# Some Inequalities of Hermite–Hadamard Type for MT-h-Convex Functions via Classical and Generalized Fractional Integrals

Authors: Hengxiao Qi; Waqas Nazeer; Fatima Abbas; Wenbo Liao
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Convexity plays a vital role in pure and applied mathematics specially in optimization theory, but the classical convexity is not enough to fulfil the needs of modern mathematics; hence, it is important to study generalized notion of convexity. Fraction integral operators also become an important tool for solving problems of model physical and engineering processes that are found to be best described by fractional differential equations. The aim of this paper is to study MT-h-convex functions via fractional integral operators. We establish several Hermite–Hadamard-type inequalities for MT-h-convex function via classical and generalized fractional integrals. We also obtain special means related to our results and present some error estimates for the trapezoidal formulas.


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## Body

## 1. Introduction

One of the most important notions in mathematics is convex functions which are very important for both pure and applied mathematicians. Convex functions are helpful in solving problems of optimization theory and many other problems of applied nature.Definition 1.
LetJ⊆R and Jo be interior of J, a mapping ψ:J⟶R said to be convex on J, if the following inequality holds for all c,d∈J and λ∈0,1,
(1)ψλc+1−λd≤λψc+1−λψd.The mappingψ is said to be concave if −ψ is convex.The theory of inequalities got the attention of many researchers, and the new inequalities are always appreciable not only in real analysis but also the researchers working in applied sciences use inequalities as a very effective tool for analyzing different practical problems and to study various properties of solution of different equations [1]. Jensen-type inequalities, Hardy-type inequalities [2], Gagliardo-Nirenberg-type inequalities [3], Grüss-type inequalities [4], Ostrowski-type inequality [5, 6], etc. are extensively studied in the literature. The most famous inequality in literature is known as Hermite–Hadamard inequality, which has fundamental role in convex analysis. The Hermite–Hadamard-type inequalities for different classes of convex function can be found in [7, 8] and references therein.Letψ:J⊆R⟶R be a convex mapping defined on the interval J of real numbers and c,d∈J. Then, the Hermite–Hadamard inequality is as follows:
(2)ψc+d2≤1c−d∫cdψλdλ≤ψc+ψd2,withc<d and c,d∈J. If ψ is concave, then both the inequalities reverse their direction.Hermite–Hadamard inequality is the most important inequality so far in inequality theory, and several extensions of this inequality are given by researchers in recent years [9]. Our motivation is to establish a generalized version of Hermite–Hadamard-type inequality for MT-h-convex functions. It is worthy to mention here that several results of literature can be obtained from our established results as a particular case by taking suitable values of involved parameters.Since classical notion of convexity is not enough for solving today’s problems, so this notion has been generalized by several researchers to meet the needs of modern mathematics. Now, we present some generalized notions of convexity.Definition 2 (see [10]).
Leth:I⟶R be a nonnegative mapping, h≠0. The mapping ψ:J⟶R is said to be h-convex, if ψ is nonnegative and for all c,d∈J, λ∈0,1, the following inequality holds:
(3)ψλc+1−λd≤hλψc+h1−λψd.The mapping is said to beh-concave if inequality (3) is reversed.Definition 3 (see [11, 12]).
A mappingψ:J⊆R⟶R is said to be MT-convex on J if it is nonnegative and satisfies the following inequality:
(4)ψλc+1−λd≤λ21−λψc+1−λ2λψd.Motivated by the above two notions, we introduce the following notion of MT-h-convex function.Definition 4.
A mappingψ:J⊆R⟶R is said to be MT-h-convex on J, if for all c,d∈J and hλ∈0,1, it is nonnegative and satisfies the following inequality:
(5)ψλc+1−λd≤hλ2h1−λψc+h1−λ2hλ.Fraction integral operators also become an important tool for solving problems of model physical and engineering processes that are found to be best described by fractional differential equations. Fractional calculus creates a diversity in inequality theory of convex analysis [13, 14].The following generalized fractional integral operators were introduced by Ertugral and Sariaya [15] as follows:Letρ:0,∞⟶0,∞ be such that
(6)∫01ρλλdλ<∞,then the left-hand-sided and right-hand-sided generalized fractional integral operators are defined as:
(7)c+Jρψv=∫cvρv−λv−λψλ,v>c,(8)d−Jρψv=∫vdρλ−vλ−vψλ,v<d,respectively.The following remark justifies the generality of the above fractional integral operators.Remark 5.
(1) Ifρλ=λ, then (7) and (8) convert to usual Riemann fractional integral, respectively
(9)Jc+ψv=∫cvψλdλ,v>c,Jd−ψv=∫vdψλdλ,v<d.
(2) Ifρλ=λη/Γη, then (7) and (8) reduce to the Riemann-Liouville integral [14, 16]
(10)Jc+ηψv=1Γη∫cvv−λη−1ψλdλ,v>c,Jd−ηψv=1Γη∫vdλ−vη−1ψλdλ,v<d.
Here,Γη=∫0∞t−mmη−1dm and Jc+0ψv=Jd−0ψv=ψv.Note that, forη=1, the Riemann-Liouville integral converts to the classical integrals. For the other interesting special cases of (7) and (8), we refer to the readers [17, 18].The aim of this paper is to establish Hermite–Hadamard-type inequalities for the proposed notion of MT-h-convex function in the setting of classical and generalized fractional integral operators. As applications of our results, we present special means related with our results. We also establish some error estimates for the trapezoidal formula.


## 2. Hermite–Hadamard-Type Inequalities via Classical Fractional Integral

First, we present the following identity that has been obtained in [19] which will play a crucial role in the proof of our main results.Lemma 6.
Letψ:J⊂R⟶R be a differentiable function on Jo and c,d∈J with c<d. If ψ′∈L1c,d, then the following inequality holds:
(11)d−vψd+v−cψcd−c−1d−c∫cdψsds=v−c2d−c∫011−λψ′λv+1−λcdλ+d−v2d−c∫011−λψ′λv+1−λddλ,for each v∈c,d.Theorem 7.
Letψ:J⊂R+⟶R be a differentiable function on Jo such that ψ′∈L1c,d where c,d∈J. If ψ′ is MT-h-convex function on c,d and ψ′≤Q where v∈c,d, then we have
(12)d−vψd+v−cψcd−c−1d−c∫cdψsds≤Qv−c2+d−v2d−c×12S1+S2,where
(13)S1=∫011−λhλh1−λdλ,S2=∫011−λh1−λhλdλ,
withhλ/h1−λ<∞ is finite.Proof.
From Lemma6, we have
(14)d−vψd+v−cψcd−c−1d−c∫cdψsds=v−c2d−c∫011−λψ′λv+1−λcdλ+d−v2d−c∫011−λψ′λv+1−λddλ.
Applying mode on both sides,(15)d−vψd+v−cψcd−c−1d−c∫cdψsds≤v−c2d−c∫011−λψ′λv+1−λcdλ+d−v2d−c∫011−λψ′λv+1−λddλ.
Employing MT-h-convexity ofψ′, we have
(16)≤v−c2d−c∫011−λhλ2h1−λψ′v+h1−λ2hλψ′cdλ+d−v2d−c∫011−λhλ2h1−λψ′v+h1−λ2hλψ′ddλ.
Sinceψ′v≤Q, so
(17)≤Qv−c2d−c∫011−λhλ2h1−λdλ+∫011−λh1−λ2hλdλ+Qd−v2d−c∫011−λhλ2h1−λdλ+∫011−λh1−λ2hλdλ≤Qv−c2d−c12S1+S2+Qd−v2d−c12S1+S2≤Qv−c2+d−v2d−c×12S1+S2.
The proof is completed.Corollary 8.
In Theorem7, if we substitute hλ=λ, we get [20] Theorem 7.Remark 9.
If we takev=c+d/2 in Theorem 7, then we obtain
(18)d−vψd+v−cψcd−c−1d−c∫cdψsds≤14d−cQ×S1+S2.Theorem 10.
Letψ:J⊂R+⟶R be a differentiable function on Jo such that ψ′∈L1c,d where c,d∈J. If ψ′q is MT-h-convex mapping on c,d and q>1,1/p+1/q=1 also ψ′≤Q and v∈c,d, then we have
(19)d−vψd+v−cψcd−c−1d−c∫cdψsds≤v−c2+d−v2d−c×1p+11/p×Qq∫01hλh1−λdλ1/q.Proof.
Assume thatp>1 and using Lemma 6, we have
(20)d−vψd+v−cψcd−c−1d−c∫cdψsds≤v−c2d−c∫011−λψ′λv+1−λcdλ+d−v2d−c∫011−λψ′λv+1−λddλ.
Using Hölder inequality,(21)≤v−c2d−c∫011−λpdλ1/p∫01ψ′λv+1−λcqdλ1/q+d−v2d−c∫011−λpdλ1/p∫01ψ′λv+1−λdqdλ1/q.
Sinceψ′q is MT-h-convex mapping and ψv≤Q, so we have
(22)≤v−c2d−c×1p+11/p×∫01hλ2h1−λψ′vq+h1−λ2hλ+ψ′cq1/q+d−v2d−c×1p+11/p×∫01hλ2h1−λψ′vq+h1−λ2hλ+ψ′dq1q.
Some small calculations yield that(23)∫01hλh1−λdλ=∫01h1−λhλdλ.This implies that
(24)≤v−c2d−c×1p+11/p×Qq∫01hλh1−λdλ1/q+d−v2d−c×1p+11/p×Qq∫01hλh1−λdλ1/q=v−c2+d−v2d−c×1p+11/p×Qq∫01hλh1−λdλ1/q.
The proof is completed.Corollary 11.
In Theorem10, if we substitute hλ=λ, we get [20] Theorem 2.4.Remark 12.
If we takev=c+d/2 in Theorem 10, then we obtain
(25)d−vψd+v−cψcd−c−1d−c∫cdψsds≤d−c2×1p+11/p×Qq∫01hλh1−λdλ1/q.Theorem 13.
Letψ:J⊂R+⟶R be a differentiable function on Jo such that ψ′∈L1c,d with c<d where c,d∈J. If ψ′q is MT-h-convex function on c,d with q>1 and ψ′v≤Q where v∈c,d, then we have
(26)d−vψd+v−cψcd−c−1d−c∫cdψsds≤v−c2+d−v2d−c×121−1/q12QqS1+12QqS21/q,where
(27)S1=∫011−λhλh1−λdλ,S2=∫011−λh1−λhλdλ,
withhλ/h1−λ<∞ is finite.Proof.
Using Lemma6, Hölder inequality, and MT-h-convexity of ψ′q, we have
(28)d−vψd+v−cψcd−c−1d−c∫cdψsds≤v−c2d−c∫011−λψ′λv+1−λcdλ+d−v2d−c∫011−λψ′λv+1−λddλ≤v−c2d−c∫011−λdλ1−1/q∫011−λψ′λv+1−λcqdλ1/q+d−v2d−c∫011−λdλ1−1/q∫011−λψ′λv+1−λdqdλ1/q≤v−c2d−c×121−1/q∫011−λhλ2h1−λψ′vqdλ+∫011−λhλ2h1−λψ′cqdλ1/q+d−v2d−c×121−1/q×∫011−λhλ2h1−λψ′vqdλ+∫011−λhλ2h1−λψ′cqdλ1/q≤v−c2d−c×121−1/q12QqS1+1QqS21/q+d−v2d−c×121−1/q12QqS1+1QqS21/q≤v−c2+d−v2d−c×121−1/q12QqS1+12QqS21/q.
The proof is completed.Corollary 14.
In Theorem13, if we substitute hλ=λ, we get [20] Theorem 2.6.Remark 15.
If we takev=c+d/2 in Theorem 13, then we obtain
(29)d−vψd+v−cψcd−c−1d−c∫cdψsds≤d−c2×121−1/q12QqS1+12QqS21/q.


## 3. Hermite–Hadamard-Type Inequalities via Generalized Fractional Operators

To establish Hermite–Hadamard-type inequalities via generalized fractional operators for MT-h-convex function, we need the following identity [21].Lemma 16.
Letψ:[c,d]⟶R be differentiable function on c,d with c<d such that ψ∈L1c,d. Then, for each t∈0,1, we have
(30)1−tηΩ1+tη∇1d−cψv−1d−c1−tηw−Iρψc+tηv+Iρψd=1−tη+1∫01Ωλψ′λv+1−λcdλ−tη+1∫01∇λψ′λv+1−λddλ,where v=tc+1−td, and
(31)Ωλ=∫0λρv−cssds<∞,∇λ=∫0λρd−vssds<∞.Theorem 17.
Letψ:c,d⟶R be a differentiable function on c,d and ψ′∈L1c,d with 0≤c<d and η>0. Then, the following inequality holds for each t∈0,1. If ψ′ is MT-h-convex on c,d,
(32)1−tηΩ1+tη∇1d−cψv−1d−c1−tηv−Iρψc+tηv+Iρψd≤1−tη+12M1ψ′v+N1ψ′c+tη+12M1ψ′v+N1ψ′d,
where the constantsM1,M2,N1, and N2 are
(33)M1=∫01hλh1−λΩλdλ,M2=∫01hλh1−λ∇λdλ,N1=∫01h1−λhλΩλdλ,N2=∫01h1−λhλ∇λdλ.Proof.
From Lemma16, we have
(34)1−tηΩ1+tη∇1d−cψv−1d−c1−tηv−Iρψc+tηv+Iρψd=1−tη+1∫01Ωλψ′λv+1−λcdλ−tη+1∫01∇λψ′λv+1−λddλ.
Using mode property on both sides, we obtain(35)1−tηΩ1+tη∇1d−cψv−1d−c1−tηv−Iρψc+tηv+Iρψd≤1−tη+1∫01Ωλψ′λv+1−λcdλ+tη+1∫01∇λψ′λv+1−λddλ.
Sinceψ′ is MT-h-convex, so we have
(36)≤1−tη+1∫01Ωλhλ2h1−λψ′vh1−λ2hλψ′c+tη+1∫01∇λhλ2h1−λψ′v+h1−λ2hλψ′ddλ.
After simplification, we obtain desired result(37)=1−tη+12M1ψ′v+N1ψ′c+tη+12M1ψ′v+N1ψ′d.Corollary 18.
In Theorem17, if we substitute hλ=λ, we get [21] Theorem 2.1.Theorem 19.
Letψ:c,d⟶R be a differentiable function over c,d and ψ′εL1c,d with 0≤c<d and η>0. If ψ′q is MT-h-convex with q>1. Then for each t∈0,1, the following inequality holds:
(38)1−tηΩ1+tη∇1d−cψv−1d−c1−tηv−Iρψc+tηv+Iρψd≤1−tη+1∫01Ωλpdλ1/p12ψ′vq+ψ′cq∫01hλh1−λdλ1/q+tη+1∫01Ωλpdλ1/p12ψ′vq+ψ′dq∫01hλh1−λdλ1/q,where 1/p+1/q=1 and p>1.Proof.
By using Lemma16, Hölder’s inequality, and MT-h-convexity of ψ′q, we get
(39)1−tηΩ1+tη∇1d−cψv−1d−c1−tηv−Iρψc+tηv+Iρψd≤1−tη+1∫01Ωλψ′λv+1−λcdλ+tη+1∫01∇λψ′λv+1−λddλ≤1−tη+1∫01Ωλpdλ1/p∫01ψ′λv+1−λcqdλ1/q+tη+1∫01∇λpdλ1/p∫01ψ′λv+1−λdqdλ1/q≤1−tη+1∫01Ωλpdλ1/p∫01hλ2h1−λψ′vq+h1−λ2hλψ′cqdλ1/q+tη+1∫01∇λpdλ1/p∫01hλ2h1−λψ′vq+h1−λ2hλψ′dqdλ1/q.
Since,(40)∫01hλh1−λdλ=∫01h1−λhλdλ,this implies that
(41)≤1−tη+1∫01Ωλpdλ1/p12ψ′vq+ψ′cq∫01hλh1−λdλ1/q+tη+1∫01Ωλpdλ1/p12ψ′vq+ψ′dq∫01hλh1−λdλ1/q.
The proof is completed.Corollary 20.
If we substitutehλ=λ in Theorem 19, we obtain [21] Theorem 7.Theorem 21.
Letψ:c,d⟶R be a differentiable function on c,d and ψ′∈L1c,d with 0≤c<d and η>0. If function ψ′q is MT-h-convex on c,d for q>1, then for each tε0,1, we have
(42)1−tηΩ1+tμ∇1d−cψv−1d−c1−tηv−Iρψc+tηv+Iρψη≤121/q1−tη+1∫01Ωλdλ1−1/qM1ψ′vq+N1ψ′cq1/q+121/qtη+1∫01∇λdλ1−1/qM2ψ′vq+N2ψ′dq1/q.Proof.
By using Lemma16, power mean integral inequality, and MT-h-convexity of ψ′q, we have
(43)1−tηΩ1+tμ∇1d−cψv−1d−c1−tηv−Iρψc+tηv+Iρψη≤1−tη+1∫01Ωλψ′λv+1−λcdλ+tη+1∫01∇λψ′λv+1−λddλ≤1−tη+1∫01Ωλdλ1−1/q∫01Ωλψ′λv+1−λcqdλ1/q+tη+1∫01∇λdλ1−1/q∫01∇λψ′λv+1−λdqdλ1/q≤1−tη+1∫01Ωλdλ1−1/q∫01Ωλhλ2h1−λψ′vq+h1−λ2hλψ′vqdλ1/q+tη+1∫01∇λdλ1−1/q×∫01∇λhλ2h1−λψ′vq+h1−λ2hλψ′dqdλ1/q≤121/q1−tη+1∫01Ωλdλ1−1/qM1ψ′vq+N1ψ′cq1/q+121/qtη+1∫01∇λdλ1−1/qM2ψ′vq+N2ψ′dq1/q.The proof is completed.Corollary 22.
If we substitute identity functionhλ=λ in Theorem 21, we obtain [21] Theorem 2.3.


## 4. Application to Special Means

In this section, we present applications of our results in special means. Firstly, we give definitions of special means.(1)
The arithmetic mean(44)A=Ac,d=c+d2;c,d∈R.(2)
The logarithmic mean(45)Lc,d=d−clnd−lnc;c≠d,cd≠0,c,d∈R.(3)
The generalized logarithmic mean(46)Lnc,d=dn+1−cn+1d−cn+11/n;n∈ℤ−1,0,c,d∈R,c≠d.Now, using the results of §2, we give some applications to special means of real numbers.Proposition 23.
Assume thatc,d∈R,0<c<d and n∈ℤ,n≥2. Then, ∀q≥1 following inequality holds:
(47)Acn,dn−Lnnc,d≤d−c2×1p+11/p×Qq∫01hλh1−λdλ1/qAcn,dn−Lnnc,d≤d−c2×121−1/q12QqS1+12QqS21/q.Proof.
Forψv=vn the statement follows by Remark 12 and 15 where v∈R,n∈ℤ,n≥2.Proposition 24.
Assume thatc,d∈R,0<a<d. Then, ∀q≥1, we have
(48)Ac−1,d−1−L−1c,d≤d−c2×1p+11/p×Qq∫01hλh1−λdλ1/qAc−1,d−1−L−1c,d≤d−c2×121−1/q12QqS1+12QqS21/q.Proof.
The statement follows by Remark12 and 15 for ψv=1/v.


## 5. Estimates of Error for Trapezoidal Formula

Assume thatf is a division c=v0<v1<⋯<vn−1<vn=d of interval c,d and consider the quadrature formula
(49)∫01ψvdv=Trψ,f+Erψ,f,where
(50)Trψ,f=∑k=0n−1ψvkψvk+12vk+1−vk,for the trapezoidal versionErψ,f donates the associated approximation error.Proposition 25.
Letψ:J⊆R⟶R be a differentiable function on Jo such that ψ′∈L1c,d, where c,d∈J with c<d and ψ′q is MT-h-convex on c,d. Then, in 24, for every division f of c,d and ψ′v≤Q,v∈c,d, the trapezoidal error estimate satisfies
(51)Erψ,f≤12×1p+11/p×Qq∫01hλh1−λdλ1/q∑k=0n−1vk+1−vk2,where p>1,1/p+1/q=1.Proof.
On applying Remark12 on the subinterval vk,vk+1k=0,1,2,⋯,n−1 of the division, we have
(52)∫cdψvdv−Trψ,f=∑k=0n−1∫vkvk+1ψvdv−ψvk+ψvk+12vk+1−vk≤∑k=0n−1∫vkvk+1ψvdv−ψvk+ψvk+12vk+1−vk≤12×1p+11/p×Qq∫01hλh1−λdλ1/q∑k=0n−1vk+1−vk2.With this, the proof is completed.Proposition 26.
Letψ:J⊆R⟶R be a differentiable function on Jo such that ψ′∈L1c,d, where c,d∈J with c<d and ψ′q is MT-h-convex on c,d where q≥1, for every division f of c,d and ψ′v≤Q,v∈c,d, the trapezoidal error estimate satisfies
(53)Erψ,f≤122−1/q12QqS1+12QqS21/q∑k=0n−1vk+1−vk2.Proof.
By using Remark15, the proof comes the same as Proposition 25.


## 6. Conclusion

Convexity and fractional integral operators are the most important notions to deal with the problems of today’s world. In the present paper, we introduced a more general notion of convexity, called as MT-h-convexity. The classical and generalized fractional integral operators are used to establish the most famous and most studied Hermite–Hadamard-type inequalities for the proposed class of convex functions. Applications of presented results to special means are also given. Corollaries and remarks presented in this paper justify the generality of our results. It is interesting to establish Hermite–Hadamard-type inequalities for the other variants of fractional integral operators, like Caputo fractional integral operators and Atangana fractional integral operators.



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