be defined on [0, ∞). Assume that Ψ is an increasing, positive monotone function on [0, ∞) having derivative Ψ′ and is continuous on [0, ∞) with Ψ(0) = 0 such that, for all and If for ς, Ω ∈ ℝ+ and for all η ∈ [0, λ], then
with and
Proof: Under the assumptions, we have the subsequent identity:
Multiplying both sides of (4.10) by and integrating w.r.t η over [0, λ], one obtains
Accordingly, it can be written as
In contrast, as it follows
Again, taking the product of both sides of (4.13) with and integrating w.r.t η over [0, λ], one obtains
Considering Young's inequality,
If we multiply both sides of (4.15) with and integrate w.r.t η over [0, λ], we obtain
Invoking (4.12) and (4.14) into (4.16), we obtain
Using the inequality (μ + ν)z ≤ 2z−1(μz + νz), z > 1, μ, ν > 0, one obtains
and
The desired (4.9) can be established from (4.17), (4.18) and (4.19) jointly.
Theorem 4.3. For and let two positive functions be defined on [0, ∞). Assume that Ψ is an increasing positive monotone function on [0, ∞) having derivative Ψ′ and is continuous on [0, ∞) with Ψ(0) = 0 such that, for all and If for ς, Ω ∈ ℝ+ and for all η ∈ [0, λ], then
Proof: Using the hypothesis 0 < ζ < ς ≤ Ω, we get
It can be concluded that
Further, we have that
implies that
Again, we have that
implies that
If we multiply both sides of (4.21) with and integrate w.r.t η over [0, λ], we obtain
Accordingly, it can be written as
In a similar way with (4.22), one obtains
The desired inequality (4.20) can be established by adding (4.23) and (4.24).
□
Theorem 4.4. For and let two positive functions be defined on [0, ∞). Assume that Ψ is an increasing positive monotone function on [0, ∞) having derivative Ψ′ and is continuous on [0, ∞) with Ψ(0) = 0 such that, for all and If and for ς, Ω ∈ ℝ+ and for all η ∈ [0, λ], then
with
Proof: Under the assumptions, it pursues that
Taking the product between (4.26) and we have
From (4.27), we get
and
If we multiply both sides of (4.28) with and integrate w.r.t η over [0, λ], we obtain
Likewise, it can be composed as
In the same way with (4.29), we have
The desired inequality (4.25) can be established by adding (4.30) and (4.31).
□
Theorem 4.5. For and let two positive functions be defined on [0, ∞). Assume that Ψ is an increasing positive monotone function on [0, ∞) having derivative Ψ′ and is continuous on [0, ∞) with Ψ(0) = 0 such that, for all and If for ς, Ω ∈ ℝ+ and for all η ∈ [0, λ], then
Proof: Using it follows that
Also, it follows that which yields
Finding the product between (4.33) and (4.34), we have
If we multiply both sides of (4.28) with and integrate w.r.t η over [0, λ], we obtain
with
Likewise, the required outcome (4.32) can be finished up.
□
Theorem 4.6. For and let two positive functions be defined on [0, ∞). Assume that Ψ is an increasing positive monotone function on [0, ∞) having derivative Ψ′ and is continuous on [0, ∞) with Ψ(0) = 0 such that, for all and If for ς, Ω ∈ ℝ+ and for all η ∈ [0, λ], then
where
Proof: Under the given conditions can be written as
and
From (4.35) and (4.38), we obtain
where
From hypothesis, it also follows that implies that
and
From (4.40) and (4.41), we obtain
which can be composed as
We can compose from (4.40) and (4.43)
Multiplying both sides of (4.44) by and integrating w.r.t η over [0, λ], one obtains
Likewise, it can be composed as
Repeating the same procedure as above, for (4.45), we have
The desired inequality (4.36) is obtained from (4.46) and (4.47).
□
5. Conclusion
This article succinctly expresses the newly defined fractional integral operator. We characterize the strategy of generalized -fractional integral operators for the generalization of reverse Minkowski inequalities. The outcomes presented in section 3 are the generalization of the existing work done by Dahmani [31] for the RL-fractional integral operator. Also, the consequences in section 3 under certain conditions are reduced to the special cases proved in Set al. [41]. The variants built in section 4 are the generalizations of the existing results derived in Sulaiman [42]. Additionally, our consequences will reduce to the classical results established by Sroysang [43]. Our consequences with this new integral operator have the capacities to be used for the assessment of numerous scientific issues as utilizations of the work, which incorporates existence and constancy for the fractional-order differential equations.
Statements
Author contributions
All authors contributed to each part of this work equally, read, and approved the final manuscript.
Funding
This work was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11301127, 11701176, 11626101, 11601485) and the Natural Science Foundation of Huzhou City (Grant No. 2018YZ07).
Acknowledgments
The authors are thankful to the referees for their useful suggestions and comments.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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Summary
Keywords
Minkowski inequality, fractional integral inequality, generalized K-fractional integrals, holder inequalitiy, generalized Riemann-Liouville fractional integral
Citation
Rashid S, Hammouch Z, Kalsoom H, Ashraf R and Chu YM (2020) New Investigation on the Generalized -Fractional Integral Operators. Front. Phys. 8:25. doi: 10.3389/fphy.2020.00025
Received
15 December 2019
Accepted
27 January 2020
Published
28 February 2020
Volume
8 - 2020
Edited by
Devendra Kumar, University of Rajasthan, India
Reviewed by
Haci Mehmet Baskonus, Harran University, Turkey; Muhammad Bilal Riaz, University of the Free State, South Africa; Sushila Rathore, Vivekananda Global University, India
Updates
Copyright
© 2020 Rashid, Hammouch, Kalsoom, Ashraf and Chu.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Yu Ming Chu chuyuming2005@126.com
This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics
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