# Further on Inequalities forα,h−m-Convex Functions via k-Fractional Integral Operators

Authors: Tao Yan; Ghulam Farid; Ayşe Kübra Demirel; Kamsing Nonlaopon
Journal: Journal of Mathematics
 (2022)

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## Abstract

The purpose of this article is to demonstrate new generalizedk-fractional Hadamard and Fejér–Hadamard integral inequalities for α,h−m-convex functions. To prove these inequalities, k-fractional integral operators including the generalization of the Mittag–Leffler function are used. The results presented in this article can be considered an important advancement of previously published inequalities.


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## Body

## 1. Introduction

Theory of convexity offers an effective and charming area of research and is also a theory that featured prominently and surprisingly in distinct disciplines such as mathematical analysis, optimization, economics, finance, engineering and game theory. Convexity theory is very closely related with the theory of inequalities. Many inequalities well known in the literature are direct applications of the properties of convex functions. The usage of fractional integral operators for getting the generalized types of classic inequalities has become an important method in advanced mathematical studies of inequalities.One of the convexity theory studies in the literature belongs to Gao et al. [1]. They presented a new type of functions called n-polynomial harmonically exponential type convex, and specified some of their algebraic features. Mehrez and Agarwal [2] established new type of integral inequalities for convex functions and indicated new inequalities for some special and q-special functions. Tariq [3] defined the concept of p–harmonic exponential type convex functions. Also, they investigated some integral inequalities in the form of applications for some means. Another study on convexity theory and inequalities was presented by Butt et al. [4]. They presented the notion of m–polynomial p-harmonic exponential type convex functions and demonstrated various new integral inequalities. Srivastava et al. [5] obtained a new class of the bi-close-to-convex functions described in the open unit disk by using the Borel distribution series of the Mittag–Leffler type. Also, the authors demonstrated the Fekete–Szego type inequalities via the bi-close-to-convex function class.Fractional calculus, which is the study of integrals and derivatives of fractional order, has expanded significantly over the late nineteenth century. It ranges from chemical, viscoelasticity, and statistical physics to electrical and mechanical engineering. The fundamental working doctrine of fractional analysis is to present new fractional derivative and integral operators, and to analyze the benefits of these operators through the instrument of modeling studies, and collations. Integral operators, which form a significant part of fractional calculus, are now resources of many fields such as inequality theory, engineering, statistics, mathematical biology, and modeling, which take advantage of fractional analysis. Many inequalities have been generalized through the instrument of fractional integral operators and provide construction of new approximations.One of the fractional calculus studies in the literature belongs to Abdeljawad et al. [6]. They obtained generalized Hermite–Hadamard type inequalities and generalized Simpson type inequalities for s,m-convex functions with the help of local fractional integration. Akdemir et al. [7] used generalized fractional integral operators. By using these operators, they proved new and general variants of Chebyshev’s inequality. Butt et al. [8] established a general integral identity to acquire new integral inequalities of several Hadamard types. For this purpose, they used a new version of the Atangana–Baleanu integral operator. Khan et al. [9] explored two fractional integral operators related to Fox H-function owing to Saxena and Kumbhat. They proved series expansion of the images of the M-series with the help of these fractional operators. Another study to k-fractional integrals was presented by Qi et al. [10]. They constructed some generalized fractional integral inequalities of the Hermite–Hadamard type via α,m-convex functions. Also, they demonstrated that one can get and expand some Riemann–Liouville fractional integral inequalities and classical integral inequalities of Hermite–Hadamard’s type. Tunc et al. [11] presented the generalized k-fractional integrals of a function with respect to the another function that generalizes many several types of fractional integrals. Also, they studied trapezoid inequalities for the functions whose derivatives in absolute value are convex. Önalan et al. [12] proved many Hermite–Hadamard type integral inequalities for functions whose absolute values of the second derivatives are s-convex and s-concave using fractional integral operators with the Mittag–Leffler kernel. Zhu et al. [13] explored a weighted integral identity of Simpson-like type. Relying on this identity, they obtained some estimation-type results connected with the weighted Simpson-like type integral inequalities for the first order differentiable functions. Srivastava et al. [14] established the homogeneous q-shift operator and the homogeneous q-difference operator. Based on these operators, they searched generalized Cauchy and Hahn polynomials.


## 2. Preliminaries

Now let us define some important functions.Definition 1 (see [15]).
A functionφ:a,b⟶ℝis called a convex function, if(1)φηι+1−ηκ≤ηφι+1−ηφκ
holds for allι,κ∈a,b and η∈0,1.Definition 2 (see [16]).
The functionφ:0,b⟶ℝ,b>0, is called theα,m-convex function, if(2)φηι+m1−ηκ≤ηαφι+m1−ηαφκ
holds for allι,κ∈0,b,η∈0,1 and α,m∈0,12.Definition 3 (see [17]).
A functionφ:0,b⟶ℝis said to bes,m-convex, if(3)φηι+m1−ηκ≤ηsφι+m1−ηsφκ
holds for allι,κ∈0,b, η∈0,1 and s,m∈0,1.Definition 4 (see [18]).
A functionφ:0,b⟶ℝis said to bes,m-convex in the second sense, if(4)φηι+m1−ηκ≤ηsφι+m1−ηsφκ
holds for allι,κ∈0,b , η∈0,1 and (s,m)∈0,12.Definition 5 (see [19]).
LetJ⊆ℝbe an interval including0,1and leth:J⟶ℝbe a nonnegative function. Then the functionφ:0,b⟶ℝis called theh−m-convex function, if(5)φηι+m1−ηκ≤hηφι+mh1−ηφκ
holds for allι,κ∈0,b, η∈0,1 and m∈0,1.Definition 6 (see [20]).
LetJ⊆ℝbe an interval including0,1and leth:J⟶ℝbe a nonnegative function. Then the functionφ:0,b⟶ℝis called theα,h−m-convex function, if(6)φηι+m1−ηκ≤hηαφι+mh1−ηαφκ
holds for allι,κ∈0,b, η∈0,1 and α,m∈0,12.Remark 1.
(i)
By takingm=α=1andhη=ηin (6), we obtain the definition of convex function (1).(ii)
By takinghη=ηin (6), we obtain the definition ofα,m-convex function (2).(iii)
By takinghη=ηandα=sin (6), we obtain the definition ofs,m-convex function (3).(iv)
By takinghη=ηsandα=1in (6), we obtain the definition ofs,m-convex function in the second sense (4).(v)
By takingα=1in(6), we obtain the definition ofh,m-convex function (5).(vi)
By takingα=m=hη=1in (6), we obtain the definition ofp-function described by Dragomir et al. in [21].Now let us represent some definitions of fractional integral operators that will form the basis for this article.Definition 7 (see [22]).
Letγ,c,w,α,l,∈ℂ,ℜl,ℜα>0,ℜc>ℜγ>0withμ,δ>0,p˜≥0, and0<v≤δ+μ. Letφ∈L1a,b,ι∈a,b. In that case, the generalized fractional operators are defined by(7)Fμ,α,l,w,a+γ,δ,v,cφι;p˜=∫aιι−ηα−1Eμ,α,lγ,δ,v,cwι−ημ;p˜φηdη,Fμ,α,l,w,b−γ,δ,v,cφι;p˜=∫ιbη−ια−1Eμ,α,lγ,δ,v,cwη−ιμ;p˜φηdη,where(8)Eμ,α,lγ,δ,v,cη;p˜=∑n=0∞βp˜γ+nv,c−γβγ,c−γcnvΓμn+αηnlnδ
is generalized extended Mittag–Leffler function, andβp˜ is the expansion of beta function described as below:(9)βp˜ι,κ=∫01ηι−11−ηκ−1e−p˜/η1−ηdη,where ℜι,ℜκ,ℜp˜>0.Definition 8 (see [23]).
Letφ,ψ:a,b⟶ℝ with 0<a<b, be the functions, φ be positive, φ∈L1a,b and ψ be differentiable and strictly increasing. Let ϕ/ι be an increasing on a,∞, γ,c,w,α,l∈ℂ,ℜl,ℜα>0,ℜc>ℜγ>0 with μ,δ>0,p˜≥0, and 0<v≤μ+δ. In that case, for ι∈a,b, the fractional operators are described by(10)ψFμ,α,l,w,a+ϕ,γ,δ,v,cφι;p˜=∫aιϕψι−ψηψι−ψηEμ,α,lγ,δ,v,cwψι−ψημ;p˜ψ′ηφηdη,ψFμ,α,l,w,b−ϕ,γ,δ,v,cφι;p˜=∫ιbϕψη−ψιψη−ψιEμ,α,lγ,δ,v,cwψη−ψιμ;p˜ψ′ηφηdη.Definition 9 (see [23]).
Letφ,ψ:a,b⟶ℝ with 0<a<b, be the functions such that φ be positive and φ∈L1a,b and ψ be differentiable and strictly increasing. Let γ,c,w,α,l,∈ℂ,ℜl,ℜα>0,ℜc>ℜγ>0, μ,δ>0,p˜≥0, and 0<v≤μ+δ. In that case, for ι∈a,b, the united operators are described by(11)ψFμ,α,l,w,a+γ,δ,v,cφι;p˜=∫aι(ψι−ψη)α−1Eμ,α,lγ,δ,v,cwψι−ψημ;p˜ψ′ηφηdη,ψFμ,α,l,w,b−ϕ,γ,δ,v,cφι;p˜=∫ιb(ψη−ψι)α−1Eμ,α,lγ,δ,v,cwψη−ψιμ;p˜ψ′ηφηdη.Recently, Yue et al. defined generalizedk-fractional operators including a further extension of Mittag–Leffler function in [24] as noted below:Definition 10.
Letφ,ψ:a,b⟶ℝwith0<a<b; be the functions such thatφbe positive andφ∈L1a,bandψbe differentiable and strictly increasing. Letγ,c,w,α,l,∈ℝandα>k,l,α>0,c>γ>0with0<v≤δ+μ,p˜≥0andμ,δ>0. In that case, forι∈a,b, the right-left generalizedk-fractional operatorsψkFμ,α,l,w,a+γ,δ,v,cφandψkFμ,α,l,w,b−γ,δ,v,cφare defined by(12)ψkFμ,α,l,w,a+γ,δ,v,cφι;p˜=∫aι(ψι−ψη)α/k−1Eμ,α,lγ,δ,v,cwψι−ψημ;p˜ψ′ηφηdη,(13)gkFμ,α,l,w,b−γ,δ,v,cφι;p˜=∫ιb(ψη−ψι)α/k−1Eμ,α,lγ,δ,v,cwψη−ψιμ;p˜ψ′ηφηdη.The following inequality is the admitted Hadamard inequality.Theorem 1.
Letφ:a,b⟶ℝwitha<b, be a convex function. In that case, the below inequality occurs:(14)φa+b2≤1b−a∫abφιdι≤φa+φb2.Theorem 2.
Letφ:a,b⟶ℝbe a convex andψ:a,b⟶ℝbe nonnegative and symmetric in respect ofa+b/2and integrable. In that case, the below inequality occurs:(15)φa+b2∫abψιdι≤∫abφιψιdι≤φa+φb2∫abψιdι.This inequality in [25] presented by Fejér is known as a weighted type of Hadamard’s inequality.Many authors have been established several refinements and extensions of the Hadamard and the Fejér–Hadamard inequalities for various fractional integral operators (for details see, [2, 7, 11, 16, 17, 19–21, 26–34] and references therein). This article aims to derive the Hadamard and Fejér–Hadamard inequalities about generalized k-fractional integrals involving Mittag–Leffler functions via α,h−m-convex functions. In the upcoming section, we will utilize k-fractional integral operators and α,h−m-convexity to prove the two versions of the Hadamard inequality and the Fejér–Hadamard inequality.


## 3. Thek-Fractional Inequalities of Hadamard and Fejér–Hadamard Type

In this section, we first describe the below generalizedk-fractional Hadamard’s inequality.Theorem 3.
Leth:J⟶ℝis nonnegative, nonzero and integrable function andφ,ψ:a,b⟶ℝ,0≤a<mb, be the functions such thatφ∈L1a,bandφbe positive andψbe differentiable and strictly increasing. Ifφisα,h−m-convex, the below inequalities fork-fractional operators (12) and (13) occur:(16)φψa+mψb2ψkFμ,τ,l,w¯,a+γ,δ,v,c1mψb;p˜≤h12αψkFμ,τ,l,w¯,a+γ,δ,v,cφ°ψmψb;p˜+mτ/k+1h2α−12αψkFμ,τ,l,w¯mμ,b−γ,δ,v,cφ°ψψam;p˜≤h12αφψa+mτ/k+1h2α−12αφψb∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜hηαdη+mh12αφψb+mτ/k+1h2α−12αφψam2∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜h1−ηαdη,where w¯=w/mψb−ψaμ for all η∈a,b.Proof.
Sinceφ is α,h−m-convex on a,b, for all ι,κ∈a,b, we have(17)φψι+mψκ2≤h12αφψι+mh2α−12αφψκ.Setting ψι=ηψa+m1−ηψb and ψκ=ψa/m1−η+ηψb in above inequality, we have(18)φψa+mψb2≤h12αφηψa+m1−ηψb+mh2α−12αφψam1−η+ηgb.
Multiplying both sides of (18) by ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜, then integrating over 0,1, we have(19)φψa+mψb2∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜dη≤h12α∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜φηψa+m1−ηψbdη+mh2α−12α∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜φψam1−η+ηψbdη.By specifying ψι=ηψa+m1−ηψb and ψκ=ψa/m1−η+ηψb in (19), we have(20)φψa+mψb2∫aψ−1mψb(mψb−ψι)τ/k−1Eμ,τ,lγ,δ,v,cw¯mψb−ψιμ;p˜ψ′ιdι≤h12α∫aψ−1mψb(mψb−ψι)τ/k−1Eμ,τ,lγ,δ,v,cw¯mψb−ψιμ;p˜φψιψ′ιdι+mτ/k+1h2α−12α∫ψ−1ψa/mb(ψκ−ψam)τ/k−1Eμ,τ,lγ,δ,v,cw¯mμψκ−ψamμ;p˜φψκψ′κdκ.
By usagek-fractional operators (12) and (13), the first side of (16) is achieved.
To evidence the second side of (16), once again α,h−m-convexity of φ over a,b, for η∈0,1, we achieve(21)h12αφηψa+m1−ηψb+mτ/k+1h2α−12αφ1−ηψam+ηψb≤hηαh12αφψa+mτ/k+1h2α−12αφψb+mh1−ηαh12αφψb+mτ/k+1h2α−12αφψam2.
Multiplying both sides of (21) by ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜, next integrating over 0,1, we achieve(22)h12α∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜φηψa+m1−ηψbdη+mτ/k+1h2α−12α∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜φ1−ηψam+ηψbdη≤h12αφψa+mτ/k+1h2α−12αφψb∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜hηαdη+mh12αφψb+mτ/k+1h2α−12αφψam2∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜h1−ηαdη.
Settingψι=ηψa+m1−ηψb and ψκ=1−ηψa/m+ηψb in (22), in that case by utilizing k-fractional operators (12) and (13), the second side of (16) is achieved.Corollary 1.
By usage (16), anymorek-fractional inequalities are offered as noted below:(i)
By choosingψ=Iandp˜=w=0, we obtain(23)φa+mb2∫ambmb−ιτ/k−1dι≤h12α∫ambmb−ιτ/k−1φιdι+mτ/k+1h2α−12α∫ambκ−amτ/k−1φκdκ≤h12αφa+mτ/k+1h2α−12αφb∫01ητ/k−1hηαdη+mh12αφb+mτ/k+1h2α−12αφam2∫01ητ/k−1h1−ηαdη.(ii)
By choosingψ=Iandp˜=0, we obtain(24)φa+mb2∫ambmb−ιτ/k−1Eμ,τ,lγ,δ,v,cw¯mb−ιμdι≤h12α∫ambmb−ιτ/k−1Eμ,τ,lγ,δ,v,cw¯mb−ιμφιdι+mτ/k+1h2α−12α∫a/mbκ−amτ/k−1Eμ,τ,lγ,δ,v,cw¯mμκ−amμφκdκ≤h12αφa+mτ/k+1h2α−12αφb∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημhηαdη+mh12αφb+mτ/k+1h2α−12αφam2∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημh1−ηαdη.(iii)
By settingm=1andψ=I, we obtain(25)φa+b2∫abb−ιτ/k−1Eμ,τ,lγ,δ,v,cw¯b−ιμ;p˜dι≤h12α∫abb−ιτ/k−1Eμ,τ,lγ,δ,v,cw¯b−ιμ;p˜φιdι+h2α−12α∫abκ−aτ/k−1Eμ,τ,lγ,δ,v,cw¯κ−aμ;p˜φκdκ≤h12αfa+h2α−12αφb∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜hηαdη+h12αφb+h2α−12αφa∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜h1−ηαdη.(iv)
By choosinghη=ηandp˜=w=0, we obtain(26)φψa+mψb2∫aψ−1mψb(mψb−ψι)τ/k−1ψ′ιdι≤12α∫aψ−1mψb(mψb−ψι)τ/k−1φψιψ′ιdι+mτ/k+12α−12α∫ψ−1ψa/mb(ψκ−ψam)τ/k−1φψκψ′κdκ≤12αφψa+mτ/k+12α−12αφψbkτ+αk+m12αφψb+mτ/k+12α−12αφψam2αk2ττ+αk.(v)
By settingα=1andψ=I, we get(27)φa+mb2∫ambmb−ιτ/k−1Eμ,τ,lγ,δ,v,cw¯mb−ιμ;p˜dι≤h12∫ambmb−ιτ/k−1Eμ,τ,lγ,δ,v,cw¯mb−ιμ;p˜φιdι+mτ/k+1h12∫a/mbκ−amτ/k−1Eμ,τ,lγ,δ,v,cw¯mμκ−amμ;p˜φκdκ≤h12φa+mτ/k+1h12φb∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜hηdη+mh12φb+mτ/k+1h12φam2∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜h1−ηdη.(vi)
By settingα=m=1,hη=ηandψ=I, we get(28)φa+b2∫abb−ιτ/k−1Eμ,τ,lγ,δ,v,cw¯b−ιμ;p˜dι≤12∫abb−ιτ/k−1Eμ,τ,lγ,δ,v,cw¯b−ιμ;p˜φιdι+∫abκ−aτ/k−1Eμ,τ,lγ,δ,v,cw¯κ−aμ;p˜φκdκ≤φa+φb2∫01ητ/kEμ,τ,lγ,δ,v,cwημ;p˜dη+∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜1−ηdη.Remark 2.
The abovek-fractional inequalities are farther in line with already known conclusions as noted below: (i) By choosingk=1in Corollary 1(v), an inequality for extended generalized fractional integrals is acquired. (ii) By choosingk=1andp˜=0in Corollary 1(v), Theorem 2.1 of [28] is acquired. (iii) By choosingm=1, andhη=ηin Corollary 1(v), Theorem 2.1 of [27] is acquired. (iv) By choosingp˜=w=0in Corollary 1(v), Theorem 2.1 of [20] is acquired.Remark 3.
(i) By choosingk=1andp˜=0in Remark 1(iii), an inequality for extended generalized fractional integrals is acquired. (ii) By choosingk=1andp˜=w=0in Remark 1(iii), Theorem 2 of [29] is acquired. (iii) By choosingk=1in Remark 1(iv), Corollary 2.2 of [20] is acquired.The below lemma is beneficial to offer the Fejér–Hadamard’s inequality for generalizedk-fractional integrals.Lemma 1.
Letφ,ψ:a,b⟶ℝwith0≤a<mb, be the functions such thatφ∈L1a,bandφpositive andψbe differentiable and strictly increasing. Ifφψι=φψa+mψb−ψι, in that case for generalizedk-fractional operators (11) and (12), we get(29)ψkFμ,τ,l,w¯,a+γ,δ,v,cφ°ψmψb;p˜=ψkFμ,τ,l,w¯mμ,b−γ,δ,v,cφ°ψψam;p˜=12ψkFμ,τ,l,w¯,a+γ,δ,v,cφ°gmψb;p˜+ψkFμ,τ,l,w¯mμ,b−γ,δ,v,cφ°ψψam;p˜,for all η∈a,b.Proof.
By description of generalizedk-fractional operators (12) and (13), we get(30)ψkFμ,τ,l,w¯,a+γ,δ,v,cφ°ψmψb;p˜=∫aψ−1mψb(mψb−ψι)τ/k−1Eμ,τ,lγ,δ,v,cw¯mψb−ψιμ;p˜φ°ψιψ′ιdι=∫aψ−1mψb(mψb−ψι)τ/k−1Eμ,τ,lγ,δ,v,cw¯mψb−ψιμ;p˜φψιψ′ιdι=∫ag−1mψb(mψb−ψι)τ/k−1Eμ,τ,lγ,δ,v,cw¯mψb−ψιμ;p˜φψa+mψb−ψιψ′ιdι.
Settingψη=ψa+mψb−ψι in the above equation and using φψι=φψa+mψb−ψι, we have(31)ψkFμ,τ,l,w¯,a+γ,δ,v,cφ°ψmψb;p˜=∫ψ−1ψa/mb(mψη−ψa)τ/k−1Eμ,τ,lγ,δ,v,cw¯mψη−ψaμ;p˜φψηψ′ηdη=∫ψ−1ψa/mb(mψη−ψa)τ/k−1Eμ,τ,lγ,δ,v,cw¯mψη−ψaμ;p˜φ°ψηψ′ηdη.
This implies(32)ψkFμ,τ,l,w¯,a+γ,δ,v,cφ°ψmψb;p˜=ψkFμ,τ,l,w¯mμ,b−γ,δ,v,cφ°ψψam;p˜.
By addingψkFμ,τ,l,w¯,a+γ,δ,v,cφ°ψmψb;p˜ on both sides of (32), we have(33)2ψkFμ,τ,l,w¯,a+γ,δ,v,cφ°ψmψb;p˜=ψkFμ,τ,l,w¯mμ,b−γ,δ,v,cφ°ψψam;p˜+ψkFμ,τ,l,w¯,a+γ,δ,v,cφ°ψmψb;p˜.
From equations (32) and (33), the result can be obtained.The first type of Fejér–Hadamard inequality is endued through generalizedk-fractional integrals as noted below:Theorem 4.
Leth:J⟶ℝbe nonnegative, nonzero, and integrable function andφ,ψ:a,b⟶ℝ,0≤a<mb, be the functions such thatφ∈L1a,bandφbe positive andψbe differentiable and strictly increasing,ris a nonnegative and integrable function. Ifφisα,h−m-convex andφψι=φψa+mψb−ψι, in that case the below inequalities for generalizedk-fractional operators (12) and (13) occur:(34)φψa+mψb2ψkFμ,τ,l,w¯,a+γ,δ,v,cr°ψmψb;p˜+ψkFμ,τ,l,w¯mμ,b−γ,δ,v,cr°ψψam;p˜≤2h12αψkFμ,τ,l,w¯,a+γ,δ,v,cφ°r°ψmψb;p˜+2mτ/k+1h2α−12αψkFμ,τ,l,w¯mμ,b−γ,δ,v,cφ°r°ψψam;p˜≤2h12αφψa+mτ/k+1h2α−12αφψb×∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜rηψa+m1−ηψbhηαdη+2mh12αφψb+mτ/k+1h2α−12αφψam2×∫01tτ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜rηψa+m1−ηψbh1−ηαdη,where w¯=w/mψb−ψaμ for all η∈a,b.Proof.
We demonstrate the claim as follows:
Multiplying both sides of (18) by ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜rηψa+m1−ηψb and then integrating over 0,1, we have(35)φψa+mψb2∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜rηψa+m1−ηψbdη≤h12α∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜φηψa+m1−ηψbrηψa+m1−ηψbdη+mh2α−12α∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜φ1−ηψam+ηψbrηψa+m1−ηψbdη.
By specifyingψι=ηψa+m1−ηψb and ψκ=1−ηψa/m+ηψb, that is ψa+mψb−ψι=1−ηψa+mηψb, in (35), then using φψι=φψa+mψb−ψι, we have(36)φψa+mψb2∫aψ−1mψb(mψb−ψι)τ/k−1Eμ,τ,lγ,δ,v,cw¯mψb−ψιμ;p˜r°ψιψ′ιdι≤h12α∫aψ−1mψb(mψb−ψι)τ/k−1Eμ,τ,lγ,δ,v,cw¯mψb−ψιμ;p˜φ°ψιr°ψιψ′ιdι+mh2α−12α∫ψ−1ψa/mb(ψι−ψam)τ/k−1Eμ,τ,lγ,δ,v,cw¯mμψι−ψamμ;p˜φ°ψιr°ψιψ′ιdι.
This implies(37)φψa+mψb2ψkFμ,τ,l,w¯,a+γ,δ,v,cr°ψmψb;p˜≤h12αψkFμ,τ,l,w¯,a+γ,δ,v,cφ°r°ψmψb;p˜+mτ/k+1h2α−12αψkFμ,τ,l,w¯mμ,b−γ,δ,v,cφ°r°ψψam;p˜.
Using Lemma1 in the above inequality, we have the first side of (34).
To demonstrate second side of (34), multiplying both parts of (21) by 2ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜rψa+m1−ηψb and then integrating over 0,1, we have(38)2h12α∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜rηψa+m1−ηψbφηψa+m1−ηψbdη+2mτ/k+1h2α−12α∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜rηψa+m1−ηψbφ1−ηψam+ψbdη≤2h12αφψa+mτ/k+1h2α−12αφψb×∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜rηψa+m1−ηψbhηαdη+2mh12αφψb+mτ/k+1h2α−12αφψam2×∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜rηψa+m1−ηψbh1−ηαdη.
Settingψι=ηψa+m1−ηψb and ψκ=1−ηψa/m+ηψb, then using φψι=φψa+mψb−ψι in (38), we have(39)2h12αψkFμ,τ,l,w¯,a+γ,δ,v,cφ°r°ψmψb;p˜+2mτ/k+1h2α−12αψkFμ,τ,l,w¯mμ,b−γ,δ,v,cφ°r°ψψam;p˜≤2h12αφψa+mτ/k+1h2α−12αφψb×∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜rηψa+m1−ηψbhηαdη+2mh12αφψb+mτ/k+1h2α−12αφψam2×∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜rηψa+m1−ηψbh1−ηαdη.
By usage Lemma1 in the above inequality, we have the second side of (34).Corollary 2.
By using (34), some morek-fractional inequalities are offered as noted below:(i)
By choosingψ=Iandp˜=w=0, we obtain(40)φa+mb2∫ambmb−ιτ/k−1rιdι≤2h12α∫ambmb−ιτ/k−1φ∘rιdι+2mτ/k+1h2α−12α∫a/mbκ−amτ/k−1φ∘rκdκ≤2h12αφa+mτ/k+1h2α−12αφb∫01ητ/k−1rηa+m1−ηbhηαdη+2mh12αφb+mτ/k+1h2α−12αφam2∫01ητ/k−1rηa+m1−ηbh1−ηαdη.(ii)
By choosingp˜=0andψ=I, we obtain(41)φa+mb2∫ambmb−ιτ/k−1Eμ,τ,lγ,δ,v,cw¯mb−ιμrιdι≤2h12α∫ambmb−ιτ/k−1Eμ,τ,lγ,δ,v,cw¯mb−ιμφ∘rιdι+2mτ/k+1h2α−12α∫a/mbκ−amτ/k−1Eμ,τ,lγ,δ,v,cw¯mμκ−amμφ∘rκdκ≤2h12αφa+mτ/k+1h2α−12αφb∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημrηa+m1−ηbhηαdη+2mh12αφb+mτ/k+1h2α−12αφam2∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημrηa+m1−ηbh1−ηαdη.(iii)
By choosingm=1andψ=I, we obtain(42)φa+b2∫abb−ιτ/k−1Eμ,τ,lγ,δ,v,cw¯b−ιμ;p˜rιdι≤2h12α∫abb−ιτ/k−1Eμ,τ,lγ,δ,v,cw¯b−ιμ;p˜φ∘rιdι+2h2α−12α∫abκ−aτ/k−1Eμ,τ,lγ,δ,v,cw¯κ−aμ;p˜φ∘rκdκ≤2h12αφa+h2α−12αφb∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜rηa+1−ηbhηαdη+2h12αφb+h2α−12αφa∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜rηa+1−ηbh1−ηαdη.(iv)
By choosingp˜=w=0andhη=η, we obtain(43)φψa+mψb2∫aψ−1mψb(mψb−ψι)τ/k−1r∘ψιψ′ιdι≤12α−1∫aψ−1mψb(mψb−ψι)τ/k−1φ∘r∘ψιψ′ιdι+mτ/k+12α−12α−1∫ψ−1ψamb(ψκ−ψam)τ/k−1φ∘r∘ψκψ′κdκ≤212αφψa+mτ/k+12α−12αφψb∫01ητ/k−1rηψa+m1−ηψbηαdη+2m12αφψb+mτ/k+12α−12αφψam2∫01ητ/k−1rηψa+m1−ηψb1−ηαdη.(v)
By choosingα=1andψ=I, we obtain(44)φa+mb2∫ambmb−ιτ/k−1Eμ,τ,lγ,δ,v,cw¯mb−ιμ;p˜rιdι≤2h12∫ambmb−ιτ/k−1Eμ,τ,lγ,δ,v,cw¯mb−ιμ;p˜φ∘rιdι+2mτ/k+1h12∫a/mbκ−amτ/k−1Eμ,τ,lγ,δ,v,cw¯mμκ−amμ;p˜φ∘rκdκ≤2h12φa+mτ/k+1h12φb∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜rηa+m1−ηbhηdη+2mh12φb+mτ/k+1h12φam2∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜rηa+m1−ηbh1−ηdη.(vi)
By choosingα=m=1,hη=ηand, we obtain(45)φa+b2∫abb−ιτ/k−1Eμ,τ,lγ,δ,v,cw¯b−ιμ;p˜rιdι≤∫abb−ιτ/k−1Eμ,τ,lγ,δ,v,cw¯b−ιμ;p˜φ∘rιdι+∫abκ−aτ/k−1Eμ,τ,lγ,δ,v,cw¯κ−aμ;p˜φ∘rκdκ≤φa+φb[∫01ητ/kEμ,τ,lγ,δ,v,cwημ;p˜rηa+1−ηbdη+∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜rηa+1−ηb1−ηdη].(vii)
By choosingα=k=1andψ=I, we obtain(46)φa+mb2∫ambmb−ιτ−1Eμ,τ,lγ,δ,v,cw¯mb−ιμ;p˜rιdι≤2h12∫ambmb−ιτ−1Eμ,τ,lγ,δ,v,cw¯mb−ιμ;p˜φ∘rιdι+2mτ+1h12∫a/mbκ−amτ−1Eμ,τ,lγ,δ,v,cw¯mμκ−amμ;p˜φ∘rκdκ≤2h12φa+mτ+1h12φb∫01ητ−1Eμ,τ,lγ,δ,v,cwημ;p˜rηa+m1−ηbhηdη+2mh12φb+mτ+1h12φam2∫01ητ−1Eμ,τ,lγ,δ,v,cwημ;p˜rηa+m1−ηbh1−ηdη.Remark 4.
The abovek-fractional inequalities are farther in line with foreknown conclusions as noted below: (i) By choosingk=1in Corollary 2(vi), Theorem 2.2 of [27] is acquired. (ii) By choosingp˜=0in in Corollary 2(vii), Theorem 2.5 of [28] is acquired. (iii) By choosingk=I,p˜=w=0andhη=ηin Corollary 2(v), an inequality form-convex functions via Riemann–Liouville integrals is acquired. (iv) By choosingk=1andp˜=0in in Corollary 2(vi), an inequality for extended generalized fractional integrals is acquired. (v) By choosingk=1andp˜=w=0in in Corollary 2(vi), Theorem 4 of [26] is acquired. (vi) By choosinghη=ηin in Corollary 3.2 (vii), Theorem 3.1 of [27] is acquired.In the subsequent theorem, we offer another type of Hadamard’s inequality.Theorem 5.
Leth:J⟶ℝis nonnegative, nonzero and integrable function andφ,ψ:a,b⟶ℝ,0≤a<mb, be the functions such thatφ∈L1a,bandφbe positive andψbe differentiable and strictly increasing. Ifφisα,h−m-convex, in that case for generalizedk-fractional operators (12) and (13), we acquire(47)φψa+mψb2ψkFμ,τ,l,w¯,ψ−1mψb+ψa/2+γ,δ,v,c1mψb;p˜≤h12αψkFμ,τ,l,w̳,ψ−1mψb+ψa/2+γ,δ,v,cφ°ψmψb;p˜+mτ/k+1h2α−12αψkFμ,τ,l,w̳mμ,ψ−1mψb+ψa/2m−γ,δ,v,cφ°ψψam;p˜≤h12αφψa+mτ/k+1h2α−12αφψb∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜hηα2αdη+mh12αφψb+mτ/k+1h2α−12αφψam2∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜h2α−ηα2αdη,where w¯¯=2μw/mψb−ψaμ for all η∈a,b.Proof.
Settingψι=η/2ψa+m2−η/2ψb and ψκ=2−η/2ψa/m+η/2ψb in (3.2), we have(48)φψa+mψb2≤h12αφη2ψa+m2−η2ψb+mh2α−12αφ2−η2ψam+η2ψb.
Multiplying both parts of (48) by ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜ and then integrating over 0,1, we have(49)φψa+mψb2∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜dη≤h12α∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜φη2ψa+m2−η2ψbdη+mh2α−12α∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜φ2−η2ψam+η2ψbdη.
By takingψι=η/2ψa+m2−η/2ψb and ψκ=2−η/2ψa/m+η/2ψb in (49), in that case by usage k-fractional operators (2.12) and (2.13), the first side of (47) is acquired.
To demonstrate the second side of (47), once again α,h−m-convexity of φ over a,b, for η∈0,1, we get(50)h12αφη2ψa+m2−η2ψb+mτ/k+1h2α−12αφ2−η2ψam+η2ψb≤hηα2αh12αφψa+mτ/k+1h2α−12αφψb+mh2α−ηα2αh12αφψb+mτ/k+1h2α−12αφψam2.
Multiplying both sides of (50) by ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜, then integrating over 0,1, we acquire(51)h12α∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜φη2ψa+m2−η2ψbdη+mτ/k+1h2α−12α∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜φ2−η2ψam+η2ψbdη≤h12αφψa+mτ/k+1h2α−12αφψb∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜hηα2αdη+mh12αφψb+mτ/k+1h2α−12αφψam2∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜h2α−ηα2αdη.
Choosingψι=η/2ψa+m2−η/2ψb and ψκ=2−η/2ψa/m+η/2ψb in (51), in that case by usage k-fractional operators (12) and (13), the second side of (47) is acquired.Corollary 3.
By using (47), anymorek-fractional inequalities are offered as noted below:(i)
By choosingψ=Iandp˜=w=0, we have(52)φa+mb2∫a+mb/2mbmb−ιτk−1dι≤h12α∫a+mb/2mbmb−ιτk−1φιdι+mτk+1h2α−12α∫a/ma+mb/2κ−amτk−1φκdκ≤h12αφa+mτk+1h2α−12αφb∫01ητk−1hηα2αdη+mh12αφb+mτk+1h2α−12αφam2∫01ητk−1h2α−ηα2αdη.(ii)
By choosingp˜=0andψ=I, we have(53)φa+mb2∫a+mb/2mbmb−ιτk−1Eμ,τ,lγ,δ,v,cw¯¯mb−ιμdι≤h12α∫a+mb/2mbmb−ιτk−1Eμ,τ,lγ,δ,v,cw¯¯mb−ιμφιdι+mτk+1h2α−12α∫ama+mb2κ−amτk−1Eμ,τ,lγ,δ,v,cw¯¯mμκ−amμφκdκ≤h12αφa+mτk+1h2α−12αφb∫01ητk−1Eμ,τ,lγ,δ,v,cwημhηα2αdη+mh12αφb+mτk+1h2α−12αφam2∫01ητk−1Eμ,τ,lγ,δ,v,cwημh2α−ηα2αdη.(iii)
By choosingm=1andψ=I, we acquire(54)φa+b2∫a+b/2bb−ιτ/k−1Eμ,τ,lγ,δ,v,cw¯¯b−ιμ;p˜dι≤h12α∫a+b/2bb−ιτ/k−1Eμ,τ,lγ,δ,v,cw¯¯b−ιμ;p˜φιdι+h2α−12α∫abκ−aτ/k−1Eμ,τ,lγ,δ,v,cw¯¯κ−aμ;p˜φκdκ≤h12αφa+h2α−12αφb∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜hηα2αdη+h12αφb+h2α−12αφa∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜h2α−ηα2αdη.(iv)
By choosingp˜=w=0andhη=η, we have(55)φψa+mψb2∫ψ−1ψa+mψb/2ψ−1mψb(mψb−ψι)τ/k−1ψ′ιdι≤12α∫ψ−1ψa+mψb/2ψ−1mψb(mψb−ψι)τ/k−1φψιψ′ιdι+mτ/k+12α−12α∫ψ−1ψa/mψ−1ψa+mψb/2m(ψκ−ψam)τ/k−1φψκψ′κdκ≤12αφψa+mτ/k+12α−12αφψbk2ατ+αk+m12αφψb+mτ/k+12α−12αφψam2kτ−k2ατ+αk.(v)
By choosingα=1andψ=I, we have(56)φa+mb2∫a+mb/2mbmb−ιτ/k−1Eμ,τ,lγ,δ,v,cw¯¯mb−ιμ;p˜dι≤h12∫a+mb/2mbmb−ιτ/k−1Eμ,τ,lγ,δ,v,cw¯¯mb−ιμ;p˜φιdι+mτ/k+1h12∫a/ma+mb/2κ−amτ/k−1Eμ,τ,lγ,δ,v,cw¯¯mμκ−amμ;p˜φκdκ≤h12φa+mτ/k+1fb∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜hη2dη+mh12φb+mτ/k+1φam2∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜h2−η2dη.(vi)
By choosingα=m=1,hη=ηandψ=I, we have(57)φa+b2∫a+b/2bb−ιτ/k−1Eμ,τ,lγ,δ,v,cw¯¯b−ιμ;p˜dι≤12[∫a+b/2bb−ιτ/k−1Eμ,τ,lγ,δ,v,cw¯¯b−ιμ;p˜φιdι+∫aa+b/2κ−aτ/k−1Eμ,τ,lγ,δ,v,cw¯¯κ−aμ;p˜φκdκ]≤φa+φb2∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜dη.Remark 5.
The abovek-fractional inequalities are farther in line with foreknown conclusions as noted below: (i) By choosingk=1in Corollary 3(v), an inequality for extended generalized fractional integrals is acquired. (ii) By choosingk=1andp˜=0in Corollary 3(v), Theorem 2.2 of [28] is acquired.The second type of the Fejér–Hadamard’s inequality for generalizedk-fractional integrals is dedicated as noted below:Theorem 6.
Leth:J⟶ℝis nonnegative, nonzero and integrable function andφ,ψ:a,b⟶ℝ,0≤a<mb, be the functions such thatφ∈L1a,bandφbe positive andψbe differentiable and strictly increasing,ris a nonnegative and integrable function. Ifφisα,h−m-convex andφψι=φψa+mψb−ψι, in that case the below inequalities for generalizedk-fractional operators (12) and (13) occur:(58)φψa+mψb2ψkFμ,τ,l,w̳,ψ−1mψb+ψa/2+γ,δ,v,cr°ψmψb;p˜≤h12αψkFμ,τ,l,w¯¯,ψ−1mψb+ψa/2+γ,δ,v,cφ°r°ψmψb;p˜+mτ/k+1h2α−12αψkFμ,τ,l,w̳mμ,ψ−1mψb+ψa/2m−γ,δ,v,cφ°r°ψψam;p˜≤h12αφψa+mτ/k+1h2α−12αφψb×∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜rη2ψa+m2−η2ψbhηα2αdη+mh12αφψb+mτ/k+1h2α−12αφψam2×∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜rη2ψa+m2−η2ψbh2α−ηα2αdη,where w¯¯=2μw/mψb−ψaμ for all η∈ap,bp.Proof.
We demonstrate the claim as follows:
Multiplying (48) by ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜rη/2ψa+m2−η/2ψb and then integrating over 0,1, we have(59)φψa+mψb2∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜rη2ψa+m2−η2ψbdη≤h12α∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜φη2ψa+m2−η2ψbrη2ψa+m2−η2ψbdη+mh2α−12α∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜φ2−η2ψam+η2ψbrη2ψa+m2−η2ψbdη.
By settingψι=η/2ψa+m2−η/2ψb and ψκ=2−η/2ψa/m+η/2ψb, that is, ψa+mψb−ψι=2−η/2ψa+mη/2ψb, in (59), in that case by usage φψι=φψa+mψb−ψι and k-fractional integral operators (12) and (13), the first side of (58) is acquired.
To demonstrate the second side of (58), multiplying both parts of (50) by(60)ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜rη2ψa+m2−η2ψb,and then integrating over 0,1, we have(61)h12α∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜φη2ψa+m2−η2ψbrη2ψa+m2−η2ψbdη+mτ/k+1h2α−12α∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜φ2−η2ψam+η2ψbrη2ga+m2−η2ψbdη≤h12αφψa+mτ/k+1h2α−12αφψb×∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜rη2ψa+m2−η2ψbhηα2αdη+mh12αφψb+mτ/k+1h2α−12αφψam2×∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜rη2ψa+2−η2ψbh2α−ηα2αdη.
Settingψι=η/2ψa+m2−η/2ψb and ψκ=2−η/2ψa/m+η/2ψb in (59), then by using φψι=φψa+mψb−ψι and k-fractional integral operators (12) and (13), the second inequality of (58) is obtained.Corollary 4.
By using (58), some morek-fractional inequalities are offered as noted below:(i)
By choosingψ=Iandp˜=w=0, we obtain(62)φa+mb2∫a+mb/2mbmb−ιτ/k−1rιdι≤h12α∫a+mb/2mbmb−ιτ/k−1φ°rιdι+mτ/k+1h2α−12α∫a/ma+mb/2κ−amτ/k−1φ°rκdκ≤h12αφa+mτ/k+1h2α−12αφb×∫01ητ/k−1rη2a+m2−η2bhηα2αdη+mh12αφb+mτ/k+1h2α−12αφam2×∫01ητ/k−1rη2a+m2−η2bh2α−ηα2αdη.(ii)
By choosingψ=Iandp˜=0, we obtain(63)φa+mb2∫a+mb/2mbmb−ιτ/k−1Eμ,τ,lγ,δ,v,cw¯¯mb−ιμrιdι≤h12α∫a+mb/2mbmb−ιτ/k−1Eμ,τ,lγ,δ,v,cw¯¯mb−ιμφ∘rιdι+mτ/k+1h2α−12α∫a/ma+mb/2κ−amτ/k−1Eμ,τ,lγ,δ,v,cw¯¯mμκ−amμφ∘rκdκ≤h12αφa+mτ/k+1h2α−12αφb×∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημrη2a+m2−η2bhηα2αdη+mh12αφb+mτ/k+1h2α−12αφam2×∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημrη2a+m2−η2bh2α−ηα2αdη.(iii)
By choosingm=1andψ=I, we obtain(64)φa+b2∫a+b/2bb−ιτ/k−1Eμ,τ,lγ,δ,v,cw¯¯b−ιμ;p˜rιdι≤h12α∫a+b/2bb−ιτ/k−1Eμ,τ,lγ,δ,v,cw¯¯b−ιμ;p˜φ∘rιdι+h2α−12α∫abκ−aτ/k−1Eμ,τ,lγ,δ,v,cw¯¯κ−aμ;p˜φ∘rκdκ≤h12αφa+h2α−12αφb∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜rη2a+2−η2bhηα2αdη+h12αφb+h2α−12αφa∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜rη2a+2−η2bh2α−ηα2αdη.(iv)
By choosingp˜=w=0andhη=η, we obtain(65)φψa+mψb2∫ψ−1ψa+mψb/2ψ−1mψb(mψb−ψι)τ/k−1rψιψ′ιdι≤12α∫ψ−1ψa+mψb/2ψ−1mψb(mψb−ψι)τ/k−1φ∘r∘ψιψ′ιdι+mτ/k+12α−12α∫ψ−1ψa/2ψ−1ψa+mψb/2m(ψκ−ψam)τ/k−1φ∘r∘ψκψ′κdκ≤12αφψa+mτ/k+12α−12αφψb+m12αφψb+mτ/k+12α−12αφψam2×∫01ητ/k−1rη2ψa+m2−η2ψbh2α−ηα2αdη.(v)
By choosingα=1andψ=I, we get(66)φa+mb2∫a+mb/2mbmb−ιτ/k−1Eμ,τ,lγ,δ,v,cw¯¯mb−ιμ;p˜rιdι≤h12∫a+mb/2mbmb−ιτ/k−1Eμ,τ,lγ,δ,v,cw¯¯mb−ιμ;p˜φ∘rιdι+mτ/k+1h12∫a/ma+mb/2κ−amτ/k−1Eμ,τ,lγ,δ,v,cw¯¯mμκ−amμ;p˜φ∘rκdκ≤h12φa+mτ/k+1φb∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜rη2a+m2−η2bhη2dη+mh12fb+mτ/k+1φam2∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜rη2a+m2−η2bh2−η2dη.(vi)
By choosingα=m=1,hη=ηandψ=I, we get(67)φa+b2∫a+b/2bb−ιτ/k−1Eμ,τ,lγ,δ,v,cw¯¯b−ιμ;p˜rιdι≤12[∫a+b/2bb−ιτ/k−1Eμ,τ,lγ,δ,v,cw¯¯b−ιμ;p˜φ∘rιdι+∫aa+b/2κ−aτ/k−1Eμ,τ,lγ,δ,v,cw¯¯κ−aμ;p˜φ∘rκdκ]≤φa+φb2∫01ητ/k−1Eμ,τ,lγ,δ,v,cwημ;p˜rη2a+2−η2bdη.Remark 6.
Those as mentioned abovek-fractional inequalities are farther in line with foreknown conclusions as by choosingk=1in Corollary 4(v), an inequality for extended generalized fractional integrals is obtained.



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