# Hadamard and Fejér–Hadamard Inequalities forα,h−m−p-Convex Functions via Riemann–Liouville Fractional Integrals

Authors: Wenyan Jia; Muhammad Yussouf; Ghulam Farid; Khuram Ali Khan
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, we introduceα,h−m−p-convex function and some related functions. By applying this generalized definition, new versions of Hadamard and Fejér–Hadamard fractional integral inequalities for Riemann–Liouville fractional integrals are given. The presented results hold at the same time for different types of convexities.


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## Body

## 1. Introduction and Preliminary Results

Convex functions play an important role in mathematical inequalities. Many inequalities for convex and related functions have been studied in recent decades and consequently published in well-reputed journals (see [1–15]). The utilization of fractional integral operators for establishing the generalized versions of classical inequalities has become a fashion in modern study of mathematical inequalities. In this regard, the Hadamard inequality is studied extensively for fractional integral operators.The aim of this paper is to present Hadamard-type inequalities and their weighted versions called Fejér–Hadamard-type inequalities for Riemann–Liouville fractional integral operators of generalized convex functions. These fractional inequalities will hold simultaneously for convex,p-convex, harmonically convex, m-convex, h-convex, α,m-convex, h−m-convex, p,h-convex, and s,m-convex functions.First, we give the definition of Riemann–Liouville fractional integrals (see [16]).Definition 1.
The left and right sided Riemann–Liouville fractional integrals for a functionf∈L1a,b of order τ∈ℝτ>0 are defined by(1)Ia+τfx=1Γτ∫axx−tτ−1ftdt,x>a,(2)Ib−τfx=1Γτ∫xbt−xτ−1ftdt,x<b,where Γτ=∫0∞e−ttτ−1dt.Next, we give definitions of convex and related functions which are useful for this study.Definition 2 (see [17]).
A functionf:a,b⟶ℝ is said to be convex if(3)ftx+1−ty≤tfx+1−tfy,∀x,y∈a,b,t∈0,1holds. If the inequality in (3) is reversed, then f is called concave function.Definition 3 (see [18, 19]).
LetI⊂0,∞ be an interval and p∈ℝ∖0. Then, a function f:I⟶ℝ is said to be p-convex, if(4)ftap+1−tbp1/p≤tfa+1−tfbholds for a,b∈I and t∈0,1. If the inequality in (4) is reversed, then f is called p-concave function.Definition 4 (see [3]).
Leth:J⊆ℝ⟶ℝ be a positive function. Let I⊂0,∞ be an interval and p∈ℝ∖0. A function f:I⟶ℝ is said to be p,h-convex, if(5)ftap+1−tbp1/p≤htfa+h1−tfbholds for a,b∈I and t∈0,1. If the inequality in (5) is reversed, then f is called p,h-concave function.The following definition unifies several kinds of convex functions for exampleh−m-convex, s,m-convex, and α,m-convex functions in a single inequality.Definition 5 (see [20]).
LetJ⊆ℝ be an interval containing 0,1 and let h:J⟶ℝ be a non-negative function. A function f:0,b⟶ℝ is called α,h−m-convex function, if f is non-negative and for all x,y∈0,b,t∈0,1 and α,m∈0,12, one has(6)ftx+m1−ty≤htαfx+mh1−tαfy.In the upcoming section, we define a new generalized notion which will be calledα,h−m−p-convex function. It will generate all kinds of convex functions discussed in aforementioned premise. By using this new definition and Riemann–Liouville fractional integrals, we will prove two versions of the Hadamard inequality. In Section 3, we give the weighted versions of the Hadamard inequalities presented in Section 2. Also, results for h−m−p-convex, α,m−p-convex, and α,h−p-convex functions are explicitly given in the form of corollaries.


## 2. Hadamard-Type Inequalities forα,h−m-p-Convex Function

We defineα,h−m-p-convex function as follows.Definition 6.
LetJ⊆ℝ be an interval containing 0,1 and let h:J⟶ℝ be a non-negative function. Let I⊂0,∞ be an interval and p∈ℝ∖0. A function f:I⟶ℝ is said to be α,h−m-p-convex, if(7)ftap+m1−tbp1/p≤htαfa+mh1−tαfbholds provided tap+m1−tbp1/p∈I for t∈0,1 and α,m∈0,12.For different choices ofα,h,m, and p, the outcomes of Definition 6 are given in the remark as follows.Remark 1.
(i)
Fixingp=1 in Definition 6, it gives α,h−m-convexity (see 20, [Definition 4.5]).(ii)
Fixingp=−1 and ht=t in Definition 6, it gives α,m-HA-convexity (see 4, Definition 2.1]).(iii)
Fixingα=m=1 in Definition 6, it gives p,h-convex function (see [3]).(iv)
Fixingα=1=p and ht=ts in Definition 6, it gives s,m-convex function (see [21]).(v)
Fixingα=1=p and ht=t in Definition 6, it gives m-convex function (see [17]).We defineh−m−p-convex function by setting α=1 in Definition 6 as follows.Definition 7.
LetJ⊆ℝ be an interval containing 0,1 and let h:J⟶ℝ be a non-negative function. Let I⊂0,∞ be an interval and p∈ℝ∖0. A function f:I⟶ℝ is said to be h−m−p-convex, if(8)ftap+m1−tbp1/p≤htfa+mh1−tfbholds provided tap+m1−tbp1/p∈I for t∈0,1 and α,m∈0,12.We defineα,m−p-convex function by setting ht=t in Definition 6 as follows.Definition 8.
LetI⊂0,∞ be an interval and p∈ℝ∖0. A function f:I⟶ℝ is said to be α,m−p-convex, if(9)ftap+m1−tbp1/p≤tαfa+m1−tαfbholds provided tap+m1−tbp1/p∈I for t∈0,1 and α,m∈0,12.We defineα,h−p-convex function by setting m=1 in Definition 6 as follows.Definition 9.
LetJ⊆ℝ be an interval containing 0,1 and let h:J⟶ℝ be a non-negative function. Let I⊂0,∞ be an interval and p∈ℝ∖0. A function f:I⟶ℝ is said to be α,h−p-convex, if(10)ftap+1−tbp1/p≤htαfa+h1−tαfbholds for a,b∈I, t∈0,1,α∈0,1.We defineα,h−m-HA-convex function by setting p=−1 in Definition 6 as follows.Definition 10.
LetJ⊆ℝ be an interval containing 0,1 and let h:J⟶ℝ be a non-negative function. Let I⊂0,∞ be an interval and p∈ℝ∖0. A function f:I⟶ℝ is said to be α,h−m-HA-convex, if(11)fabam1−t+bt≤htαfa+mh1−tαfbholds for a, b, ∈I, t∈0,1 and α,m∈0,12.We defines,m−p-convex function by setting α=1 and ht=ts in Definition 6 as follows.Definition 11.
LetJ⊆ℝ be an interval containing 0,1 and let h:J⟶ℝ be a non-negative function. Let I⊂0,∞ be an interval. A function f:I⟶ℝ is said to be s,m−p-convex, if(12)ftap+m1−tbp1/p≤tsfa+m1−tsfbholds provided tap+m1−tbp1/p∈I for t∈0,1 and s,m∈0,12.We defines,m−p-Godunova–Levin function by setting α=1 and ht=t−s in Definition 6 as follows.Definition 12.
LetJ⊆ℝ be an interval containing 0,1 and let h:J⟶ℝ be a non-negative function. Let I⊂0,∞ be an interval and p∈ℝ∖0. A function f:I⟶ℝ is said to be s,m−p-Godunova–Levin, if(13)ftap+m1−tbp1/p≤fats+mfb1−tsholds provided tap+m1−tbp1/p∈I for t∈0,1 and s,m∈0,12.In the next results, it is assumed that functionsf,g, and h are integrable and all involved integrals are finite.Theorem 1.
Letf:I⟶ℝ be a positive α,h−m−p-convex function as given in Definition 6 with tbp+1−tap/m1/p∈I,m≠0,ap<mbp. Then, the following inequality for fractional integral operators (1) and (2) holds:(14)fap+mbp21/p≤Γτ+1mbp−apτh12αIap+τf∘ξmbp+mτ+1h2α−12αIbp−τf∘ξapm≤τh12αfa+mh2α−12αfb∫01tτ−1htαdt+mτh12αfb+mh1−12αfam2∫01tτ−1h1−tαdt,ξt=t1/p.Proof.
Byα,h−m−p-convexity of f, one can have(15)fxp+myp21/p≤h12αfx+mh2α−12αfy.
By settingx=tap+m1−tbp1/p and y=tbp+1−tap/m1/p in (15), we get(16)fap+mbp21/p≤h12αftap+m1−tbp1/p+mh2α−12αftbp+apm1−t1/p.
Multiplying both sides of (16) by tτ−1 and integrating over 0,1, we get(17)1τfap+bp21/p≤h12α∫01tτ−1ftap+m1−tbp1/pdt+mh2α−12α∫01tτ−1ftbp+apm1−t1/pdt.
Settingtap+m1−tbp=x, that is, t=mbp−x/mbp−ap and tbp+ap/m1−t=y, that is, t=y−ap/m/bp−ap/m, on the right hand side of (17) and after simplification, one can get first inequality of (14).
Again by using theα,h−m−p-convexity of f, the right hand side of (16) leads to the following inequality:(18)h12αftap+m1−tbp1/p+mh2α−12αftbp+apm1−t1/p≤h12αfa+mh2α−12αfbhtα+mh12αfb+mh2α−12αfam2h1−tα.
Multiplyingtτ−1 on both sides of (18) and integrating over 0,1, we have(19)h12α∫01tτ−1ftap+m1−tbp1/pdt+mh2α−12α∫01tτ−1ftbp+apm1−t1/pdt≤h12αfa+mh2α−12αfb∫01tτ−1htαdt+mh12αfb+mh2α−12αfam2∫01tτ−1h1−tαdt.
Settingtap+m1−tbp=x, that is, t=mbp−x/mbp−ap and 1−tap/m+tbp=y, that is, t=y−ap/m/bp−ap/m, in (19) and after calculation, one can get the second inequality of (14).The inequality established in (14) gives the Hadamard inequality for α,h−m−p-convex function by fixing τ=1. The readers can also obtain the Hadamard inequality for h−m−p-convex, α,m−p-convex, α,h−p-convex, and all deducible functions by fixing parameters and the function h of their choice. Also, inequality (14) provides fractional versions of the Hadamard inequality for Riemann–Liouville fractional integrals of functions stated in the introduction section.Further consequences of the above theorem are stated in the following.Remark 2.
(i)
On fixingα=m=1, ht=t, and p=1, [[14], Theorem 2] is obtained.(ii)
On fixingα=m=1, p=1, ht=t, and τ=1, classical Hadamard inequality [22] is obtained.(iii)
On fixingα=m=1, ht=t, and p=−1, [[23], Theorem 2.4] is obtained.Now, we give another variant of the Hadamard inequality as follows.Theorem 2.
Let the assumptions of Theorem1 hold. Then, we have the following inequality:(20)fap+mbp21/p≤Γτ+12mbp−apτ×h12αIap+mbp/2+τf∘ξmbp+mh2α−12αIap+mbp/2−τf∘ξapm≤τh12αfa+mh2α−12αfb∫01tτ−1ht2αdt+mh12αfb+mh2α−12αfam2∫01tτ−1h2−t2αdt,ξt=t1/p.Proof.
By settingx=t/2ap+m2−t/2bp1/p and y=t/2bp+ap/m2−t/21/p in (15), we get(21)fap+mbp21/p≤h12αft2ap+m2−t2bp1/p+mh2α−12αft2bp+2−t2apm1/p.
Multiplying both sides of (21) by tτ−1 and integrating over 0,1, we get(22)1τfap+mbp21/p≤h12α∫01tτ−1ft2ap+m2−t2bp1/pdt+mh2α−12α∫01tτ−1ft2bp+apm2−t21/pdt.
Settingt/2ap+m2−t/2bp=x, that is, t/2=mbp−x/mbp−ap and 2−t/2ap/m+t/2bp=y, that is, t/2=y−ap/m/mbp−ap/m, on the right hand side of (22) and after simplification, one can get first inequality of (20).
Again by usingα,h−m−p-convexity of f, the right hand side of (21) leads to the following inequality:(23)h12αft2ap+m2−t2bp1/p+mh2α−12αft2bp+2−t2apm1/p≤h12αfa+mh2α−12αfbht2α+mh12αfb+mh2α−12αfamh1−t2α.
Multiplyingtτ−1 on both sides of (23) and integrating over 0,1, we have(24)h12α∫01tτ−1ft2ap+m2−t2bp1/pdt+mh2α−12α∫01tτ−1ft2bp+apm2−t21/pdt≤h12αfa+mh2α−12αfb∫01tτ−1ht2αdt+mh12αfb+mh2α−12αfam2∫01tτ−1h1−t2αdt.
Settingt/2ap+m2−t/2bp=x, that is, t/2=mbp−x/mbp−x and 2−t/2ap+t/2bp=y, that is, t/2=y−ap/m/bp−ap/m, in (24) and after calculation, one can get second inequality of (20).The above version of the Hadamard inequality gives the corresponding inequalities forh−m−p-convex, α,m−p-convex, α,h−p-convex, and all deducible functions by fixing parameters and the function h.Further consequences of the above theorem are stated in the following.Remark 3.
(i)
On fixingα=1=m, p>0, and ht=t in Theorem 2, [[24], Theorem 2.1 (i)] is obtained.(ii)
On fixingα=1=m, p<0, and ht=t in Theorem 2, [[24], Theorem 2.1 (ii)] is obtained.(iii)
On fixingα=1=m, p=1, and ht=t in Theorem 2, [[24], Corollary 2.1] is obtained.Remark 4.
From Theorems1 and 2, one can deduce results for convex, p-convex, m-convex, h-convex, α,m-convex, h−m-convex, s,m-convex, and p,h-convex functions.


## 3. Fejér–Hadamard-Type Inequalities

In this section, we present the Fejér–Hadamard-type inequalities forα,h−m−p-convex functions by applying definition of Riemann–Liouville fractional integral operators. We also give results for new definitions obtained in Section 2.Theorem 3.
Letf:I⟶ℝ be a positive α,h−m−p-convex function as given in Definition 6 and fap+mbp−x=fx. If g:I⟶ℝ is a positive function, then the following inequality for fractional integral operators (1) and (2) holds:(25)fap+mbp21/pIap+τg∘ξmbp≤h12αIap+τfg∘ξmbp+mμ+1h2α−12αIbp−τfg∘ξapm≤mbp−apτΓτh12αfa+mh2α−12αfb∫01tτ−1gtap+m1−tbp1phtαdt+mh12αfb+mh2α−12αfam2∫01tτ−1gtap+m1−tbp1/ph1−tαdtξt=t1/p,fg∘ξ=f∘ξg∘ξ.Proof.
Multiplying (16) by tτ−1gtap+m1−tbp1/p and integrating over 0,1, we get(26)fap+mbp21/p∫01tτ−1gtap+m1−tbp1/pdt≤h12α∫01tτ−1ftap+m1−tbp1/pgtap+m1−tbp1/pdt+mh2α−12α∫01tτ−1ftbp+1−tapm1/pgtap+m1−tbp1/pdt.
Settingtap+m1−tbp=x, that is, 1−tap+mtbp=ap+mbp−x, in (26) and utilizing condition fx=fap+mbp−x, by fractional integral operators (1) and (2), one can get first inequality of (25).
Now, multiplyingtτ−1gtap+m1−tbp1/p in (18) and integrating over 0,1, we have(27)h12α∫01tτ−1ftap+m1−tbp1/pgtap+m1−tbp1/pdt+mh2α−12α∫01tτ−1ftbp+apm1−t1/pgtap+m1−tbp1/pdt≤h12αfa+mh2α−12αfb∫01tτ−1gtap+m1−tbp1/phtαdt+mh12αfb+mh2α−12αfam2∫01tτ−1gtap+m1−tbp1/ph1−tαdt.
Again settingtap+m1−tbp=x, that is, 1−tap+mtbp=ap+mbp−x, in (27) and utilizing condition fx=fap+mbp−x, by using definitions of fractional integral operators (1) and (2), one can get second inequality of (25).The consequences of the above theorem are stated in the following.Remark 5.
(i)
On fixingα=m=1, ht=t, gx=1, and p=1, [[14], Theorem 2] is obtained.(ii)
On fixingα=m=1, p=1, ht=t, gx=1, and τ=1, the classical Hadamard inequality [25] is obtained.(iii)
On fixingα=m=1, p=1, ht=t, and τ=1, the Fejér–Hadamard inequality [26] is obtained.Now, we give another variant of the Fejér–Hadamard inequality as follows.Theorem 4.
Let the assumptions of Theorem3 hold. Then, we have the following inequality:(28)fap+mbp21/pIap+mbp/2+τg∘ξmbp≤h12αIap+mbp/2+τfg∘ξmbp+mμ+1h2α−12αIap+mbp/2−τfg∘ξapm≤1Γτmbp−ap2τh12αfa+mh2α−12αfb×∫01tτ−1gt2ap+m2−t2bp1/pht2αdt+mh12αfb+mh2α−12αfam2∫01tτ−1gt2ap+m2−t2bp1/ph2α−tα2αdtξt=t1/p,fg∘ξ=f∘ξg∘ξ.Proof.
Multiplying (21) by tτ−1gt/2ap+m2−t/2bp1/p and integrating over 0,1, we get(29)fap+mbp21/p∫01tτ−1gt2ap+m2−t2bp1/pdt≤h12α∫01tτ−1ft2ap+m2−t2bp1/pgt2ap+m2−t2bp1/pdt+mh2α−12α∫01tτ−1ft2bp+apm2−t21/pgt2ap+m2−t2bp1/pdt.
Settingt/2ap+m2−t/2bp=x, that is, m2−t/2ap+t/2bp=ap+mbp−x, in (29) and utilizing condition fx=fap+mbp−x, by fractional integral operators (1) and (2), one can get first inequality of (28).
Multiplyingtτ−1gt/2ap+m2−t/2bp1/p on both sides of (23) and integrating over 0,1, we have(30)h12α∫01tτ−1ft2ap+m2−t2bp1/pgt2ap+m2−t2bp1/pdt+mh2α−12α∫01tτ−1ft2bp+2−t2apm1/pgt2ap+m2−t2bp1/pdt≤h12αfa+mh2α−12αfb∫01tτ−1gt2ap+m2−t2bp1/pht2αdt+mh12αfb+mh2α−12αfam2∫01tτ−1gt2ap+m2−t2bp1/ph2α−tα2αdt.
Again settingt/2ap+m2−t/2bp=x, that is, m2−t/2ap+t/2bp=ap+mbp−x, in (30) and utilizing condition fx=fap+mbp−x, by fractional integral operators (1) and (2), one can get second inequality of (28).The consequences of the above theorem are stated in the following.Remark 6.
(i)
On fixingα=1=m, p>0, gx=1, and ht=t in Theorem 4, [[24], Theorem 2.1 (i)] is obtained.(ii)
On fixingα=1=m, p<0, gx=1, and ht=t in Theorem 4, [[24], Theorem 2.1 (ii)] is obtained.(iii)
On fixingα=1=m, p=1, gx=1, and ht=t in Theorem 4, [[24], Corollary 2.1] is obtained.Remark 7.
From Theorems3 and 4, one can deduce results for convex, p-convex, m-convex, h-convex, α,m-convex, h−m-convex, and p,h-convex functions.


### 3.1. Results forh−m−p-Convex Functions

By settingα=1 in Theorems 1–4, the results for h−m−p-convex functions are obtained as follows.Theorem 5.
Under the assumptions of Theorem1, the following inequality holds for h−m−p-convex functions:(31)1h12fap+mbp21/p≤Γτ+1mbp−apτIap+τf∘ξmbp+mμ+1Ibp−τf∘ξapm≤τfa+mfb∫01tτ−1htdt+mfb+mfam2∫01tτ−1h1−tdt.Theorem 6.
Under the assumptions of Theorem2, the following inequality holds for h−m−p-convex functions:(32)1h1/2fap+mbp21/p≤Γτ+12mbp−apτIap+mbp/2+τf∘ξmbp+mμ+1Iap+mbp/2−τf∘ξapm≤τfa+mfb∫01tτ−1ht2dt+mfb+mfam2∫01tτ−1h2−t2dt.Theorem 7.
Under the assumptions of Theorem3, the following inequality holds for h−m−p-convex functions:(33)1h1/2fap+mbp21/pIap+τg∘ξmbp≤Iap+τfg∘ξmbp+mIbp−τfg∘ξapm≤mbp−apτΓτfa+mfb∫01tτ−1gtap+m1−tbp1/phtdt+mfb+mfam2∫01tτ−1gtap+m1−tbp1/ph1−tdt.Theorem 8.
Under the assumptions of Theorem4, the following inequality holds for h−m−p-convex functions:(34)1h1/2fap+mbp21/pIap+mbp/2+τg∘ξmbp≤Iap+mbp/2+τfg∘ξbp+mIap+bp/2−τfg∘ξapm≤1Γτmbp−ap2τfa+mfb∫01tτ−1gt2ap+2−t2bp1/pht2dt+mfb+mfam2∫01tτ−1gt2ap+m2−t2bp1/ph2−t2dt.


### 3.2. Results forα,m−p-Convex Functions

By settinght=t in Theorems 1, 2, 3, and 4, the results for α,h−p-convex functions are obtained as follows.Theorem 9.
Under the assumptions of Theorem1, the following inequality holds for α,m−p-convex functions:(35)2αfap+mbp21/p≤Γτ+1mbp−apτIap+τf∘ξmbp+mμ+12α−1Ibp−τf∘ξapm≤ττ+αfa+m2α−1fb+mατ+αfb+m2α−1fam2.Theorem 10.
Under the assumptions of Theorem2, the following inequality holds for α,m−p-convex functions:(36)2αfap+mbp21/p≤Γτ+12mbp−apτIap+mbp/2+τf∘ξmbp+m2α−1Iap+mbp/2−τf∘ξapm,≤τ2ατ+αfa+m2α−1fb+m1−τ2ατ+αfb+m2α−1fam2.Theorem 11.
Under the assumptions of Theorem3, the following inequality holds for α,m−p-convex functions:(37)2αfap+mbp21/pIap+τg∘ξmbp≤Iap+τfg∘ξmbp+m2α−1Ibp−τfg∘ξapm≤mbp−apτΓτfa+m2α−1fb∫01tτ+α−1gtap+m1−tbp1/pdt+mfb+m2α−1fam2∫01tτ−1−tτ+α−1gtap+m1−tbp1/pdt.Theorem 12.
Under the assumptions of Theorem4, the following inequality holds for α,m−p-convex functions:(38)2αfap+mbp21/pIap+mbp/2+τg∘ξmbp≤Iap+mbp/2+τfg∘ξmbp+m2α−1Iap+mbp/2−τfg∘ξapm≤1Γτmbp−ap2τ12αfa+m2α−1fb∫01tτ+α−1gt2ap+2−t2bp1/pdt+mfb+m2α−1fam2∫01tτ−1−tτ+α−12αgt2ap+m2−t2bp1/pdt.


### 3.3. Results forα,h−p-Convex Functions

By settingm=1 in Theorems 1, 2, 3, and 4, the results for α,h−p-convex functions are obtained as follows.Theorem 13.
Under the assumptions of Theorem1, the following inequality holds for α,h−p-convex functions:(39)fap+bp21/p≤Γτ+1bp−apτh12αIap+τf∘ξbp+h1−12αIbp−τf∘ξap,≤τh12αfa+h1−12αfb∫01tτ−1htαdt+h12αfb+h1−12αfa∫01tτ−1h1−tαdt,ξt=t1/p.Theorem 14.
Under the assumptions of Theorem2, the following inequality holds for α,h−p-convex functions:(40)fap+bp21/p≤Γτ+12bp−apτh12αIap+bp/2+τf∘ξbp+h1−12αIap+bp/2−τf∘ξap,≤τh12αfa+h1−12αfb∫01tτ−1ht2αdt+h12αfb+h1−12αfa∫01tτ−1h1−t2αdt,ξt=t1/p.Theorem 15.
Under the assumptions of Theorem3, the following inequality holds for α,h−p-convex functions:(41)fap+bp21/pIap+τg∘ξbp≤h12αIap+τfg∘ξbp+h1−12αIbp−τfg∘ξap≤bp−apτΓτh12αfa+h1−12αfb∫01tτ−1gtap+1−tbp1/phtαdt+h12αfb+h1−12αfa∫01tτ−1gtap+1−tbp1/ph1−tαdt,ξt=t1/p,fg∘ξ=f∘ξg∘ξ.Theorem 16.
Under the assumptions of Theorem4, the following inequality holds for α,h−p-convex functions:(42)fap+bp21/pIap+bp/2+τg∘ξbp≤h12αIap+bp/2+τfg∘ξbp+h1−12αIap+bp/2−τfg∘ξap≤1Γτbp−ap2τh12αfa+h1−12αfb∫01tτ−1gt2ap+1−t2bp1/p×ht2αdt+h12αfb+1−12αfa×∫01tτ−1gt2ap+1−t2bp1/ph1−t2αdt,ξt=t1/p,fg∘ξ=f∘ξg∘ξ.


## 3.1. Results forh−m−p-Convex Functions

By settingα=1 in Theorems 1–4, the results for h−m−p-convex functions are obtained as follows.Theorem 5.
Under the assumptions of Theorem1, the following inequality holds for h−m−p-convex functions:(31)1h12fap+mbp21/p≤Γτ+1mbp−apτIap+τf∘ξmbp+mμ+1Ibp−τf∘ξapm≤τfa+mfb∫01tτ−1htdt+mfb+mfam2∫01tτ−1h1−tdt.Theorem 6.
Under the assumptions of Theorem2, the following inequality holds for h−m−p-convex functions:(32)1h1/2fap+mbp21/p≤Γτ+12mbp−apτIap+mbp/2+τf∘ξmbp+mμ+1Iap+mbp/2−τf∘ξapm≤τfa+mfb∫01tτ−1ht2dt+mfb+mfam2∫01tτ−1h2−t2dt.Theorem 7.
Under the assumptions of Theorem3, the following inequality holds for h−m−p-convex functions:(33)1h1/2fap+mbp21/pIap+τg∘ξmbp≤Iap+τfg∘ξmbp+mIbp−τfg∘ξapm≤mbp−apτΓτfa+mfb∫01tτ−1gtap+m1−tbp1/phtdt+mfb+mfam2∫01tτ−1gtap+m1−tbp1/ph1−tdt.Theorem 8.
Under the assumptions of Theorem4, the following inequality holds for h−m−p-convex functions:(34)1h1/2fap+mbp21/pIap+mbp/2+τg∘ξmbp≤Iap+mbp/2+τfg∘ξbp+mIap+bp/2−τfg∘ξapm≤1Γτmbp−ap2τfa+mfb∫01tτ−1gt2ap+2−t2bp1/pht2dt+mfb+mfam2∫01tτ−1gt2ap+m2−t2bp1/ph2−t2dt.


## 3.2. Results forα,m−p-Convex Functions

By settinght=t in Theorems 1, 2, 3, and 4, the results for α,h−p-convex functions are obtained as follows.Theorem 9.
Under the assumptions of Theorem1, the following inequality holds for α,m−p-convex functions:(35)2αfap+mbp21/p≤Γτ+1mbp−apτIap+τf∘ξmbp+mμ+12α−1Ibp−τf∘ξapm≤ττ+αfa+m2α−1fb+mατ+αfb+m2α−1fam2.Theorem 10.
Under the assumptions of Theorem2, the following inequality holds for α,m−p-convex functions:(36)2αfap+mbp21/p≤Γτ+12mbp−apτIap+mbp/2+τf∘ξmbp+m2α−1Iap+mbp/2−τf∘ξapm,≤τ2ατ+αfa+m2α−1fb+m1−τ2ατ+αfb+m2α−1fam2.Theorem 11.
Under the assumptions of Theorem3, the following inequality holds for α,m−p-convex functions:(37)2αfap+mbp21/pIap+τg∘ξmbp≤Iap+τfg∘ξmbp+m2α−1Ibp−τfg∘ξapm≤mbp−apτΓτfa+m2α−1fb∫01tτ+α−1gtap+m1−tbp1/pdt+mfb+m2α−1fam2∫01tτ−1−tτ+α−1gtap+m1−tbp1/pdt.Theorem 12.
Under the assumptions of Theorem4, the following inequality holds for α,m−p-convex functions:(38)2αfap+mbp21/pIap+mbp/2+τg∘ξmbp≤Iap+mbp/2+τfg∘ξmbp+m2α−1Iap+mbp/2−τfg∘ξapm≤1Γτmbp−ap2τ12αfa+m2α−1fb∫01tτ+α−1gt2ap+2−t2bp1/pdt+mfb+m2α−1fam2∫01tτ−1−tτ+α−12αgt2ap+m2−t2bp1/pdt.


## 3.3. Results forα,h−p-Convex Functions

By settingm=1 in Theorems 1, 2, 3, and 4, the results for α,h−p-convex functions are obtained as follows.Theorem 13.
Under the assumptions of Theorem1, the following inequality holds for α,h−p-convex functions:(39)fap+bp21/p≤Γτ+1bp−apτh12αIap+τf∘ξbp+h1−12αIbp−τf∘ξap,≤τh12αfa+h1−12αfb∫01tτ−1htαdt+h12αfb+h1−12αfa∫01tτ−1h1−tαdt,ξt=t1/p.Theorem 14.
Under the assumptions of Theorem2, the following inequality holds for α,h−p-convex functions:(40)fap+bp21/p≤Γτ+12bp−apτh12αIap+bp/2+τf∘ξbp+h1−12αIap+bp/2−τf∘ξap,≤τh12αfa+h1−12αfb∫01tτ−1ht2αdt+h12αfb+h1−12αfa∫01tτ−1h1−t2αdt,ξt=t1/p.Theorem 15.
Under the assumptions of Theorem3, the following inequality holds for α,h−p-convex functions:(41)fap+bp21/pIap+τg∘ξbp≤h12αIap+τfg∘ξbp+h1−12αIbp−τfg∘ξap≤bp−apτΓτh12αfa+h1−12αfb∫01tτ−1gtap+1−tbp1/phtαdt+h12αfb+h1−12αfa∫01tτ−1gtap+1−tbp1/ph1−tαdt,ξt=t1/p,fg∘ξ=f∘ξg∘ξ.Theorem 16.
Under the assumptions of Theorem4, the following inequality holds for α,h−p-convex functions:(42)fap+bp21/pIap+bp/2+τg∘ξbp≤h12αIap+bp/2+τfg∘ξbp+h1−12αIap+bp/2−τfg∘ξap≤1Γτbp−ap2τh12αfa+h1−12αfb∫01tτ−1gt2ap+1−t2bp1/p×ht2αdt+h12αfb+1−12αfa×∫01tτ−1gt2ap+1−t2bp1/ph1−t2αdt,ξt=t1/p,fg∘ξ=f∘ξg∘ξ.



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*Source: 9945114-2021-08-16.xml*