# Conformable Integral Inequalities of the Hermite-Hadamard Type in terms of GG- and GA-Convexities

Authors: Yousaf Khurshid; Muhammad Adil Khan; Yu-Ming Chu
Journal: Journal of Function Spaces
 (2019)

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## Abstract

In the article, we present several conformable fractional integrals’ versions of the Hermite-Hadamard type inequalities for GG- and GA-convex functions and provide their applications in special bivariate means.


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## Body

## 1. Introduction

LetI∈R be an interval and let h:I→R be a convex function. Then the well known HH (Hermite-Hadamard) inequality [1] states that(1)hκ1+κ22≤1κ2-κ1∫κ1κ2hxdx≤hκ1+hκ22for all κ1,κ2∈I. It is well known that the convexity has been playing a key role in mathematical programming, engineering, and optimization theory. Recently, many generalizations and extensions for the classical convexity can be found in the literature [2–14]. In [15, 16], Niculescu defined the GA- and GG-convex functions as follows.Definition 1 (see [15]).
A functionh:I→[0,∞) is said to be GA-convex if the inequality (2)hκ1tκ21-t≤thκ1+1-thκ2holds for all κ1,κ2∈I and t∈[0,1].Definition 2 (see [16]).
A functionh:I→[0,∞) is said to be GG-convex if the inequality (3)hκ1tκ21-t≤hκ1thκ21-tholds for all κ1,κ2∈I and t∈[0,1].Zhang, Ji, and Qi established Lemma3 and Theorems 4–7.Lemma 3 (see [17]).
Letκ1,κ2∈R+ and h:[κ1,κ2]→R be a differentiable function on (κ1,κ2). Then the identity(4)κ2hκ2-κ1hκ1-∫κ1κ2hxdx=log⁡κ2-log⁡κ1∫01κ22tκ121-th′κ2tκ11-tdtholds if h′∈L([κ1,κ2]).Theorem 4 (see [17]).
If the functionh satisfies the conditions of Lemma 3 and, additionally, if h′x is GA-convex, then we have the following inequality:(5)κ2hκ2-κ1hκ1-∫κ1κ2hxdx≤h′κ22κ22-Lκ22,κ12+h′κ12Lκ22,κ12-κ12,where and in what follows L(κ1,κ2)=(κ2-κ1)/(log⁡κ2-log⁡κ1) is the logarithmic mean of κ1 and κ2.Theorem 5 (see [17]).
If the functionh satisfies the conditions of Lemma 3 and, additionally, if h′xγ(γ>1) is GA-convex, then one has(6)κ2hκ2-κ1hκ1-∫κ1κ2hxdx≤log⁡κ2-log⁡κ11-1/γLκ22,κ121-1/γ×h′κ2γκ22-Lκ22,κ12+h′κ1γLκ22,κ12-κ1221/γ.Theorem 6 (see [17]).
If the functionh satisfies the conditions of Lemma 3, then the inequality(7)κ2hκ2-κ1hκ1-∫κ1κ2hxdx≤log⁡κ2-log⁡κ11-1/γ×h′κ2γκ22γ-Lκ22γ,κ12γ+h′κ1γLκ22γ,κ12γ-κ12γ2γ1/γholds if h′xγ(γ>1) is GA-convex.Theorem 7 (see [17]).
Ifϑ,γ>1 with ϑ-1+γ-1=1 and the function h satisfies the conditions of Lemma 3 and, additionally, if h′γ is GA-convex, then we have(8)κ2hκ2-κ1hκ1-∫κ1κ2hxdx≤log⁡κ2-log⁡κ1Lκ22ϑ,κ12ϑ1/ϑAh′κ2γ,h′κ1γ1/γ,where A(x,y)=(x+y)/2 is the arithmetic mean of x and y.The conformable fractional derivative of order0<α≤1 at s>0 for a function h:[0,∞)→R was defined in [18] as follows: (9)Dαhs=limϵ→0hs+ϵs1-α-hsϵ.h is said to be α-fractional differentiable if the conformable fractional derivative of h of order α exists. The fractional derivative at 0 is defined as hα(0)=lims→0+hα(s). Theorem 8 for the conformable fractional derivative can be found in the literature [18].Theorem 8 (see [18]).
Letα∈(0,1] and h1,h2 be α-differentiable at t>0. Then (i)
dα/dαttn=ntn-α for all n∈R(ii)
dα/dαt(c)=0 if c∈R is a constant(iii)
dα/dαtκ1h1(t)+κ2h2(t)=κ1dα/dαt(h1(t))+κ2dα/dαt(h2(t)) for all κ1,κ2∈R(iv)
dα/dαth1(t)h2(t)=h1(t)dα/dαt(h2(t))+h2(t)dα/dαt(h1(t))(v)
dα/dαth1(t)/h2(t)=h2tdα/dαth1t-h1tdα/dαth2t/(h2(t))2(vi)
dα/dαt(h1∘h2)(t)=h1′(h2(t))dα/dαt(h2(t)) if h1 is differentiable at h2(t).
In addition,(10)dαdαth1t=t1-αddth1t if h1 is differentiable.Definition 9 (see [18], conformable fractional integral).
Letα∈(0,1] and 0≤κ1<κ2. A function h:[κ1,κ2]→R is α-fractional integrable on [κ1,κ2] if the integral(11)∫κ1κ2hxdαx≔∫κ1κ2hxxα-1dxexists and is finite. All α-fractional integrable functions on [κ1,κ2] are indicated by Lα([κ1,κ2]).Remark 10.
(12) I α κ 1 h 1 s = I 1 κ 1 s α - 1 h 1 = ∫ κ 1 s h 1 x x 1 - α d x ,where the integral is the usual Riemann improper integral and α∈(0,1].Recently, the conformable integrals and derivatives have been the subject of intensive research; many remarkable properties and inequalities involving the conformable integrals and derivatives can be found in the literature [19–37].Anderson [38] provided the conformable integral version of the HH inequality as follows.Theorem 11 (see [38]).
Ifα∈(0,1] and h:[κ1,κ2]→R is an α-fractional differentiable function such that Dαh is increasing, then the inequality(13)ακ2α-κ1α∫κ1κ2hxdαx≤hκ1+hκ22holds. Moreover if h is decreasing on [κ1,κ2], then we have(14)hκ1+κ22≤ακ2α-κ1α∫κ1κ2hxdαx.If α=1, then this reduces to the classical HH inequality.In this paper, we shall establish the Hermite-Hadamard type inequalities for GA and GG-convex functions via conformable fractional integrals and give their applications in the special bivariate means.


## 2. Main Results

In order to establish our main results, we need a lemma which we present in this section.Lemma 12.
Letκ1,κ2∈R+, α∈(0,1], and h:[κ1,κ2]→R be an α-fractional differentiable function on (κ1,κ2). Then the identity(15)κ2αhκ2-κ1αhκ1-α∫κ1κ2hxdαx=log⁡κ2-log⁡κ1∫01κ2tκ11-t2αDαhκ2tκ11-tt1-αdαtholds if Dα(h)∈Lα([κ1,κ2]).Proof.
Using integration by parts, we have(16)I=∫01κ2tκ11-t2αDαhκ2tκ11-tdt=∫01κ2tκ11-tα+1h′κ2tκ11-tdt. By the change of the variable x=κ2tκ11-t and integration by parts, we have (17)I=1log⁡κ2-log⁡κ1∫κ1κ2xαh′xdx=1log⁡κ2-log⁡κ1κ2αhκ2-κ1αhκ1-α∫κ1κ2hxdαx.Now multiplying by (log⁡κ2-log⁡κ1), we obtain the required result.Remark 13.
Letα=1, then Lemma 12 reduces to Lemma 3.Theorem 14.
If the functionh satisfies the conditions of Lemma 12 and, additionally, if h′x is GG-convex, then we have(18)κ2αhκ2-κ1αhκ1-α∫κ1κ2hxdαx≤log⁡κ2-log⁡κ1Lκ2α+1h′κ2,κ1α+1h′κ1.Proof.
It follows from theGG-convexity and Lemma 12 that(19)κ2αhκ2-κ1αhκ1-α∫κ1κ2hxdαx≤log⁡κ2-log⁡κ1∫01κ2tκ11-tα+1h′κ2tκ11-tdt≤log⁡κ2-log⁡κ1∫01κ2tκ11-tα+1h′κ2th′κ11-tdt.The desired result can be obtained by evaluating the above integral.Theorem 15.
Ifϑ,γ>1 with ϑ-1+γ-1=1 and the function h satisfies the conditions of Lemma 12 and, additionally, if h′γ is GG-convex, then one has(20)κ2αhκ2-κ1αhκ1-α∫κ1κ2hxdαx≤log⁡κ2-log⁡κ1Lκ2α+1ϑ,κ1α+1ϑ1/ϑLh′κ2γ,h′κ1γ1/γ.Proof.
It follows from Lemma12, the property of the modulus, the GG-convexity of h′γ, and the Hölder inequality that(21)κ2αhκ2-κ1αhκ1-α∫κ1κ2hxdαx≤log⁡κ2-log⁡κ1∫01κ2tκ11-tα+1h′κ2tκ11-tdt≤log⁡κ2-log⁡κ1∫01κ2tκ11-tα+1ϑdt1/ϑ∫01h′κ2tκ11-tγdt1/γ≤log⁡κ2-log⁡κ1∫01κ2tκ11-tα+1ϑdt1/ϑ∫01h′κ2γth′κ11-tγdt1/γ.The desired result can be obtained by evaluating the above integral.Theorem 16.
If the functionh satisfies the conditions of Lemma 12 and, additionally, if h′xγ(γ>1) is GG-convex, then we have the inequality(22)κ2αhκ2-κ1αhκ1-α∫κ1κ2hxdαx≤log⁡κ2-log⁡κ1Lκ2α+1γh′κ2γ,κ1α+1γh′κ1γ1/γ.Proof.
By using Lemma12 we clearly see that(23)κ2αhκ2-κ1αhκ1-α∫κ1κ2hxdαx≤log⁡κ2-log⁡κ1∫01κ2tκ11-tα+1h′κ2tκ11-tdt.Since γ>1, we can choose ϑ>1 such that ϑ-1+γ-1=1. Applying the Hölder integral inequality and the GG-convexity of h′γ we have(24)κ2αhκ2-κ1αhκ1-α∫κ1κ2hxdαx≤log⁡κ2-log⁡κ1∫01dt1/ϑ∫01κ2tκ11-tα+1γh′κ2tκ11-tγdt1/γ≤log⁡κ2-log⁡κ1∫01dt1/ϑ∫01κ2tκ11-tα+1γh′κ2γth′κ11-tγdt1/γ. The desired result can be obtained by evaluating the above integral.Theorem 17.
If the functionh satisfies the conditions of Lemma 12 and, additionally, if h′xγ(γ>1) is GG-convex, then we have(25)κ2αhκ2-κ1αhκ1-α∫κ1κ2hxdαx≤log⁡κ2-log⁡κ1Lκ2α+1,κ1α+11-1/γLκ2α+1h′κ2γ,κ1α+1h′κ1γ1/γ.Proof.
From the GG-convexity ofh′γ, the power mean inequality, and the property of the modulus together with Lemma 12 we get(26)κ2αhκ2-κ1αhκ1-α∫κ1κ2hxdαx≤log⁡κ2-log⁡κ1∫01κ2tκ11-tα+1h′κ2tκ11-tdt≤log⁡κ2-log⁡κ1∫01κ2tκ11-tα+1dt1-1/γ∫01κ2tκ11-tα+1h′κ2tκ11-tγdt1/γ≤log⁡κ2-log⁡κ1∫01κ2tκ11-tα+1dt1-1/γ∫01κ2tκ11-tα+1h′κ2γth′κ11-tγdt1/γ.The desired result can be obtained by evaluating the above integral.Theorem 18.
If the functionh satisfies the conditions of Lemma 12 and, additionally, if h′x is GA-convex, then(27)κ2αhκ2-κ1αhκ1-α∫κ1κ2hxdαx≤h′κ2α+1κ2α+1-Lκ2α+1,κ1α+1+h′κ1α+1Lκ2α+1,κ1α+1-κ1α+1.Proof.
It follows from theGA-convexity of h′ and the property of the modulus together with Lemma 12 that(28)κ2αhκ2-κ1αhκ1-α∫κ1κ2hxdαx≤log⁡κ2-log⁡κ1∫01κ2tκ11-tα+1h′κ2tκ11-tdt≤log⁡κ2-log⁡κ1∫01κ2tκ11-tα+1th′κ2+1-th′κ1dt=log⁡κ2-log⁡κ1∫01κ2tκ11-tα+1th′κ2dt+log⁡κ2-log⁡κ1∫01κ2tκ11-tα+11-th′κ1dt.The desired result can be obtained by evaluating the above integrals.Remark 19.
By settingα=1 in inequality (27), we regain inequality (5).Theorem 20.
If the functionh satisfies the conditions of Lemma 12 and, additionally, if h′xγ(γ>1) is GA-convex, then we have the following inequality:(29)κ2αhκ2-κ1αhκ1-α∫κ1κ2hxdαx≤log⁡κ2-log⁡κ11-1/γLκ2α+1,κ1α+11-1/γ×h′κ2γκ2α+1-Lκ2α+1,κ1α+1+h′κ1γLκ2α+1,κ1α+1-κ1α+1α+11/γ.Proof.
From the GA-convexity ofh′γ, the power mean inequality, the property of the modulus, and Lemma 12 we clearly see that(30)κ2αhκ2-κ1αhκ1-α∫κ1κ2hxdαx≤log⁡κ2-log⁡κ1∫01κ2tκ11-tα+1h′κ2tκ11-tdt≤log⁡κ2-log⁡κ1∫01κ2tκ11-tα+1dt1-1/γ∫01κ2tκ11-tα+1h′κ2tκ11-tγdt1/γ≤log⁡κ2-log⁡κ1∫01κ2tκ11-tα+1dt1-1/γ×∫01κ2tκ11-tα+1th′κ2γ+1-th′κ1γdt1/γ.The desired result can be obtained by evaluating the above integrals.Remark 21.
By settingα=1 in inequality (29), we regain inequality (6).Theorem 22.
If the functionh satisfies the conditions of Lemma 12 and, additionally, if h′x(γ>1) is GA-convex, then we have the following inequality:(31)κ2αhκ2-κ1αhκ1-α∫κ1κ2hxdαx≤log⁡κ2-log⁡κ11-1/γ×h′κ2γκ2γα+1-Lκ2γα+1,κ1γα+1+h′κ1γLκ2γα+1,κ1γα+1-κ1γα+1γα+11/γ.Proof.
With the help of the GA-convexity ofh′γ, the power mean inequality, the property of the modulus, and Lemma 12, we can write(32)
κ
2
α
h
κ
2
-
κ
1
α
h
κ
1
-
α
∫
κ
1
κ
2
h
x
d
α
x
≤
log
⁡
κ
2
-
log
⁡
κ
1
∫
0
1
κ
2
t
κ
1
1
-
t
α
+
1
h
′
κ
2
t
κ
1
1
-
t
d
t
≤
log
⁡
κ
2
-
log
⁡
κ
1
∫
0
1
d
t
1
-
1
/
γ
∫
0
1
κ
2
t
κ
1
1
-
t
γ
α
+
1
h
′
κ
2
t
κ
1
1
-
t
γ
d
t
1
/
γ
≤
log
⁡
κ
2
-
log
⁡
κ
1
∫
0
1
d
t
1
-
1
/
γ
∫
0
1
κ
2
t
κ
1
1
-
t
γ
α
+
1
t
h
′
κ
2
γ
+
1
-
t
h
′
κ
1
γ
d
t
1
/
γ
.The desired result can be obtained by evaluating the above integrals.Remark 23.
By settingα=1 in inequality (31), we regain inequality (7).Theorem 24.
Ifϑ,γ>1 with ϑ-1+γ-1=1 and the function h satisfies the conditions of Lemma 12 and, additionally, if h′γ is GA-convex, then we have the following inequality:(33)κ2αhκ2-κ1αhκ1-α∫κ1κ2hxdαx≤log⁡κ2-log⁡κ1Lκ2α+1ϑ,κ1α+1ϑ1/ϑAh′κ2γ,h′κ1γ1/γ,where A(x,y) represents the arithmetic mean of x and y.Proof.
With the help of theGA-convexity of h′γ, the Hölder integral inequality, the property of the modulus, and Lemma 12, we can write(34)κ2αhκ2-κ1αhκ1-α∫κ1κ2hxdαx≤log⁡κ2-log⁡κ1∫01κ2tκ11-tα+1h′κ2tκ11-tdt≤log⁡κ2-log⁡κ1∫01κ2tκ11-tα+1ϑdt1/ϑ∫01th′κ2γ+1-th′κ1γdt1/γ=log⁡κ2-log⁡κ1κ1α+1ϑ∫01κ2α+1ϑκ1α+1ϑtdt1/ϑ∫01th′κ2γ+1-th′κ1γdt1/γ.The desired result can be obtained by evaluating the above integrals.Remark 25.
By settingα=1 in inequality (33), we regain inequality (8).


## 3. Applications to Special Means

A bivariate functionM:(0,∞)×(0,∞)↦(0,∞) is said to be a bivariate mean if min⁡{κ1,κ2}≤M(κ1,κ2)≤max⁡{κ1,κ2} for all κ1,κ2∈(0,∞). Recently, the bivariate mean has attracted the attention of many researchers; in particular, many remarkable inequalities for the bivariate means and their related special functions can be found in the literature [39–42].In this section, we use the results obtained in Section2 to present several applications to the arithmetic mean(35)Aκ1,κ2=κ1+κ22,κ1,κ2>0,logarithmic mean (36)Lκ1,κ2=κ2-κ1log⁡κ2-log⁡κ1,κ1≠κ2,κ1,κ2∈R+,and (α,μ)-th generalized logarithmic mean (37)Lα,μ+1κ1,κ2=ακ2μ+α+1-κ1μ+α+1κ2α-κ1αμ+α+1μ+11/μ+1,κ1≠κ2,μ≠-1,-α-1,α∈0,1,μ∈R.Proposition 26.
Letα∈(0,1], κ1,κ2∈R+, and μ>0. Then(38)κ2α-κ1ααμ+1Lα,μ+1μ+1κ1,κ2≤κ2μα+1κ2α+1-Lκ2α+1,κ1α+1+κ1μα+1Lκ2α+1,κ1α+1-κ2α+1.Proof.
Let(39)hx=xμ+1μ+1for x>0. Then h′xγ is a GA-convex function on R+ for γ≥1. Let γ=1. Then making use of function (39) in Theorem 18, we obtain the required result.Proposition 27.
Letκ1,κ2∈R+, γ>1, μ>0, and α∈(0,1]. Then (40)κ2α-κ1ααμ+1Lα,μ+1μ+1κ1,κ2≤log⁡κ2-log⁡κ11-1/γLκ2α+1,κ1α+11-1/γ×κ2μγκ2α+1-Lκ2α+1,κ1α+1+κ1μγLκ2α+1,κ1α+1-κ1α+1α+11/γ.Proof.
Using function (39) in Theorem 20, we obtain the required result.Proposition 28.
Letκ1,κ2∈R+, γ>1, μ>0, and α∈(0,1]. Then (41)κ2α-κ1ααμ+1Lα,μ+1μ+1κ1,κ2≤log⁡κ2-log⁡κ11-1/γ×κ2μγκ2γα+1-Lκ2γα+1,κ1γα+1+κ1μγLκ2γα+1,κ1γα+1-κ1γα+1γα+11/γ.Proof.
Using function (39) in Theorem 22, we obtain the required result.Proposition 29.
Letκ1,κ2∈R+, μ>0, α∈(0,1], and ϑ,γ>1 with ϑ-1+γ-1=1. Then (42)κ2α-κ1ααμ+1Lα,μ+1μ+1κ1,κ2≤log⁡κ2-log⁡κ1Lκ2ϑα+1,κ1ϑα+11/ϑAκ2μγ,κ1μγ1/γ.Proof.
Using function (39) in Theorem 24, we obtain the required result.


## 4. Conclusions

In the article, we derive the conformable fractional integrals’ versions of the Hermite-Hadamard type inequalities for GG- and GA-convex functions. Our approach is based on an identity involving the conformable fractional integrals, the Hölder inequality, and the power mean inequality. The proven results generalized some previously obtained results. As applications, we provide several inequalities for some special bivariate means. The present idea may stimulate further research in the theory of inequalities for other generalized integrals, for example, as presented in [35–37].



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