# Some Fractional Integral Inequalities Involving Mittag-Kernels

Authors: Xiujun Zhang; Ghulam Farid; Jiraporn Reunsumrit; Ayyaz Ahmad; Thanin Sitthiwirattham
Journal: Journal of Mathematics
 (2022)

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## Abstract

This paper aims to present fractional versions of Minkowski-type integral inequalities via integral operators involving Mittag-Leffler functions in their kernels. Inequalities for various kinds of well-known integral operators can be deduced by selecting specific values of involved parameters. Some particular cases of main results provide connection with the inequalities which have been published in recent years.


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## Body

## 1. Introduction

The fractional calculus deals with the study of differentiation and integration of noninteger order. Fractional integral and derivative operators are the key factors of this subject. It is fair to say that fractional calculus tools and procedures have penetrated practically in almost every subject of modern engineering and science. Assaleh and Ahmed have discussed modeling of speech signals using fractional calculus in [1]. Douglas has discussed some applications of fractional calculus to polymer science in [2]. Fellah and Depollier have discussed applications of fractional calculus to the sound waves propagation in rigid porous materials in [3]. For further applications of fractional calculus in different fields one can see [4].Fractional calculus operators with different kinds of kernels are useful tools for generalizing the classical mathematical inequalities. The kernels involving in other mathematical notions play very vital role in their existence and applications [5,6]. By using fractional integral operators, researchers are continuously working on generalizations of well-known classical integral inequalities. For example, Andrić et al. established Opial-type inequalities for Riemann-Liouville fractional derivatives in [7]. Again, Andrić et al. established composition identities for the Caputo fractional derivatives and gave applications to Opial-type inequalities in [8]. Sarikaya et al. established Hadamard-type inequalities for Riemann-Liouville fractional integrals via convex function in [9,10]. Mehreen and Anwar presented Hadamard- and Fejér-Hadamard-type inequalities for p-convex functions via conformable fractional integrals in [11]. Set and Celik presented fractional Hadamard-type inequalities for quasi-convex function in [12]. Ntouyas et al. established Pólya-Szegö- and Chebyshev-type inequalities for Riemann-Liouville fractional integrals in [13]. Rahman et al. established Minkowski-type inequalities for generalized proportional fractional integral operators in [14,15]. Andrić et al. established Minkowski-type inequalities types inequalities via an extended generalized Mittag-Leffler function in [16]. Fractional derivative operators are also important in the formation mathematical models of real-world problems. The fractional inequalities provide suitable constraints in the presentation of solutions of fractional initial value and fractional boundary value problems. For example, Li et al. have discussed time-fractional nonlinear problems in [17].We aim to develop integral versions of Minkowski-type inequalities. We employ the generalized integral operator in collaboration with the unified Mittag-Leffler function [18,19]. Various fractional integral versions of Minkowski-type inequalities can be deduced from the main results of this paper. The unified Mittag-Leffler function plays a key role in the formation of results of this paper.Gosta Mittag-Leffler introduced an important function (well known as Mittag-Leffler function) [20] as follows:(1)Eαt=∑l=0∞tlΓαl+1,where t,α∈ℂ,ℜα>0 and Γ. is the well-known gamma function.Many scientists have made numerous generalizations after the appearance of Mittag-Leffler function. Wiman generalized this function using two parameters [21] as follows:(2)Eα,βt=∑l=0∞tlΓαl+β,where t,α,β∈ℂandℜα>0.Prabhakar defined Mittag-Leffler function of three parameters [22] as follows:(3)Eα,βγt=∑l=0∞γltll!Γαl+β,where t,α,β,γ∈ℂwithℜα>0andγl=Γγ+l/Γγ is the Pochhammer symbol. Shukla and Parajapati further extended Mittag-Leffler function [23] as follows:(4)Eα,βγ,kt=∑l=0∞γlktll!Γαl+β,where t,α,β,γ∈ℂ,minℜα,ℜβ,ℜγ>0andk∈0,1∪ℕ.Salim and Faraj defined the generalized Mittag-Leffler function [24] as follows:(5)Eα,β,γδ,μ,kt=∑l=0∞δlktlμγlΓαl+β,where t,α,β,δ,μ∈ℂ,minℜα,ℜβ,ℜδ,ℜμ>0andk,γ>0.Rahman et al. further extended the Mittag-Leffler function [25] as follows:(6)Eα,βγ,k,δt;p=∑l=0∞δlktll!Γαl+βBpγ+lk,δ−γBγ,δ−γ,where t,α,β,γ,δ∈ℂ,ℜα,ℜγ,ℜδ>0withp≥0,q>0 and B represents well-known beta function while Bp represents p-beta function defined by(7)Bpm,n=∫01tm−11−tn−1e−p/t1−tdt,where ℜm,ℜn,ℜp>0. Bhatnagar et al. introduced the generalization of Mittag-Leffler function with the name of generalized Q function [26] as follows:(8)Qα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯=∑l=0∞Πi=1nBbi,lλρlθkltlΠi=1nBai,lγδlμνlΓαl+β,where a¯=a1,a2,…,an,b¯=b1,b2,…,bn,α,β,γ,δ,μ,ν,λ,ρ,θ,ai,bi∈ℂ, min {ℜα,ℜβ,ℜγ,ℜθ,ℜλ,ℜδ,ℜρ} >0, k∈0,1∪ℕ.Andrić et al. further defined an extended generalized Mittag-Leffler function [27] as follows:(9)Eα,β,γλ,θ,k,rt;p=∑l=0∞Bpλ+lk,θ−λθlktlBλ,θ−λγlrΓαl+β,where t,α,β,γ,θ,λ∈ℂ,ℜα,ℜβ,ℜγ,ℜθ,ℜλ>0, ℜθ>ℜλ with p≥0, r>0, 0<k≤r+ℜα and θlk=Γθ+lk/Γθ.The integral operators associated to generalizedQ function and extended generalized Mittag-Leffler function are defined as follows.Definition 1 (see [28]).
Letf∈L1u,v. Then, ∀ξ∈u,v, and the fractional integral operators corresponding to (8) are given by(10)IQu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,nfξ;a¯,b¯=∫uξξ−tβ−1Qα,β,γ,δ,μ,νλ,ρ,θ,k,nωξ−tα;a¯,b¯ftdt,(11)IQv−,α,β,γ,δ,μ,νω,λ,ρ,θ,k,nfξ;a¯,b¯=∫ξvt−ξβ−1Qα,β,γ,δ,μ,νλ,ρ,θ,k,nωt−ξα;a¯,b¯ftdt.Definition 2 (see [27]).
Letf∈L1u,v. Then, ∀ξ∈u,v, and the fractional integral operators corresponding to (9) are given by(12)εu+,α,β,γω,λ,θ,k,rfξ;p=∫uξξ−tβ−1Eα,β,γ,λ,θ,k,rωξ−tα;pftdt,(13)εv−,α,β,γω,λ,θ,k,rfξ;p=∫ξvt−ξβ−1Eα,β,γ,λ,θ,k,rωξ−tα;pftdt.Recently, the integral operator based on extended generalized Mittag-Leffler function (9) is defined as follows.Definition 3 (see [29]).
Letω,β,γ,λ,θ∈ℂ,ℜβ,ℜγ>0,ℜθ>ℜλ>0 with p≥0,α,r>0 and 0<k≤r+α. Let ϕ∈L1u,v,0<u<v<∞, be a positive function. Let g:u,v⟶ℝ be a differentiable function, strictly increasing. Also, let ζ/ξ be an increasing function on u,∞ and ξ∈u,v. Then, the unified integral operator is given by(14)Fu+,α,β,γω,λ,θ,k,rgζϕξ;p=∫uξζgξ−gtgξ−gtEα,β,γλ,θ,k,rωgξ−gtα;pg′tϕtdt.Recently, we introduced the unified Mittag-Leffler function and the associated fractional integral operator as follows.Definition 4 (see [18]).
Fora¯=a1,a2,…,an,b¯=b1,b2,…,bn,c¯=c1,c2,…,cn, where ai,bi,ci∈ℂ;i=1,…,n such that ℜai,ℜbi,ℜci>0∀i. Also, let α,β,γ,δ,μ,ν,λ,ρ, θ,t∈ℂ, min ℜα,ℜβ,ℜγ,ℜδ,ℜλ,ℜθ>0, and k∈0,1∪ℕ with p≥0. Let k+ℜρ<ℜδ+ν+α with Imρ=Imδ+ν+α. Then, the unified Mittag-Leffler function is given by(15)Mα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=∑l=0∞Πi=1nBpbi,aiλρlθklΠi=1nBci,aiγδlμνltlΓαl+β.Definition 5 (see [30]).
Letω,a¯=a1,a2,…,an,b¯=b1,b2,…,bn,c¯=c1,c2,…,cn, where ai,bi,ci∈ℂ;i=1,…,n such that ℜai,ℜbi,ℜci>0∀i. Also, let β,γ,μ,λ, θ,t∈ℂ, min ℜβ,ℜγ,ℜμ,ℜλ,ℜθ>0, ρ,δ,ν,α>0, and k∈0,1∪ℕ. Let k+ρ<δ+ν+α with p≥0. Let ϕ∈L1u,v, 0<u<v<∞ be a positive function and let g:u,v⟶ℝ be a differentiable and strictly increasing function. Also let ζ/ξ be an increasing function on u,∞∀ξ∈u,v. Then, the unified integral operator in its generalized form is defined by the following integral:(16)Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζϕξ;p=∫uξζgξ−gtgξ−gtMα,β,γ,δ,μ,νλ,ρ,θ,k,nωgξ−gtα;pg′tϕtdt.We use the Mittag-Leffler function given in (15) and the associated integral operator (16) to establish the extended versions of Minkowski-type integral inequalities in Section 2. Further, we give the reverse Minkowski-type integral inequalities in Section 3. The consequences of main results are also given in the form of remarks and corollaries.


## 2. Integral Versions of Minkowski-Type Inequalities

Theorem 1.
Letω,a¯=a1,a2,…,an,b¯=b1,b2,…,bn,c¯=c1,c2,…,cn, where ai,bi,ci∈ℂ;i=1,…,n such that ℜai,ℜbi,ℜci>0∀i. Also, let β,γ,δ,μ,λ, θ,t∈ℂ, min ℜβ,ℜγ,ℜλ,ℜθ>0, ρ,δ,ν,α>0 and k∈0,1∪ℕ. Let k+ρ<δ+ν+α with w≥0. Let g:u,v⟶ℝ be a differentiable and strictly increasing function and let ϕ,ψ,ζ1,ζ2 be m−power integrable and positive functions on u,v such that the quotient ϕξ/ψξ is bounded above by ζ2ξ and bounded below by ζ1ξ∀ξ∈u,v. For m>1 such that 1/m+1/n=1, the following inequality holds:(17)Ωgζu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,nϕξ;w1/mΩgζu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,nψξ;w1/n≤Ωgζu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,nζ21/nϕ1/mψ1/nξ;w1/mΩgζu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,nζ1−1/mϕ1/mψ1/nξ;w1/n.Proof.
Under the conditions defined onϕ/ψ in the statement of theorem, one can have the following inequalities:(18)ψt≤1ζ11/mtϕ1/mtψ1/nt,ϕt≤ζ2t1/nϕ1/mtψ1/nt.Multiplying both sides of the above inequalities with(19)ζgξ−gtgξ−gtMα,β,γ,δ,μ,νλ,ρ,θ,k,nωgξ−gtα;wg′t,and then integrating on u,ξ with respect to t, we get the following inequalities, respectively:(20)Ωgζu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,nψξ;w≤Ωgζu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,nζ1−1/mϕ1/mψ1/nξ;w,(21)Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζϕξ;w≤Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζϕζ21/nϕ1/mψ1/nξ;w.By taking power 1/n of (20) and 1/m of (21), respectively, one can have(22)Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζψξ;w1/n≤Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζζ1−1/mϕ1/mψ1/nξ;w1/n,(23)Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζϕξ;w1/m≤Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζζ21/nϕ1/mψ1/nξ;w1/m.
The inequalities (22) and (23) produce the inequality (17).Corollary 1.
Under the assumptions of Theorem1, for β>1, the following inequality holds:(24)Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngϕξ;w1/mΩu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngψξ;w1/n≤Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζ21/nϕ1/mψ1/nξ;w1/mΩu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζ1−1/mϕ1/mψ1/nξ;w1/n.Proof.
Letζξ=ξβ. Then, ζ/ξ is increasing for β>1, and the inequality (17) reduces to the inequality (24).Corollary 2.
The following inequality holds for the integral operator given in (14):(25)Fu+,α,β,γω,λ,θ,k,rgζϕξ;w1/mFu+,α,β,γω,λ,θ,k,rgζψξ;w1/n≤Fu+,α,β,γω,λ,θ,k,rgζζ21/nϕ1/mψ1/nξ;w1/mFu+,α,β,γω,λ,θ,k,rgζζ1−1/mϕ1/mψ1/nξ;w1/n.Proof.
By settingn=1, b1=λ+lk, a1=θ−λ, c1=λ, ρ=ν=0, δ>0 in (17), the inequality (25) is obtained.Remark 1.
(i)
By settingζ1ξ=k1, ζ2ξ=k2,∀ξ∈u,v with 0<k1<k2, the inequality (24) reproduces [[30], Corollary 2].(ii)
By setting suitable values to the parameters of unified Mittag-Leffler function, takingg as identity function, then we have [[30], Corollary 3] and [[30], Remark 2].In the proof of next result, we will use the following Young’s inequality; forx,y≥0 with m>1, satisfying m−1+n−1=1(26)xy≤m−1xm+n−1yn.Also, the following inequality will be required:(27)x+ym≤2m−1xm+ym;x,y≥0andm>1.Theorem 2.
Suppose that the assumptions of Theorem1 hold true. Then, the following inequality holds:(28)Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζϕψξ;w≤m−12m−1Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζζ2ζ2+1mϕm+ψmξ;w+n−12n−1Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζ1ζ1+1nϕn+ψnξ;w.Proof.
From the given assumptions, one can have(29)ζ1tψt≤ϕt.
Addingψt on both sides of the above inequality and by simplifying, we get(30)ψt≤ζ1t+1−1ϕt+ψt.
By taking powern, we have(31)ψnt≤ζ1t+1−nϕt+ψtn.
Multiplying both sides by the quantity given in (19) and integrating over u,ξ, the above inequality takes the following form:(32)n−1Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζψnξ;w≤n−1Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζζ1+1−nϕ+ψnξ;w.
Also, on the other hand, we have(33)ϕtζ2t≤ψt.
Addingϕt on both sides and taking power m after rearranging terms, we get(34)ϕmt≤ζ2tζ2t+1mϕt+ψtm.
Multiplying both sides of the above inequality by the quantity given in (19) and integrating over u,ξ, we get(35)m−1Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζϕmξ;w≤m−1Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζζ2ζ2+1mψ+ϕmξ;w.
By Young’s inequality, we have(36)ϕtψt≤m−1ϕmt+n−1ψnt.
Multiplying both sides by the quantity given in (19) and integrating over u,ξ, the above inequality takes the form(37)Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζξ;w≤m−1Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζϕmξ;w+n−1Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζψnξ;w.
Adding (32) and (35) and then using (37), we get the following inequality:(38)Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζϕψξ;w≤m−1Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζζ2ζ2+1mϕ+ψmξ;w+n−1Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζ1ζ1+1nϕ+ψnξ;w.
By using (27) in (38), one can obtain (28).Corollary 3.
Under the assumptions of Corollary1, we have(39)Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngϕψξ;w≤m−12m−1Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζ2ζ2+1mϕm+ψmξ;w+n−12n−1Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ng1ζ1+1nϕn+ψnξ;w.Proof.
Letζξ=ξβ. Then, ζ/ξ is increasing for β>1, and the inequality (28) reduces to the inequality (39).Corollary 4.
The following inequality holds for the integral operator given in (14):(40)Fu+,α,β,γω,λ,θ,k,rgζϕψξ;w≤m−12m−1Fu+,α,β,γω,λ,θ,k,rgζζ2ζ2+1mϕm+ψmξ;w+n−12n−1Fu+,α,β,γω,λ,θ,k,rgζ1ζ1+1nϕn+ψnξ;w.Proof.
By settingn=1, b1=λ+lk, a1=θ−λ, c1=λ, ρ=ν=0, δ>0 in (28), the inequality (39) is obtained.Remark 2.
(i)
By settingζ1ξ=k1, ζ2ξ=k2,∀ξ∈u,v with 0<k1<k2, the inequality (39) reproduces [[30], Corollary 4].(ii)
By setting suitable values to the parameters of unified Mittag-Leffler function, takingg as identity function, we have [[30], Corollary 5] and [[30], Remark 3].Theorem 3.
The following inequalities hold under the assumptions of Theorem1:(41)Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζ1ζ2ϕψξ;w≤Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζ1ζ1+1ζ2+1ϕ+ψ2ξ;w≤Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζ1ζ1ϕψξ;w.Proof.
From the inequality (33), we have the following inequalities:(42)ϕt+ψt≤ζ2t+1ψt,(43)ζ2−1tζ2t+1ϕt≤ϕt+ψt.
On the other hand, we also have(44)ϕt+ψt≥ζ1t+1ψt,(45)ζ1−1tζ1t+1ϕt≥ϕt+ψt.
From (42) and (44), the following inequality holds:(46)ζ1t+1ψt≤ϕt+ψt≤ζ2t+1ψt.From (43) and (45), we get(47)ζ2−1tζ2t+1ϕt≤ϕt+ψt≤ζ1−1tζ1t+1ϕt.Multiplying (46) and (47), we obtain(48)ζ2−1tϕtψt≤1ζ1t+1ζ2t+1ϕt+ψt2≤ζ1−1tϕtψt.Now, multiplying the above inequality by (19) and integrating over u,ξ, we get (41).Corollary 5.
Under the assumptions of Corollary1, we get(49)Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ng1ζ2ϕψξ;w≤Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ng1ζ1+1ζ2+1ϕ+ψ2ξ;w≤Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ng1ζ1ϕψξ;w.Proof.
Letζξ=ξβ. Then, ζ/ξ is increasing for β>1, and the inequality (41) reduces to the inequality (49).Corollary 6.
The following inequality holds for the integral operator given in (14)(50)Fu+,α,β,γω,λ,θ,k,rgζ1ζ2ϕψξ;w≤Fu+,α,β,γω,λ,θ,k,rgζ61ζ1+1ζ2+1ϕ+ψ2ξ;w≤Fu+,α,β,γω,λ,θ,k,rgζ1ζ1ϕψξ;w.Proof.
By settingn=1, b1=λ+lk, a1=θ−λ, c1=λ, ρ=ν=0, δ>0 in (41), the inequality (50) is obtained.Remark 3.
(i)
By settingζ1ξ=k1, ζ2ξ=k2,∀ξ∈u,v with 0<k1<k2, (49) reproduces [[30], Corollary 6].(ii)
By setting suitable values to the parameters of unified Mittag-Leffler function, takingg as identity function, then we have [[30], Corollary 7] and [[30], Remark 4].Theorem 4.
Suppose that the assumptions of Theorem1 are valid and let ϕ,ψ,ζ1,ζ2,f be m−power integrable and positive functions on u,v such that 0<ft<ζ1t≤ϕt/ψt≤ζ2t∀t∈u,v, then the following inequalities hold(51)Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζϕ−fψζ2−fmξ;w1/m+Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζζ2ϕ−fψζ2−fmξ;w1/m≤Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζϕmξ;w1/m+Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζψmξ;w1/m≤Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζϕ−fψζ1−fmξ;w1/m+Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζζ1ϕ−fψζ1−fmξ;w1/m.Proof.
By the given assumptions, we have(52)0<ft<ζ1t≤ϕtψt≤ζ2t,t∈u,v.
The above inequality can be arranged as(53)ζ1t−ft≤ϕt−ftψtψt≤ζ2t−ft.
By taking reciprocal of the above inequality, we get(54)1ζ2t−ft≤ψtϕt−ftψt≤1ζ1t−ft,from which we can write(55)ϕt−ftψtmζ2t−ftm≤ψmt≤ϕt−ftψtmζ1t−ftm.
The following inequality follows by multiplying the above inequality with the quantity given in (19) and integrating over u,ξ:(56)Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζϕ−fψζ2−fmξ;w≤Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζψmξ;w≤Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζϕ−fψζ1−fmξ;w.
Taking power1/m, we get(57)Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζϕ−fψζ2−fmξ;w1/m≤Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζψmξ;w1/m≤Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζϕ−fψζ1−fmξ;w1/m.
Also from (52), we have(58)1ζ2t≤ψtϕt≤1ζ1t.
By multiplying the above inequality by−ft, we get(59)−ftζ1t≤−ftψtϕt≤−ftζ2t.
This gives the following inequality by adding 1 in each terms of the above inequality(60)ζ1t−ftζ1t≤ϕt−ftψtϕt≤ζ2t−ftζ2t.
By taking reciprocal and simplifying, we obtain(61)ζ2tϕt−ftψtζ2t−ft≤ϕt≤ζ1tϕt−ftψtζ1t−ft.
Taking powerm of the above inequality and multiplying by the quantity given in (19), and integrating over u,ξ, the following inequality is obtained:(62)Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζζ2ϕ−fψζ2−fmξ;w≤Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζϕmξ;w≤Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζζ1ϕ−fψζ1−fmξ;w.
Taking power1/m, we have(63)Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζζ2ϕ−fψζ2−fmξ;w1/m≤Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζϕmξ;w1/m≤Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζζ1ϕ−fψζ1−fmξ;w1/m.
Adding (57) and (63), we obtain (50).Corollary 7.
Under the assumptions of Corollary1, the following inequalities hold(64)Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngϕ−fψζ2−fmξ;w1/m+Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζ2ϕ−fψζ2−fmξ;w1/m≤Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngϕmξ;w1/m+Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngψmξ;w1/m≤Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngϕ−fψζ1−fmξ;w1/m+Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζ1ϕ−fψζ1−fmξ;w1/m.Proof.
Letζξ=ξβ. Then, ζ/ξ is increasing for β>1, and the inequality (51) reduces to the inequality (63).Corollary 8.
The following inequality holds for the integral operator given in (14)(65)Fu+,α,β,γω,λ,θ,k,rgζϕ−fψζ2−fmξ;w1/m+Fu+,α,β,γω,λ,θ,k,rgζζ2ϕ−fψζ2−fmξ;w1/m≤Fu+,α,β,γω,λ,θ,k,rgζϕmξ;w1/m+Fu+,α,β,γω,λ,θ,k,rgζψmξ;w1/m≤Fu+,α,β,γω,λ,θ,k,rgζϕ−fψζ1−fmξ;w1/m+Fu+,α,β,γω,λ,θ,k,rgζζ1ϕ−fψζ1−fmξ;w1/m.Proof.
By settingn=1, b1=λ+lk, a1=θ−λ, c1=λ, ρ=ν=0, δ>0 in (51), the inequality (64) is obtained.Remark 4.
(i)
By settingfξ=m, ζ1ξ=k1, ζ2ξ=k2,∀ξ∈u,v with 0<k1<k2, the inequality (66) reproduces [[30], Corollary 8].(ii)
By setting suitable values to the parameters of unified Mittag-Leffler function, takingg as identity function, we have [[30], Corollary 9] and [[30], Remark 5].


## 3. Reverse Minkowski-Type Integral Inequalities

Here, we present certain reverse versions of Minkowski-type inequalities.Theorem 5.
Let the assumptions of Theorem1 hold, then the following inequality holds for m≥1(66)Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζϕmξ;w1/m+Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζψmξ;w1/m≤Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζζ21+ζ2mϕ+ψmξ;w1/m+Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζ11+ζ1mϕ+ψmξ;w1/m.Proof.
From given assumptions, we haveζ1t≤ϕt/ψt from which we have(67)ζ1tψt≤ϕt.
This gives the following inequality by addingψt and simplifying after taking power m(68)ψmt≤11+ζ1tmϕt+ψtm.
Multiplying both sides of the inequality by quantity given in (19) and integrating on u,ξ, the above inequality becomes(69)Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζψmξ;w≤Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζ11+ζ1tmψ+ϕmξ;w.
This takes the following form on taking power1/m(70)Ωgζu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,nψmξ;w1/m≤Ωgζu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,n11+ζ1tmψ+ϕmξ;w1/m.
Also, by the assumptions of the theorem, we have(71)ϕtζ2t≤ψt,from which we can easily obtain(72)ϕt≤ζ2t1+ζ2tψt+ϕt.
This gives the following inequality(73)ϕmt≤ζ2t1+ζ2tmψt+ϕtm.
Multiplying both sides of the above inequality by the quantity given in (19) and integrating over u,ξ, we get(74)Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζϕmξ;w≤Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζζ21+ζ2mψ+ϕmξ;w.
This takes the following form(75)Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζϕmξ;w1/m≤Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζζ21+ζ2mψ+ϕmξ;w1/m.
The required inequality (66) can easily be obtained by adding (70) and (75).Corollary 9.
Suppose that the assumptions of Corollary1 hold true, then we have(76)Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngϕmξ;w1/m+Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngψmξ;w1/m≤Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζ21+ζ2mϕ+ψmξ;w1/m+Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ng11+ζ1mϕ+ψmξ;w1/m.Proof.
Letζξ=ξβ. Then, ζ/ξ is increasing for β>1, and the inequality (66) reduces to the inequality (76).Corollary 10.
The following inequality holds for the integral operator given in (14)(77)Fu+,α,β,γω,λ,θ,k,rgζϕmξ;w1/m+Fu+,α,β,γω,λ,θ,k,rgζψmξ;w1/m≤Fu+,α,β,γω,λ,θ,k,rgζζ21+ζ2mϕ+ψmξ;w1/m+Fu+,α,β,γω,λ,θ,k,rgζ11+ζ1mϕ+ψmξ;w1/m.Proof.
By settingn=1, b1=λ+lk, a1=θ−λ, c1=λ, ρ=ν=0, δ>0 in (66), the inequality (76) is obtained.Remark 5.
(i)
By settingζ1ξ=k1, ζ2ξ=k2,∀ξ∈u,v with 0<k1<k2, the inequality (76) reproduces [[30], Corollary 10].(ii)
By setting suitable values to the parameters of unified Mittag-Leffler function, takingg as identity function, then we have [[30], Corollary 11] and [[30], Remark 6].Theorem 6.
Under the assumptions of Theorem1, for m≥1, we have(78)Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζϕmξ;w2/m+Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζψmξ;w2/m≥Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζ1+ζ2ζ2mϕmξ;w1/mΩu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζ1+ζ1mψmξ;w1/m−2Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζϕmξ;w1/mΩu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζψmξ;w1/m.Proof.
We follow the same method as in previous theorem to get the inequalities (67) and (73). Rearranging (68), we get(79)1+ζ1tmψmt≤ϕt+ψtm.
By rearranging terms of (73), we have(80)1+ζ2tζ2tmϕmt≤ψt+ϕtm.
Multiplying (79) with the quantity given in (19) and integrating over u,ξ, we get the following form(81)Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζ1+ζ1mψmξ;w≤Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζϕ+ψmξ;w.
Taking power1/m, we have(82)Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζ1+ζ1mψmξ;w1/m≤Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζϕ+ψmξ;w1/m.
Also, multiplying (79) with the quantity in (19) and integrating over u,ξ, we obtain(83)Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζ1+ζ2ζ2mϕmξ;w≤Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζψ+ϕmξ;w.
By taking power1/m, we get(84)Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζ1+ζ2ζ2mϕmξ;w1/m≤Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζψ+ϕmξ;w1/m.
Taking product of (82) and (84), the following inequality is obtained(85)Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζ1+ζ1mψmξ;w1/mΩu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζ1+ζ2ζ2mϕmξ;w1/m≤Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngζψ+ϕmξ;w1/m2.
From above inequality, one can easily obtain (78) by the help of Minkowski inequality, and the formula a+b2=a2+2ab+b2.Corollary 11.
Under the assumptions of Corollary1, we obtain(86)Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngϕmξ;w2/m+Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngψmξ;w2/m≥Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ng1+ζ2ζ2mϕmξ;w1/mΩu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ng1+ζ1mψmξ;w1/m−2Ωu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngϕmξ;w1/mΩu+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,ngψmξ;w1/m.Proof.
Letζξ=ξβ. Then, ζ/ξ is increasing for β>1, and the inequality (78) reduces to the inequality (85).Corollary 12.
The following inequality holds for the integral operator given in (14)(87)Fu+,α,β,γω,λ,θ,k,rgζϕmξ;w2/m+Fu+,α,β,γω,λ,θ,k,rgζψmξ;w2/m≥Fu+,α,β,γω,λ,θ,k,rgζ1+ζ2ζ2mϕmξ;w1/mFu+,α,β,γω,λ,θ,k,rgζ1+ζ1mψmξ;w1/m−2Fu+,α,β,γω,λ,θ,k,rgζϕmξ;w1/mFu+,α,β,γω,λ,θ,k,rgζψmξ;w1/m.Proof.
By settingn=1, b1=λ+lk, a1=θ−λ, c1=λ, ρ=ν=0, δ>0 in (78), the inequality (87) is obtained.Remark 6.
(i)
By settingζ1ξ=k1, ζ2ξ=k2,∀ξ∈u,v with 0<k1<k2, the inequality (86) reproduces [[30], Corollary 12].(ii)
By setting suitable values to the parameters of unified Mittag-Leffler function, takingg as identity function, we have [[30], Corollary 13] and [[30], Remark 7].


## 4. Conclusion

The generalized integral versions of Minkowski-type inequalities are established using unified integral operators which are associated with unified Mittag-Leffler function. Integral operators utilized to obtain these inequalities are directly connected with many fractional integral operators defined by different authors in recent decades. The reader can get fractional integral versions of the Minkowski-type inequalities by setting specific values to the parameters involved in Mittag-Leffler function. Some generalized fractional integral versions of Minkowski-type inequalities introduced in our previous work are also deduced. We are interested to utilize the unified Mittag-Leffler function and associated integrals to formulate generalized results regarding classical mathematical inequalities in our future work.



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