# Some New Hermite–Hadamard-Type Inequalities Associated with Conformable Fractional Integrals and Their Applications

Authors: Arshad Iqbal; Muhammad Adil Khan; Sana Ullah; Yu-Ming Chu
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this article, we establish some new Hermite–Hadamard-type inequalities involving the conformable fractional integrals. As applications, several inequalities for the approximation error in the midpoint formula and certain bivariate means are derived.


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## Body

## 1. Introduction

LetI⊆ℝ be an interval. Then, a real-valued function ϕ:I⟶ℝ is said to be convex (concave) on I if the inequality(1)ϕϑd1+1−ϑd2≤≥ϑϕd1+1−ϑϕd2,holds for all d1,d2∈I with d1≠d2 and ϑ∈0,1.It is well known that the convex functions have wide applications in pure and applied mathematics [1–27]. In particular, many remarkable inequalities and properties for the convex functions can be found in the literature [28–64]. Recently, a great deal of generalizations, extensions, and variants have been made for the convexity [65–72].Letd1,d2∈ℝ with d1<d2 and ϕ:d1,d2⟶ℝ be a convex function. Then, the classical Hermite–Hadamard inequality [73–77] states that the double inequality(2)ϕd1+d22≤1d2−d1∫d1d2ϕzdz≤ϕd1+ϕd22,holds.Alomari et al. [78] and Chen and Feng [79] proved the following.Theorem 1 (see [78]).
LetI⊆ℝ be an interval, I∘ be the interior of I, and ϕ:I∘⟶ℝ be a differentiable mapping. Then, the identity(3)ϕd1+d22−1d2−d1∫d1d2ϕzdz=d2−d14∫01ϕ′d1+d2z2+d11−zzdz+∫01ϕ′d1+d21−z2+d2zz−1dz,holds for d1,d2∈I∘ with d1<d2 if ϕ′∈Ld1,d2.Theorem 2 (see [79]).
LetI⊆ℝ be an interval, I∘ be the interior of I, and ϕ:I∘⟶ℝ be a differentiable mapping. Then, the identity(4)ϕλd1+1−λd2−1d2−d1∫d1d2ϕzdz=λ2d2−d1∫01ϕ′λd1+1−λd21−z+d2zz−1dz+1−λ2d2−d1∫01ϕ′λd1+1−λd2z+d11−zzdz,holds for d1,d2∈I∘ with d1<d2 and λ∈0,1 if ϕ′∈Ld1,d2.Now, we recall the definitions of the conformable fractional derivative and integral.Definition 1 (see [80]).
Letϕ:0,∞⟶ℝ be a real-valued function and α∈0,1. Then, the α-order conformable fractional derivative Dαϕz of ϕ at z∈0,∞ is defined by(5)Dαϕz=limν⟶0ϕz+νz1−α−ϕzν,where ϕ is said to be α-differentiable at z if Dαϕz exists. In particular, Dαϕ0 is defined by(6)Dαϕ0=limz⟶0+Dαϕz,and we use ϕαz or dα/dαzϕ to denote Dαϕz.
The following formulas can be derived from Definition1 immediately.Theorem 3.
Letα∈0,1 and ϕ1 and ϕ2 be two α-differentiable real-valued functions at m>0. Then, one has(7)dαdαmmμ=μmμ−α,dαdαmμ=0,dαdαmμ1ϕ1m+μ2ϕ2m=μ1dαdαmϕ1m+μ2dαdαmϕ2m,dαdαmϕ1mϕ2m=ϕ1mdαdαmϕ2m+ϕ2mdαdαmϕ1m,dαdαmϕ1mϕ2m=ϕ2mdα/dαmϕ1m−ϕ1mdα/dαmϕ2mϕ2m2,for any constants μ,μ1,μ2∈ℝ. If ϕ1 is differentiable at ϕ2m, then(8)dαdαmϕ1ϕ2m=ϕ1′ϕ2mdαdαmϕ2m.Moreover, ifϕ1 is differentiable, then(9)dαdαmϕ1m=m1−αddmϕ1m.Definition 2.
Letα∈0,1 and d1,d2∈0,∞ with d1<d2. Then, the real-valued function ϕ1:d1,d2⟶ℝ is said to be α-fractional integrable if the integral(10)∫d1d2ϕ1ydαy≕∫d1d2ϕ1yyα−1dy,exists and is finite. All α-fractional integrable functions on d1,d2 are denoted by Lα1d1,d2.Remark 1.
(11)
I
α
d
1
ϕ
1
m
=
I
1
d
1
m
α
−
1
ϕ
1
=
∫
d
1
m
ϕ
1
y
y
1
−
α
d
y
,where the integral is the usual Riemann improper integral and α∈0,1.
Anderson [81] established the Hermite–Hadamard inequality for the conformable fractional integral as follows.Theorem 4.
Letα∈0,1, d1,d2∈ℝ with d1<d2, and ϕ:d1,d2⟶ℝ be an α-fractional differentiable function such that Dαϕ is increasing. Then, one has(12)αd2α−d1α∫d1d2ϕydαy≤ϕd1+ϕd22.Moreover, if the functionϕ is decreasing on d1,d2, then(13)ϕd1+d22≤αd2α−d1α∫d1d2ϕydαy.Remark 2.
Ifα=1, then we clearly see that inequalities (12) and (13) reduce to inequality (2).
The aim of this article is to establish some new Hermite–Hadamard-type inequalities for the conformable fractional integrals and present their applications in the error estimations of the midpoint formula and certain bivariate means.


## 2. Main Results

In order to establish our main results, we need a lemma which we present in this section.Lemma 1.
Letα∈0,1, d1,d2∈0,∞ with d1<d2, and ϕ:d1,d2⟶ℝ be an α-fractional differentiable function. Then, the identity(14)ϕλd1+1−λd2−αd2α−d1α∫d1d2ϕsdαs=d1−d2λ−1d2α−d1α∫01zλd1+1−λd2+1−zd12α−1−d1αzλd1+1−λd2+1−zd1α−1×Dαϕzλd1+1−λd2+1−zd1z1−αdαz+d2−d1λd2α−d1α∫01zd2+1−zλd1+1−λd22α−1−d2αzd2+1−zλd1+1−λd2α−1×Dαϕzd2+1−zλd1+1−λd2z1−αdαz,holds for any λ∈0,1 if Dαϕ∈Lα1d1,d2.Proof.
Integrating by parts, we have(15)I1=∫01zλd1+1−λd2+1−zd12α−1−d1αzλd1+1−λd2+1−zd1α−1×Dαϕzλd1+1−λd2+1−zd1z1−αdαz=∫01zλd1+1−λd2+1−zd1α−d1αϕ′zλd1+1−λd2+1−zd1dz=zλd1+1−λd2+1−zd1α−d1αϕzλd1+1−λd2+1−zd1d1−d2λ−101−α∫01zλd1+1−λd2+1−zd1α−1ϕzλd1+1−λd2+1−zd1dz=λd1+1−λd2α−d1αϕλd1+1−λd2d1−d2λ−1−αd1−d2λ−1∫d1λd1+1−λd2ϕsdαs,I2=∫01zd2+1−zλd1+1−λd22α−1−d2αzd2+1−zλd1+1−λd2α−1×Dαϕzd2+1−zλd1+1−λd2z1−αdαz=∫01zd2+1−zλd1+1−λd2α−d2αϕ′zd2+1−zλd1+1−λd2dz=zd2+1−zλd1+1−λd2α−d2αϕzd2+1−zλd1+1−λd2d2−d1λ01−α∫01zd2+1−zλd1+1−λd2α−1ϕzd2+1−zλd1+1−λd2dz=d2α−λd1+1−λd2αϕλd1+1−λd2d2−d1λ−αd2−d1λ∫λd1+1−λd2d2ϕsdαs,where we use the changes of variables s=zλd1+1−λd2+1−zd1 in I1 and s=zd2+1−zλd1+1−λd2 in I2. Then, multiply I1 by d1−d2λ−1/d2α−d1α and I2 by d2−d1λ/d2α−d1α to get the desired result.Remark 3.
Letα=1, then identity (14) reduces to (4).Remark 4.
Letα=1 and λ=1/2, then identity (14) reduces to (3).Theorem 5.
Letα∈0,1, d1,d2∈0,∞ with d1<d2, and ϕ:d1,d2⟶ℝ be an α-fractional differentiable function. Then, the Hermite–Hadamard-type inequality(16)ϕλd1+1−λd2−αd2α−d1α∫d1d2ϕsdαs≤d2−d11−λd2α−d1αA1αϕ′d1+B1αϕ′λd1+1−λd2+d2−d1λd2α−d1αA2αϕ′d2+B2αϕ′λd1+1−λd2,holds for any λ∈0,1 if Dαh∈Lα1d1,d2 and ϕ′ is convex on d1,d2, where(17)A1α=112λ+1λd1+1−λd2d1α−1+d2α−1+λd1d1α−1−d2α−1+d1d2α−1−5d1α−1,B1α=λd1+1−λd2123d2α−11−λ+d1α−11+3λ+d1αλ−5,A2α=λd2α−d1α6,B2α=λd2α−d1α3.Proof.
It follows from Lemma1 and the convexity of the functions ψy=yα−1 and θy=−yα for y>0 together with ϕ′ that(18)ϕλd1+1−λd2−αd2α−d1α∫d1d2ϕsdαs≤d2−d11−λd2α−d1α∫01zλd1+1−λd2+1−zd1α−d1α×ϕ′zλd1+1−λd2+1−zd1dz+d2−d1λd2α−d1α∫01d2α−zd2+1−zλd1+1−λd2α×ϕ′zd2+1−zλd1+1−λd2dz≤d2−d11−λd2α−d1α∫01zλd1+1−λd2+1−zd1α−1×zλd1+1−λd2+1−zd1−aαϕ′zλd1+1−λd2+1−zd1dz+d2−d1λd2α−d1α∫01d2α−zd2α+1−zλd1α+1−λd2α×ϕ′zd2+1−zλd1+1−λd2dz≤d2−d11−λd2α−d1α∫01zλd1α−1+1−λd2α−1+1−zd1α−1×zλd1+1−λd2+1−zd1−d1αzϕ′λd1+1−λd2+1−zϕ′d1dz+d2−d1λd2α−d1α∫01d2α−zd2α+1−zλd1α+1−λd2α×zϕ′d2+1−zϕ′λd1+1−λd2dz=d2−d11−λd2α−d1αA1αϕ′d1+B1αϕ′λd1+1−λd2+d2−d1λd2α−d1αA2αϕ′d2+B2αϕ′λd1+1−λd2.Remark 5.
Letα=1. Then, inequality (16) leads to(19)ϕλd1+1−λd2−αd2−d1∫d1d2ϕsds≤1−λA11ϕ′d1+B11ϕ′λd1+1−λd2+λA21ϕ′d2+B21ϕ′λd1+1−λd2,where A1α, B1α, A2α, and B2α are defined as in Theorem 5.Theorem 6.
Letα∈0,1, d1,d2∈0,∞ with d1<d2, and p,q>1 such that 1/p+1/q=1 and ϕ:d1,d2⟶ℝ be an α-fractional differentiable function. Then, the Hermite–Hadamard-type inequality(20)ϕλd1+1−λd2−αd2α−d1α∫d1d2ϕsdαs≤d2−d11−λd2α−d1αA1α,p1/pE11/q+d2−d1λd2α−d1αA2α,p1/pE21/q,is valid for all λ∈0,1 if Dαϕ∈Łα1d1,d2 and ϕ′q is convex on d1,d2, where(21)A1α,p=∫01zλd1+1−λd2+1−zd1α−d1αpdz=1λ−1d1−d2∫d1λd1+1−zd2sα−d1αpds,A2α,p=∫01d2α−zd2+1−zλd1+1−λd2αpdz=1λd2−d1∫λd1+1−zd2d2d2α−sαpds,E1α=ϕ′λd1+1−λd2q+ϕ′d1q2,E2α=ϕ′d2q+ϕ′λd1+1−λd2q2.Proof.
Making use of Lemma1, the convexity of ϕ′q on d1,d2, Hölder inequality, and the property of the modulus, we get(22)ϕλd1+1−λd2−αd2α−d1α∫d1d2ϕsdαs≤d2−d11−λd2α−d1α∫01zλd1+1−λd2+1−zd1α−d1α×ϕ′λd1+1−λd2+1−zd1dz+d2−d1λd2α−d1α∫01d2α−zd2+1−zλd1+1−λd2α×ϕ′zd2+1−zλd1+1−λd2dz≤d2−d11−λd2α−d1α∫01zλd1+1−λd2+1−zd1α−d1αpdz1/p×∫01ϕ′zλd1+1−λd2+1−zd1qdz1/q+d2−d1λd2α−d1α∫01d2α−zd2+1−zλd1+1−λd2αpdz1/p×ϕ′zd2+1−zλd1+1−λd2qdz1/q≤d2−d11−λd2α−d1α∫01zλd1+1−λd2+1−zd1α−d1αpdz1/p×∫01zϕ′λd1+1−λd2q+1−zϕ′d1qdz1/q+d2−d1λd2α−d1α∫01d2α−zd2+1−zλd1+1−λd2αpdz1/p×zϕ′d2q+1−zϕ′λd1+1−λd2qdz1/q=d2−d11−λd2α−d1αA1α,p1/pE11/p+d2−d1λd2α−d1αA2α,p1/pE21/q.Remark 6.
Letα=1. Then, inequality (20) leads to(23)ϕλd1+1−λd2−1d2−d1∫d1d2ϕsds≤1−λA11,p1/pE11/q+λA21,p1/pE21/q,where A1α,P, A2α,P, E1α, and E2α are defined as in Theorem 6.Theorem 7.
Letα∈0,1, d1,d2∈0,∞ with d1<d2, q≥1, and ϕ:d1,d2⟶ℝ be an α-fractional differentiable function. Then, the Hermite–Hadamard-type inequality(24)ϕλd1+1−λd2−αd2α−d1α∫d1d2ϕsdαs≤d2−d11−λd2α−d1αP1α1−1/qP2αϕ′λd1+1−λd2q+P3αϕ′d1q1/q+d2−d1λd2α−d1αQ1α1−1/qQ2αϕ′d2q+Q3αϕ′λd1+1−λd2q1/q,takes place for all λ∈0,1 if Dαϕ∈Lα1d1,d2 and ϕ′q is convex on d1,d2, where(25)P1α=1λ−1d1−d2λd1+1−λd2α+1−d1α+1α+1−d1αλ−1d1−d2,Q1α=1λd2−d1λd1+1−λd2α+1−d2α+1α+1−d2α2d2−λd1+d2,P2α=1λ−1d2−d12λd1+1−λd2α+2−d1α+2α+2+d1d1α+1−λd1+1−λd2α+1α+1+d1αd12−λd1+1−λd222+d1α+1d1−d21−λ,Q2α=1λ2d2−d12d2d2α+1−λd1+1−λd2α+1α+1+λd2−d1λd1+1−λd2α+1−d2α+1α+1+λd1+1−λd2α+2−d2α+2α+2+d2α+1λd1+1−λd2−d2+d2αd22−λd1+1−λd222+λ2d2αd2−d12,P3α=1λ−1d2−d12λd1+1−λd2α+2−d1α+2α+2+d1d1α+1−λd1+1−λd2α+1α+1+d1αd12−λd1+1−λd222+d1α+1d1−d2λ−1,Q3α=1λ2d2−d12d2λd1+1−λd2α+1−d2α+1α+1+d2α+2−λd1+1−λd2α+2α+2+d2αλd1+1−λd22−d222+d2α+1λd2−d1.Proof.
It follows from Lemma1, convexity of the function ϕ′q, power mean inequality, and the property of the modulus that(26)ϕλd1+1−λd2−αd2α−d1α∫d1d2ϕsdαs≤d2−d11−λd2α−d1α∫01zλd1+1−λd2+1−zd1α−d1α×ϕ′zλd1+1−λd2+1−zd1dz+d2−d1λd2α−d1α∫01d2α−zd2+1−zλd1+1−λd2α×ϕ′zd2+1−zλd1+1−λd2dz≤d2−d11−λd2α−d1α∫01zλd1+1−λd2+1−zd1α−d1αdz1−1/q×∫01zλd1+1−λd2+1−zd1α−d1αϕ′zλd1+1−λd2+1−zd1qdz1/q+d2−d1λd2α−d1α∫01d2α−zd2+1−zλd1+1−λd2αdz1−1/q×∫01d2α−zd2+1−zλd1+1−λd2αϕ′zd2+1−zλd1+1−λd2qdz1/q≤d2−d11−λd2α−d1α∫01zλd1+1−λd2+1−zd1α−d1αdz1−1/q×∫01zλd1+1−λd2+1−zd1α−d1αzϕ′λd1+1−λd2q+1−zϕ′d1qdz1/q+d2−d1λd2α−d1α∫01d2α−zd2+1−zλd1+1−λd2αdz1−1/q×∫01d2α−zd2+1−zλd1+1−λd2αzϕ′d2q+1−zϕ′λd1+1−λd2qdz1/q=d2−d11−λd2α−d1αP1α1−1/qP2αϕ′λd1+1−λd2q+P3αϕ′d1q1/q+d2−d1λd2α−d1αQ1α1−1/qQ2αϕ′d2q+Q3αϕ′λd1+1−λd2q1/q.Remark 7.
Letα=1. Then, from inequality (24), we get(27)ϕλd1+1−λd2−1d2−d1∫d1d2ϕsds≤1−λP111−1/qP21ϕ′λd1+1−λd2q+P31ϕ′d1q1/q+λQ111−1/qQ21ϕ′d2q+Q31ϕ′λd1+1−λd2q1/q,where P1α, P2α, P3α, Q1α, Q2α, and Q3α are defined as in Theorem 7.Theorem 8.
Letα∈0,1, d1,d2∈0,∞ with d1<d2, q≥1, and ϕ:d1,d2⟶ℝ be an α-fractional differentiable function. Then, the Hermite–Hadamard-type inequality(28)ϕλd1+1−λd2−αd2α−d1α∫d1d2ϕsdαs≤d2−d11−λd2α−d1αM1α1−1/qM2αϕ′λd1+1−λd2q+M3αϕ′d1q1/q+d2−d1λd2α−d1αN1α1−1/qN2αϕ′d2q+N3αϕ′λd1+1−λd2q1/q,takes place for all λ∈0,1 if Dαϕ∈Lα1d1,d2 and ϕ′q is convex on d1,d2, where(29)M1α=d1αλ2+λ−2+d2αλ−123−d2d1α−12λ2+λ−3+d1d2α−12λ2−λ−16,N1α=λd2α−d1α2=3N2α=32N3α,M2α=λ−1d1α−d2α3=2M3α.Proof.
From Lemma1 and convexity of ϕ′q, we get(30)ϕλd1+1−λd2−αd2α−d1α∫d1d2ϕsdαs≤d2−d11−λd2α−d1α∫01zλd1+1−λd2+1−zd1α−d1α×ϕ′zλd1+1−λd2+1−zd1dz+d2−d1λd2α−d1α∫01d2α−zd2+1−zλd1+1−λd2α×ϕ′zd2+1−zλd1+1−λd2dz≤d2−d11−λd2α−d1α∫01zλd1+1−λd2+1−zd1α−d1αdz1−1/q×∫01zλd1+1−λd2+1−zd1α−d1αϕ′zλd1+1−λd2+1−zd1qdz1/q+d2−d1λd2α−d1α∫01d2α−zd2+1−zλd1+1−λd2αdz1−1/q×∫01d2α−zd2+1−zλd1+1−λd2αϕ′zd2+1−zλd1+1−λd2sqdz1/q≤d2−d11−λd2α−d1α∫01zλd1+1−λd2+1−zd1α−1×zλd1+1−λd2+1−zd1−d1αdz1−1/q×∫01zλd1+1−λd2+1−zd1α−d1αzϕ′λd1+1−λd2q+1−zϕ′d1qdz1/q+d2−d1λd2α−d1α∫01d2α−zd2+1−zλd1+1−λd2αdz1−1/q×∫01d2α−zd2+1−zλd1+1−λd2αzϕ′d2q+1−zϕ′λd1+1−λd2qdz1/q≤d2−d11−λd2α−d1α∫01zλd1α−1+1−λd2α−1+1−zd1α−1×zλd1+1−λd2+1−zd1−d1αdz1−1/q×∫01zλd1α+1−λd2α+1−zd1α−d1αzϕ′λd1+1−λbq+1−zϕ′d1qdz1/q+d2−d1λd2α−d1α∫01d2α−zd2α+1−zλd1α+1−λd2αdz1−1/q×∫01d2α−zd2α+1−zλd1α+1−λd2αzϕ′d2q+1−zϕ′λd1+1−λd2qdz1/q=d2−d11−λd2α−d1αM1α1−1/qM2αϕ′λd1+1−λd2q+M3αϕ′d1q1/q+d2−d1λd2α−d1αN1α1−1/qN2αϕ′d2q+N3αϕ′λd1+1−λd2q1/q.Remark 8.
Letα=1. Then, inequality (28) becomes(31)ϕλd1+1−λd2−1d2−d1∫d1d2ϕsds≤1−λM111−1/qM21ϕ′λd1+1−λd2q+M31ϕ′d1q1/q+λN111−1/qN21ϕ′d2q+N31ϕ′λd1+1−λd2q1/q,where M1α, M2α, M3α, N1α, N2α, and N3α are defined as in Theorem 8.Theorem 9.
Letα∈0,1, d1,d2∈0,∞ with d1<d2, q>1, and ϕ:d1,d2⟶ℝ be an α-fractional differentiable function. Then, the Hermite–Hadamard-type inequality(32)ϕλd1+1−λd2−αd2α−d1α∫d1d2ϕsdαs≤d2−d11−λd2α−d1αA1αϕ′C1αA1α+d2−d1λd2α−d1αB1αϕ′C2αB1α,holds for any λ∈0,1 if Dαϕ∈Lα1d1,d2 and ϕ′q is concave on d1,d2, where(33)A1α=1d1−d2λ−1λd1+1−λd2α+1−d1α+1α+1+d1αλ−1d2−d1,B1α=1λd2−d1d2αλd2−d1+λd1+1−λd2α+1−d2α+1α+1,C1α=1d1−d2λ−1λd1+1−λd2α+2−d1α+2α+2+d1αd12−λd1+1−λd222,C2α=1λd2−d1λd1+1−λd2α+2−d2α+2α+2+d2αd22−λd1+1−λd222.Proof.
Letγ∈0,1. Then, the power mean inequality and the concavity of ϕ′q lead to the conclusion that(34)γϕ′d1+1−γϕ′d2q≤γϕ′d1q+1−γϕ′d2q≤ϕ′γd1+1−γd2q,which shows that ϕ′ is also concave.
Making use of Lemma1, we get(35)ϕλd1+1−λd2−αd2α−d1α∫d1d2ϕsdαs≤d2−d11−λd2α−d1α∫01zλd1+1−λd2+1−zd1α−d1α×ϕ′λd1+1−λd2+1−zd1dz+d2−d1λd2α−d1α∫01d2α−zd2+1−zλd1+1−λd2α×ϕ′zd2+1−zλd1+1−λd2dz.
Applying Jensen integral inequality, we have(36)∫01zλd1+1−λd2+1−zd1α−d1αϕ′λd1+1−λd2+1−zd1dz≤∫01zλd1+1−λd2+1−zd1α−d1αdz×ϕ′∫01zλd1+1−λd2+1−zd1α−d1αzλd1+1−λd2+1−zd1dz∫01zλd1+1−λd2+1−zd1α−d1αdz=A1αϕ′C1αA1α.
Similarly, one has(37)∫01d2α−zd2+1−zλd1+1−λd2αϕ′d2α−zd2+1−zλd1+1−λd2dz≤∫01d2α−zd2+1−zλd1+1−λd2αdz×ϕ′∫01d2α−zd2+1−zλd1+1−λd2αzd2+1−zλd1+1−λd2dz∫01d2α−zd2+1−zλd1+1−λd2αdz=B1αϕ′C2αB1α.Remark 9.
Letα=1. Then, inequality (32) gives(38)ϕλd1+1−λd2−1d2−d1∫d1d2ϕsds≤1−λA11ϕ′C11A11+λB11ϕ′C21B11,where A1α, C1α, B1α, and C2α are defined as in Theorem 9.Theorem 10.
Letα∈0,1, d1,d2∈0,∞ with d1<d2, and ϕ:d1,d2⟶ℝ be an α-fractional differentiable function. Then, the Hermite-Hadamard-type inequality(39)ϕλd1+1−λd2−αd2α−d1α∫d1d2ϕsdαs≤d2−d11−λd2α−d1αA1αϕ′d1+B1αϕ′λd1+1−λd2+d2−d1λd2α−d1αA2αϕ′d2+B2αϕ′λd1+1−λd2,holds for all λ∈0,1 if Dαϕ∈Lα1d1,d2 and ϕ′ is convex on d1,d2, where A1α, B1α, A2α, and B2α are defined as in Theorem 5.Proof.
It follows from Lemma1 and the convexity of ϕ′ that(40)ϕλd1+1−λd2−αd2α−d1α∫d1d2ϕsdαs≤d2−d11−λd2α−d1α∫01zλd1+1−λd2+1−zd1α−d1α×ϕ′zλd1+1−λd2+1−zd1dz+d2−d1λd2α−d1α∫01d2α−zd2+1−zλd1+1−λd2α×ϕ′zd2+1−zλd1+1−λd2dz≤d2−d11−λd2α−d1α∫01zλd1+1−λd2+1−zd1α−d1α×zϕ′λd1+1−λd2+1−zϕ′d1dz+d2−d1λd2α−d1α∫01d2α−zd2+1−zλd1+1−λd2α×zϕ′d2+1−zϕ′λd1+1−λd2dz=d2−d11−λd2α−d1αA1αϕ′d1+B1αϕ′λd1+1−λd2+d2−d1λd2α−d1αA2αϕ′d2+B2αϕ′λd1+1−λd2.Remark 10.
Letα=1. Then, inequality (39) leads to (41)ϕλd1+1−λd2−αd2−d1∫d1d2ϕsds≤1−λA11ϕ′d1+B11ϕ′λd1+1−λd2+λA21ϕ′d2+B21ϕ′λd1+1−λd2.


## 3. Applications to Bivariate Means

A real-valued functionM:0,∞×0,∞⟶0,∞ is said to be a bivariate mean [82] if mina,b≤Ma,b≤maxa,b for all a,b∈0,∞. Recently, the properties and applications for the bivariate means and their related special functions have attracted the attention of many researchers [83–93].Letd1,d2>0 with d1≠d2, α∈−1,0∪0,1, and r≠0. Then, the arithmetic mean Ad1,d2, logarithmic mean Ld1,d2, and α,r-th generalized logarithmic mean Lα,rd1,d2 are defined by(42)Ad1,d2=d1+d22,Ld1,d2=d2−d1logd2−logd1,Lα,rd1,d2=αd2r+α−d1r+αd2α−d1αr+α1/r,respectively.In this section, we shall present several inequalities involving the arithmetic, logarithmic, and generalized logarithmic means.Letx>0 and ϕx=xr. Then, Theorems 5 and 6 lead to Theorems 11 and 12 immediately.Theorem 11.
Letd1,d2>0 with d1<d2, r>1, and α∈0,1. Then, the inequalities(43)Ard1,d2−Lα,rrd1,d2≤d2−d11−λd2α−d1αA1αd1r−1+B1αλd1+1−λd2r−1+d2−d1λd2α−d1αA2αd2r−1+B2αλd1+1−λd2r−1,A−1d1,d2−Lα,−1−1d1,d2≤d2−d11−λd2α−d1αA1αd1−2+B1αλd1+1−λd2−2+d2−d1λd2α−d1αA2αd2−2+B2αλd1+1−λd2−2,hold for all λ∈0,1, where A1α, A2α, B1α, and B2α are defined as in Theorem 5.Theorem 12.
Letd1,d2>0 with d1<d2, r>1, α∈0,1, and p,q>1 with 1/p+1/q=1. Then, the inequalities(44)Ard1,d2−Lα,rrd1,d2≤d2−d11−λd2α−d1αA1α,p1/pλd1+1−λd2r−1q+d1r−1q21/p+d2−d1λd2α−d1αA2α,p1/pd2r−1qλd1+1−λd2r−1q21/q,A−1d1,d2−Lα,−1−1d1,d2≤d2−d11−λd2α−d1αA1α,p1/pλd1+1−λd2−2q+d1−2q21/q+d2−d1λd2α−d1αA2α,p1/pd2−2q+λd1+1−λd2−2q21/q,hold for all λ∈0,1, where A1α,p and A2α,p are defined as in Theorem 6.Letx>0 and ϕx=1/x. Then, Theorem 6 leads to Theorems 13 immediately.Theorem 13.
Letd1,d2>0 with d1<d2, α∈0,1, and q>1. Then, the inequalities(45)Ard1,d2−Lα,rrd1,d2≤d2−d11−λd2α−d1αM1α1−1/qM2αλd1+1−λd2r−1q+M3αd1r−1q1/q+d2−d1λd2α−d1αN1α1−1/qN2αd2r−1q+N3αλd1+1−λd2r−1q1/q,A−1d1,d2−Lα,−1−1d1,d2≤d2−d11−λd2α−d1αM1α1−1/qM2αλd1+1−λd2−2q+M3αd1−2q1/q+d2−d1λd2α−d1αN1α1−1/qN2αd2−2q+N3αλd1+1−λd2−2q1/q,hold, where M1α, M2α, M3α, N1α, N2α, and N3α are defined as in Theorem 8.


## 4. Applications to Midpoint Formula

LetP be the partition of the points d1=y0<y1<⋯<yn−1<yn=d2 of the interval d1,d2 and consider the quadrature formula(46)∫d1d2ϕydαy=Tαϕ,P+Eαϕ,P,where(47)Tαϕ,P=∑i=0n−1ϕyi+yi+12yi+1α−yiαα,is the midpoint version and Eαϕ,P denotes the associated approximation error. In this section, we are going to derive some new estimates for the midpoint formula.Theorem 14.
Letd1,d2>0 with d1<d2, α∈0,1, and ϕ:d1,d2⟶ℝ be an α-differentiable function. Then, one has(48)Eαϕ,P≤∑i=0n−1yi+1−yi1−λyi+1α−yiαAiαϕ′yi+Biαϕ′λyi+1−λyi+1+∑i=0n−1yi+1−yiλyi+1α−yiαA¯iαϕ′yi+1+B¯iαϕ′λyi+1−λyi+1,if Dαϕ∈Lα1d1,d2 and ϕ′ is convex on d1,d2, where(49)Aiα=112λyi+1−λyi+1λ+1yiα−1+yi+1α−1+λyiyiα−1−yi+1α−1+yiyi+1α−1−5yiα−1,Biα=λyi+1−λyi+1123λyiα−1−yi+1α−1+3yi+1α−1+yiα−1+yiαλ−5,A¯iα=λyi+1α−yiα6,B¯iα≔λyi+1α−yiα3.Proof.
Applying Theorem5 on the subintervals yi,yi+1i=0,1,…,n−1 of the partition P, we have(50)ϕλyi+1−λyi+1yi+1α−yiαα−∫yiyi+1ϕydαy≤yi+1−yi1−λyi+1α−yiαAiαϕ′yi+Biαϕ′λyi+1−λyi+1+yi+1−yiλyi+1α−yiαA¯iαϕ′yi+1+B¯iαϕ′λyi+1−λyi+1,∫d1d2ϕydαy−Tαϕ,P=∑i=0n−1∫yiyi+1ϕydαy−ϕλyi+1−λyi+1yi+1α−yiαα≤∑i=0n−1∫yiyi+1ϕydαy−ϕλyi+1−λyi+1yi+1α−yiαα≤∑i=0n−1yi+1−yi1−λyi+1α−yiαAiαϕ′yi+Biαϕ′λyi+1−λyi+1+∑i=0n−1yi+1−yiλyi+1α−yiαA¯iαϕ′yi+1+B¯iαϕ′λyi+1−λyi+1.
Making use of Theorems6–8 and similar arguments as in the proof of Theorem 14, we get Theorems 15 and 16 immediately.Theorem 15.
Letd1,d2>0 with d1<d2, α∈0,1, and ϕ:d1,d2⟶ℝ be an α-differentiable function and p,q>1 with 1/p+1/q=1. Then, the inequality(51)Eαϕ,P≤∑i=0n−1yi+1−yi1−λyi+1α−yiαAiα,p1/pEi1/q+∑i=0n−1yi+1−yiλyi+1α−yiαA¯iα,p1/pE¯i1/q,holds for all λ∈0,1 if Dαϕ∈Lα1d1,d2 and ϕ′q is convex on d1,d2, where(52)Eiα=ϕ′λyi+1−λyi+1q+ϕ′yiq2,E¯iα=ϕ′yi+1q+ϕ′λyi+1−λyi+1q2,Aiα,p=1λ−1yi−yi+1∫yiλyi+1−zyi+1sα−yiαpds,A¯iα,p=1λyi+1−yi∫λyi+1−zyi+1yi+1yi+1α−sαpds.Theorem 16.
Letd1,d2>0 with d1<d2, α∈0,1, and ϕ:d1,d2⟶ℝ be an α-differentiable function and q≥1. Then, the inequality(53)Eαϕ,P≤∑i=0n−1yi+1−yi1−λyi+1α−yiαMiα1−1/qM¯iαϕ′λyi+1−λyi+1q+M¯iαϕ′yiq1/q+∑i=0n−1yi+1−yiλyi+1α−yiαNiα1−1/qN¯iαϕ′yi+1q+Ni¯αϕ′λyi+1−λyi+1q1/q,holds for all λ∈0,1 if Dαϕ∈Lα1d1,d2 and ϕ′q is convex on d1,d2, where(54)Miα=yiαλ2+λ−2+yi+1αλ−123−yi+1yiα−12λ2+λ−3+yiyi+1α−12λ2−λ−16,Niα=λyi+1α−yiα2=3N¯iα=32N¯iα,M¯iα=λ−1yiα−yi+1α3=2M¯iα.



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