# Generalization of Hermite-Hadamard Type Inequalities via Conformable Fractional Integrals

Authors: Muhammad Adil Khan; Yousaf Khurshid; Ting-Song Du; Yu-Ming Chu
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We establish a Hermite-Hadamard type identity and several new Hermite-Hadamard type inequalities for conformable fractional integrals and present their applications to special bivariate means.


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## Body

## 1. Introduction

In the field of nonlinear programming and optimization theory, no one can ignore the role of convex sets and convex functions. For the class of convex functions, many inequalities have been introduced such as Jensen’s, Hermite-Hadmard, and Slater’s inequalities. Among those inequalities, the most famous and important inequality is the Hermite-Hadamard’s inequality [1] which can be stated as follows.LetI⊆R be an interval and h:I⊆R→R be a convex function defined on I. Then the double inequality(1)ha1+a22≤1a2-a1∫a1a2hxdx≤ha1+ha22holds for all a1,a2∈I with a1<a2. Both inequalities in (1) hold in the reverse direction if the function h is concave on I.In the last 60 years, many efforts have gone on generalizations, extensions, variants, and applications for the Hermite-Hadamard’s inequality (see [2–13]). Anderson [14] and Sarikaya et al. [15] provide the important variants for the Hermite-Hadamard’s inequality.Recently, the author in [16] gave a new definition for the (conformable) fractional derivative as follows.Let0<α≤1 and h:[0,∞)→R be a real-valued function. Then the α-order (conformable) fractional derivative of h at s>0 is defined by (2)Dαhs=limϵ→0⁡hs+ϵs1-α-hsϵ.h is said to be α-differentiable if the α-order (conformable) fractional derivative of h exists, and the α-order (conformable) fractional derivative of h at 0 is defined as hα(0)=lims→0+hα(s).Now we discuss some theorems for the (conformable) fractional derivative.Theorem 1.
Letα∈(0,1] and h1,h2 be α-differentiable at s>0. Then one has the following:(i)
(dα/dαs)sn=nsn-α for all n∈R.(ii)
(dα/dαs)(c)=0 for all constants c∈R.(iii)
(dα/dαs)a1h1s+a2h2s=a1(dα/dαs)(h1(s))+a2(dα/dαs)(h2(s)) for all constants a1,a2∈R.(iv)
(dα/dαs)h1sh2s=h1(s)(dα/dαs)(h2(s))+h2(s)(dα/dαs)(h1(s)).(v)
(dα/dαs)h1(s)/h2(s)=(h2sdα/dαsh1s-h1sdα/dαsh2s)/h2s2.(vi)
(dα/dαs)(h1(h2(s)))=h1′(h2(s))(dα/dαs)(h2(s)) if h1 is differentiable at h2(s).
In addition,(3)dαdαsh1s=s1-αddsh1sif h1 is differentiable.Definition 2 (conformable fractional integral).
Letα∈(0,1] and 0≤a1<a2. Then the function h1:[a1,a2]→R is said to be α-fractional integrable on [a1,a2] if the integral(4)∫a1a2h1xdαx≔∫a1a2h1xxα-1dxexists and is finite. All α-fractional integrable functions on [a1,a2] are indicated by Lα([a1,a2]).Remark 3.
(5)Iαa1h1s=I1a1sα-1h1=∫a1sh1xx1-αdx,where the integral is the usual Riemann improper integral and α∈(0,1].Recently, the conformable integrals and derivatives have attracted the attention of many researchers, and many remarkable properties and inequalities for the conformable integrals and derivatives can be found in the literature [17–24]. Anderson [14] found the conformable integral version of the Hermite-Hadamard inequality as follows.Theorem 4 (see [14]).
Ifα∈(0,1] and h1:[a1,a2]→R is an α-fractional differentiable function such that Dα(h) is increasing, then we have the following inequality:(6)αa2α-a1α∫a1a2hxdαx≤ha1+ha22.Moreover if the function h is decreasing on [a1,a2], then we have(7)ha1+a22≤αa2α-a1α∫a1a2hxdαx.If α=1, then inequalities (6) and (7) reduce to the classical Hermite-Hadamard’s inequalities.The main purpose of the article is to present the conformable fractional integrals version of the Hermite-Hadamard’s inequality. We first establish an identity for the conformable fractional integrals (Lemma5) and discuss their special cases. Then applying Jensen’s inequality, power mean inequality, Hölder inequality, the convexity of the functions xα-1 and -xα(x>0,α∈(0,1]), and the identity given by Lemma 5, we obtain inequalities for conformable fractional integrals version of the Hermite-Hadamard’s inequality. At last, using particular classes of convex functions we find several new inequalities for some special bivariate means. For some related results, see [25, 26].


## 2. Main Results

The main results of our work can be calculated with the help of the following lemma associated with inequality (8).Lemma 5.
Leta1,a2∈R+ with a1<a2, α∈(0,1], and h:[a1,a2]→R be an α-fractional differentiable function on (a1,a2). Then the identity(8)a2α-ηαha2+ηα-a1αha1a2α-a1α-αa2α-a1α∫a1a2hsdαs=η-a1a2α-a1α∫011-ta1+tη2α-1-ηα1-ta1+tηα-1×Dαh1-ta1+tηt1-αdαt+a2-ηa2α-a1α∫011-ta2+tη2α-1-ηα1-ta2+tηα-1×Dαh1-ta2+tηt1-αdαtholds for any η∈[a1,a2] if Dα(h)∈Lα([a1,a2]).Proof.
It follows from Theorem1, Definition 2, and integrating by parts that (9)
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,where we have used the changes of variable s=(1-t)a1+tη and s=(1-t)a2+tη to get the desired result.Remark 6.
Letα=1. Then identity (8) becomes (11)a2-ηha2+η-a1ha1a2-a1-1a2-a1∫a1a2hsds=η-a12a2-a1∫01t-1h′1-ta1+tηdt+a2-η2a2-a1∫011-th′1-ta2+tηdt,which was proved by Kavurmaci et al. in [2].Theorem 7.
Leta1,a2∈R+ with a1<a2, α∈(0,1], and h:[a1,a2]→R be an α-differentiable function on (a1,a2). Then the inequality (12)a2α-ηαha2+ηα-a1αha1a2α-a1α-αa2α-a1α∫a1a2hsdαs≤η-a1a2α-a1α12ηαh′a1-13a1αh′a1-16ηαh′a1+12ηαh′η-16a1αh′η-13ηαh′η+a2-ηa2α-a1α14a2αh′a2+112ηα-1a2h′a2+112ηa2α-1h′a2+112ηαh′a2-12ηαh′a2+112a2αh′η+112ηα-1a2h′η+112ηa2α-1h′η+14ηαh′η-12ηαh′ηholds for any η∈[a1,a2] if Dα(h)∈Lα([a1,a2]) and h′ is convex on [a1,a2].Proof.
Letx>0, φ1(x)=xα-1, and φ2(x)=-xα. Then we clearly see that both the functions φ1(x) and φ2(x) are convex. From Lemma 5 and the convexity of φ1, φ2, and h′, we have(13)
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.From the final upper bound above, we have the following: (14)a2α-ηαha2+ηα-a1αha1a2α-a1α-αa2α-a1α∫a1a2hsdαs≤η-a1a2α-a1α12ηαh′a1-13a1αh′a1-16ηαh′a1+12ηαh′η-16a1αh′η-13ηαh′η+a2-ηa2α-a1α14a2αh′a2+112ηα-1a2h′a2+112ηa2α-1h′a2+112ηαh′a2-12ηαh′a2+112a2αh′η+112ηα-1a2h′η+112ηa2α-1h′η+14ηαh′η-12ηαh′η.Remark 8.
Letα=1. Then inequality (12) leads to (15)η-a1ha1+a2-ηha2a2-a1-1a2-a1∫a1a2hxdx≤η-a12a2-a1h′η+2h′a16+a2-η2a2-a1h′η+2h′a26,which was proved by Kavurmaci et al. in [2].Theorem 9.
Leta1,a2∈R+ with a2>a1, α∈(0,1], p,q>1 such that 1/p+1/q=1 and h:[a1,a2]→R be an α-differentiable function on (a1,a2). Then the inequality(16)a2α-ηαha2+ηα-a1αha1a2α-a1α-αa2α-a1α∫a1a2hsdαs≤η-a1a2α-a1αA1α,p1/ph′a1q+h′ηq21/q+a2-ηa2α-a1αB1α,p1/ph′a2q+h′ηq21/qholds for any η∈[a1,a2] if Dα(h)∈Lα([a1,a2]) and h′q is convex on [a1,a2], where (17)A1α,p=∫01ηα-1-ta1α+tηαpdt,B1α,p=∫011-ta2α-1+tηα-11-ta2+tη-ηαpdt.Proof.
It follows from inequality (13) that (18)a2α-ηαha2+ηα-a1αha1a2α-a1α-αa2α-a1α∫a1a2hsdαs≤η-a1a2α-a1α∫01ηα-1-ta1α+tηαh′1-ta1+tηdt+a2-ηηα-a1α∫011-ta2α-1+tηα-11-ta2+tη-ηαh′1-ta2+tηdt.Making use of Hölder’s inequality, one has(19)∫01ηα-1-ta1α+tηαh′1-ta1+tηdt≤∫01ηα-1-ta1α+tηαpdt1/p∫01h′1-ta1+tηqdt1/q≤A1α,p1/p∫011-th′a1q+th′ηqdt1/q=A1α,p1/ph′a1q+h′ηq21/q.Similarly, we have (20)∫011-ta2α-1+tηα-11-ta2+tη-ηαh′1-ta2+tηdt≤∫011-ta2α-1+tηα-11-ta2+tη-ηαpdt1/p×∫01h′1-ta2+tηqdt1/q≤B1α,p1/p∫011-th′a2q+th′ηqdt1/q≤B1α,p1/ph′a2q+h′ηq21/q.
Hence, we have the result in (16).Remark 10.
By puttingα=1 in (16), we obtain the inequality (21)η-a1ha1+a2-ηha2a2-a1-1a2-a1∫a1a2hxdx≤1p+11/p121/qη-a12h′ηq+h′a1q1/q+a2-η2h′ηq+h′a2q1/qa2-a1,which was proved by Kavurmaci et al. in [2].Theorem 11.
Leta1,a2∈R+ with a1<a2, α∈(0,1], q>1 and h:[a1,a2]→R be an α-differentiable function on (a1,a2). Then the inequality(22)a2α-ηαha2+ηα-a1αha1a2α-a1α-αa2α-a1α∫a1a2hsdαs≤η-a1a2α-a1αA1α1-1/qA2αh′a1q+A3αh′ηq1/q+a2-ηa2α-a1αB1α1-1/qB2αh′a2+B3αh′η1/qholds for any η∈[a1,a2] if Dα(h)∈Lα([a1,a2]) and h′q is convex on [a1,a2], where (23)A1α=ηα-a1α2,B1α=2a2α+ηα-1a2+ηa2α-1-4ηα6,A2α=ηα-a1α3,A3α=ηα-a1α6,B2α=3a2α+ηα-1a2+ηa2α-1-5ηα12,B3α=a2α+ηα-1a2+ηa2α-1-3ηα12.Proof.
It follows from inequality (13) that (24)a2α-ηαha2+ηα-a1αha1a2α-a1α-αa2α-a1α∫a1a2hsdαs≤η-a1a2α-a1α∫01ηα-1-ta1α+tηαh′1-ta1+tηdt+a2-ηηα-a1α∫011-ta2α-1+tηα-11-ta2+tη-ηαh′1-ta2+tηdt.Making use of the power mean inequality, we get (25)∫01ηα-1-ta1α+tηαh′1-ta1+tηdt≤∫01ηα-1-ta1α+tηαdt1-1/q×∫01ηα-1-ta1α+tηαh′1-ta1+tηqdt1/q.Similarly, we have (26)∫011-ta2α-1+tηα-11-ta2+tη-ηαh′1-ta2+tηdt≤∫011-ta2α-1+tηα-11-ta2+tη-ηαdt1-1/q×∫011-ta2α-1+tηα-11-ta2+tη-ηαh′1-ta2+tηqdt1/q.From the convexity of h′q, we have (27)∫01ηα-1-ta1α+tηαh′1-ta1+tηqdt≤∫01ηα-1-ta1α+tηα1-th′a1q+th′ηqdt=h′a1q∫01ηα-1-ta1α+tηα1-tdt+h′ηq∫01ηα-1-ta1α+tηαtdt=f′a1q12ηα-13a1α-16ηα+h′ηq12ηα-16a1α-13ηαand (28)
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,where we have also used the facts that (29)∫01ηα-1-ta1α+tηαdt=A1α=ηα-12a1α-12ηα,∫011-ta2α-1+tηα-11-ta2+tη-ηαdt=B1α=13a2α+16ηα-1a2+16ηa2α-1+13ηα-ηα.Hence, we have the result in (22).Remark 12.
Ifα=1, then inequality (22) becomes (30)η-a1ha1+a2-ηha2a2-a1-1a2-a1∫a1a2hxdx≤12131/qη-a12h′ηq+2h′a1q1/q+a2-η2h′ηq+2h′a2q1/qa2-a1,which can be found in [2].Theorem 13.
Leta1,a2∈R+ with a1<a2, α∈(0,1], q>1 and h:[a1,a2]→R be an α-differentiable function on (a1,a2). Then the inequality(31)a2α-ηαha2+ηα-a1αha1a2α-a1α-αa2α-a1α∫a1a2hsdαs≤η-a1a2α-a1αA1α1-1/qA2αh′a1q+A3αh′ηq1/q+a2-ηa2α-a1αB1α1-1/qB2αh′a2q+B3αh′ηq1/qis valid for any η∈[a1,a2] if Dα(h)∈Lα([a1,a2]) and h′q is convex on [a1,a2], where (32)A1α=ηα-ηα+1-a1α+1α+1η-a1,B1α=ηα+1-a2α+1α+1a2-η-ηα,A2α=12ηα+a1α+1α+1η-a1α+2η-a1-a1α+2η-a1-ηα+2α+1η-a12α+2,B2α=-a2α+1α+1a2-ηα+2a2-η+a2α+2a2-η+ηα+2α+1a2-η2α+2-12ηα,A3α=12ηα-ηα+1α+1η-a1α+2η-a1-ηα+2η-a1-a1α+2α+1η-a12α+2,B3α=ηα+1α+1a2-ηα+2a2-η-ηα+2a2-η-a2α+2α+1a2-η2α+2-12ηα.Proof.
From Theorem1, Definition 2, and Lemma 5, we get (33)a2α-ηαha2+ηα-a1αha1a2α-a1α-αa2α-a1α∫a1a2hsdαs=η-a1a2α-a1α∫011-ta1+tη2α-1-ηα1-ta1+tηα-1Dαh1-ta1+tηdt+a2-ηa2α-a1α∫011-ta2+tη2α-1-ηα1-ta2+tηα-1Dαh1-ta2+tηdt≤η-a1a2α-a1α∫01ηα-1-ta1+tηαh′1-ta1+tηdt+a2-ηa2α-a1α∫011-ta2+tηα-ηαh′1-ta2+tηdt.Making use of power mean inequality, we get (34)∫01ηα-1-ta1+tηαh′1-ta1+tηαdt≤∫01ηα-1-ta1+tηαdt1-1/q×∫01ηα-1-ta1+tηαh′1-ta1+tηqdt1/q.Similarly, we have (35)∫011-ta2+tηα-ηαh′1-ta2+tηdt≤∫011-ta2+tηα-ηαdt1-1/q×∫011-ta2+tηα-ηαh′1-ta2+tηqdt1/q.It follows from the convexity of h′q that (36)∫01ηα-1-ta1+tηαh′1-ta1+tηqdt≤∫01ηα-1-ta1+tηα1-th′a1q+th′ηqdt=h′a1q∫01ηα-1-ta1+tηα1-tdt+h′ηq∫01ηα-1-ta1+tηαtdt=h′a1q12ηα+a1α+1α+1η-a1α+2η-a1-a1α+2η-a1-ηα+2α+1η-a12α+2+h′ηq12ηα-ηα+1α+1η-a1α+2η-a1-ηα+2η-a1-a1α+2α+1η-a12α+2and(37)∫011-ta2+tηα-ηαh′1-ta2+tηqdt≤∫011-ta2+tηα-ηα1-th′a2q+th′ηqdt=h′a2q∫011-ta2+tηα-ηα1-tdt+h′ηq∫011-ta2+tηα-ηαtdt=h′a2q-a2α+1α+1a2-ηα+2a2-η+a2α+2a2-η+ηα+2α+1a2-η2α+2-12ηα+h′ηqηα+1α+1a2-ηα+2a2-η-ηα+2a2-η-a2α+2α+1a2-η2α+2-12ηα,where we have used the identities (38)∫01ηα-1-ta1+tηαdt=ηα-ηα+1-a1α+1α+1η-a1,∫011-ta2+tηα-ηαdt=ηα+1-a2α+1α+1a2-η-ηα.Hence, we have the result in (31).Theorem 14.
Leta1,a2∈R+ with a1<a2, α∈(0,1] and h:[a1,a2]→R be an α-differentiable function on (a1,a2). Then the inequality(39)a2α-ηαha2+ηα-a1αha1a2α-a1α-αa2α-a1α∫a1a2hsdαs≤η-a1a2α-a1αA1αh′C1αA1α+a2-ηa2α-a1αB1αh′C2αB1αholds for any η∈[a1,a2] if Dα(h)∈Lα1([a1,a2]) and h′q is concave on [a1,a2] for some q>1, where (40)A1α=ηα-a1α2,B1α=2a2α+ηα-1a2+ηa2α-1-4ηα6,C1α=a1ηα-2a1α+1+ηα+16,C2α=3a2α+1+ηα-1a22+2ηa2α-4a2ηα+ηα+1a2α-1-3ηα+112.Proof.
We clearly see thath′ is concave because h′q is concave for some q>1 (see [27]). From Theorem 1, Definition 2, Lemma 5, the concavity of h′, and Jensen’s integral inequality, we have (41)a2α-ηαha2+ηα-a1αha1a2α-a1α-αa2α-a1α∫a1a2hsdαs≤η-a1a2α-a1α∫01ηα-1-ta1α+tηαh′1-ta1+tηdt+a2-ηa2α-a1α∫011-ta2α-1+tηα-11-ta2+tη-ηαh′1-ta2+tηdt,∫01ηα-1-ta1α+tηαh′1-ta1+tηdt≤∫01ηα-1-ta1α+tηαdt×h′∫01ηα-1-ta1α+tηα1-ta1+tηdt∫01ηα-1-ta1α+tηαdt=A1αh′C1αA1α,∫011-ta2α-1+tηα-11-ta2+tη-ηαh′1-ta2+tηdt≤∫011-ta2α-1+tηα-11-ta2+tη-ηαdt×h′∫011-ta2α-1+tηα-11-ta2+tη-ηα1-ta2+tηdt∫011-ta2α-1+tηα-11-ta2+tη-ηαdt=B1αh′C2αB1α,where we have used the identities (42)∫01ηα-1-ta1α+tηαdt=A1α=ηα-12a1α-12ηα,∫011-ta2α-1+tηα-11-ta2+tη-ηαdt=B1α=13a2α+16ηα-1a2+16ηa2α-1+13ηα-ηα,∫01ηα-1-ta1α+tηα1-ta1+tηdt=C1α=12a1ηα-13a1α+1-16a1ηα+12ηα+1-16ηa1α-13ηα+1and(43)∫011-ta2α-1+tηα-11-ta2+tη-ηα1-ta2+tηdt=C2α=14a2α+1+112ηα-1a22+16ηa2α-13a2ηα+112ηα+1a2α-1-14ηα+1.Remark 15.
Ifα=1, then inequality (39) becomes(44)η-a1ha1+a2-ηha2a2-a1-1a2-a1∫a1a2hxdx≤12η-a12h′η+2a1/3+a2-η2h′η+2a2/3a2-a1.


## 3. Applications to Special Means

A bivariate functionM:(0,∞)×(0,∞)↦(0,∞) is said to be a mean if min⁡{x,y}≤M(x,y)≤max⁡{x,y} for all x,y∈(0,∞). Recently, the mean value theory has been the subject of intensive research, and many remarkable inequalities and properties for various bivariate means can be found in the literature [28–33].In this section, we use the results obtained in Section2 to give some applications to the weighted arithmetic mean (45)Aa1,a2;w1,w2=w1a1+w2a2w1+w2a1,a2>0and (α,r)-th generalized logarithmic mean (46)Lα,ra1,a2=αa2α+r-a1α+rα+ra2α-a1α1/ra1,a2>0,a1≠a2,r∈R,r≠0,α,-α∈0,1.Proposition 16.
Leta1,a2∈R+ with a1<a2 and r>1. Then the inequality (47)Aa1r,a2r;ηα-a1α,a2α-ηα-Lα,rra1,a2≤rη-a1a2α-a1α12ηαa1r-1-13a1αa1r-1-16ηαa1r-1+12ηαηr-1-16a1αηr-1-13ηαηr-1+ra2-ηa2α-a1α14a2αa2r-1+112ηα-1a2a2r-1+112ηa2α-1a2r-1+112ηαa2r-1-12ηαa2r-1+112a2αηr-1+112ηα-1a2ηr-1+112ηa2α-1ηr-1+14ηαηr-1-12ηαηr-1holds for any α∈(0,1] and η∈[a1,a2].Proof.
Leth(x)=xr. Then the result follows easily from Theorem 7 and the convexity of h(x) on the interval [a1,a2].Proposition 17.
Leta1,a2∈R+ with a1<a2 and r>1. Then the inequality (48)Aa1-1,a2-1;ηα-a1α,a2α-ηα-Lα,-1ra1,a2≤η-a1a2α-a1α12ηαa1-2-13a1αa1-2-16ηαa1-2+12ηαη-2-16a1αη-2-13ηαη-2+a2-ηa2α-a1α14a2αa2-2+112ηα-1a2a2-2+112ηa2α-1a2-2+112ηαa2-2-12ηαa2-2+112a2αη-2+112ηα-1a2η-2+112ηa2α-1η-2+14ηαη-2-12ηαη-2holds for any α∈(0,1] and η∈[a1,a2].Proof.
Leth(x)=1/x. Then Proposition 17 follows from Theorem 7 and the convexity of the function h(x) on the interval [a1,a2].Proposition 18.
Leta1,a2∈R+ with a1<a2 and r>1. Then the inequality (49)Aa1r,a2r;ηα-a1α,a2α-ηα-Lα,rra1,a2≤rη-a1a2α-a1αA1α1-1/qA2αa1r-1q+A3αηr-1q1/q+ra2-ηa2α-a1αB1α1-1/qB2αa2r-1q+B3αηr-1q1/qholds for all α∈(0,1] and η∈[a1,a2], where Ai(α) and Bi(α)(i=1,2,3) are defined as in Theorem 11.Proof.
Leth(x)=xr. Then Proposition 18 follows from Theorem 11 and the convexity of h(x) on [a1,a2] immediately.Proposition 19.
Leta1,a2∈R+ with a1<a2 and r>1. Then the inequality (50)Aa1-1,a2-1;ηα-a1α,a2α-ηα-Lα,-1ra1,a2≤η-a1a2α-a1αA1α1-1/qA2αa1-2q+A3αη-2q1/q+a2-ηa2α-a1αB1α1-1/qB2αa2-2q+B3αη-2q1/qholds for all α∈(0,1] and η∈[a1,a2], where Ai(α) and Bi(α)(i=1,2,3) are defined as in Theorem 11.Proof.
Leth(x)=1/x. Then the result follows easily from Theorem 11 and the convexity of h(x) on the interval [a1,a2].


## 4. Conclusion

In the article, we establish an identity and several new inequalities of Hermite-Hadamard type for conformable fractional integrals by use of the convexity theory and Jesen’s inequality, Hölder inequality, and power mean inequality and present their applications to special bivariate means. The given Hermite-Hadamard type inequalities for conformable fractional integrals are the generalizations of the corresponding results established by Kavurmaci, Avci, and Özdemir in [2], and the idea may stimulate further research in the theory of Hermite-Hadamard’s inequalities, conformable fractional integrals, and generalized convex functions.



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