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Original Research ARTICLE

Front. Phys., 28 February 2020 | https://doi.org/10.3389/fphy.2020.00025

New Investigation on the Generalized K-Fractional Integral Operators

  • 1Department of Mathematics, Government College University, Faisalabad, Pakistan
  • 2Faculty of Science and Techniques Moulay Ismail University of Meknes, Errachidia, Morocco
  • 3School of Mathematical Sciences, Zhejiang University, Hangzhou, China
  • 4Department of Mathematics, Lahore College Women University, Lahore, Pakistan
  • 5Department of Mathematics, Huzhou University, Huzhou, China

The main objective of this paper is to develop a novel framework to study a new fractional operator depending on a parameter K > 0, known as the generalized K-fractional integral operator. To ensure appropriate selection and with the discussion of special cases, it is shown that the generalized K-fractional integral operator generates other operators. Meanwhile, we derived notable generalizations of the reverse Minkowski inequality and some associated variants by utilizing generalized K-fractional integrals. Moreover, two novel results correlate with this inequality, and other variants associated with generalized K-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 [18]). 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 [913]). 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 [1618]). 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 K-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 [2330].

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 K-functions and also established Gronwall inequalities involving the generalized Riemann-Liouville and Hadamard K-fractional derivatives with applications. In Dahmani [25], Dahmani explored variants on intervals that are known as generalized (K,s)-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 K 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 (K,s)-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 K-fractional integrals. Taking into account the novel ideas, we provide a new version for reverse Minkowski inequality in the frame of the generalized K-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 K-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 K-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 K-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 Q1(τ) is said to be in Lp, u[0, ∞] space if

Lp,u[0,)={Q1:Q1Lp,u[0,)=(υ1υ2|Q1(η)|pξudη)1p<,1p<,u0}.

For r = 0,

Lp[0,)={Q1:Q1Lp[0,)=(υ1υ2|Q1(η)|pdη)1p<,1p<}.

Definition 2.2. ([38]) “Let Q1L1[0,) 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 χΨp(0,)(1p<) of those real-valued Lebesgue measureable functions Q1 on [0, ∞) for which

Q1χΨp=(0|Q1(η)|pΨ(η)dη)1p<,    1p<

and for the case p = ∞

Q1χΨ=ess    sup0η<[Ψ(η)Q1(η)].

In particular, when Ψ(λ) = λ (1 ≤ p < ∞), the space χΨp(0,) matches with the Lp[0, ∞)-space and, furthermore, if we take Ψ(λ) = ln λ (1 ≤ p < ∞), the space χΨp(0,) concurs with Lp, u[1, ∞)-space.

Now, we present a new fractional operator that is known as the generalized K-fractional integral operator of a function in the sense of another function Ψ.

Definition 2.3. Let Q1χΨq(0,), 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 K-fractional integral operators of a function Q1 in the sense of another function Ψ of order η > 0 are stated as:

( ΨTυ1+,τρ,KQ1)(λ)=1KΓK(ρ)υ1λΨ(η)(Ψ(λ)-Ψ(η))ρK-1Q1(η)dη,    υ1<λ    (2.1)

and

( ΨTυ2-,τρ,KQ1)(λ)=1KΓK(ρ)λυ2Ψ(η)(Ψ(η)-Ψ(λ))ρK-1Q1(η)dη,    λ<υ2,    (2.2)

where ρ ∈ ℂ, ℜ(ρ) > 0, and ΓK(λ)=0ηλ-1e-ηKKdη,    (λ)>0 is the K-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 K=1, it turns into the both sided generalized RL-fractional integral operator [20].

(2) Choosing Ψ(λ) = λ, it turns into the both-sided K-fractional integral operator [21].

(3) Choosing Ψ(λ) = λ along with K=1, it turns into the both-sided RL-fractional integral operators.

(4) Choosing Ψ(λ) = logλ along with K=1, it turns into the both-sided Hadamard fractional integral operators [9, 20].

(5) Choosing Ψ(λ)=λββ,β>0, along with K=1, it turns into the both-sided Katugampola fractional integral operators [17].

(6) Choosing Ψ(λ)=(λ-a)ββ,β>0 along with K=1, it turns into the both-sided conformable fractional integral operators defined by Jarad et al. [2].

(7) Choosing Ψ(λ)=λu+vu+v along with K=1, it turns into the both-sided generalized conformable fractional integrals defined by Khan et al. [40].

Definition 2.4. Let Q1χΨq(0,), 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 K-fractional integral operator of a function Q1 in the sense of another function Ψ of order η > 0 is stated as:

( ΨT0+,λρ,KQ1)(λ)=1KΓK(ρ)0λΨ(η)(Ψ(λ)-Ψ(η))ρK-1Q1(η)dη,    η>0,    (2.3)

where ΓK is the K-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 Q1,Q2 be two positive functions on [0, ∞). If 0<ςQ1(η)Q2(η)Ω,λ[υ1,υ2], then

(υ1υ2Q1s(λ)dλ)1s+(υ1υ2Q2s(λ)dλ)1s1+Ω(ς+2)(ς+1)(Ω+1)(υ1υ2(Q1+Q2)s(λ)dλ)1s.

Theorem 2.6. Set et al. [41] For s ≥ 1, let Q1,Q2 be two positive functions on [0, ∞). If 0<ςQ1(η)Q2(η)Ω,λ[υ1,υ2], then

(υ1υ2Q1s(λ)dλ)2s+(υ1υ2Q2s(λ)dλ)2s((1+Ω)(ς+1)Ω-2)(υ1υ2Q1s(λ)dλ)1s(υ1υ2Q2s(λ)dλ)1s.

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 Q1,Q2 be two positive functions on [0, ∞) such that, for all λ>0,Tυ1+ρQ1s(λ)<,Tυ1+ρQ2s(λ)<. If 0<ςQ1(λ)Q2(λ)Ω,η[υ1,λ], then

(Tυ1+ρQ1s(λ))1s+(Tυ1+ρQ2s(λ))1s1+Ω(ς+2)(ς+1)(Ω+1)(Tυ1+ρ(Q1+Q2)s(λ))1s.

Theorem 2.8. Dahmani [31] For ρ ∈ ℂ, ℜ(ρ) > 0, s ≥ 1, and let Q1,Q2 be two positive functions on [0, ∞) such that, for all λ>0,Tυ1+ρQ1s(λ)<,Tυ1+ρQ2s(λ)<. If 0<ςQ1(λ)Q2(λ)Ω,η[υ1,λ], then

(Tυ1+ρQ1s(λ))2s+(Tυ1+ρQ2s(λ))2s((1+Ω)(ς+2)Ω-2)(Tυ1+ρQ1s(λ))1s(Tυ1+ρQ2s(λ))1s.

3. Reverse Minkowski Inequality via Generalized K-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 K-fractional integrals defined in section (2.4).

Theorem 3.1. For K>0,ρ,(ρ)>0 and s ≥ 1, and let two positive functions Q1,Q2 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 λ>0,ΨT0+,λρ,KQ1s(λ)< and  ΨT0+,λρ,KQ2s(λ)<. If 0<ςQ1(η)Q2(η)Ω for ς, Ω ∈ ℝ+ and for all η ∈ [0, λ], then

( ΨT0+,λρ,KQ1s(λ))1s+( ΨT0+,λρ,KQ2s(λ))1sθ1( ΨT0+,λρ,K(Q1+Q2)s(λ))1s    (3.1)

with θ1=Ω(ς+1)+(Ω+1)(ς+1)(Ω+1).

Proof: Under the given conditions Q1(η)Q2(η)Ω,0ηλ, it can written as

Q1(η)Ω(Q1(η)+Q2(η))-ΩQ1(η),

which implies that

(Ω+1)sQ1s(η)Ωs(Q1(η)+Q2(η))s.    (3.2)

If we multiply both sides of (3.2) by 1KΓK(ρ)Ψ(η)(Ψ(λ)-Ψ(η))ρK-1 and integrate w.r.t η over [0, λ], one obtains

(Ω+1)sKΓK(ρ)0λΨ(η)(Ψ(λ)-Ψ(η))ρK-1Q1s(η)dηΩsKΓK(ρ)0λΨ(η)(Ψ(λ)-Ψ(η))ρK-1(Q1(η)+ Q2(η))sdη.    (3.3)

Accordingly, it can be written as

( ΨT0+,λρ,KQ1s(λ))1sΩΩ+1( ΨT0+,λρ,K(Q1+Q2)s(λ))1s.    (3.4)

In contrast, as ςQ2(λ)Q1(λ),, it follows

(1+1ς)sQ2s(η)(1ς)s(Q1(η)+Q2(η))s.    (3.5)

Again, taking the product of both sides of (3.5) with 1KΓK(ρ)Ψ(η)(Ψ(λ)-Ψ(η))ρK-1 and integrating w.r.t η over [0, λ], we obtain

( ΨT0+,λρ,KQ2s(λ))1s1ς+1( ΨT0+,λρ,K(Q1+Q2)s(λ))1s.    (3.6)

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 K-fractional integral.

Theorem 3.2. For K>0,ρ,(ρ)>0 and s ≥ 1, let two positive functions Q1,Q2 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 λ>0,ΨT0+,λρ,KQ1s(λ)< and  ΨT0+,λρ,KQ2s(λ)<. If 0<ςQ1(η)Q2(η)Ω for ς, Ω ∈ ℝ+ and for all η ∈ [0, λ], then

( ΨT0+,λρ,KQ1s(λ))2s+( ΨT0+,λρ,KQ2s(λ))2sθ2( ΨT0+,λρ,KQ1s(λ))1s( ΨT0+,λρ,KQ2s(λ))1s    (3.7)

with θ2=(ς+1)(Ω+1)Ω-2.

Proof: Multiplying 3.4 and 3.6 results in

(ς+1)(Ω+1)Ω( ΨT0+,λρ,KQ1s(λ))1s( ΨT0+,λρ,KQ2s(λ))1s( ΨT0+,λρ,K(Q1+Q2)s(λ))2s.    (3.8)

Involving the Minkowski inequality, on the right side of (3.8), we get

(ς+1)(Ω+1)Ω( ΨT0+,λρ,KQ1s(λ))1s( ΨT0+,λρ,KQ2s(λ))1s(( ΨT0+,λρ,KQ1s(λ))1s+( ΨT0+,λρ,KQ2s(λ))1s)2.    (3.9)

From 3.9, we conclude that

( ΨT0+,λρ,KQ1s(λ))2s+( ΨT0+,λρ,KQ2s(λ))2s((ς+1)(Ω+1)Ω-2)( ΨT0+,λρ,KQ1s(λ))1s( ΨT0+,λρ,KQ2s(λ))1s.

4. Certain Associated Inequalities via the Generalized K-Fractional Integral Operator

Theorem 4.1. For K>0,ρ,(ρ)>0,s,r1,1s+1r=1 and let two positive functions Q1,Q2 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 λ>0,ΨT0+,τη,KQ1s(λ)< and  ΨT0+,λρ,KQ2s(λ)<. If 0<ςQ1(η)Q2(η)Ω for ς, Ω ∈ ℝ+ and for all η ∈ [0, λ], then

( ΨT0+,λρ,KQ1(λ))1s( ΨT0+,λρ,KQ2(λ))1r(Ως)1sr(( ΨT0+,λρ,KQ11s(λ)Q21r(λ)).    (4.1)

Proof: Under the given condition Q1(η)Q2(η)Ω,0ηλ, it can be expressed as

Q1(η)ΩQ2(η),

which implies that

Q21r(η)Ω-1rQ11r(η).    (4.2)

Taking the product of both sides of (4.2) by Q11s(η), we are able to rewrite it as follows:

Q11s(η)Q21r(η)Ω-1rQ1(η).    (4.3)

Multiplying both sides of (4.3) with 1KΓK(ρ)Ψ(η)(Ψ(λ)-Ψ(η))ρK-1 and integrating w.r.t η over [0, λ], one obtains

Ω-1rKΓK(ρ)0λΨ(η)(Ψ(λ)-Ψ(η))ρK-1Q1(η)dη1KΓK(ρ)0λΨ(η)(Ψ(λ)-Ψ(η))ρK-1Q11s(η)Q21r(η)dη.    (4.4)

As a consequence, we can rewrite as follows

Ω-1sr( ΨT0+,λρ,KQ1(λ))1s( ΨT0+,λρ,KQ11s(λ)Q11r(λ))1s.    (4.5)

Similarly, as ςQ2(η)Q1(η), it follows that

ς1sQ21s(η)Q11s(η).    (4.6)

Again, taking the product of both sides of (4.6) by Q21s(η) and using the relation 1s+1r=1 gives

ς1sQ2(η)Q11s(η)Q21s(η).    (4.7)

If we multiply both sides of (4.7) by 1KΓK(ρ)Ψ(η)(Ψ(λ)-Ψ(η))ρK-1 and integrate w.r.t η over [0, λ], we obtain

ς1sr( ΨT0+,λρ,KQ2(λ))1r( ΨT0+,λρ,KQ11s(λ)Q11r(λ))1r.    (4.8)

Finding the product between (4.5) and (4.8) and using the relation 1s+1r=1, we get the desired inequality (4.1).

Theorem 4.2. For K>0,ρ,(ρ)>0,s,r1,1s+1r=1, and let two positive functions Q1,Q2 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 λ>0,ΨT0+,τη,KQ1s(λ)< and  ΨT0+,λρ,KQ2s(λ)<. If 0<ςQ1(η)Q2(η)Ω for ς, Ω ∈ ℝ+ and for all η ∈ [0, λ], then

( ΨT0+,λρ,KQ1(λ)Q1(λ))θ3( ΨT0+,λρ,K(Q1s+Q2s)(λ))+ θ4 ( ΨT0+,λρ,K(Q1r+Q2r)(λ))    (4.9)

with θ3=2s-1Ωss(Ω+1)s and θ4=2r-1r(ς+1)r.

Proof: Under the assumptions, we have the subsequent identity:

(Ω+1)sQ1s(η)Ωs(Q1+Q2)s(η).    (4.10)

Multiplying both sides of (4.10) by 1KΓK(ρ)Ψ(η)(Ψ(λ)-Ψ(η))ρK-1 and integrating w.r.t η over [0, λ], one obtains

(Ω+1)sKΓK(ρ)0λΨ(η)(Ψ(λ)-Ψ(η))ρK-1Q1s(η)dηΩsKΓK(ρ)0λΨ(η)(Ψ(λ)-Ψ(η))ρK-1(Q1+Q2)s(η)dη.    (4.11)

Accordingly, it can be written as

 ΨT0+,λρ,KQ1s(λ)Ωs(Ω+1)s ΨT0+,λρ,K(Q1+Q2)s(λ).    (4.12)

In contrast, as 0<ςQ1(η)Q2(η),0<η<λ, it follows

(ς+1)rQ2r(η)(Q1+Q2)r(η).    (4.13)

Again, taking the product of both sides of (4.13) with 1KΓK(ρ)Ψ(η)(Ψ(λ)-Ψ(η))ρK-1 and integrating w.r.t η over [0, λ], one obtains

 ΨT0+,λρ,KQ2r(λ)1(ς+1)r ΨT0+,λρ,K(Q1+Q2)r(λ).    (4.14)

Considering Young's inequality,

Q1(η)Q2(η)Q1s(η)s+Q2r(η)r.    (4.15)

If we multiply both sides of (4.15) with 1KΓK(ρ)Ψ(η)(Ψ(λ)-Ψ(η))ρK-1 and integrate w.r.t η over [0, λ], we obtain

 ΨT0+,λρ,K(Q1Q2)(λ) ΨT0+,λρ,KQ1s(λ)s+ ΨT0+,λρ,KQ2r(λ)r.    (4.16)

Invoking (4.12) and (4.14) into (4.16), we obtain

 ΨT0+,λρ,K(Q1Q2)(λ) ΨT0+,λρ,KQ1s(λ)s+ ΨT0+,λρ,KQ2r(λ)rΩs(Ω+1)s ΨT0+,λρ,K(Q1+Q2)s(λ)+1(ς+1)r ΨT0+,λρ,K(Q1+Q2)r(λ).    (4.17)

Using the inequality (μ + ν)z ≤ 2z−1z + νz), z > 1, μ, ν > 0, one obtains

 ΨT0+,λρ,K(Q1+Q2)s(λ)2s-1ΨT0+,λρ,K(Q1s+Q2s)(λ)    (4.18)

and

 ΨT0+,λρ,K(Q1+Q2)r(λ)2r-1ΨT0+,λρ,K(Q1r+Q2r)(λ).    (4.19)

The desired (4.9) can be established from (4.17), (4.18) and (4.19) jointly.

Theorem 4.3. For K>0,ρ,(ρ)>0,s,r1,1s+1r=1 and let two positive functions Q1,Q2 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 λ>0,ΨT0+,τη,KQ1s(λ)< and  ΨT0+,λρ,KQ2s(λ)<. If 0<ζ<ςQ1(η)Q2(η)Ω for ς, Ω ∈ ℝ+ and for all η ∈ [0, λ], then

Ω+1Ω-ζ(𝔍Ψλ(Q1(λ)-Q2(λ)))(𝔍ΨλQ1(λ))1s+(𝔍ΨλQ2(λ))1sς+1ς-ζ(𝔍Ψλ(Q1(λ)-Q2(λ)))1s.    (4.20)

Proof: Using the hypothesis 0 < ζ < ς ≤ Ω, we get

ςζΩζςζ+ςςζ+ΩΩζ+Ω                  (Ω+1)(ς-ζ)(ς+1)(Ω-ζ).

It can be concluded that

Ω+1Ω-ζς+1ς-ζ.

Further, we have that

ς-ζQ1(η)-ζQ2(η)Q2(η)Ω-ζ

implies that

(Q1(η)-ζQ2(η))s(Ω-ζ)sQ2s(η)(Q1(η)-ζQ2(η))s(ς-ζ)s.    (4.21)

Again, we have that

1ΩQ2(η)Q1(η)1ςς-ζζςQ1(η)-ζQ2(η)ζQ1(η)Ω-ζζΩ

implies that

(ΩΩ-ζ)s(Q1(η)-ζQ2(η))sQ1s(η)(ςς-ζ)s(Q1(η)-ζQ2(η))s.    (4.22)

If we multiply both sides of (4.21) with 1KΓK(ρ)Ψ(η)(Ψ(λ)-Ψ(η))ρK-1 and integrate w.r.t η over [0, λ], we obtain

1KΓK(ρ)(Ω-ζ)s0λΨ(η)(Ψ(λ)- Ψ(η))ρK-1(Q1(η)-ζQ2(η))sdη1KΓK(ρ)0λΨ(η)(Ψ(λ)-Ψ(η))ρK-1Q2s(η)dη1KΓK(ρ)(ς-ζ)s0λΨ(η)(Ψ(λ)- Ψ(η))ρK-1(Q1(η)-ζQ2(η))sdη.

Accordingly, it can be written as

1Ω-ζ( ΨT0+,λρ,K(Q1(λ)-ζQ2(λ))s)1s( ΨT0+,λρ,KQ1s(λ))1s1ς-ζ(𝔍Ψλ(Q1(λ)-ζQ2(λ))s)1s.    (4.23)

In a similar way with (4.22), one obtains

ΩΩ-ζ( ΨT0+,λρ,K(Q1(λ)-ζQ2(λ))s)1s( ΨT0+,λρ,KQ1s(λ))1sςς-ζ( ΨT0+,λρ,K(Q1(λ)-ζQ2(λ))s)1s.    (4.24)

The desired inequality (4.20) can be established by adding (4.23) and (4.24).

Theorem 4.4. For K>0,ρ,(ρ)>0,s,r1,1s+1r=1 and let two positive functions Q1,Q2 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 λ>0,ΨT0+,τη,KQ1s(λ)< and  ΨT0+,λρ,KQ2s(λ)<. If 0dQ1(η)D and 0fQ2(η)F for ς, Ω ∈ ℝ+ and for all η ∈ [0, λ], then

( ΨT0+,λρ,KQ1s(λ))1s+( ΨT0+,λρ,KQ2s(λ))1sθ5(𝔍Ψλ(Q1+Q2)s(λ))1s    (4.25)

with θ5=D(d+F)+F(D+f)(D+f)(d+F).

Proof: Under the assumptions, it pursues that

1F1Q2(λ)1f.    (4.26)

Taking the product between (4.26) and 0dQ1(η)D, we have

dFQ1(λ)Q2(λ)Df.    (4.27)

From (4.27), we get

Q2s(η)(Fd+F)s(Q1(η)+Q2(η))s    (4.28)

and

Q1s(η)(Df+D)s(Q1(η)+Q2(η))s.    (4.29)

If we multiply both sides of (4.28) with 1KΓK(ρ)Ψ(η)(Ψ(λ)-Ψ(η))ρK-1 and integrate w.r.t η over [0, λ], we obtain

1KΓK(ρ)0λΨ(η)(Ψ(λ)-Ψ(η))ρK-1Q2s(η)dηFs(d+F)sKΓK(ρ)0λΨ(η)(Ψ(λ)- Ψ(η))ρK-1(Q1(η)+Q2(η))sdη.

Likewise, it can be composed as

( ΨT0+,λρ,KQ2s(λ))1sFd+F( ΨT0+,λρ,K(Q1+Q2)s(λ))1s.    (4.30)

In the same way with (4.29), we have

( ΨT0+,λρ,KQ1s(λ))1sDf+D( ΨT0+,λρ,K(Q1+Q2)s(λ))1s.    (4.31)

The desired inequality (4.25) can be established by adding (4.30) and (4.31).

Theorem 4.5. For K>0,ρ,(ρ)>0,s1, and let two positive functions Q1,Q2 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 λ>0,ΨT0+,τη,KQ1s(λ)< and  ΨT0+,λρ,KQ2s(λ)<. If 0<θ<ςQ1(η)Q2(η)Ω for ς, Ω ∈ ℝ+ and for all η ∈ [0, λ], then

1Ω( ΨT0+,λρ,KQ1(λ)Q2(λ))1(ς+1)(Ω+1)( ΨT0+,λρ,K(Q1+Q2)2(λ))1ς( ΨT0+,λρ,KQ1(λ)Q2(λ)).    (4.32)

Proof: Using 0<ςQ1(η)Q2(η)Ω, it follows that

(ς+1)Q2(η)Q1(η)+Q2(η)Q2(η)(Ω+1).    (4.33)

Also, it follows that 1ΩQ2(η)Q1(η)1ς, which yields

Q1(η)(Ω+1Ω)Q1(η)+Q2(η)Q1(η)(ς+1ς).    (4.34)

Finding the product between (4.33) and (4.34), we have

Q1(η)Q2(η)Ω(Q1(η)+Q2(η))2(ς+1)(Ω+1)Q1(η)Q2(η)ς.    (4.35)

If we multiply both sides of (4.28) with 1KΓK(ρ)Ψ(η)(Ψ(λ)-Ψ(η))ρK-1 and integrate w.r.t η over [0, λ], we obtain

1ΩKΓK(ρ)0λΨ(η)(Ψ(λ)-Ψ(η))ρK-1Q1(η)Q2(η)dηθ61KΓK(ρ)0λΨ(η)(Ψ(λ)-Ψ(η))ρK-1(Q1(η)+ Q2(η))2dη1ςKΓK(ρ)0λΨ(η)(Ψ(λ)-Ψ(η))ρK-1Q1(η)Q2(η)dη

with θ6=1(ς+1)(Ω+1).

Likewise, the required outcome (4.32) can be finished up.

Theorem 4.6. For K>0,ρ,(ρ)>0,s1, and let two positive functions Q1,Q2 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 λ>0,ΨT0+,τη,KQ1s(λ)< and  ΨT0+,λρ,KQ2s(λ)<. If 0<θ<ςQ1(η)Q2(η)Ω for ς, Ω ∈ ℝ+ and for all η ∈ [0, λ], then

( ΨT0+,λρ,KQ1s(λ))1s+( ΨT0+,λρ,KQ2s(λ))1s2( ΨT0+,λρ,KHs(Q1(λ),Q2(λ)))1s,    (4.36)

where H(Q1(η),Q2(η))=max{Ω(Ως+1)Q1(λ)-ΩQ2(λ),(ς+Ω)Q2(λ)-Q1(λ)ς}.

Proof: Under the given conditions 0<ςQ1(η)Q2(η)Ω,0ηλ, can be written as

0<ςΩ+ς-Q1(η)Q2(η),    (4.37)

and

Ω+ς-Q1(η)Q2(η)Ω.    (4.38)

From (4.35) and (4.38), we obtain

Q2(η)<(Ω+ς)Q2(η)-Q1(η)ςH(Q1(η),Q2(η)),    (4.39)

where H(Q1(η),Q2(η))=max{Ω(Ως+1)Q1(λ)-ΩQ2(λ),(ς+Ω)Q2(λ)-Q1(λ)ς}.

From hypothesis, it also follows that 0<1ΩQ2(η)Q1(η)1ς implies that

1Ω1Ω+1ς-Q2(η)Q1(η)    (4.40)

and

1Ω+1ς-Q2(η)Q1(η)1ς.    (4.41)

From (4.40) and (4.41), we obtain

1Ω(1Ω+1ς)Q1(η)-Q2(η)Q1(η)1ς,    (4.42)

which can be composed as

Q1(η)Ω(1Ω+1ς)Q1(η)-ΩQ2(η)            =Ω(Ω+ς)Q1(η)-Ω2ςQ2(η)ςΩ            =(Ως+1)Q1(η)-ΩQ2(η)            Ω[(Ως+1)Q1(η)-ΩQ2(η)]            H(Q1(η),Q2(η)).    (4.43)

We can compose from (4.40) and (4.43)

Q1s(η)Hs(Q1(η),Q2(η)),    (4.44)
Q2s(η)Hs(Q1(η),Q2(η)).    (4.45)

Multiplying both sides of (4.44) by 1KΓK(ρ)Ψ(η)(Ψ(λ)-Ψ(η))ρK-1 and integrating w.r.t η over [0, λ], one obtains

1KΓK(ρ)0λΨ(η)(Ψ(λ)-Ψ(η))ρK-1Q1s(η)dη1KΓK(ρ)0λΨ(η)(Ψ(λ)-Ψ(η))ρK-1Hs(Q1(η),Q2(η))dη.

Likewise, it can be composed as

( ΨT0+,λρ,KQ1s(λ))1s( ΨT0+,λρ,KHs(Q1(λ),Q2(λ)))1s.    (4.46)

Repeating the same procedure as above, for (4.45), we have

( ΨT0+,λρ,KQ2s(λ))1s( ΨT0+,λρ,KHs(Q1(λ),Q2(λ)))1s.    (4.47)

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 K-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.

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).

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.

Acknowledgments

The authors are thankful to the referees for their useful suggestions and comments.

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Keywords: Minkowski inequality, fractional integral inequality, generalized K-fractional integrals, holder inequalitiy, generalized Riemann-Liouville fractional integral, 26D15, 26D10, 90C23

Citation: Rashid S, Hammouch Z, Kalsoom H, Ashraf R and Chu YM (2020) New Investigation on the Generalized K-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.

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

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