# (p,h)-Convex Functions Associated with Hadamard and Fejér-Hadamard Inequalities via k-Fractional Integral Operators

Authors: Xiujun Zhang; Ghulam Farid; Ayşe Kübra Demirel; Chahn Yong Jung
Journal: Journal of Function Spaces
 (2022)

---

## Abstract

In this article, generalized versions of thek-fractional Hadamard and Fejér-Hadamard inequalities are constructed. To obtain the generalized versions of these inequalities, k-fractional integral operators including the well-known Mittag-Leffler function are utilized. The class of (p,h)-convex functions for Hadamard-type inequalities give the generalizations of results which have been proved in literature for p-convex, h-convex, and several functions deducible from these two classes.


---

## Body

## 1. Introduction

The subject of fractional calculus which includes the study of fractional-order integrals and derivatives has become a very prominent research topic in recent years. It has got the attention of researchers working in almost all fields of science and engineering. Integral operators are a very important part of fractional calculus; they play an important role in the mathematical treatment of different kinds of real-world problems. In the theory of mathematical inequalities, fractional integral operators appeared as a tool for the generalizations of classical inequalities. In the last two decades, a lot of such generalizations have been published by several authors.Iscan and Wu [1] have generalized Hermite-Hadamard inequalities for harmonically convex functions by applying fractional integrals. Akkurt et al. [2] have obtained inequalities for k−h-Riemann-Liouville fractional integrals with the aid of synchronous and monotonic functions. Mubeen and Habibullah [3] have introduced k-analogue of Riemann-Liouville fractional integrals which are used for establishing k-fractional versions of well-known inequalities. A study of k-fractional integrals has been presented by Tunç et al. in [4], and fractional inequalities have been proved. Farid et al. in [5] demonstrated some novel Fejér-Hadamard and Hadamard inequalities for harmonically convex functions by using fractional integrals containing the Mittag-Leffler function.Kunt and Iscan [6] presented Hermite-Hadamard-Fejér-type inequalities for p-convex functions using fractional integrals. Rashid et al. [7] demonstrated the Hadamard and the Fejér-Hadamard-type inequalities for the generalized fractional integral operator involving Mittag-Leffler functions for preinvexity and m-preinvexity. Klnç et al. [8] proved Hadamard and Fejér-Hadamard inequalities for (h,m)-strongly convex functions via generalized fractional integrals with Mittag-Leffler functions. Jia et al. [9] demonstrated new types of Hadamard and Fejér-Hadamard fractional integral inequalities for Riemann-Liouville fractional integrals. Yussouf et al. [10] presented generalized types of Hadamard and Fejér-Hadamard-type fractional integral inequalities by utilizing generalized fractional integrals including Mittag-Leffler functions. Faisal et al. [11] proved new Hermite-Hadamard-Jensen-Mercer-type inequalities for convex functions by using Riemann-Liouville fractional integrals. The other recent studies in this area were presented by Zhao et al. [12–14] and Chu et al. [15].The purpose of this article is to investigate the Hadamard and the Fejér-Hadamard-type inequalities for (p,h)-convex functions via generalized k-fractional integrals including Mittag-Leffler functions. Inequalities for several kinds of convex functions are special cases of results of this paper. Also, several new inequalities can be deduced from presented results in specific settings. First, we give definitions of integral operators and some results which will set the foundation for this article.Definition 1 (see [16]).
Letw,α,l,γ,c,∈ℂ, and Rl,Rα>0,Rc>Rγ>0 with p~≥0,δ,μ>0,and0<v≤δ+μ. Also, let σ∈L1ε,vandζ∈ε,v. In that case, the generalized fractional operators Fμ,α,l,w,εγ,δ,v,c+σ and Fμ,α,l,w,vγ,δ,v,c−σ are defined by
(1)Fμ,α,l,w,ε+γ,δ,v,cσζ;p~=∫εζζ−tα−1Eμ,α,lγ,δ,v,cwζ−tμ;p~σtdt,Fμ,α,l,w,υ−γ,δ,v,cσζ;p~=∫ζυt−ζα−1Eμ,α,lγ,δ,v,cwt−ζμ;p~σtdt,where
(2)Eμ,α,lγ,δ,v,ct;p~=∑n=0∞βp~γ+nv,c−γβγ,c−γcnvΓμn+αtnlnδis the enlarged generalized Mittag-Leffler function and βp~ is the expansion of beta function described as noted below:
(3)βp~ζ,η=∫01tζ−11−tη−1e−p~/t1−tdt,where Rζ,Rη,Rcp~>0.Definition 2 (see [16]).
Letσ,ς:ε,υ⟶ℝ with 0<ε<υ be the functions such that σ is positive and σ∈L1ε,υ and ς be differentiable and absolutely increasing. Also, let ϕ/ζ be an increasing function on ε,∞, and γ,c,w,α,l,∈ℂ, Rl,Rα>0,Rc>Rγ>0 with p~≥0,δ,μ>0, and 0<v≤μ+δ. In that case, for ζ∈ε,υ, the fractional operators are defined by
(4)ςFμ,α,l,w,ε+ϕ,γ,δ,v,cσζ;p~=∫εζϕςζ−ςtςζ−ςtEμ,α,lγ,δ,v,cwςζ−ςtμ;p~ς′tσtdt,ςFμ,α,l,w,υ−ϕ,γ,δ,v,cσζ;p~=∫ζυϕςt−ςζςt−ςζEμ,α,lγ,δ,v,cwςt−ςζμ;p~ς′tσtdt.Definition 3 (see [17]).
Letσ,ς:ε,υ⟶ℝ with 0<ε<υ be the functions such that σ is positive and σ∈L1ε,υ and ς be differentiable and absolutely increasing. Also, let w,γ,c,α,l,∈ℂandRl,Rα>0,Rc>Rγ>0withp~≥0,δ,μ>0and0<v≤μ+δ. In that case, for ζ∈ε,υ, the unified integral operators are defined by
(5)ςFμ,α,l,w,ε+γ,δ,v,cσζ;p~=∫εζςζ−ςtα−1Eμ,α,lγ,δ,v,cwςζ−ςtμ;p~ς′tσtdt,ςFμ,α,l,w,υ−γ,δ,v,cσζ;p~=∫ζυςt−ςζα−1Eμ,α,lγ,δ,v,cwςt−ςζμ;p~ς′tσtdt.Recently, Yue et al. [18] described generalized k-fractional operators including the further extension of the Mittag-Leffler function as noted below.Definition 4.
Letσ,ς:ε,υ⟶ℝ with 0<ε<υ be the functions such that σ be positive and σ∈L1ε,υ and ς be differentiable and absolutely increasing. Let α>k and γ,c,w,α,l,∈ℝ,α,l>0, and c>γ>0, with p~≥0,δ,μ>0and0<v≤μ+δ. In that case, for ζ∈ε,υ, the left-right generalized k-fractional operators ςkFμ,α,l,w,ε+γ,δ,v,cσ and ςkFμ,α,l,w,υ−γ,δ,v,cσ are defined by
(6)ςkFμ,α,l,w,ε+γ,δ,v,cσζ;p~=∫εζςζ−ςtα/k−1Eμ,α,lγ,δ,v,cwςε−ςtμ;p~ς′tσtdt,(7)ςkFμ,α,l,w,υ−γ,δ,v,cσζ;p~=∫ζυςt−ςζα/k−1Eμ,α,lγ,δ,v,cwςt−ςζμ;p~ς′tσtdt.Next, we define convex and related functions.Definition 5 (see [19]).
A functionσ:ε,υ⟶ℝ is said to be convex, if
(8)σtζ+1−tη≤tσζ+1−tση,for all ζ,η∈ε,υ and t∈0,1.Definition 6 (see [20]).
LetI⊂0,∞ be a real interval and p∈ℝ\0. A function σ:I⟶ℝ is said to be a p-convex function, if
(9)σtζp+1−tηp1/p≤tσζ+1−tση,for all ζ,η∈I and t∈0,1.Definition 7 (see [6]).
Letp∈ℝ\0. In that case, a function σ:ε,υ⊂0,∞⟶ℝ is called p-symmetric in accordance with εp+υp/21/p, if
(10)σt1/p=σεp−υp−t1/p,for t∈0,1.Definition 8 (see [21]).
Leth:J⟶ℝ be a nonnegative and nonzero function and p∈ℝ\0. We say that σ:I⟶ℝ is a p,h-convex function, if σ is nonnegative and
(11)σtζp+1−tηp1/p≤htσζ+h1−tση,for all ζ,η∈I and t∈0,1. If the above inequality is reversed, then σ is said to be a p,h-concave function.Remark 9.
(i)
By takinght=t in Definition 8, we obtain the definition of p-convex function(ii)
By takinght=t and p=1 in Definition 8, we obtain the definition of convex function(iii)
By takinght=ts and p=1 in Definition 8, we obtain the definition of s-convex function(iv)
By takinght=t−1 and p=1 in Definition 8, we obtain the definition of Godunova-Levin-type convex function(v)
By takingp=1 in Definition 8, we obtain the definition of h-convex function(vi)
By takinght=1 and p=1 in Definition 8, we obtain the definition of p-functionThe following inequality is the well-known Hadamard inequality.Theorem 10 (see [22]).
Letσ:ε,υ⟶ℝ be a convex function for ε<υ. Then, the following inequality holds:
(12)σε+υ2≤1υ−ε∫ευσζdζ≤σε+συ2.The Fejér-Hadamard inequality is a weighted type of the Hadamard inequality presented by Fejér in [23].Theorem 11.
Letσ:ε,υ⟶ℝ be a convex function and ς:ε,υ⟶ℝ be nonnegative, integrable, and symmetric with respect to ε+υ/2. In that case, the below inequality takes:
(13)σε+υ2∫ευςζdζ≤∫ευσζςζdζ≤σε+συ2∫ευςζdζ.For other classes of functions defined after motivating from convex function, the above inequalities have been studied extensively (see [1, 5, 6, 24–26]).In the upcoming section, first, we construct the Hadamard inequality forp,h-convex function via generalized k-fractional integrals. Then, an identity is used to construct the Fejér-Hadamard inequality for p,h-convex function via generalized k-fractional integrals. After, another version of the Hadamard inequality and the Fejér-Hadamard inequality is presented. Moreover, the presented results generalize many already published results.


## 2.k-Fractional Integral Inequalities of Hadamard and Fejér-Hadamard Type

The generalizedk-fractional Hadamard inequality is stated and proved in the following theorem.Theorem 12.
Leth:J⟶ℝ be nonnegative, nonzero, and integrable function. Also, let σ,ς:ε,υ⟶ℝ with 0<ε<υ be the functions such that σ is positive and σ∈L1ε,υ, and ς is differentiable and absolutely increasing. If σ is p,h-convex, and p∈R\0, in that case, the below inequalities for k-fractional operators (6) and (7) occur:
(i)
Ifp>0, in that case,(14)σςpε+ςpυ21/pςkFμ,α,l,w¯,ς−1ςpε+γ,δ,v,c1ς−1ςpυ;p~≤h12ςkFμ,α,l,w¯,ς−1ςpε+γ,δ,v,cσ∘θς−1ςpυ;p~+ςkFμ,α,l,w¯,ς−1ςpυ−γ,δ,v,cσ∘θς−1ςpε;p~≤h12σςε+σςυςkFμ,α,l,w¯,1−γ,δ,v,ch0;p~+ςkFμ,α,l,w¯,0+γ,δ,v,ch1;p~,where w¯=w/ςpυ−ςpεμ and θt=ς1/pt for t∈εp,υp(ii)
Ifp<0, in that case,(15)σςpε+ςpυ21/pςkFμ,α,l,w¯,ς−1ςpε−γ,δ,v,c1ς−1ςpυ;p~≤h12ςkFμ,α,l,w¯,ς−1ςpυ+γ,δ,v,cσ∘ςς−1ςpυ;p~+ςkFμ,α,l,w¯,ς−1ςpε−γ,δ,v,cσ∘ςς−1ςpε;p~≤h12σςε+σςυςkFμ,α,l,w¯,0−γ,δ,v,ch0;p~+ςkFμ,α,l,w¯,1+γ,δ,v,ch1;p~,where w¯=w/ςpε−ςpυμ and θt=ς1/pt for t∈υp,εpProof.
We demonstrate claim (i) as noted below:(i)
Sinceσ is p,h-convex function on ε,υ, for ζ,η∈I, we get(16)σςpζ+ςpη21/p≤h12σςζ+σςη.
Takingςζ=tςpε+1−tςpυ1/p and ςη=tςpυ+1−tςpε1/p in the above inequality, we get
(17)σςpε+ςpυ21/p≤h12σtςpε+1−tςpυ1/p+σtςpυ+1−tςpε1/p.
Multiplying both sides of (17) by tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~ and integrating over 0,1, we get
(18)σςpε+ςpυ21/p∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~dt≤h12∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~σtςpε+1−tςpυ1/pdt+h12∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~σtςpυ+1−tςpε1/pdt.
By choosingςζ=tςpε+1−tςpυ and ςη=tςpυ+1−tςpε in (18), we get
(19)σςpε+ςpυ21/p∫ς−1ςpες−1ςpυςpυ−ςζα/k−1Eμ,α,lγ,δ,v,cw¯ςpυ−ςζμ;p~ς′ζdζ≤h12∫ς−1ςpες−1ςpυςpυ−ςζα/k−1Eμ,α,lγ,δ,v,cw¯ςpυ−ςζμ;p~σς1/pζς′ζdζ+h12∫ς−1ςpες−1ςpυςη−ςpεα/k−1Eμ,α,lγ,δ,v,cw¯ςη−ςpεμ;p~σς1/pης′ηdη.
By utilizingk-fractional operators (6) and (7), the first side of (14) is acquired.
Now, to demonstrate the second side of (14), once again, p,h-convexity of f over ε,υ, and for t∈0,1, we get
(20)σtςpε+1−tςpυ1/p+σtςpυ+1−tςpε1/p≤ht+h1−tσςε+σςυ.
Multiplying both sides of (20) by h1/2tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~ and integrating over 0,1, we get
(21)h12∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~σtςpε+1−tςpυ1/pdt+∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~σtςpυ+1−tςpε1/pdt≤h12σςε+σςυ×∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~htdt+∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~h1−tdt.
Takingςζ=tςpε+1−tςpυ and ςη=tςpυ+1−tςpε in (21), in that case, by utilizing k-fractional operators (6) and (7), the second side of (14) is acquired.
(ii)
Proof is the same as the proof of (i)Corollary 13.
By utilizing (14) and (15), some more k-fractional inequalities are offered as noted below:
(i)
By choosingς=I and p~=w=0, we acquire(22)σεp+υp21/p∫εpυpυp−ζα/k−1dζ≤h12∫εpυpυp−ζα/k−1σζ1/pdζ+∫εpυpη−εpα/k−1ση1/pdη≤h12σε+συ∫01tα/k−1htdt+∫01tα/k−1h1−tdt(ii)
By choosingς=I, p~=0,p=−1, we acquire(23)σ2ευε+υ∫1/ε1/υ1υ−ζα/k−1Eμ,α,lγ,δ,v,cw¯1υ−ζμdζ≤h12∫1/ε1/υ1υ−ζα/k−1Eμ,α,lγ,δ,v,cw¯1υ−ζμσ1ζdζ+∫1/ε1/υη−1εα/k−1Eμ,α,lγ,δ,v,cw¯η−1εμσ1ηdη≤h12σε+συ∫01tα/k−1Eμ,α,lγ,δ,v,cwtμhtdt+∫01tα/k−1Eμ,α,lγ,δ,v,cwtμh1−tdt(iii)
By choosingp=−1 and ς=I, we acquire(24)σ2ευε+υ∫1/ε1/υ1υ−ζα/k−1Eμ,α,lγ,δ,v,cw¯1υ−ζμ;p~dζ≤h12∫1/ε1/υ1υ−ζα/k−1Eμ,α,lγ,δ,v,cw¯1υ−ζμ;p~σ1ζdζ+∫1/ε1/υη−1εα/k−1Eμ,α,lγ,δ,v,cw¯η−1εμ;p~σ1ηdη≤h12σε+συ∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~htdt+∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~h1−tdt(iv)
By choosingp=−1, w=p~=0, and ς=I, we acquire(25)σ2ευε+υ∫1/ε1/υ1υ−ζα/k−1dζ≤h12∫1/ε1/υ1υ−ζα/k−1σ1ζdζ+∫1/ε1/υη−1εα/k−1σ1ηdη≤h12σε+συ∫01tα/k−1htdt+∫01tα/k−1h1−tdt(v)
By choosingp=−1, we acquire(26)σς−1ε+ς−1υ2−1×∫ς−1ς−1ες−1ς−1υς−1υ−ςζα/k−1Eμ,α,lγ,δ,v,cw¯ς−1υ−ςζμ;p~ς′ζdζ≤h12∫ς−1ς−1ες−1ς−1υς−1υ−ςζα/k−1Eμ,α,lγ,δ,v,cw¯ς−1υ−ςζμ;p~σς−1ζς′ζdζ+∫ς−1ς−1ες−1ς−1υςη−ς−1εα/k−1Eμ,α,lγ,δ,v,cw¯ςη−ς−1εμ;p~σς−1ης′ηdη≤h12σε+συ∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~htdt+∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~h1−tdtRemark 14.
The well-knownk-fractional inequalities are further noted as below:
(i)
By choosinght=t and k=1 in Corollary 13 (i), Theorem 9 of [6] is acquired(ii)
By choosinght=t and k=1 in Corollary 13 (ii), Theorem 2.1 of [24] is acquired(iii)
By choosinght=t and k=1 in Corollary 13 (iii), Theorem 2.1 of [5] is acquired(iv)
By choosinght=t and k=1 in Corollary 13 (iv), Theorem 4 of [1] is acquired(v)
By choosinght=t and k=1 in Corollary 13 (v), Theorem 2.1 of [27] is acquiredThe below lemma is beneficial to offer the Fejér-Hadamard inequality for generalizedk-fractional integrals.Lemma 15 (see [18]).
Letσ,ς:ε,υ⟶ℝ with 0<ε<υ be the functions such that σ is positive and σ∈L1ε,υ, and ς is differentiable and absolutely increasing. If p∈R\0 and σς1/pζ=σςpε+ςpυ−ςζ1/p, in the case for generalized k-fractional operators (6) and (7), we have
(i)
Ifp>0, in that case,(27)ςkFμ,α,l,w¯,ς−1ςpε+γ,δ,v,cσ∘θς−1ςpυ;p~=ςkFμ,α,l,w¯,ς−1ςpυ−γ,δ,v,cσ∘θς−1ςpε;p~=12ςkFμ,α,l,w¯,ς−1ςpε+γ,δ,v,cσ∘θς−1ςpυ;p~+ςkFμ,α,l,w¯,ς−1ςpυ−γ,δ,v,cσ∘θς−1ςpε;p~,with θt=ς1/pt for all t∈εp,υp(ii)
Ifp<0, in that case,(28)ςkFμ,α,l,w¯,ς−1ςpυ+γ,δ,v,cσ∘θς−1ςpε;p~=ςkFμ,α,l,w¯,ς−1ςpε−γ,δ,v,cσ∘θς−1ςpυ;p~=12ςkFμ,α,l,w¯,ς−1ςpυ+γ,δ,v,cσ∘θς−1ςpε;p~+ςkFμ,α,l,w¯,ς−1ςpε−γ,δ,v,cσ∘θς−1ςpυ;p~,with θt=ς1/pt for all t∈υp,εpThe Fejér-Hadamard inequality for generalizedk-fractional integrals is stated and proved in the following theorem.Theorem 16.
Leth:J⟶ℝ be a nonnegative and nonzero function. Also, let σ,ς,r:ε,υ⟶ℝ, 0<ε<υ, be the functions such that σ is positive and σ∈L1ε,υ, ς is differentiable and absolutely increasing, and r is a nonnegative and integrable function. If σ is p,h-convex with p∈R\0, and σς1/pζ=σςpε+ςpυ−ςζ1/p, then the following inequalities for generalized k-fractional integral operators (6) and (7) hold:
(i)
Ifp>0, in that case,(29)σςpε+ςpυ21/pςkFμ,α,l,w¯,ς−1ςpε+γ,δ,v,cr∘θς−1ςpυ;p~+ςkFμ,α,l,w¯,ς−1ςpυ−γ,δ,v,cr∘θς−1ςpε;p~≤h12ςkFμ,α,l,w¯,ς−1ςpε+γ,δ,v,cσ∘r∘θς−1ςpυ;p~+ςkFμ,α,l,w¯,ς−1ςpυ−γ,δ,v,cσ∘r∘θς−1ςpε;p~≤h12σςε+σςυ×∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~rtςpε+1−tςpυ1/phtdt+∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~rtςpε+1−tςpυ1/ph1−tdt,where w¯=w/ςpυ−ςpεμ, and θt=ς1/pt for t∈εp,υp(ii)
Ifp<0, in that case,(30)σςpε+ςpυ21/pςkFμ,α,l,w¯,ς−1ςpυ+γ,δ,v,cr∘θς−1ςpε;p~+ςkFμ,α,l,w¯,ς−1ςpε−γ,δ,v,cr∘θς−1ςpυ;p~≤h12ςkFμ,α,l,w¯,ς−1ςpυ+γ,δ,v,cσ∘r∘θς−1ςpε;p~+ςkFμ,α,l,w¯,ς−1ςpε−γ,δ,v,cσ∘r∘θς−1ςpυ;p~≤h12σςε+σςυ∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~rtςpε+1−tςpυ1/phtdt+∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~rtςpε+1−tςpυ1/ph1−tdt,where w¯=w/ςpε−ςpυμ, and θt=ς1/pt for t∈υp,εpProof.
We demonstrate claim (i) as noted below:(i)
Multiplying both sides of (17) by tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~rtςpε+1−tςpυ1/p and then integrating over 0,1, we have(31)σςpε+ςpυ21/p∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~rtςpε+1−tςpυ1/pdt≤h12∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~σtςpε+1−tςpυ1/prtςpε+1−tςpυ1/pdt+∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~σtςpυ+1−tςpε1/prtςpε+1−tςpυ1/pdt.
By choosingςx=tςpε+1−tςpυ, that is, ςpε+ςpυ−ςζ=tςpυ+1−tςpε, in (31) and using σς1/pζ=σςpε+ςpυ−ςζ1/p, we have
(32)σςpε+ςpυ21/p×∫ς−1ςpες−1ςpυςpυ−ςζα/k−1Eμ,α,lγ,δ,v,cw¯ςpυ−ςζμ;p~r∘θζς′ζdζ≤h12∫ς−1ςpες−1ςpυςpυ−ςζα/k−1Eμ,α,lγ,δ,v,cw¯ςpυ−ςζμ;p~σ∘θζr∘θζς′ζdζ+∫ς−1ςpες−1ςpυςζ−ςpεα/k−1Eμ,α,lγ,δ,v,cw¯ςζ−ςpεμ;p~σ∘θζr∘θζς′ζdζ.
This implies(33)σςpε+ςpυ21/pςkFμ,α,l,w¯,ς−1ςpε+γ,δ,v,cr∘θς−1ςpυ;p~≤h12ςkFμ,α,l,w¯,ς−1ςpε+γ,δ,v,cσ∘r∘θς−1ςpυ;p~+ςkFμ,α,l,w¯,ς−1ςpυ−γ,δ,v,cσ∘r∘θς−1ςpε;p~.
By using Lemma15 (i) in the above inequality, we get the first inequality of (29).
Now, to demonstrate the second side of (29), multiplying both sides of (20) with tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~rtςpε+1−tςpυ1/p, and next integrating over 0,1, we obtain
(34)∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~rtςpε+1−tςpυ1/pσtςpε+1−tςpυ1/pdt+∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~rtςpε+1−tςpυ1/pσtςpυ+1−tςpε1/pdt≤h12σςε+σςυ∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~rtςpε+1−tςpυ1/pht+h1−tdt.
Settingςζ=tςpε+1−tςpυ and using σς1/pζ=σςpε+ςpυ−ςζ1/p in (34), we have
(35)ςkFμ,α,l,w¯,ς−1ςpε+γ,δ,v,cσ∘r∘θς−1ςpυ;p~+ςkFμ,α,l,w¯,ς−1ςpυ−γ,δ,v,cσ∘r∘θς−1ςpε;p~≤h12σςε+σςυ×∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~rtςpε+1−tςpυ1/pht+h1−tdt.
By utilizing Lemma15 (i) in the above inequality, we get the second side of (29).
(ii)
The proof is similar to the proof of (i)Corollary 17.
By utilizing (29) and (30), some more k-fractional inequalities are offered as noted below:
(i)
By choosingς=I and p~=0, we acquire(36)σεp+υp21/p∫εpυpυp−ζα/k−1Eμ,α,lγ,δ,v,cw¯υp−ζμr∘θζdζ≤h12∫εpυpυp−ζα/k−1Eμ,α,lγ,δ,v,cw¯υp−ζμσ∘θζr∘θζdζ+∫εpυpζ−εpα/k−1Eμ,α,lγ,δ,v,cw¯ζ−εpμσ∘θζr∘θζdζ≤h12σε+συ∫01tα/k−1Eμ,α,lγ,δ,v,cwtμrtεp+1−tυp1/phtdt+∫01tα/k−1Eμ,α,lγ,δ,v,cwtμrtεp+1−tυp1/ph1−tdt(ii)
By choosingς=I and w=p~=0, we acquire(37)σεp+υp21/p∫εpυpυp−ζα/k−1r∘θζdζ≤h12∫εpυpυp−ζα/k−1σ∘θζr∘θζdζ+∫εpυpζ−εpα/k−1σ∘θζr∘θζdζ≤h12σε+συ∫01tα/k−1rtεp+1−tυp1/phtdt+∫01tα/k−1rtεp+1−tυp1/ph1−tdt(iii)
By choosingp~=0, rζ=1, ς=I, and p=−1, we acquire(38)σ2ευε+υ∫1/ε1/υ1υ−ζα/k−1Eμ,α,lγ,δ,v,cw¯1υ−ζμθζdζ≤h12∫1/ε1/υ1υ−ζα/k−1Eμ,α,lγ,δ,v,cw¯1υ−ζμσ∘θζθζdζ+∫1/ε1/υζ−1εα/k−1Eμ,α,lγ,δ,v,cw¯ζ−1εμσ∘θζθζdζ≤h12σε+συ∫01tα/k−1Eμ,α,lγ,δ,v,cwtμhtdt+∫01tα/k−1Eμ,α,lγ,δ,v,cwtμh1−tdt(iv)
By choosingς=I, p=−1, and rζ=1, we acquire(39)σ2ευε+υ∫1/ε1/υ1υ−ζα/k−1Eμ,α,lγ,δ,v,cw¯1υ−ζμ;p~θζdζ≤h12∫1/ε1/υ1υ−ζα/k−1Eμ,α,lγ,δ,v,cw¯1υ−ζμ;p~σ∘θζθζdζ+∫1/ε1/υζ−1εα/k−1Eμ,α,lγ,δ,v,cw¯ζ−1εμ;p~σ∘θζθζdζ≤h12σε+συ∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~htdt+∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~h1−tdt(v)
By choosingw=p~=0, rζ=1, p=−1, and ς=I, we acquire(40)σ2ευε+υ∫1/ε1/υ1υ−ζα/k−1θζdζ≤h12∫1/ε1/υ1υ−ζα/k−1σ∘θζθζdζ+∫1/ε1/υζ−1εα/k−1σ∘θζθζdζ≤h12σε+συ∫01tα/k−1htdt+∫01tα/k−1h1−tdt(vi)
By choosingp=−1, we acquire(41)σς−1ε+ς−1υ2−1×∫ς−1ς−1ες−1ς−1υς−1υ−ςζα/k−1Eμ,α,lγ,δ,v,cw¯ς−1υ−ςζμ;p~r∘θζς′ζdζ≤h12∫ς−1ς−1ες−1ς−1υς−1υ−ςζα/k−1Eμ,α,lγ,δ,v,cw¯ς−1υ−ςζμ;p~σ∘θζr∘θζς′ζdζ+∫ς−1ς−1ες−1ς−1υςζ−ς−1εα/k−1Eμ,α,lγ,δ,v,cw¯ςζ−ς−1εμ;p~σ∘θζr∘θζς′ζdζ≤h12σςε+σςυ∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~rςεςυtςυ+1−tςεhtdt+∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~rςεςυtςυ+1−tςεh1−tdtRemark 18.
The mentionedk-fractional inequalities are further connected with foreknown conclusions as noted below:
(i)
By settinght=t and k=1 in Corollary 17 (ii), Theorem 9 of [6] is obtained(ii)
By settinght=t and k=1 in Corollary 17 (iii), Theorem 2.1 of [24] is obtained(iii)
By settinght=t and k=1 in Corollary 17 (iv), Theorem 2.1 of [5] is obtained(iv)
By settinght=t and k=1 in Corollary 17 (v), Theorem 4 of [1] is obtained(v)
By settinght=t and k=1 in Corollary 17 (vi), Theorem 2.5 of [27] is obtainedIn the next theorem, we offer another type of the Hadamard inequality.Theorem 19.
Leth:J⟶ℝ is a nonnegative and nonzero function. Also, let σ,ς:ε,υ⟶ℝ, 0<ε<υ, be the functions such that σ is positive and σ∈L1ε,υ and ς is differentiable and absolutely increasing. If σ is p,h-convex, p∈R\0, then for generalized k-fractional integral operators (6) and (7), we have the following:
(i)
Ifp>0, in that case,(42)σςpε+ςpυ21/pςkFμ,α,l,w¯¯,ς−1ςpε+ςpυ/2+γ,δ,v,c1ς−1ςpυ;p~≤h12ςkFμ,α,l,w¯¯,ς−1ςpε+ςpυ/2+γ,δ,v,cσ∘θς−1ςpυ;p~+ςkFμ,α,l,w¯¯,ς−1ςpε+ςpυ/2−γ,δ,v,cσ∘θς−1ςpε;p~≤h12σςε+σςυ×∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~ht2dt+∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~h2−t2dt,where w¯¯=2μw/ςpυ−ςpεμ, and θt=ς1/pt for t∈εp,υp(ii)
Ifp<0, in that case,(43)σςpε+ςpυ21/pςkFμ,α,l,w¯¯,ς−1ςpε+ςpυ/2−γ,δ,v,c1ς−1ςpυ;p~≤h12ςkFμ,α,l,w¯¯,ς−1ςpε+ςpυ/2+γ,δ,v,cσ∘θς−1ςpε;p~+ςkFμ,α,l,w¯¯,ς−1ςpε+ςpυ/2−γ,δ,v,cσ∘θς−1ςpυ;p~≤h12σςε+σςυ×∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~ht2dt+∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~h2−t2dt,where w¯¯=2μw/ςpε−ςpυμ, and θt=ς1/pt for t∈υp,εpProof.
We demonstrate claim (i) as noted below:(i)
Choosingςζ=t/2ςpε+2−t/2ςpυ1/p and ςη=t/2ςpυ+2−t/2ςpε1/p in (16), we have(44)σςpε+ςpυ21/p≤h12σt2ςpε+2−t2ςpυ1/p+σt2ςpυ+2−t2ςpε1/p.
Multiplying both sides of (44) by tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~ and then integrating over 0,1, we have
(45)σςpε+ςpυ21/p∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~dt≤h12∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~σt2ςpε+2−t2ςpυ1/pdt+∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~σt2ςpυ+2−t2ςpε1/pdt.
By choosingςζ=t/2ςpε+2−t/2ςpυ1/p and ςη=t/2ςpυ+2−t/2ςpε1/p in (45), then by using k-fractional integral operators (6) and (7), the first inequality of (42) is obtained.
Now, to demonstrate the second side of (42), once again p,h-convexity of σ over ε,υ and for t∈0,1, we obtain
(46)σt2ςpε+2−t2ςpυ1/p+σt2ςpυ+2−t2ςpε1/p≤σςε+σςυht2+h2−t2.
Multiplying both sides of (46) by h1/2tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~ and then integrating over 0,1, we have
(47)h12∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~σt2ςpε+2−t2ςpυ1/pdt+∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~σt2ςpυ+2−t2ςpε1/pdt≤h12σςε+σςυ×∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~ht2dt+∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~h2−t2dt.
Choosingςζ=t/2ςpε+2−t/2ςpυ and ςη=t/2ςpυ+2−t/2ςpε in (47) and by utilizing k-fractional operators (6) and (7), the second side of (42) is acquired.
(ii)
Proof is the same as the proof of (i)Corollary 20.
By utilizing (42) and (43), some more k-fractional inequalities are offered as noted below:
(i)
By choosingς=I and p~=0,p=−1, we acquire(48)σ2ευε+υ∫2ευ/ε+υ1/υ1υ−ζα/k−1Eμ,α,lγ,δ,v,cw¯¯1υ−ζμdζ≤h12∫2ευ/ε+υ1/υ1υ−ζα/k−1Eμ,α,lγ,δ,v,cw¯¯1υ−ζμσζ−1dζ+∫1/ε2ευ/ε+υη−1εα/k−1Eμ,α,lγ,δ,v,cw¯¯η−1εμση−1dη≤h12σε+συ×∫01tα/k−1Eμ,α,lγ,δ,v,cwtμht2dt+∫01tα/k−1Eμ,α,lγ,δ,v,cwtμh2−t2dt(ii)
By choosingp=−1 and ς=I, we acquire(49)σ2ευε+υ∫2ευ/ε+υ1/υ1υ−ζα/k−1Eμ,α,lγ,δ,v,cw¯¯1υ−ζμ;p~dζ≤h12∫2ευ/ε+υ1/υ1υ−ζα/k−1Eμ,α,lγ,δ,v,cw¯¯1υ−ζμ;p~σζ−1dζ+∫1/ε2ευ/ε+υη−1εα/k−1Eμ,α,lγ,δ,v,cw¯¯η−1εμ;p~ση−1dη≤h12σε+συ×∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~ht2dt+∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~h2−t2dt(iii)
By choosingp=−1, we acquire(50)σς−1ε+ς−1υ2−1×∫ς−1ς−1ε+ς−1υ/2ς−1ς−1υς−1υ−ςζα/k−1Eμ,α,lγ,δ,v,cw¯¯ς−1υ−ςζμ;p~ς′ζdζ≤h12∫ς−1ς−1ε+ς−1υ/2ς−1ς−1υς−1υ−ςζα/k−1Eμ,α,lγ,δ,v,cw¯¯ς−1υ−ςζμ;p~σς−1ζς′ζdζ+∫ς−1ς−1ες−1ς−1ε+ς−1υ/2ςη−ς−1εα/k−1Eμ,α,lγ,δ,v,cw¯¯ςη−ς−1εμ;p~σς−1ης′ηdη≤h12σςε+σςυ×∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~ht2dt+∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~h2−t2dtRemark 21.
The mentionedk-fractional inequalities are farther connected with foreknown conclusions as noted below:
(i)
By choosinght=t and k=1 in Corollary 20 (i), Theorem 2.3 of [24] is acquired(ii)
By choosinght=t and k=1 in Corollary 20 (ii), Theorem 2.3 of [5] is acquired(iii)
By choosinght=t and k=1 in Corollary 20 (iii), Theorem 2.7 of [27] is acquiredThe second type of the Fejér-Hadamard inequality for generalizedk-fractional integrals is given as noted below.Theorem 22.
Leth:J⟶ℝ be a nonnegative and nonzero function. Also, let σ,ς,r:ε,υ⟶ℝ with 0<ε<υ be the functions such that σ is positive and σ∈L1ε,υ, ς is differentiable and absolutely increasing, and r is a nonnegative and integrable function. If σ is p,h-convex, p∈R\0 and σς1/pζ=σςpε+ςpυ−ςζ1/p, then the following inequalities for generalized k-fractional integral operators (6) and (7) hold:
(i)
Ifp>0, in that case,(51)σςpε+ςpυ21/pςkFμ,α,l,w¯¯,ς−1ςpε+ςpυ/2+γ,δ,v,cr∘θς−1ςpυ;p~≤h12ςkFμ,α,l,w¯¯,ς−1ςpε+ςpυ/2+γ,δ,v,cr∘σ∘θς−1ςpυ;p~+ςkFμ,α,l,w¯¯,ς−1ςpε+ςpυ/2−γ,δ,v,cr∘σ∘θς−1ςpε;p~≤h12σςε+σςυ×∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~rt2ςpε+2−t2ςpυ1/pht2dt+∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~rt2ςpε+2−t2ςpυ1/ph2−t2dt,where w¯¯=2μw/ςpυ−ςpεμ, and θt=ς1/pt for t∈εp,υp(ii)
Ifp<0, in that case,(52)σςpε+ςpυ21/pςkFμ,α,l,w¯¯,ς−1ςpε+ςpυ/2−γ,δ,v,cr∘θς−1ςpυ;p~≤h12ςkFμ,α,l,w¯¯,ς−1ςpε+ςpυ/2+γ,δ,v,cr∘f∘θς−1ςpε;p~+ςkFμ,α,l,w¯¯,ς−1ςpε+ςpυ/2−γ,δ,v,cr∘σ∘θς−1ςpυ;p~≤h12σςε+σςυ×∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~rt2ςpε+2−t2ςpυ1/pht2dt+∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~rt2ςpε+2−t2ςpυ1/ph2−t2dt,where w¯¯=2μw/ςpε−ςpυμ, and θt=ς1/pt for t∈υp,εpProof.
We demonstrate the first claim as noted below:(i)
Multiplying (44) by tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~rt/2ςpε+2−t/2ςpυ1/p and then integrating over 0,1, we have(53)σςpε+ςpυ21/p∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~rt2ςpε+2−t2ςpυ1/pdt≤h12∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~rt2ςpε+2−t2ςpυ1/pσt2ςpε+2−t2ςpυ1/pdt+∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~rt2ςpε+2−t2ςpυ1/pσt2ςpυ+2−t2ςpε1/pdt.
By choosingςζ=t/2ςpε+2−t/2ςpυ and ςη=t/2ςpυ+2−t/2ςpε, that is, ςpε+ςpυ−ςζ=t/2ςpυ+2−t/2ςpε, in (53), in that case, by utilizing σς1/pζ=σςpε+ςpυ−ςζ1/p and k-fractional integral operators (6) and (7), the first side of (51) is obtained.
Now, to demonstrate the second side of (51), multiplying both sides of (46) with h1/2tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~rt/2ςpε+2−t/2ςpυ1/p and integrating over 0,1, we obtain
(54)h12∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~rt2ςpε+2−t2ςpυ1/pσt2ςpε+2−t2ςpυ1/pdt+∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~rt2ςpε+2−t2ςpυ1/pσt2ςpυ+2−t2ςpε1/pdt≤h12σςε+σςυ×∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~rt2ςpε+2−t2ςpυ1/pht2dt+∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~rt2ςpε+2−t2ςpυ1/ph2−t2dt.
By choosingςζ=t/2ςpε+2−t/2ςpυ and ςη=t/2ςpυ+2−t/2ςpε, that is, ςpε+ςpυ−ςζ=t/2ςpυ+2−t/2ςpε, in (54), then by using σς1/pζ=σςpε+ςpυ−ςζ1/p and k-fractional integral operators (6) and (7), the second inequality of (51) is acquired.
(ii)
Proof is the same as the proof of (i)Corollary 23.
By utilizing (51) and (52), some more k-fractional inequalities are offered as noted below:
(i)
By choosingς=I and p~=0,p=−1, we acquire(55)σ2ευε+υ∫2ευ/ε+υ1/υ1υ−ζα/k−1Eμ,α,lγ,δ,v,cw¯¯1υ−ζμrζ−1dζ≤h12∫2ευ/ε+υ1/υ1υ−ζα/k−1Eμ,α,lγ,δ,v,cw¯¯1υ−ζμσζ−1rζ−1dζ+∫1/ε2ευ/ε+υζ−1εα/k−1Eμ,α,lγ,δ,v,cw¯¯ζ−1εμσζ−1rζ−1dζ≤h12σε+συ∫01tα/k−1Eμ,α,lγ,δ,v,cwtμr2ευtυ+2−tεht2dt+∫01tα/k−1Eμ,α,lγ,δ,v,cwtμr2ευtυ+2−tεh2−t2dt(ii)
By choosingp=−1 and ς=I, we acquire(56)σ2ευε+υ∫2ευ/ε+υ1/υ1υ−ζα/k−1Eμ,α,lγ,δ,v,cw¯¯1υ−ζμ;p~rζ−1dζ≤h12∫2ευ/ε+υ1/υ1υ−ζα/k−1Eμ,α,lγ,δ,v,cw¯¯1υ−ζμ;p~σζ−1rζ−1dζ+∫1/ε2ευ/ε+υζ−1εα/k−1Eμ,α,lγ,δ,v,cw¯¯ζ−1εμ;p~σζ−1rζ−1dζ≤h12σε+συ∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~r2ευtυ+2−tεht2dt+∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~r2ευtυ+2−tεh2−t2dt(iii)
By choosingp=−1, we acquire(57)σς−1ε+ς−1υ2−1×∫ς−1ς−1ε+ς−1υ/2ς−1ς−1υς−1υ−ςζα/k−1Eμ,α,lγ,δ,v,cw¯¯ς−1υ−ςζμ;p~rς−1ζς′ζdζ≤h12∫ς−1ς−1ε+ς−1υ/2ς−1ς−1υς−1υ−ςζα/k−1Eμ,α,lγ,δ,v,cw¯¯ς−1υ−ςζμ;p~σς−1ζrς−1ζς′ζdζ+∫ς−1ς−1ες−1ς−1ε+ς−1υ/2ςζ−ς−1εα/k−1Eμ,α,lγ,δ,v,cw¯¯ςζ−ς−1εμ;p~σς−1ζrς−1ζς′ζdζ≤h12σςε+σςυ∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~r2ςεςυtςυ+2−tςεht2dt+∫01tα/k−1Eμ,α,lγ,δ,v,cwtμ;p~r2ςεςυtςυ+2−tςεh2−t2dtRemark 24.
The well-knownk-fractional inequalities are further noted as below:
(i)
By choosinght=t and k=1 in Corollary 23 (i), Theorem 2.6 of [24] is acquired(ii)
By choosinght=t and k=1 in Corollary 23 (ii), Theorem 2.6 of [5] is acquired(iii)
By choosinght=t and k=1 in Corollary 23 (iii), Theorem 2.10 of [27] is acquired


## 3. Conclusion

In this article, generalized versions ofk-fractional Hadamard and Fejér-Hadamard inequalities are presented. To obtain these inequalities, k-fractional integral operators including the Mittag-Leffler function have been used for p,h-convex functions. Many published results in the literature are directly connected with the findings of this paper. Some corollaries have been formulated as new k-fractional Hadamard and Fejér-Hadamard inequalities.



---

*Source: 3832330-2022-04-28.xml*