# New Generalized Riemann–Liouville Fractional Integral Versions of Hadamard and Fejér–Hadamard Inequalities

Authors: Kamsing Nonlaopon; Ghulam Farid; Ammara Nosheen; Muhammad Yussouf; Ebenezer Bonyah
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, a new class of functions, namely, exponentiallyα,h−m−p-convex functions is introduced to unify various classes of functions already defined in the subject of convex analysis. By using this class of functions, generalized versions of well known fractional integral inequalities of Hadamard and Fejér–Hadamard type are obtained. The results of this paper generate fractional integral inequalities of Hadamard and Fejér–Hadamard type for various types of convex and exponentially convex functions simultaneously.


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## Body

## 1. Introduction and Preliminary Results

Inequalities are important tools for mathematical modeling of problems that occur in the diverse fields of science and engineering. Convex functions are very useful in establishing new and generalized inequalities. For example, Jensen’s inequality for convex functions is one of the most celebrating inequalities in the literature. Many classical inequalities are direct consequences of Jensen’s inequality. Motivated from the properties and representations of convex functions, researchers have published a lot of new definitions of functions which are usually utilized for extensions, refinements, and generalizations of well known inequalities. In recent decades, it becomes a fashion for authors to generalize the classical concepts related to ordinary derivatives and integrals via fractional integral/derivative operators. These techniques are used frequently in generalizing the classical mathematical inequalities. For a detailed study, we refer the readers to [1–13].The goal of this paper is to establish general Riemann–Liouville fractional integral inequalities of Hadamard and Fejér–Hadamard type by defining a new class of functions which will concurrently hold for many kinds of convex and exponentially convex functions. Next, we give definitions of Riemann–Liouville fractional integrals which we will utilize to establish main results. After that we will give definition of convex function with a detailed discussion on related definitions.Definition 1 (see [14]).
Letf∈L1a,b. Then, the left- and right-sided Riemann–Liouville fractional integrals of f of order τ∈ℝτ>0 are given as follows:(1)Ia+τfx=1Γτ∫axx−tτ−1ftdt,x>a,(2)Ib−τfx=1Γτ∫xbt−xτ−1ftdt,x<b,where Γ⋅ is the gamma function.Definition 2 (see [15]).
A real-valued functionf:a,b⟶ℝ is called convex if the following inequality holds:(3)ftx+1−ty≤tfx+1−tfy,∀x,y∈a,b,t∈0,1.There are many kinds of functions which have been defined inspiring by inequality (3). For example, functions, namely, p-convex [16], h-convex [17], m-convex [15], s-convex [18], harmonically convex [6], and many others are defined just by convenient possible modifications in the inequality (3). Moreover, s,m-convex [19], α,m-convex [20], h−m-convex [21], α,h−m-convex [22], and p,h-convex [3] functions have been defined elegantly after the definition of convex function. Further, in [23], the notion of α,h−m−p-convex function is defined which unifies all the aforementioned convexities.There also exists a class of exponentially convex functions stated as follows.Definition 3 (see [24]).
A real-valued functionf:J⊂ℝ⟶ℝ+ is called exponentially convex on J if the following inequality holds:(4)ftx+1−ty≤tfxeηx+1−tfyeηy,t∈0,1,∀x,y∈J,η∈ℝ.The term exponentially convex function is used likewise to convex function, and notions of exponentiallyp-convex [25], exponentially h-convex [26], exponentially s-convex [25] have been introduced. Also, definitions of exponentially s,m-convex [27], exponentially α,m-convex [26], exponentially h−m-convex [26], exponentially α,h−m-convex [28], and exponentially p,h-convex [29] functions have been published.The exponentiallyα,h−m-convex function is defined as follows.Definition 4 (see [28]).
LetJ⊆ℝ be an interval containing 0,1, and let h:J⟶ℝ be a nonnegative function. Then, a function f:I⟶ℝ on an interval of real line is said to be exponentially α,h−m-convex, if for all x,y∈I, t∈0,1, α,m∈0,1, and η∈ℝ, the following inequality holds:(5)ftx+m1−ty≤htαfxeηu1+mh1−tαfyeηy.The following example is important to distinguish an exponentially convex function from convex function.Example 1 (see [30]).
The functionfx=xexp−x is exponentially 1,Id−1-convex function but not 1,Id−1-convex function. More precisely the function f is exponentially convex function on 0,∞ but not a convex function on this domain.All the aforementioned definitions have been used to derive Hadamard and Fejér–Hadamard type inequalities. We are motivated to combine all types of convexities and exponential convexities in a single definition. We will define exponentiallyα,h−m−p-convex function and prove Hadamard and Fejér–Hadamard type inequalities which will unify a plenty of classical inequalities.The paper is organized as follows: In Section2, a new class of functions will be called exponentially α,h−m−p-convex function. Some new definitions will be deduced in connection with existing definitions in the literature of mathematical inequalities. In Section 3, we will present the Hadamard and Fejér–Hadamard inequalities for newly defined functions via Riemann–Liouville fractional integrals. We will identify a number of implications of the results established in this section.


## 2. Exponentiallyα,h−m−p-Convex Function and Deduced Definitions

We define exponentiallyα,h−m−p-convex function as follows.Definition 5.
LetJ⊆ℝ be an interval containing 0,1, and let h:J⟶ℝ be a nonnegative function. Let I⊂0,∞ be a real interval and p∈ℝ∖0. A function f:I⟶ℝ is called exponentially α,h−m−p-convex if for t∈0,1, η∈ℝ and α,m∈0,12, the following inequality holds:(6)ftap+m1−tbp1/p≤htαfaeηa+mh1−tαfbeηb,where a,b∈I provided tap+m1−tbp1/p∈I.Remark 1.
The following convex functions are reproduced from above definition:(i)
In Definition5, if we put p=−1, m=α=1, and η=0, we have harmonically h-convex function reproduced (see Definition 2.10 in [31]).(ii)
In Definition5, for p=1 and η=0, we have α,h−m-convex function reproduced (see Definition 4.5 in [20]).(iii)
In Definition5, for α=m=1 and η=0, we have p,h-convex function reproduced (see [3]).(iv)
In Definition5, for p=1, exponentially α,h−m-convex function is reproduced (see Definition 1 in [26]). For further deduced functions, see Remark 1 in [26].(v)
In Definition5, for α=p=1, exponentially h−m-convex function is reproduced (see Definition 2 in [26]).(vi)
In Definition5, for p=1 and ht=t, exponentially α,m-convex function is reproduced (see Definition 3 in [26]).(vii)
In Definition5, for p=−1, α=m=1, ht=ts, and η=0, we have harmonic s-convex function in second sense reproduced (see Remark 1 in [32]).(viii)
In Definition5, for p=−1, α=m=1, ht=t, and η=0, we have harmonic convex function reproduced (see [33]).(ix)
In Definition5, for p=1, α=1, ht=ts, and η=0, we have s,m-convex function in second sense reproduced (see Definition 1.2 in [19]).(x)
In Definition5, for p=−1, α=1, ht=t, and η=0, we have m-HA-convex function reproduced (see Definition 2 in [34]).(xi)
In Definition5, for p=−1, ht=t, and η=0, α,m-HA-convex function is reproduced (see Definition 2.1 in [35]).Forα=1 in (6), we get exponentially h−m−p-convex function as follows:(7)ftap+m1−tbp1/p≤htfaeηa+mh1−tfbeηb.Forht=t in (6), we get exponentially α,m−p-convex function as follows:(8)ftap+m1−tbp1/p≤tαfaeηa+m1−tαfbeηb.Form=1 in (6), we get exponentially α,h−p-convex function as follows:(9)ftap+1−tbp1/p≤htαfaeηa+h1−tαfbeηb.Forα=1 and ht=ts in (6), we get exponentially s,m−p-convex function as follows:(10)ftap+m1−tbp1/p≤tsfaeηa+m1−tsfbeηb.Forht=ts in (6), we get exponentially s,m−p-Godunova–Levin function of second kind as follows:(11)ftap+m1−tbp1/p≤1tsfaeηa+m1−tsfbeηb.Form=α=1 and ht=1 in (6), we get exponentially p,P-convex function as follows:(12)ftap+1−tbp1/p≤faeηa+fbeηb.Forα=m=1, p=−1, and ht=1/t in (6), we get exponentially Godunova–Levin type exponentially harmonic convex function as follows:(13)fabtb+1−ta≤1tfaeηa+11−tfbeηb.Forα=m=1, p=−1, and ht=1/ts in (6), we get exponentially harmonic convex function as follows:(14)fabtb+1−ta≤1tsfaeηa+11−tsfbeηb.Forp=−1 in (6), we get exponentially α,h−m-HA-convex function as follows:(15)fabtb+m1−ta≤htαfaeηa+mh1−tαfbeηb.Forp=−1 and m=1 in (6), we get exponentially α,h-HA-convex function as follows:(16)fabtb+1−ta≤htαfaeηa+h1−tαfbeηb.Forp=−1, m=α=1, and ht=t in (6), we get exponentially HA-convex function as follows:(17)fabtb+1−ta≤tfaeηa+1−tfbeηb.Forp=−1 and ht=t in (6), we get exponentially α,m-HA-convex function as follows:(18)fabtb+m1−ta≤tαfaeηa+m1−tαfbeηb.From now to onward, we will use the notationEpα,h−m for exponentially α,h−m−p-convex function.


## 3. Inequalities of Hadamard Type forEpα,h−m Function

Theorem 1.
Letf:I⊂0,∞⟶ℝ be an Epα,h−m positive function as defined in Definition 5 and f∈L1a,b,a,b∈I,a<b. Then, for α,m∈0,12, one can have fractional integral inequalities for operators (1) and (2) as follows.(i) Forp>0, we have(19)fap+mbp21/p≤Γτ+1mbp−apτD1ηh12αIap+τf°ξmbp+D2ηmτ+1h2α−12αIbp−τf°ξapm≤τD1ηh12αfaeηa+D2ηmh2α−12αfbeηb∫01tτ−1htαdt+mD1ηh12αfbeηb+D2ηmh2α−12αfa/m2eηa/m2∫01tτ−1h1−tαdt,where ξz=z1/p, z∈ap,mbp, D1η=e−ηbm1/p for η<0, D1η=e−ηa for η≥0, D2η=e−ηa/m1/p for η>0, and D2η=e−ηb for η≤0.(ii) Forp<0, we have(20)fap+mbp21/p≤Γτ+1ap−mbpτD3ηh12αIap−τf°ξmbp+D4ηmτ+1h2α−12αIbp+τf°ξapm≤τD3ηh12αfaeηa+D4ηmh2α−12αfbeηb∫01tτ−1htαdt+mD3ηh12αfbeηb+D4ηmh2α−12αfa/m2eηa/m2∫01tτ−1h1−tαdt,where ξz=z1/p, z∈mbp,ap, D3η=e−ηbm1/p for η<0, D3η=e−ηa for η≥0, D4η=e−ηa/m1/p for η<0, and D4η=e−ηb for η≥0.Proof.
(i) By using (6), one can have the following inequality:(21)fxp+myp21/p≤h12αfxeηx+mh2α−12αfyeηy.
Forx=tap+m1−tbp1/p and y=tbp+1−tap/m1/p in (21), we get(22)fap+mbp21/p≤h12αftap+m1−tbp1/peηtap+m1−tbp1/p+mh2α−12αftbp+1−tap/m1/peηtbp+1−tap/m1/p.
Multiplying the above inequality withtτ−1 on both sides and integrating over 0,1, we have(23)fap+mbp21/p∫01tτ−1dt≤h12α∫01tτ−1ftap+m1−tbp1/peηtap+m1−tbp1/pdt+mh2α−12α∫01tτ−1ftbp+1−tap/m1/peηtbp+1−tap/m1/pdt.
Settap+m1−tbp=x, that is, t=mbp−x/mbp−ap and tbp+1−tap/m=y, that is, t=y−ap/m/bp−ap/m in right hand side of the above inequality. Then, after some calculations, one can obtain the first inequality of (19).
On the other hand, by using (6) on the right hand side of (22), one can obtain the inequality as follows:(24)h12αftap+m1−tbp1/peηtap+m1−tbp1/p+mh2α−12αftbp+1−tap/m1/peηtbp+1−tap/m1/p≤h1/2αeηtap+m1−tbp1/phtαfaeηa+mh1−tαfbeηb+mh2α−1/2αeηtbp+1−tap/m1/phtαfbeηb+mh1−tαfa/m2eηa/m2.
Multiplying the above inequality withtτ−1, by integrating over 0,1, one can get(25)D1ηh12α∫01tτ−1ftap+m1−tbp1/pdt+D2ηmh2α−12α∫01tτ−1ftbp+1−tapm1/pdt≤D1ηh12αfaeηa+D2ηmh2α−12αfbeηb∫01tτ−1htαdt+mD1ηh12αfbeηb+D2ηmh2α−12αfa/m2eηa/m2∫01tτ−1h1−tαdt.
Settap+m1−tbp=x, that is. t=mbp−x/mbp−ap and tbp+1−tap/m=y, that is, t=y−ap/m/bp−ap/m in (25). Then, after some calculations, the second inequality of (19) is obtained.
(ii) Proof is similar as (i).Remark 2.
(i)
In Theorem1 (i), if we put α=m=1, ht=t, η=0, and p=1, then Theorem 2 in [12] is reproduced.(ii)
In Theorem1 (i), if we put α=m=1, p=1, ht=t, η=0, and τ=1, then classical Hadamard inequality is reproduced.(iii)
In Theorem1 (ii), if we put α=m=1, ht=t, η=0, and p=−1, then Theorem 4 in [8] is reproduced.The other variant of the Hadamard inequality is stated and proved as follows.Theorem 2.
Let the assumptions of Theorem1 hold. Then, we have the following inequalities.(i) Forp>0, we have(26)fap+mbp21/p≤Γτ+12mbp−apτ×D1ηh12αIap+mbp/2+τf∘ξmbp+D2ηmτ+1h2α−12αIap+mbp/2m−τf°ξapm≤τD1ηh12αfaeηa+D2ηmh2α−12αfbeηb∫01tτ−1ht2αdt+mD1ηh12αfbeηb+D2ηmh2α−12αfa/m2eηa/m2∫01tτ−1h1−t2αdt,where ξz=z1/p, z∈ap,mbp, D1η,andD2η are same as given in Theorem 1 (i).(ii) Forp<0, we have(27)fap+mbp21/p≤Γτ+12ap−mbpτ×D3ηh12αIap+mbp/2−τf∘ξmbp+D4ηmτ+1h2α−12αIap+mbp/2m+τf°ξapm≤τD3ηh12αfaeηa+D4ηmh2α−12αfbeηb∫01tτ−1ht2αdt+mD3ηh12αfbeηb+D4ηmh2α−12αfa/m2eηa/m2∫01tτ−1h1−t2αdt,where ξz=z1/p, z∈mbp,ap, D3η,andD4η are same as given in Theorem 1 (ii).Proof.
(i) Forx=t/2ap+m1−t/2bp1/p and y=t/2bp+1−t/2ap/m1/p in (21), we get(28)fap+mbp21/p≤h12αft/2ap+m1−t/2bp1/peηt/2ap+m1−t/2bp1/p+mh2α−12αft/2bp+1−t/2ap/m1/peηt/2bp+1−t/2ap/m1/p.
Multiplying the above inequality withtτ−1 on both sides and integrating over 0,1, we have(29)fap+mbp21/p∫01tτ−1dt≤h12α∫01tτ−1ft/2ap+m1−t/2bp1/peηt/2ap+m1−t/2bp1/pdt+mh2α−12α∫01tτ−1ft/2bp+1−t/2ap/m1/peηt/2bp+1−t/2ap/m1/pdt.
Sett/2ap+m1−t/2bp=x, that is, t/2=mbp−x/mbp−ap and 1−t/2ap/m+t/2bp=y, that is, t/2=y−ap/m/bp−ap/m in right hand side of the above inequality. Then, after some calculations, one can obtain the first inequality of (26).
On the other hand, by applying theEpα,h−m of f, from right hand side of (28), one can obtain the following inequality:(30)h12αft/2ap+m1−t/2bp1/peηt/2ap+m1−t/2bp1/p+mh2α−12αft/2bp+1−t/2ap/m1/peηt/2bp+1−t/2ap/m1/p≤h1/2αeηt/2ap+m1−t/2bp1/pht2αfaeηa+mh1−t2αfbeηb+mh2α−1/2αeηt/2bp+1−t/2ap/m1/pht2αfbeηb+mh1−t2αfa/m2eηa/m2.
Multiplyingtτ−1 on both sides of (30), then by integrating on 0,1, one can get(31)D1ηh12α∫01tτ−1ft2ap+m1−t2bp1/pdt+D2ηmh2α−12α∫01tτ−1ft2bp+1−t2apm1/pdt≤D1ηh12αfaeηa+D2ηmh2α−12αfbeηb∫01tτ−1ht2αdt+mD1ηh12αfbeηb+D2ηmh2α−12αfa/m2eηa/m2∫01tτ−1h1−t2αdt.
Sett/2ap+m1−t/2bp=x, that is, t/2=mbp−x/mbp−ap and 1−t/2ap/m+t/2bp=y, that is, t/2=y−ap/m/bp−ap/m in (31). Then, after some calculations, the second inequality of (26) is obtained.
(ii) Proof is similar as (i).Remark 3.
(i)
In Theorem2 (i), if we put α=1=m, η=0, p>0, and ht=t, then Theorem 2.1(i) in [36] is reproduced.(ii)
In Theorem2 (ii), if we put α=1=m, η=0, p<0, and ht=t, then Theorem 2.1(ii) in [36] is reproduced.(iii)
In Theorem2 (i), if we put α=1=m, p=1, η=0, and ht=t, then Corollary 2.1 in [36] is reproduced.Remark 4.
From Theorems1 and 2, one can deduce results for convex, exponentially convex, Ep1,Id−1, Ep1,Id−m, Ep1,h−1, Epα,Id−m, Ep1,h−m, and Ep1,h−1 functions.


## 4. Fejér–Hadamard Type Inequalities forEpα,h−m Function

Theorem 3.
Letf:I⟶ℝ be an Epα,h−m positive function as given in Definition 5 and fap+mbp−x/m=fx,a,b∈I,a<b, m≠0. If g:I⟶ℝ is a positive function and f,g∈L1a,b, then one can have fractional integral inequalities for operators (1) and (2) as follows.(i) Forp>0, we have(32)fap+mbp21/pIap+τg∘ξmbp≤D1ηh12αIap+τfg∘ξmbp+D2ηmτ+1h2α−12αIbp−τfg°ξapm≤mbp−apτΓτD1ηh12αfaeηa+D2ηmh2α−12αfbeηb×∫01tτ−1gtap+m1−tbp1/phtαdt+mD1ηh12αfbeηb+D2ηmh2α−12αfa/m2eηa/m2×∫01tτ−1gtap+m1−tbp1/ph1−tαdt,where ξz=z1/p, z∈ap,mbp, fg∘ξ=f∘ξg∘ξ, D1η,andD2η are same as given in Theorem 1 (i).(ii) Forp<0, we have(33)fap+mbp21/pIap−τg∘ξmbp≤D3ηh12αIap−τfg∘ξmbp+D4ηmτ+1h2α−12αIbp+τfg∘ξapm≤ap−mbpτΓτD3ηh12αfaeηa+D4ηmh2α−12αfbeηb×∫01tτ−1gtap+m1−tbp1/phtαdt+mD3ηh12αfbeηb+D4ηmh2α−12αfa/m2eηa/m2×∫01tτ−1gtap+m1−tbp1/ph1−tαdt,where ξz=z1/p, z∈mbp,ap, fg∘ξ=f∘ξg∘ξ, D3η,andD4η are same as given in Theorem 1 (ii).Proof.
(i) Multiplying (22) by tτ−1gtap+m1−tbp1/p, then making integration on 0,1, the following inequality is yielded:(34)fap+mbp21/p∫01tτ−1gtap+m1−tbp1/pdt≤D1ηh12α∫01tτ−1ftap+m1−tbp1/pgtap+m1−tbp1/pdt+D2ηmh2α−12α∫01tτ−1ftbp+1−tapm1/pgtap+m1−tbp1/pdt.
Fortap+m1−tbp=x, that is, 1−tap/m+tbp=ap+mbp−x/m in (34) and then utilizing condition fx=fap+mbp−x/m and equations (1) and (2), the first inequality of (32) can be achieved.
Now, multiplyingtτ−1gtap+m1−tbp1/p with (24) and integrating over 0,1, we have(35)D1ηh12α∫01tτ−1ftap+m1−tbp1/pgtap+m1−tbp1/pdt+D2ηmh2α−12α∫01tτ−1ftbp+1−tapm1/pgtap+m1−tbp1/pdt≤D1ηh12αfaeηa+D2ηmh2α−12αfbeηb∫01tτ−1gtap+m1−tbp1/phtαdt+mD1ηh12αfbeηb+D2ηmh2α−12αfa/m2eηa/m2∫01tτ−1gtap+m1−tbp1/ph1−tαdt.
Again, settingtap+m1−tbp=x, that is, 1−tap/m+tbp=ap+mbp−x/m in (35) and utilizing condition fx=fap+mbp−x/m, then by using definitions (1) and (2), one can get second inequality of (32).
(ii) Proof is similar as (i).Remark 5.
(i)
In Theorem3 (i), if we put α=m=1, ht=t, gx=1, η=0, and p=1 then Theorem 2 in [12] is reproduced.(ii)
In Theorem3 (i), if we put α=m=1, p=1, ht=t, gx=1, η=0, and τ=1, then the Hadamard inequality is reproduced.(iii)
In Theorem3 (i), if we put α=m=1, p=1, ht=t, η=0, and τ=1 then classical Fejér–Hadamard inequality is reproduced.(iv)
In Theorem3 (ii), if we put α=m=1, ht=t, gx=1, η=0, and p=−1, then Theorem 4 in [8] is reproduced.(v)
In Theorem3 (ii), if we put α=m=1, ht=t, η=0, and p=−1 then Theorem 5 in [8] is reproduced.The second variant of the Fejér–Hadamard inequality is stated and proved as follows.Theorem 4.
Let the assumptions of Theorem3 hold. Then, we have the following inequalities.(i) Forp>0, we have(36)fap+mbp21/pIap+mbp/2+τg∘ξmbp≤D1ηh12αIap+mbp/2+τfg∘ξmbp+D2ηmτ+1h2α−12αIap+mbp/2m−τfg∘ξapm≤1Γτmbp−ap2τD1ηh12αfaeηa+D2ηmh2α−12αfbeηb×∫01tτ−1gt2ap+m1−t2bp1/pht2αdt+mD1ηh12αfbeηb+D2ηmh2α−12αfa/m2eηa/m2∫01tτ−1gt2ap+m1−t2bp1/ph1−t2αdt,where ξz=z1/p, z∈ap,mbp, fg∘ξ=f∘ξg∘ξ, D1η,andD2η are same as given in Theorem 1 (i).(ii) Forp<0, we have(37)fap+mbp21/pIap+mbp/2−τg∘ξmbp≤D3ηh12αIap+mbp/2−τfg∘ξmbp+D4ηmτ+1h2α−12αIap+mbp/2m+τfg∘ξapm≤1Γτap−mbp2τD3ηh12αfaeηa+D4ηmh2α−12αfbeηb×∫01tτ−1gt2ap+m1−t2bp1/pht2αdt+mD3ηh12αfbeηb+D4ηmh2α−12αfa/m2e​ηa/m2∫01tτ−1gt2ap+m1−t2bp1/ph1−t2αdt,where ξz=z1/p, z∈mbp,ap, fg∘ξ=f∘ξg∘ξ, D3η,andD4η are same as given in Theorem 1 (ii).Proof.
(i) Multiplying (28) by tτ−1gt/2ap+m1−t/2bp1/p and integrating over 0,1, the following inequality is yielded:(38)fap+mbp21/p∫01tτ−1gt2ap+m1−t2bp1/pdt≤D1ηh12α∫01tτ−1ft2ap+m1−t2bp1/pgt2ap+m1−t2bp1/pdt+D2ηmh2α−12α∫01tτ−1ft2bp+1−t2apm1/pgt2ap+m1−t2bp1/pdt.
Settingt/2ap+m1−t/2bp=x, that is 1−t/2ap/m+t/2bp=ap+mbp−x/m in (38) and using condition fx=fap+mbp−x/m and the definitions (1), (2), one can get first inequality of (36).
Now, multiplyingtτ−1gt/2ap+m1−t/2bp1/p with (30) and integrating over 0,1, we have(39)D1ηh12α∫01tτ−1ft2ap+m1−t2bp1/pgt2ap+m1−t2bp1/pdt+D2ηmh2α−12α∫01tτ−1ft2bp+1−t2apm1/pgt2ap+m1−t2bp1/pdt≤D1ηh12αfaeηa+D2ηmh2α−12αfbeηb∫01tτ−1gt2ap+m1−t2bp1/pht2αdt+mD1ηh12αfbeηb+D2ηmh2α−12αfa/m2eηa/m2∫01tτ−1gt2ap+m1−t2bp1/ph1−t2αdt.
Again fort/2ap+m1−t/2bp=x, that is, 1−t/2ap/m+t/2bp=ap+mbp−x/m in (39) and the utilizing condition fx=fap+mbp−x/m and equations (1) and (2), the second inequality of (36) can be achieved.
(ii) Proof is similar as (i).Remark 6.
(i)
In Theorem4 (i), if we put α=1=m, p>0, gx=1, η=0, and ht=t, then Theorem 2.1 (i) in [36] is reproduced.(ii)
In Theorem4 (ii), if we put α=1=m, p<0, gx=1, η=0, and ht=t, then Theorem 2.1(ii) in [36] is reproduced.(iii)
In Theorem4 (i), if we put α=1=m, p=1, gx=1, η=0, and ht=t, then Corollary 2.1 in [36] is reproduced.Remark 7.
From Theorems3 and 4, one can deduce results for convex, exponentially convex, Ep1,Id−1, Ep1,Id−m, Ep1,h−1, Epα,Id−m, Ep1,h−m, and Ep1,h−1 functions.


### 4.1. Results forEph−m Function

Forα=1 in Theorems 1–4, one can obtain the results for Eph−m function:Theorem 5.
With the same conditions of Theorem1, for Eph−m functions, the following inequalities hold:(i)
Forp>0, we have(40)1h1/2fap+mbp21/p≤Γτ+1mbp−apτD1ηIap+τf∘ξmbp+D2ηmτ+1Ibp−τf∘ξapm≤τD1ηfaeηa+D2ηmfbeηb∫01tτ−1htdt+mD1ηfbeηb+D2ηmfa/m2eηa/m2∫01tτ−1h1−tdt.(ii)
Forp<0, we have(41)1h1/2fap+mbp21/p≤Γτ+1ap−mbpτD3ηIap−τf∘ξmbp+D4ηmτ+1Ibp+τf∘ξapm≤τD3ηfaeηa+D4ηmfbeηb∫01tτ−1htdt+mD3ηfbeηb+D4ηmfa/m2eηa/m2∫01tτ−1h1−tdt.Theorem 6.
With the same conditions of Theorem2, for Eph−m function, the following inequalities hold:(i)
Forp>0, we have(42)1h1/2fap+mbp21/p≤Γτ+12mbp−apτD1ηIap+mbp/2+τf∘ξmbp+D2ηmτ+1Iap+mbp/2m−τf∘ξapm≤τD1ηfaeηa+D2ηmfbeηb∫01tτ−1ht2dt+mD1ηfbeηb+D2ηmfa/m2eηa/m2∫01tτ−1h1−t2dt.(ii)
Forp<0, we have(43)1h1/2fap+mbp21/p≤Γτ+12ap−mbpτD3ηIap+mbp/2−τf∘ξmbp+D4ηmτ+1Iap+mbp/2m+τf∘ξapm≤τD3ηfaeηa+D4ηmfbeηb∫01tτ−1ht2dt+mD3ηfbeηb+D4ηmfa/m2eηa/m2∫01tτ−1h1−t2dt.Theorem 7.
With the same conditions of Theorem3, for Eph−m functions, the following inequalities hold:(i)
Forp>0, we have(44)1h1/2fap+mbp21/pIap+τg∘ξmbp≤D1ηIap+τfg∘ξmbp+D2ηmIbp−τfg∘ξapm≤mbp−apτΓτD1ηfaeηa+D2ηmfbeηb∫01tτ−1gtap+m1−tbp1/phtdt+mD1ηfbeηb+D2ηmfa/m2e​ηa/m2∫01tτ−1gtap+m1−tbp1/ph1−tdt.(ii)
Forp<0, we have(45)1h1/2fap+mbp21/pIap−τg∘ξmbp≤D3ηIap−τfg∘ξmbp+D4ηmIbp+τfg∘ξapm≤ap−mbpτΓτD3ηfaeηa+D4ηmfbeηb∫01tτ−1gtap+m1−tbp1/phtdt+mD3ηfbeηb+D4ηmfa/m2e​ηa/m2∫01tτ−1gtap+m1−tbp1/ph1−tdt.Theorem 8.
With the same conditions of Theorem4, for Eph−m functions, the following inequalities hold:(i)
Forp>0, we have(46)1h1/2fap+mbp21/pIap+mbp/2+τg∘ξmbp≤D1ηIap+mbp/2+τfg∘ξmbp+D2ηmτ+1Iap+mbp/2m−τfg∘ξapm≤1Γτmbp−ap2τD1ηfaeηa+D2ηmfbeηb∫01tτ−1gt2ap+m1−t2bp1/pht2dt+mD1ηfbeηb+D2ηmfa/m2eηa/m2∫01tτ−1gt2ap+m1−t2bp1/ph1−t2dt.(ii)
Forp<0, we have(47)1h1/2fap+mbp21/pIap+mbp/2−τg∘ξmbp≤D3ηIap+mbp/2−τfg∘ξmbp+D4ηmτ+1Iap+mbp/2m+τfg∘ξapm≤1Γτap−mbp2τD3ηfaeηa+D4ηmfbeηb∫01tτ−1gt2ap+m1−t2bp1/pht2dt+mD3ηfbeηb+D4ηmfa/m2eηa/m2∫01tτ−1gt2ap+m1−t2bp1/ph1−t2dt.


### 4.2. Results forEpα−m Functions

Forht=t in Theorems 1–4, one can obtain the results for Epα−m function as follows.Theorem 9.
With the same conditions of Theorem1, for Epα−m functions, the following inequalities hold:(1)
Forp>0, we have(48)2αfap+mbp21/p≤Γτ+1mbp−apτD1ηIap+τf∘ξmbp+D2ηmτ+12α−1Ibp−τf∘ξapm≤ττ+αD1ηfaeηa+D2ηm2α−1fbeηb+mατ+αD1ηfbeηb+m2α−1D2ηfa/m2eηa/m2.(ii)
Forp<0, we have(49)2αfap+mbp21/p≤Γτ+1ap−mbpτD3ηIap−τf∘ξmbp+D4ηmτ+12α−1Ibp+τf∘ξapm≤ττ+αD3ηfaeηa+D4ηm2α−1fbeηb+mατ+αD3ηfbeηb+m2α−1D4ηfa/m2eηa/m2.Theorem 10.
With the same conditions of Theorem2, for Epα−m functions, the following inequalities hold:(i)
Forp>0, we have(50)2αfap+mbp21/p≤Γτ+12mbp−apτ×D1ηIap+mbp/2+τf∘ξmbp+D2ηmτ+12α−1Iap+mbp/2m−τf∘ξapm≤τ2ατ+αD1ηfaeηa+D2ηm2α−1fbeηb+m1−τ2ττ+αD1ηfbeηb+D2ηm2α−1fa/m2eηa/m2.(ii)
Forp<0, we have(51)2αfap+mbp21/p≤Γτ+12ap−mbpτ×D3ηIap+mbp/2−τf∘ξmbp+D4ηmτ+12α−1Iap+mbp/2m+τf∘ξapm≤τ2ατ+αD3ηfaeηa+D4ηm2α−1fbeηb+m1−τ2ττ+αD3ηfbeηb+D4ηm2α−1fa/m2eηa/m2.Theorem 11.
Under the assumptions of Theorem3, for Epα−m functions, the following inequalities hold:(i)
Forp>0, we have(52)2αfap+mbp21/pIap+τg∘ξmbp≤D1ηIap+τfg∘ξmbp+D2ηmτ+12α−1Ibp−τfg∘ξapm≤D1ηfaeηa+D2ηm2α−1fbeηbIap+τ+αg∘ξmbp+mD1ηfbeηb+D2ηm2α−1fa/m2eηa/m2Iap+τg∘ξmbp−Iap+τ+αg∘ξmbp.(ii)
Forp<0, we have(53)2αfap+mbp21/pIap−τg∘ξmbp≤D3ηIap−τfg∘ξmbp+D4ηmτ+12α−1Ibp+τfg∘ξapm≤D3ηfaeηa+D4ηm2α−1fbeηbIap−τ+αg∘ξmbp+mD3ηfbeηb+D4ηm2α−1fa/m2eηa/m2Iap−τg∘ξmbp−Iap−τ+αg∘ξmbp.Theorem 12.
With the same conditions of Theorem4, for Epα−m functions,(i)
Forp>0, we have(54)2αfap+mbp21/pIap+mbp/2+τg∘ξmbp≤D1ηIap+mbp/2+τfg∘ξmbp+D2ηmτ+12α−1Iap+mbp/2m−τfg∘ξapm≤12αD1ηfaeηa+D2ηm2α−1fbeηbIap+mbp/2+τ+αg∘ξmbp+mD1ηfbeηb+D2ηm2α−1fa/m2eηa/m2Iap+mbp/2+τg∘ξmbp−12αIap+mbp/2+τ+αg∘ξmbp.(ii)
Forp<0, we have(55)2αfap+mbp21/pIap+mbp/2−τg°ξmbp≤D3ηIap+mbp/2−τfg∘ξmbp+D4ηmτ+12α−1Iap+mbp/2m+τfg∘ξapm≤12αD3ηfaeηa+D4ηm2α−1fbeηbIap+mbp/2−τ+αg∘ξmbp+mD3ηfbeηb+D4ηm2α−1fa/m2eηa/m2Iap+mbp/2−τg∘ξmbp−12αIap+mbp/2−τ+αg∘ξmbp.Remark 8.
From Theorems1–4, one can deduce results for exponentially α,h−p-convex function, exponentially s,m−p-convex function of second kind, exponentially s,m−p-Godunova–Levin-convex function of second kind, exponentially p,P-convex function, Godunova–Levin type exponentially harmonic convex function, s-Godunova–Levin type exponentially harmonic convex function, exponentially α,h−m-HA-convex function, exponentially α,h-HA-convex function, exponentially HA-convex function, and exponentially α,m-HA-convex function.


## 4.1. Results forEph−m Function

Forα=1 in Theorems 1–4, one can obtain the results for Eph−m function:Theorem 5.
With the same conditions of Theorem1, for Eph−m functions, the following inequalities hold:(i)
Forp>0, we have(40)1h1/2fap+mbp21/p≤Γτ+1mbp−apτD1ηIap+τf∘ξmbp+D2ηmτ+1Ibp−τf∘ξapm≤τD1ηfaeηa+D2ηmfbeηb∫01tτ−1htdt+mD1ηfbeηb+D2ηmfa/m2eηa/m2∫01tτ−1h1−tdt.(ii)
Forp<0, we have(41)1h1/2fap+mbp21/p≤Γτ+1ap−mbpτD3ηIap−τf∘ξmbp+D4ηmτ+1Ibp+τf∘ξapm≤τD3ηfaeηa+D4ηmfbeηb∫01tτ−1htdt+mD3ηfbeηb+D4ηmfa/m2eηa/m2∫01tτ−1h1−tdt.Theorem 6.
With the same conditions of Theorem2, for Eph−m function, the following inequalities hold:(i)
Forp>0, we have(42)1h1/2fap+mbp21/p≤Γτ+12mbp−apτD1ηIap+mbp/2+τf∘ξmbp+D2ηmτ+1Iap+mbp/2m−τf∘ξapm≤τD1ηfaeηa+D2ηmfbeηb∫01tτ−1ht2dt+mD1ηfbeηb+D2ηmfa/m2eηa/m2∫01tτ−1h1−t2dt.(ii)
Forp<0, we have(43)1h1/2fap+mbp21/p≤Γτ+12ap−mbpτD3ηIap+mbp/2−τf∘ξmbp+D4ηmτ+1Iap+mbp/2m+τf∘ξapm≤τD3ηfaeηa+D4ηmfbeηb∫01tτ−1ht2dt+mD3ηfbeηb+D4ηmfa/m2eηa/m2∫01tτ−1h1−t2dt.Theorem 7.
With the same conditions of Theorem3, for Eph−m functions, the following inequalities hold:(i)
Forp>0, we have(44)1h1/2fap+mbp21/pIap+τg∘ξmbp≤D1ηIap+τfg∘ξmbp+D2ηmIbp−τfg∘ξapm≤mbp−apτΓτD1ηfaeηa+D2ηmfbeηb∫01tτ−1gtap+m1−tbp1/phtdt+mD1ηfbeηb+D2ηmfa/m2e​ηa/m2∫01tτ−1gtap+m1−tbp1/ph1−tdt.(ii)
Forp<0, we have(45)1h1/2fap+mbp21/pIap−τg∘ξmbp≤D3ηIap−τfg∘ξmbp+D4ηmIbp+τfg∘ξapm≤ap−mbpτΓτD3ηfaeηa+D4ηmfbeηb∫01tτ−1gtap+m1−tbp1/phtdt+mD3ηfbeηb+D4ηmfa/m2e​ηa/m2∫01tτ−1gtap+m1−tbp1/ph1−tdt.Theorem 8.
With the same conditions of Theorem4, for Eph−m functions, the following inequalities hold:(i)
Forp>0, we have(46)1h1/2fap+mbp21/pIap+mbp/2+τg∘ξmbp≤D1ηIap+mbp/2+τfg∘ξmbp+D2ηmτ+1Iap+mbp/2m−τfg∘ξapm≤1Γτmbp−ap2τD1ηfaeηa+D2ηmfbeηb∫01tτ−1gt2ap+m1−t2bp1/pht2dt+mD1ηfbeηb+D2ηmfa/m2eηa/m2∫01tτ−1gt2ap+m1−t2bp1/ph1−t2dt.(ii)
Forp<0, we have(47)1h1/2fap+mbp21/pIap+mbp/2−τg∘ξmbp≤D3ηIap+mbp/2−τfg∘ξmbp+D4ηmτ+1Iap+mbp/2m+τfg∘ξapm≤1Γτap−mbp2τD3ηfaeηa+D4ηmfbeηb∫01tτ−1gt2ap+m1−t2bp1/pht2dt+mD3ηfbeηb+D4ηmfa/m2eηa/m2∫01tτ−1gt2ap+m1−t2bp1/ph1−t2dt.


## 4.2. Results forEpα−m Functions

Forht=t in Theorems 1–4, one can obtain the results for Epα−m function as follows.Theorem 9.
With the same conditions of Theorem1, for Epα−m functions, the following inequalities hold:(1)
Forp>0, we have(48)2αfap+mbp21/p≤Γτ+1mbp−apτD1ηIap+τf∘ξmbp+D2ηmτ+12α−1Ibp−τf∘ξapm≤ττ+αD1ηfaeηa+D2ηm2α−1fbeηb+mατ+αD1ηfbeηb+m2α−1D2ηfa/m2eηa/m2.(ii)
Forp<0, we have(49)2αfap+mbp21/p≤Γτ+1ap−mbpτD3ηIap−τf∘ξmbp+D4ηmτ+12α−1Ibp+τf∘ξapm≤ττ+αD3ηfaeηa+D4ηm2α−1fbeηb+mατ+αD3ηfbeηb+m2α−1D4ηfa/m2eηa/m2.Theorem 10.
With the same conditions of Theorem2, for Epα−m functions, the following inequalities hold:(i)
Forp>0, we have(50)2αfap+mbp21/p≤Γτ+12mbp−apτ×D1ηIap+mbp/2+τf∘ξmbp+D2ηmτ+12α−1Iap+mbp/2m−τf∘ξapm≤τ2ατ+αD1ηfaeηa+D2ηm2α−1fbeηb+m1−τ2ττ+αD1ηfbeηb+D2ηm2α−1fa/m2eηa/m2.(ii)
Forp<0, we have(51)2αfap+mbp21/p≤Γτ+12ap−mbpτ×D3ηIap+mbp/2−τf∘ξmbp+D4ηmτ+12α−1Iap+mbp/2m+τf∘ξapm≤τ2ατ+αD3ηfaeηa+D4ηm2α−1fbeηb+m1−τ2ττ+αD3ηfbeηb+D4ηm2α−1fa/m2eηa/m2.Theorem 11.
Under the assumptions of Theorem3, for Epα−m functions, the following inequalities hold:(i)
Forp>0, we have(52)2αfap+mbp21/pIap+τg∘ξmbp≤D1ηIap+τfg∘ξmbp+D2ηmτ+12α−1Ibp−τfg∘ξapm≤D1ηfaeηa+D2ηm2α−1fbeηbIap+τ+αg∘ξmbp+mD1ηfbeηb+D2ηm2α−1fa/m2eηa/m2Iap+τg∘ξmbp−Iap+τ+αg∘ξmbp.(ii)
Forp<0, we have(53)2αfap+mbp21/pIap−τg∘ξmbp≤D3ηIap−τfg∘ξmbp+D4ηmτ+12α−1Ibp+τfg∘ξapm≤D3ηfaeηa+D4ηm2α−1fbeηbIap−τ+αg∘ξmbp+mD3ηfbeηb+D4ηm2α−1fa/m2eηa/m2Iap−τg∘ξmbp−Iap−τ+αg∘ξmbp.Theorem 12.
With the same conditions of Theorem4, for Epα−m functions,(i)
Forp>0, we have(54)2αfap+mbp21/pIap+mbp/2+τg∘ξmbp≤D1ηIap+mbp/2+τfg∘ξmbp+D2ηmτ+12α−1Iap+mbp/2m−τfg∘ξapm≤12αD1ηfaeηa+D2ηm2α−1fbeηbIap+mbp/2+τ+αg∘ξmbp+mD1ηfbeηb+D2ηm2α−1fa/m2eηa/m2Iap+mbp/2+τg∘ξmbp−12αIap+mbp/2+τ+αg∘ξmbp.(ii)
Forp<0, we have(55)2αfap+mbp21/pIap+mbp/2−τg°ξmbp≤D3ηIap+mbp/2−τfg∘ξmbp+D4ηmτ+12α−1Iap+mbp/2m+τfg∘ξapm≤12αD3ηfaeηa+D4ηm2α−1fbeηbIap+mbp/2−τ+αg∘ξmbp+mD3ηfbeηb+D4ηm2α−1fa/m2eηa/m2Iap+mbp/2−τg∘ξmbp−12αIap+mbp/2−τ+αg∘ξmbp.Remark 8.
From Theorems1–4, one can deduce results for exponentially α,h−p-convex function, exponentially s,m−p-convex function of second kind, exponentially s,m−p-Godunova–Levin-convex function of second kind, exponentially p,P-convex function, Godunova–Levin type exponentially harmonic convex function, s-Godunova–Levin type exponentially harmonic convex function, exponentially α,h−m-HA-convex function, exponentially α,h-HA-convex function, exponentially HA-convex function, and exponentially α,m-HA-convex function.


## 5. Conclusion

The Hadamard and the Fejér–Hadamard inequalities for Riemann–Liouville fractional integrals are proved by applying a generalized class of functions. Two fractional versions of the Hadamard inequality lead to almost all variants of such inequalities already published by different authors using various kinds of convex functions. Hadamard type inequalities for some new classes of functions are also given. Two fractional versions of the Fejér–Hadamard inequality are also proved which appear as generalizations of the Hadamard inequalities. By using the generalized convexity defined in this paper, one can obtain extensions of other classical integral inequalities hold for convex and related functions. It is also possible to establish these inequalities for many kinds of integral operators already existing in the literature.



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*Source: 8173785-2022-04-26.xml*