# A Study of Uniform Harmonicχ-Convex Functions with respect to Hermite-Hadamard’s Inequality and Its Caputo-Fabrizio Fractional Analogue and Applications

Authors: Miguel Vivas-Cortez; Muhammad Uzair Awan; Muhammad Zakria Javed; Muhammad Aslam Noor; Khalida Inayat Noor
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we introduce the notion of uniform harmonicχ-convex functions. We show that this class relates several other unrelated classes of uniform harmonic convex functions. We derive a new version of Hermite-Hadamard’s inequality and its fractional analogue. We also derive a new fractional integral identity using Caputo-Fabrizio fractional integrals. Utilizing this integral identity as an auxiliary result, we obtain new fractional Dragomir-Agarwal type of inequalities involving differentiable uniform harmonic χ-convex functions. We discuss numerous new special cases which show that our results are quite unifying. Finally, in order to show the significance of the main results, we discuss some applications to means of positive real numbers.


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## Body

## 1. Preliminaries

In recent years, the classical concepts of convexity have been extended and generalized in different directions using novel and innovative ideas. It has also been observed that these new extensions and generalizations enjoy some nice properties which the classical concepts of convexity have. Everyone is familiar with the fact that convexity has a close relationship with the theory of inequalities. The classical concepts of convexity have played a significant role in the development of the theory of inequalities. Many famously known results in the theory of inequalities can easily be obtained using the convexity property of the functions. Thus, this becomes an interesting problem for research, of obtaining new versions of classical inequalities using new generalizations of convexity. Interested readers may find some useful details regarding convexity, its generalizations, and associated results in [1]. In 2014, Işcan [2] introduced and investigated the notion of harmonic convex functions. He defined the class of harmonic convex functions as follows.Definition 1.
A functionY:H⊂0,+∞⟶ℝ is said to be harmonically convex, if
(1)Yxyix+1−iy≤iYy+1−iYx,holds for all x,y∈H and i∈0,1.He also obtained a new variant of Hermite-Hadamard’s inequality pertaining to harmonic convex functions as follows.Theorem 2.
LetY:I=♭1,♭2⊂0,∞⟶ℝ be a harmonically convex function, and then
(2)Y2♭1♭2♭1+♭2≤♭1♭2♭2−♭1∫♭1♭2Yxdx≤Y♭1+Y♭22.Having inspiration from the research work of Sarikaya et al. [3], Işcan and Wu [4] obtained the fractional version of Hermite-Hadamard’s inequality, essentially utilizing the concept of harmonic convex functions. For some recent investigations on harmonic convex functions, see [5, 6].In [7], the class of uniformly convex functions was defined as follows.Definition 3.
LetK⊂ℝ be a convex set. A function Y:K⟶ℝ is said to be uniformly convex with modulus ψ:0,∞⟶0,∞ if ψ is increasing, ψ vanishes only at 0, and
(3)Y1−ix+iy+i1−iψy−x≤1−iYx+iYy,∀x,y∈K,i∈0,1.The main motivation of this paper is to introduce the class of uniform harmonicχ-convex functions. We show that this class contains some other new classes of uniform harmonic convex functions. Using the class of uniform harmonic χ-convex functions, we also obtain some new versions of Hermite-Hadamard’s inequality, essentially utilizing the concepts from both ordinary calculus and fractional calculus. In order to show the significance of the main results, we present some applications to special means of real numbers.Before we proceed further, let us recall some previously known concepts from fractional calculus. The Caputo-Fabrizio fractional derivative and fractional integrals are defined, respectively, as follows.LetY∈H1♭1,♭2,♭1<♭2,α∈0,1, and then the definition of the left fractional derivative is defined as follows:
(4)D♭1CFCαi=Bα1−α∫♭1iY′xexp−αi−xα1−αdx,and the associated fractional integral is defined as follows:
(5)I♭1CFαi=1−αBαYi+αBα∫♭1iYxdx.The right Caputo-Fabrizio fractional derivative is defined as follows:(6)DCFC♭2αi=−Bα1−α∫i♭2f′xexp−αx−iα1−αdx,and the associated right fractional integral is defined as follows:
(7)ICF♭2αi=1−αBαYi+αBα∫i♭2fxdx,where Bα>0 is the normalization function satisfying B0=B1=1. For more details, see [8, 9].Our calculations involve beta and hypergeometric functions. For the sake of completeness, let us recall these classical concepts. The beta and hypergeometric functions are defined as(8)Bx,y=∫01νx−11−νy−1dν,or
(9)Bx,y=ΓxΓyΓx+y,F21x,y;c;z=1By,c−y∫01νy−11−νc−y−11−zν−xdν,c>y>0,z<1.


## 2. Main Results

In this section, we discuss our main results.


### 2.1. Uniform Harmonicχ-Convex Functions

We now introduce the class of uniform harmonicχ-convex functions.Definition 4.
Letχ:0,1⟶ℝ be a real function. A function Y:♭1,♭2⊂0,∞⟶ℝ is said to be a uniform harmonic χ-convex function, if
(10)Y♭1♭21−i♭1+i♭2≤χiY♭1+χ1−iY♭2−1−iχiψ1♭1−1♭2.Some special cases of Definition4 are enlisted as follows:
(i)
If we takeχi=i, then we obtain the class of uniform harmonic convex function, which is defined as(ii)
If we takeχi=is, then we obtain the class of uniform harmonic s-convex function, which is defined as(iii)
If we takeχi=i−s, then we obtain the class of the Godunova-Levin-Dragomir (GLD) type of uniform harmonic s-convex function, which is defined as(iv)
If we takeχi=i and ψi=δi2, then we obtain the class of strongly uniform harmonic convex function, which is defined as(v)
If we takeχi=i and ψi=−δ∣i∣, then we obtain the class of approximate harmonic convex function, which is defined as(vi)
If we takeχi=i and ψi=−δip, then we obtain the class of approximate harmonic convex function of order p>0, which is defined asDefinition 5.
A functionY:♭1,♭2⊂0,∞⟶ℝ is said to be a uniform harmonic convex function, if
(11)Y♭1♭21−i♭1+i♭2≤iY♭1+1−iY♭2−1−iiψ1♭1−1♭2.Definition 6.
A functionY:♭1,♭2⊂0,∞⟶ℝ is said to be a uniform harmonic s-convex function, if
(12)Y♭1♭21−i♭1+i♭2≤isf♭1+1−isf♭2−1−iisψ1♭1−1♭2.Definition 7.
A functionY:♭1,♭2⊂0,∞⟶ℝ is said to be a Godunova-Levin-Dragomir (GLD) type of uniform harmonic s-convex functions, if
(13)Y♭1♭21−i♭1+i♭2≤i−sY♭1+1−i−sY♭2−1−ii−sψ1♭1−1♭2.Definition 8.
A functionY:♭1,♭2⊂0,∞⟶ℝ is said to be a strongly uniform harmonic convex function, if
(14)Y♭1♭21−i♭1+i♭2≤iY♭1+1−iY♭2−δ1−ii1♭1−1♭22.Definition 9.
A functionY:♭1,♭2⊂0,∞⟶ℝ is said to be an approximate uniform harmonic convex function, if
(15)Y♭1♭21−i♭1+i♭2≤iY♭1+1−iY♭2+δ1−ii1♭1−1♭2.Definition 10.
A functionY:♭1,♭2⊂0,∞⟶ℝ is said to be an approximate uniform harmonic convex function of order p>0, if
(16)Y♭1♭21−i♭1+i♭2≤iY♭1+1−iY♭2+δ1−ii1♭1−1♭2p.


### 2.2. A New Hermite-Hadamard’s Inequality

We now derive a new variant of Hermite-Hadamard’s inequality using the class of uniform harmonicχ-convex functions. We also discuss some new special cases of this result.Theorem 11.
LetY:♭1,♭2⊂0,∞⟶ℝ be a uniform harmonic χ-convex function, and then
(17)1χ1/2Y2♭1♭2♭1+♭2+∫♭1−♭2/♭1♭2♭2−♭1/♭1♭2♭1♭24♭2−♭1ψidi≤2♭1♭2♭2−♭1∫1/♭21/♭1Y∘Λidi≤Y♭1+Y♭2∫01χi+χ1−idi−2∫011−iχiψ1♭1−1♭2di,where Λi=1/i,∀i∈♭1,♭2.Proof.
SinceY is a uniform harmonic χ-convex function, then for i=1/2, we have
(18)Y2xyx+y≤χ12Yx+Yy−12ψ1x−1y.
Using the change of variable technique, we have(19)1χ1/2Y2♭1♭2♭1+♭2≤∫01Y♭1♭21−i♭1+i♭2di+∫01Y♭1♭2i♭1+1−i♭2di−∫0112ψ♭2−♭12i−1♭1♭2di.
After simplifying, we obtain(20)1χ1/2Y2♭1♭2♭1+♭2+∫♭1−♭2/♭1♭2♭2−♭1/♭1♭2♭1♭24♭2−♭1ψidi≤2♭1♭2♭2−♭1∫1/♭21/♭1Y∘Λidi.
Now, we prove our second inequality using the notion of uniform harmonicχ convexity.
(21)Y♭1♭21−i♭1+i♭2≤χiY♭1+χ1−iY♭2−1−iχiψ1♭1−1♭2,(22)Y♭1♭2i♭1+1−i♭2≤χ1−iY♭1+χiY♭2−1−iχiψ1♭1−1♭2.
Adding (21) and (22), we have
(23)Y♭1♭21−i♭1+i♭2+Y♭1♭2i♭1+1−i♭2≤Y♭1+Y♭2χi+χ1−i−21−iχiψ1♭1−1♭2.
Integrating both sides with respect toi on 0,1, then we have
(24)2♭1♭2♭2−♭1∫1/♭21/♭1Y∘Λidi≤Y♭1+Y♭2∫01χi+χ1−idi−2∫011−iχiψ1♭1−1♭2di.
This completes the proof.We now discuss some special cases of Theorem11:
(i)
If we takeχi=i in Theorem 11, then we have the result for uniform harmonic convex functions(ii)
If we takeχi=is in Theorem 11, then we have the result for uniform harmonic s-convex functions(iii)
If we takeχi=i−s in Theorem 11, then we have the result for the GLD type of uniform harmonic s-convex functionsCorollary 12.
Under the assumptions of Theorem11, if Y is a uniform harmonic convex function, then
(25)Y2♭1♭2♭1+♭2+∫♭1−♭2/♭1♭2♭2−♭1/♭1♭2♭1♭28♭2−♭1ψidi≤♭1♭2♭2−♭1∫1/♭21/♭1Y∘Λidi≤Y♭1+Y♭22−16ψ1♭1−1♭2di.Corollary 13.
Under the assumptions of Theorem11, if Y is a uniform harmonic s-convex function, then
(26)2s−1Y2♭1♭2♭1+♭2+∫♭1−♭2/♭1♭2♭2−♭1/♭1♭2♭1♭28♭2−♭1ψidi≤♭1♭2♭2−♭1∫1/♭21/♭1Y∘Λidi≤Y♭1+Y♭2s+1−1s+1s+2ψ1♭1−1♭2di.Corollary 14.
Under the assumptions of Theorem11, if Y is a GLD type of uniform harmonic s-convex functions, then
(27)21−sY2♭1♭2♭1+♭2+∫♭1−♭2/♭1♭2♭2−♭1/♭1♭2♭1♭28♭2−♭1ψidi≤♭1♭2♭2−♭1∫1/♭21/♭1Y∘Λidi≤Y♭1+Y♭21−s−11−s2−sψ1♭1−1♭2di.


### 2.3. Fractional Hermite-Hadamard’s Inequality Using Caputo-Fabrizio Fractional Integrals

We now derive a fractional version of Theorem11 by using Caputo-Fabrizio fractional integrals.Theorem 15.
LetY:♭1,♭2⟶ℝ be a uniform harmonic χ-convex function, and then
(28)1χ1/2Y2♭1♭2♭1+♭2+♭1♭24♭2−♭1∫♭1−♭2/♭1♭2♭2−♭1/♭1♭2ψ∣i∣di+4♭1♭21−αα♭2−♭1Y∘Λk≤2♭1♭2Bαα♭2−♭11/♭2CFIαY∘Λk+1/♭1CFIαY∘Λk≤4♭1♭21−αα♭2−♭1Y∘Λk+Y♭1+Y♭2∫01χi+χ1−idi−2∫011−iχiψ1♭1−1♭2di.Proof.
SinceY is a uniform harmonic χ-convex function, then
(29)1χ1/2Y2♭1♭2♭1+♭2+∫♭1−♭2/♭1♭2♭2−♭1/♭1♭2♭1♭24♭2−♭1ψidi≤2♭1♭2♭2−♭1∫1/♭21/♭1Y∘Λidi.
The above inequality can be written as(30)1χ1/2Y2♭1♭2♭1+♭2+∫♭1−♭2/♭1♭2♭2−♭1/♭1♭2♭1♭24♭2−♭1ψidi≤2♭1♭2♭2−♭1∫1/♭2kY∘Λidi+∫k1/♭1Y∘Λidi.
Multiplying both sides byα♭2−♭1/2♭1♭2Bα, adding 21−α/BαY∘Λk, and using Caputo-Fabrizio fractional integrals, we have
(31)α♭2−♭12♭1♭2Bαχ1/2Y2♭1♭2♭1+♭2+α8Bα∫♭1−♭2/♭1♭2♭2−♭1/♭1♭2ψ∣i∣di+21−αBαY∘Λk≤21−αBαY∘Λk+αBα∫1/♭2kY∘Λidi+∫k1/♭1Y∘Λidi.
This implies(32)α♭2−♭12♭1♭2Bαχ1/2Y2♭1♭2♭1+♭2+α8Bα∫♭1−♭2/♭1♭2♭2−♭1/♭1♭2ψ∣i∣di+21−αBαY∘Λk≤I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk.
Thus, we have(33)1χ1/2Y2♭1♭2♭1+♭2+♭1♭24♭2−♭1∫♭1−♭2/♭1♭2♭2−♭1/♭1♭2ψ∣i∣di+4♭1♭21−αα♭2−♭1Y∘Λk≤2♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk.
Now, we compute our second inequality. SinceY is a uniform harmonic χ-convex function, we have
(34)2♭1♭2♭2−♭1∫1/♭21/♭1Y∘Λidi≤Y♭1+Y♭2∫01χi+χ1−idi−2∫011−iχiψ1♭1−1♭2di.
Multiplying both sides byα♭2−♭1/2♭1♭2Bα, adding 21−α/BαY∘Λk, and using Caputo-Fabrizio fractional integrals, we have
(35)21−αBαY∘Λk+αBα∫1/♭2kY∘Λidi+∫k1/♭1Y∘Λidi≤21−αBαY∘Λk+α♭2−♭12♭1♭2BαY♭1+Y♭2∫01χi+χ1−idi−α♭2−♭1♭1♭2Bα∫011−iχiψ1♭1−1♭2di.
This implies(36)2♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤4♭1♭21−αα♭2−♭1Y∘Λk+Y♭1+Y♭2∫01χi+χ1−idi−2∫011−iχiψ1♭1−1♭2di.
This completes the proof.Now, we discuss some special cases of Theorem15:
(i)
If we takeχi=i in Theorem 15, then we have the result for uniform harmonic convex functions(ii)
If we takeχi=i and ψi=δ/2i2 in Theorem 15, then we have the result for strongly harmonic convex functions(iii)
If we takeχi=is in Theorem 15, then we have the result for uniform harmonic s-convex functions(iv)
If we takeχi=is and ψi=δ/2i2 in Theorem 15, then we have the result for strongly harmonic s-convex functions(v)
If we takeχi=i−s in Theorem 15, then we have the result for the GLD type of harmonic s-convex functionsCorollary 16.
Under the assumptions of Theorem15, if Y is a uniform harmonic convex function, then
(37)Y2♭1♭2♭1+♭2+♭1♭28♭2−♭1∫♭1−♭2/♭1♭2♭2−♭1/♭1♭2ψidi+2♭1♭21−αα♭2−♭1Y∘Λk≤♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤2♭1♭21−αα♭2−♭1Y∘Λk+Y♭1+Y♭22−16ψ1♭1−1♭2.Corollary 17.
Under the assumptions of Theorem15, if Y is a strongly harmonic convex function, then
(38)Y2♭1♭2♭1+♭2+♭2−♭1224♭1♭22+2♭1♭21−αα♭2−♭1Y∘Λk≤♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤2♭1♭21−αα♭2−♭1Y∘Λk+Y♭1+Y♭22−δ12♭2−♭1♭1♭22.Corollary 18.
Under the assumptions of Theorem15, if Y is a uniform harmonic s-convex function, then
(39)2s−1Y2♭1♭2♭1+♭2+♭1♭28♭2−♭1∫♭1−♭2/♭1♭2♭2−♭1/♭1♭2ψ∣i∣di+2♭1♭21−αα♭2−♭1Y∘Λk≤♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤2♭1♭21−αα♭2−♭1Y∘Λk+Y♭1+Y♭21+s−11+s2+sψ1♭1−1♭2.Corollary 19.
Under the assumptions of Theorem15, if Y is a strongly harmonic s-convex function, then
(40)2s−1Y2♭1♭2♭1+♭2+♭2−♭1224♭1♭22+2♭1♭21−αα♭2−♭1Y∘Λk≤♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤2♭1♭21−αα♭2−♭1Y∘Λk+Y♭1+Y♭21+s−δ21+s2+s♭2−♭1♭1♭22.Corollary 20.
Under the assumptions of Theorem15, if Y is a GLD type of harmonic s-convex functions, then
(41)2−s−1Y2♭1♭2♭1+♭2+♭1♭28♭2−♭1∫♭1−♭2/♭1♭2♭2−♭1/♭1♭2ψ∣i∣di+2♭1♭21−αα♭2−♭1Y∘Λk≤♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤2♭1♭21−αα♭2−♭1Y∘Λk+Y♭1+Y♭21−s−11−s2−sψ1♭1−1♭2.


### 2.4. Dragomir-Agarwal Type of Inequalities

In this section, we derive the Dragomir-Agarwal type of inequalities using the concept of uniform harmonicχ-convex functions. For this, we first derive a new lemma which will be used as an auxiliary result in obtaining our next results.Lemma 21.
LetY:♭1,♭2⊂ℝ⟶ℝ be a differentiable function on ♭1,♭2 with ♭1<♭2. If Y′∈Ł1♭1,♭2, α∈0,1, and also k∈♭1,♭2, then we have
(42)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk=♭1♭2♭2−♭12∫011−2i1−i♭1+i♭22Y′♭1♭21−i♭1+i♭2di,where Λx=1/x.Proof.
Let us consider(43)∫011−2i1−i♭1+i♭22Y′♭1♭21−i♭1+i♭2di=Y♭1+Y♭2♭1♭2♭2−♭1−2♭1♭2♭2−♭1∫01Y♭1♭21−i♭1+i♭2di=Y♭1+Y♭2♭1♭2♭2−♭1−2♭2−♭12∫1/♭21/♭1Y∘Λxdx.
Multiplying both sides byα♭2−♭12/2Bα and subtracting 21−α/BαY∘Λk, we have
(44)α♭2−♭122Bα∫011−2i1−i♭1+i♭22Y′♭1♭21−i♭1+i♭2di−21−αBαY∘Λk=α♭2−♭1Y♭1+Y♭22♭1♭2Bα−21−αBαY∘Λk−αBα∫1/♭21/♭1Y∘Λxdx=α♭2−♭1Y♭1+Y♭22♭1♭2Bα−21−αBαY∘Λk−αBα∫1/♭2kY∘Λxdx+∫k1/♭1Y∘Λxdx=α♭2−♭1Y♭1+Y♭22♭1♭2Bα−1−αBαY∘Λk+αBα∫1/♭2kY∘Λxdx+1−αBαY∘Λk+αBα∫k1/♭1Y∘Λxdx=α♭2−♭1Y♭1+Y♭22♭1♭2Bα−I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk.
This completes the proof.Theorem 22.
IfY:♭1,♭2⟶ℝ is a uniform harmonic χ-convex function with ♭1<♭2 and i∈0,1, then we have
(45)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭12∫011−2i1−i♭1+i♭22χi∣Y′♭1∣+χ1−i∣Y′♭2∣−1−iχiψ1♭2−1♭1di.Proof.
Using Lemma21, modulus property, and uniform harmonic χ-convexity of Y′, we have
(46)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭12∫011−2i1−♭1+i♭22Y′♭1♭21−i♭1+i♭2di≤♭1♭2♭2−♭12∫011−2i1−i♭1+i♭22χi∣Y′♭1∣+χ1−i∣Y′♭2∣−1−iχiψ1♭2−1♭1di.
This completes the proof.Now, we discuss some special cases of Theorem22:
(i)
If we takeχi=1 in Theorem 22, then we have the result for uniform harmonic P-convex functions(ii)
If we takeχi=i in Theorem 22, then we have the result for uniform harmonic convex functions(iii)
If we takeχi=is in Theorem 22 and by similar proceedings as in the above corollaries, then we have the result for uniform harmonic s-convex functionsCorollary 23.
Under the assumptions of Theorem22, if ∣Y′∣ is a uniform harmonic P-convex function, then
(47)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭122∫01/21−2i1−i♭1+i♭22Y′♭1+Y′♭2−1−iψ1♭2−1♭1di+∫012i−11−i♭1+i♭22∣Y′♭1∣+∣Y′♭2∣−1−iψ1♭2−1♭1di≤♭1♭2♭2−♭12s1∣Y′♭1∣+∣Y′♭2∣−s2ψ1♭2−1♭1,where
(48)s1=♭1−222F12,1,3,121−♭2♭1+♭1−22F12,2,3,1−♭2♭1−1♭1♭2,(49)s2=♭1−222F12,1,3,121−♭2♭1−♭1−222F12,2,4,121−♭2♭1+♭1−232F12,2,4,1−♭2♭1−♭1−222F12,1,3,1−♭2♭1.Corollary 24.
Under the assumptions of Theorem22, if ∣Y′∣ is a uniform harmonic convex function, then
(50)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭122∫01/21−2i1−i♭1+i♭22iY′♭1+1−iY′♭2−1−iiψ1♭2−1♭1di+∫012i−11−i♭1+i♭22iY′♭1+1−iY′♭2−1−iiψ1♭2−1♭1di≤♭1♭2♭2−♭12s3Y′♭1+s4Y′♭2−s5ψ1♭2−1♭1,where
(51)s3=♭1−262F12,2,4,121−♭2♭1+2♭1−232F12,3,4,1−♭2♭1−♭1−222F12,2,3,1−♭2♭1,(52)s4=♭1−222F12,1,3,121−♭2♭1−♭1−262F12,2,4,121−♭2♭1+♭1−232F12,2,4,1−♭2♭1−♭1−222F12,1,3,1−♭2♭1,(53)s5=♭1−2122F12,2,4,121−♭2♭1−♭1−2482F12,3,5,121−♭2♭1+♭1−262F12,3,5,1−♭2♭1−♭1−262F12,2,4,1−♭2♭1.Remark 25.
In Corollary24, if we take
(1)
ψ=δi2, then we can obtain the result for a strongly harmonic convex function(2)
ψ=−δi, then we can obtain the result for an approximate harmonic convex function(3)
ψ=−δip, then we can obtain the result for an approximate harmonic convex function of order p>0Corollary 26.
Under the assumptions of Theorem22, if ∣Y′∣ is a uniform harmonic s-convex function, then
(54)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭12s6Y′♭1+s7Y′♭2−s8ψ1♭2−1♭1,where
(55)s6=♭1−2s+1s+22F12,s+1,s+3,121−♭2♭1+2♭1−2s+22F12,s+2,s+3,1−♭2♭1−♭1−2s+12F12,s+1,s+2,1−♭2♭1,(56)s7=♭1−222F12,1,3,121−♭2♭1−♭1−22ss+1s+22F12,s+1,s+3,121−♭2♭1+2♭1−2s+1s+22F12,2,s+3,1−♭2♭1−♭1−2s+12F12,1,s+2,1−♭2♭1,(57)s8=♭1−22s1+s2+s2F12,s+1,s+3,121−♭2♭1−♭1−221+ss+2s+32F12,s+2,s+4,121−♭2♭1+2♭1−2s+3s+22F12,s+2,s+4,1−♭2♭1−♭1−2s+1s+22F12,s+2,s+3,1−♭2♭1.Remark 27.
In Corollary26, if we take
(1)
ψ=δi2, then we can obtain the result for a strongly harmonic s-convex function(2)
ψ=−δ∣i∣, then we can obtain the result for an approximate harmonic s-convex function(3)
ψ=−δip, then we can obtain the result for an approximate harmonic s-convex function of order p>0Theorem 28.
IfY:♭1,♭2⟶ℝ is a uniform harmonic χ-convex function with ♭1<♭2 and i∈0,1, then we have
(58)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭121p+11/p∫0111−i♭1+i♭22χiY′♭1q+χ1−iY′♭2q−1−iχiψ1♭2−1♭11/qdi.Proof.
Using Lemma21, modulus property, and uniform harmonic χ-convexity of Y′q, then we have
(59)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭12∫011−2i1−i♭1+i♭22Y′♭1♭21−i♭1+i♭2di≤♭1♭2♭2−♭12∫011−2ip1/p∫0111−i♭1+i♭22qχiY′♭1q+χ1−iY′♭2q−1−iχiψ1♭2−1♭11/qdi.
This completes the proof.Now, we discuss some special cases of Theorem28:
(i)
If we takeχi=1 in Theorem 28, then we have the result for uniform harmonic P-convex functions(ii)
If we takeχi=i in Theorem 22, then we have the result for uniform harmonic convex functions(iii)
If we takeχi=is in Theorem 22, then we have the result for uniform harmonic s-convex functionsCorollary 29.
Under the assumptions of Theorem28, if ∣Y′∣ is a uniform harmonic P-convex function, then
(60)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭121p+11/p♭21−2q−♭11−2qY′♭1q+Y′♭2q♭2−♭11−2q−♭122F12q,1,3,1−♭2♭1ψ1♭2−1♭11/q.Corollary 30.
Under the assumptions of Theorem22, if ∣Y′∣ is a uniform harmonic convex function, then
(61)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭121p+11/p∫0111−i♭1+i♭22qiY′♭1+1−iY′♭2−1−iiψ1♭2−1♭1di≤♭1♭2♭2−♭121p+11/ps9Y′♭1q+s10Y′♭2q−s11ψ1♭2−1♭11/q,where
(62)s9=♭1−2q22F12q,2,3,1−♭2♭1,(63)s10=♭1−2q22F12q,1,3,1−♭2♭1,(64)s11=♭1−2q22F12q,2,4,1−♭2♭1.Remark 31.
In Corollary30, if we take
(1)
ψ=δi2, then we can obtain the result for a strongly harmonic convex function(2)
ψ=−δi, then we can obtain the result for an approximate harmonic convex function(3)
ψ=−δip, then we can obtain the result for an approximate harmonic convex function of order p>0Corollary 32.
Under the assumptions of Theorem22, if ∣Y′∣ is a uniform harmonic s-convex function, then
(65)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭121p+11/p∫0111−i♭1+i♭22qisY′♭1+1−isY′♭2−1−iisψ1♭2−1♭1di≤♭1♭2♭2−♭121p+11/ps12Y′♭1q+s13Y′♭2q−s14ψ1♭2−1♭11/q,where
(66)s12=♭1−2qs+12F12q,s+1,s+2,1−♭2♭1,s13=♭1−2qs+12F12q,1,s+2,1−♭2♭1,s14=♭1−2qs+2s+12F12q,s+1,s+3,1−♭2♭1.Remark 33.
In Corollary32, if we take
(1)
ψ=δi2, then we can obtain the result for a strongly harmonic s-convex function(2)
ψ=−δ∣i∣, then we can obtain the result for an approximate harmonic s-convex function(3)
ψ=−δip, then we can obtain the result for an approximate harmonic s-convex function of order p>0Theorem 34.
IfY:♭1,♭2⟶ℝ is a uniform harmonic χ-convex function with ♭1<♭2, 1/p+1/q=1, and i∈0,1, then
(67)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭12s151/p∫011−2i1−i♭1+i♭22χiY′♭1q+χ1−iY′♭2q−1−iχiψ1♭2−1♭11/qdi,where
(68)s15=♭1−222F12,1,3,121−♭2♭1+♭1−22F12,2,3,1−♭2♭1−1♭1♭2.Proof.
Using Lemma21, modulus property, power mean inequality, and uniform harmonic χ-convexity of Y′ and 1/p+1/q=1, then we have
(69)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭12∫011−2i1−i♭1+i♭221/p∫011−2i1−i♭1+i♭22Y′♭1♭21−i♭1+i♭2qdi1/q≤♭1♭2♭2−♭12∫011−2i1−i♭1+i♭221/p∫011−2i1−i♭1+i♭22χiY′♭1q+χ1−iY′♭2q−1−iχiψ1♭2−1♭1di1/q.
This completes the proof.Now, we discuss some special cases of Theorem34:
(i)
If we takeχi=1 in Theorem 34 and the proceedings are similar to Corollary 23, then we have the result for uniform harmonic P-convex functions(ii)
If we takeχi=i in Theorem 34, then we have the result for uniform harmonic convex functions(iii)
If we takeχi=is in Theorem 34, then we have the result for uniform harmonic s-convex functionsCorollary 35.
Under the assumptions of Theorem34, if ∣Y′∣ is a uniform harmonic P-convex function, then
(70)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭12s151/ps1Y′♭1q+Y′♭2q−s2ψ1♭2−1♭11/qdi,where s1 and s2 are given by (48) and (49), respectively.Corollary 36.
Under the assumptions of Theorem34, if Y′ is a uniform harmonic convex function, then
(71)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭12s151/ps3Y′♭1q+s4Y′♭2q−s5ψ1♭2−1♭11/q,where s3, s4, and s5 are given by (51), (52), and (53), respectively.Corollary 37.
Under the assumptions of Theorem34, if ∣Y′∣ is a uniform harmonic s-convex function, then
(72)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭12s151/ps6Y′♭1q+s7Y′♭2q−s8ψ1♭2−1♭11/q,where s6, s7, and s8 are given by (55), (56), and (57), respectively.Theorem 38.
IfY:♭1,♭2⟶ℝ is a uniform harmonic χ-convex function with ♭1<♭2 and i∈0,1, then we have
(73)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭12s161/p∫01∣1−2ipχiY′♭1∣q+χ1−iY′♭2q−1−iχiψ1♭2−1♭1di1/q,where
(74)s16=♭21−2p−♭11−2p♭2−♭11−2p.Proof.
Using Lemma21, modulus property, and uniform harmonic χ-convexity of Y′q, then we have
(75)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭12∫011−2i1−i♭1+i♭22Y′♭1♭21−i♭1+i♭2di≤♭1♭2♭2−♭12∫0111−i♭1+i♭22p1/p∫011−2iqχiY′♭1q+χ1−iY′♭2q−1−iχiψ1♭2−1♭11/qdi.
This completes the proof.Now, we discuss some special cases of Theorem38:
(i)
If we takeχi=1 in Theorem 38, then we have the result for uniform harmonic P-convex functions(ii)
If we takeχi=i in Theorem 34 and use a similar technique as in Corollary 39, then we have the result for uniform harmonic convex functionsCorollary 39.
Under the assumptions of Theorem38, if ∣Y′∣ is a uniform harmonic P-convex function, then
(76)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭12s161/p∫01/21−2ipY′♭1q+Y′♭2q−1−iψ1♭2−1♭1di+∫1/212i−1pY′♭1q+Y′♭2q−1−iψ1♭2−1♭1di1/q≤♭1♭2♭2−♭12s161/p1q+1Y′♭1q+Y′♭2q−12q+1ψ1♭2−1♭11/q,where s16 is given by (74).Corollary 40.
Under the assumptions of Theorem34, if ∣Y′∣ is a uniform harmonic convex function, then
(77)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭12s161/pY′♭1q+Y′♭2q2q+1−12q+1q+3ψ1♭2−1♭11/q,where s16 is given by (74).


## 2.1. Uniform Harmonicχ-Convex Functions

We now introduce the class of uniform harmonicχ-convex functions.Definition 4.
Letχ:0,1⟶ℝ be a real function. A function Y:♭1,♭2⊂0,∞⟶ℝ is said to be a uniform harmonic χ-convex function, if
(10)Y♭1♭21−i♭1+i♭2≤χiY♭1+χ1−iY♭2−1−iχiψ1♭1−1♭2.Some special cases of Definition4 are enlisted as follows:
(i)
If we takeχi=i, then we obtain the class of uniform harmonic convex function, which is defined as(ii)
If we takeχi=is, then we obtain the class of uniform harmonic s-convex function, which is defined as(iii)
If we takeχi=i−s, then we obtain the class of the Godunova-Levin-Dragomir (GLD) type of uniform harmonic s-convex function, which is defined as(iv)
If we takeχi=i and ψi=δi2, then we obtain the class of strongly uniform harmonic convex function, which is defined as(v)
If we takeχi=i and ψi=−δ∣i∣, then we obtain the class of approximate harmonic convex function, which is defined as(vi)
If we takeχi=i and ψi=−δip, then we obtain the class of approximate harmonic convex function of order p>0, which is defined asDefinition 5.
A functionY:♭1,♭2⊂0,∞⟶ℝ is said to be a uniform harmonic convex function, if
(11)Y♭1♭21−i♭1+i♭2≤iY♭1+1−iY♭2−1−iiψ1♭1−1♭2.Definition 6.
A functionY:♭1,♭2⊂0,∞⟶ℝ is said to be a uniform harmonic s-convex function, if
(12)Y♭1♭21−i♭1+i♭2≤isf♭1+1−isf♭2−1−iisψ1♭1−1♭2.Definition 7.
A functionY:♭1,♭2⊂0,∞⟶ℝ is said to be a Godunova-Levin-Dragomir (GLD) type of uniform harmonic s-convex functions, if
(13)Y♭1♭21−i♭1+i♭2≤i−sY♭1+1−i−sY♭2−1−ii−sψ1♭1−1♭2.Definition 8.
A functionY:♭1,♭2⊂0,∞⟶ℝ is said to be a strongly uniform harmonic convex function, if
(14)Y♭1♭21−i♭1+i♭2≤iY♭1+1−iY♭2−δ1−ii1♭1−1♭22.Definition 9.
A functionY:♭1,♭2⊂0,∞⟶ℝ is said to be an approximate uniform harmonic convex function, if
(15)Y♭1♭21−i♭1+i♭2≤iY♭1+1−iY♭2+δ1−ii1♭1−1♭2.Definition 10.
A functionY:♭1,♭2⊂0,∞⟶ℝ is said to be an approximate uniform harmonic convex function of order p>0, if
(16)Y♭1♭21−i♭1+i♭2≤iY♭1+1−iY♭2+δ1−ii1♭1−1♭2p.


## 2.2. A New Hermite-Hadamard’s Inequality

We now derive a new variant of Hermite-Hadamard’s inequality using the class of uniform harmonicχ-convex functions. We also discuss some new special cases of this result.Theorem 11.
LetY:♭1,♭2⊂0,∞⟶ℝ be a uniform harmonic χ-convex function, and then
(17)1χ1/2Y2♭1♭2♭1+♭2+∫♭1−♭2/♭1♭2♭2−♭1/♭1♭2♭1♭24♭2−♭1ψidi≤2♭1♭2♭2−♭1∫1/♭21/♭1Y∘Λidi≤Y♭1+Y♭2∫01χi+χ1−idi−2∫011−iχiψ1♭1−1♭2di,where Λi=1/i,∀i∈♭1,♭2.Proof.
SinceY is a uniform harmonic χ-convex function, then for i=1/2, we have
(18)Y2xyx+y≤χ12Yx+Yy−12ψ1x−1y.
Using the change of variable technique, we have(19)1χ1/2Y2♭1♭2♭1+♭2≤∫01Y♭1♭21−i♭1+i♭2di+∫01Y♭1♭2i♭1+1−i♭2di−∫0112ψ♭2−♭12i−1♭1♭2di.
After simplifying, we obtain(20)1χ1/2Y2♭1♭2♭1+♭2+∫♭1−♭2/♭1♭2♭2−♭1/♭1♭2♭1♭24♭2−♭1ψidi≤2♭1♭2♭2−♭1∫1/♭21/♭1Y∘Λidi.
Now, we prove our second inequality using the notion of uniform harmonicχ convexity.
(21)Y♭1♭21−i♭1+i♭2≤χiY♭1+χ1−iY♭2−1−iχiψ1♭1−1♭2,(22)Y♭1♭2i♭1+1−i♭2≤χ1−iY♭1+χiY♭2−1−iχiψ1♭1−1♭2.
Adding (21) and (22), we have
(23)Y♭1♭21−i♭1+i♭2+Y♭1♭2i♭1+1−i♭2≤Y♭1+Y♭2χi+χ1−i−21−iχiψ1♭1−1♭2.
Integrating both sides with respect toi on 0,1, then we have
(24)2♭1♭2♭2−♭1∫1/♭21/♭1Y∘Λidi≤Y♭1+Y♭2∫01χi+χ1−idi−2∫011−iχiψ1♭1−1♭2di.
This completes the proof.We now discuss some special cases of Theorem11:
(i)
If we takeχi=i in Theorem 11, then we have the result for uniform harmonic convex functions(ii)
If we takeχi=is in Theorem 11, then we have the result for uniform harmonic s-convex functions(iii)
If we takeχi=i−s in Theorem 11, then we have the result for the GLD type of uniform harmonic s-convex functionsCorollary 12.
Under the assumptions of Theorem11, if Y is a uniform harmonic convex function, then
(25)Y2♭1♭2♭1+♭2+∫♭1−♭2/♭1♭2♭2−♭1/♭1♭2♭1♭28♭2−♭1ψidi≤♭1♭2♭2−♭1∫1/♭21/♭1Y∘Λidi≤Y♭1+Y♭22−16ψ1♭1−1♭2di.Corollary 13.
Under the assumptions of Theorem11, if Y is a uniform harmonic s-convex function, then
(26)2s−1Y2♭1♭2♭1+♭2+∫♭1−♭2/♭1♭2♭2−♭1/♭1♭2♭1♭28♭2−♭1ψidi≤♭1♭2♭2−♭1∫1/♭21/♭1Y∘Λidi≤Y♭1+Y♭2s+1−1s+1s+2ψ1♭1−1♭2di.Corollary 14.
Under the assumptions of Theorem11, if Y is a GLD type of uniform harmonic s-convex functions, then
(27)21−sY2♭1♭2♭1+♭2+∫♭1−♭2/♭1♭2♭2−♭1/♭1♭2♭1♭28♭2−♭1ψidi≤♭1♭2♭2−♭1∫1/♭21/♭1Y∘Λidi≤Y♭1+Y♭21−s−11−s2−sψ1♭1−1♭2di.


## 2.3. Fractional Hermite-Hadamard’s Inequality Using Caputo-Fabrizio Fractional Integrals

We now derive a fractional version of Theorem11 by using Caputo-Fabrizio fractional integrals.Theorem 15.
LetY:♭1,♭2⟶ℝ be a uniform harmonic χ-convex function, and then
(28)1χ1/2Y2♭1♭2♭1+♭2+♭1♭24♭2−♭1∫♭1−♭2/♭1♭2♭2−♭1/♭1♭2ψ∣i∣di+4♭1♭21−αα♭2−♭1Y∘Λk≤2♭1♭2Bαα♭2−♭11/♭2CFIαY∘Λk+1/♭1CFIαY∘Λk≤4♭1♭21−αα♭2−♭1Y∘Λk+Y♭1+Y♭2∫01χi+χ1−idi−2∫011−iχiψ1♭1−1♭2di.Proof.
SinceY is a uniform harmonic χ-convex function, then
(29)1χ1/2Y2♭1♭2♭1+♭2+∫♭1−♭2/♭1♭2♭2−♭1/♭1♭2♭1♭24♭2−♭1ψidi≤2♭1♭2♭2−♭1∫1/♭21/♭1Y∘Λidi.
The above inequality can be written as(30)1χ1/2Y2♭1♭2♭1+♭2+∫♭1−♭2/♭1♭2♭2−♭1/♭1♭2♭1♭24♭2−♭1ψidi≤2♭1♭2♭2−♭1∫1/♭2kY∘Λidi+∫k1/♭1Y∘Λidi.
Multiplying both sides byα♭2−♭1/2♭1♭2Bα, adding 21−α/BαY∘Λk, and using Caputo-Fabrizio fractional integrals, we have
(31)α♭2−♭12♭1♭2Bαχ1/2Y2♭1♭2♭1+♭2+α8Bα∫♭1−♭2/♭1♭2♭2−♭1/♭1♭2ψ∣i∣di+21−αBαY∘Λk≤21−αBαY∘Λk+αBα∫1/♭2kY∘Λidi+∫k1/♭1Y∘Λidi.
This implies(32)α♭2−♭12♭1♭2Bαχ1/2Y2♭1♭2♭1+♭2+α8Bα∫♭1−♭2/♭1♭2♭2−♭1/♭1♭2ψ∣i∣di+21−αBαY∘Λk≤I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk.
Thus, we have(33)1χ1/2Y2♭1♭2♭1+♭2+♭1♭24♭2−♭1∫♭1−♭2/♭1♭2♭2−♭1/♭1♭2ψ∣i∣di+4♭1♭21−αα♭2−♭1Y∘Λk≤2♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk.
Now, we compute our second inequality. SinceY is a uniform harmonic χ-convex function, we have
(34)2♭1♭2♭2−♭1∫1/♭21/♭1Y∘Λidi≤Y♭1+Y♭2∫01χi+χ1−idi−2∫011−iχiψ1♭1−1♭2di.
Multiplying both sides byα♭2−♭1/2♭1♭2Bα, adding 21−α/BαY∘Λk, and using Caputo-Fabrizio fractional integrals, we have
(35)21−αBαY∘Λk+αBα∫1/♭2kY∘Λidi+∫k1/♭1Y∘Λidi≤21−αBαY∘Λk+α♭2−♭12♭1♭2BαY♭1+Y♭2∫01χi+χ1−idi−α♭2−♭1♭1♭2Bα∫011−iχiψ1♭1−1♭2di.
This implies(36)2♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤4♭1♭21−αα♭2−♭1Y∘Λk+Y♭1+Y♭2∫01χi+χ1−idi−2∫011−iχiψ1♭1−1♭2di.
This completes the proof.Now, we discuss some special cases of Theorem15:
(i)
If we takeχi=i in Theorem 15, then we have the result for uniform harmonic convex functions(ii)
If we takeχi=i and ψi=δ/2i2 in Theorem 15, then we have the result for strongly harmonic convex functions(iii)
If we takeχi=is in Theorem 15, then we have the result for uniform harmonic s-convex functions(iv)
If we takeχi=is and ψi=δ/2i2 in Theorem 15, then we have the result for strongly harmonic s-convex functions(v)
If we takeχi=i−s in Theorem 15, then we have the result for the GLD type of harmonic s-convex functionsCorollary 16.
Under the assumptions of Theorem15, if Y is a uniform harmonic convex function, then
(37)Y2♭1♭2♭1+♭2+♭1♭28♭2−♭1∫♭1−♭2/♭1♭2♭2−♭1/♭1♭2ψidi+2♭1♭21−αα♭2−♭1Y∘Λk≤♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤2♭1♭21−αα♭2−♭1Y∘Λk+Y♭1+Y♭22−16ψ1♭1−1♭2.Corollary 17.
Under the assumptions of Theorem15, if Y is a strongly harmonic convex function, then
(38)Y2♭1♭2♭1+♭2+♭2−♭1224♭1♭22+2♭1♭21−αα♭2−♭1Y∘Λk≤♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤2♭1♭21−αα♭2−♭1Y∘Λk+Y♭1+Y♭22−δ12♭2−♭1♭1♭22.Corollary 18.
Under the assumptions of Theorem15, if Y is a uniform harmonic s-convex function, then
(39)2s−1Y2♭1♭2♭1+♭2+♭1♭28♭2−♭1∫♭1−♭2/♭1♭2♭2−♭1/♭1♭2ψ∣i∣di+2♭1♭21−αα♭2−♭1Y∘Λk≤♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤2♭1♭21−αα♭2−♭1Y∘Λk+Y♭1+Y♭21+s−11+s2+sψ1♭1−1♭2.Corollary 19.
Under the assumptions of Theorem15, if Y is a strongly harmonic s-convex function, then
(40)2s−1Y2♭1♭2♭1+♭2+♭2−♭1224♭1♭22+2♭1♭21−αα♭2−♭1Y∘Λk≤♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤2♭1♭21−αα♭2−♭1Y∘Λk+Y♭1+Y♭21+s−δ21+s2+s♭2−♭1♭1♭22.Corollary 20.
Under the assumptions of Theorem15, if Y is a GLD type of harmonic s-convex functions, then
(41)2−s−1Y2♭1♭2♭1+♭2+♭1♭28♭2−♭1∫♭1−♭2/♭1♭2♭2−♭1/♭1♭2ψ∣i∣di+2♭1♭21−αα♭2−♭1Y∘Λk≤♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤2♭1♭21−αα♭2−♭1Y∘Λk+Y♭1+Y♭21−s−11−s2−sψ1♭1−1♭2.


## 2.4. Dragomir-Agarwal Type of Inequalities

In this section, we derive the Dragomir-Agarwal type of inequalities using the concept of uniform harmonicχ-convex functions. For this, we first derive a new lemma which will be used as an auxiliary result in obtaining our next results.Lemma 21.
LetY:♭1,♭2⊂ℝ⟶ℝ be a differentiable function on ♭1,♭2 with ♭1<♭2. If Y′∈Ł1♭1,♭2, α∈0,1, and also k∈♭1,♭2, then we have
(42)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk=♭1♭2♭2−♭12∫011−2i1−i♭1+i♭22Y′♭1♭21−i♭1+i♭2di,where Λx=1/x.Proof.
Let us consider(43)∫011−2i1−i♭1+i♭22Y′♭1♭21−i♭1+i♭2di=Y♭1+Y♭2♭1♭2♭2−♭1−2♭1♭2♭2−♭1∫01Y♭1♭21−i♭1+i♭2di=Y♭1+Y♭2♭1♭2♭2−♭1−2♭2−♭12∫1/♭21/♭1Y∘Λxdx.
Multiplying both sides byα♭2−♭12/2Bα and subtracting 21−α/BαY∘Λk, we have
(44)α♭2−♭122Bα∫011−2i1−i♭1+i♭22Y′♭1♭21−i♭1+i♭2di−21−αBαY∘Λk=α♭2−♭1Y♭1+Y♭22♭1♭2Bα−21−αBαY∘Λk−αBα∫1/♭21/♭1Y∘Λxdx=α♭2−♭1Y♭1+Y♭22♭1♭2Bα−21−αBαY∘Λk−αBα∫1/♭2kY∘Λxdx+∫k1/♭1Y∘Λxdx=α♭2−♭1Y♭1+Y♭22♭1♭2Bα−1−αBαY∘Λk+αBα∫1/♭2kY∘Λxdx+1−αBαY∘Λk+αBα∫k1/♭1Y∘Λxdx=α♭2−♭1Y♭1+Y♭22♭1♭2Bα−I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk.
This completes the proof.Theorem 22.
IfY:♭1,♭2⟶ℝ is a uniform harmonic χ-convex function with ♭1<♭2 and i∈0,1, then we have
(45)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭12∫011−2i1−i♭1+i♭22χi∣Y′♭1∣+χ1−i∣Y′♭2∣−1−iχiψ1♭2−1♭1di.Proof.
Using Lemma21, modulus property, and uniform harmonic χ-convexity of Y′, we have
(46)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭12∫011−2i1−♭1+i♭22Y′♭1♭21−i♭1+i♭2di≤♭1♭2♭2−♭12∫011−2i1−i♭1+i♭22χi∣Y′♭1∣+χ1−i∣Y′♭2∣−1−iχiψ1♭2−1♭1di.
This completes the proof.Now, we discuss some special cases of Theorem22:
(i)
If we takeχi=1 in Theorem 22, then we have the result for uniform harmonic P-convex functions(ii)
If we takeχi=i in Theorem 22, then we have the result for uniform harmonic convex functions(iii)
If we takeχi=is in Theorem 22 and by similar proceedings as in the above corollaries, then we have the result for uniform harmonic s-convex functionsCorollary 23.
Under the assumptions of Theorem22, if ∣Y′∣ is a uniform harmonic P-convex function, then
(47)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭122∫01/21−2i1−i♭1+i♭22Y′♭1+Y′♭2−1−iψ1♭2−1♭1di+∫012i−11−i♭1+i♭22∣Y′♭1∣+∣Y′♭2∣−1−iψ1♭2−1♭1di≤♭1♭2♭2−♭12s1∣Y′♭1∣+∣Y′♭2∣−s2ψ1♭2−1♭1,where
(48)s1=♭1−222F12,1,3,121−♭2♭1+♭1−22F12,2,3,1−♭2♭1−1♭1♭2,(49)s2=♭1−222F12,1,3,121−♭2♭1−♭1−222F12,2,4,121−♭2♭1+♭1−232F12,2,4,1−♭2♭1−♭1−222F12,1,3,1−♭2♭1.Corollary 24.
Under the assumptions of Theorem22, if ∣Y′∣ is a uniform harmonic convex function, then
(50)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭122∫01/21−2i1−i♭1+i♭22iY′♭1+1−iY′♭2−1−iiψ1♭2−1♭1di+∫012i−11−i♭1+i♭22iY′♭1+1−iY′♭2−1−iiψ1♭2−1♭1di≤♭1♭2♭2−♭12s3Y′♭1+s4Y′♭2−s5ψ1♭2−1♭1,where
(51)s3=♭1−262F12,2,4,121−♭2♭1+2♭1−232F12,3,4,1−♭2♭1−♭1−222F12,2,3,1−♭2♭1,(52)s4=♭1−222F12,1,3,121−♭2♭1−♭1−262F12,2,4,121−♭2♭1+♭1−232F12,2,4,1−♭2♭1−♭1−222F12,1,3,1−♭2♭1,(53)s5=♭1−2122F12,2,4,121−♭2♭1−♭1−2482F12,3,5,121−♭2♭1+♭1−262F12,3,5,1−♭2♭1−♭1−262F12,2,4,1−♭2♭1.Remark 25.
In Corollary24, if we take
(1)
ψ=δi2, then we can obtain the result for a strongly harmonic convex function(2)
ψ=−δi, then we can obtain the result for an approximate harmonic convex function(3)
ψ=−δip, then we can obtain the result for an approximate harmonic convex function of order p>0Corollary 26.
Under the assumptions of Theorem22, if ∣Y′∣ is a uniform harmonic s-convex function, then
(54)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭12s6Y′♭1+s7Y′♭2−s8ψ1♭2−1♭1,where
(55)s6=♭1−2s+1s+22F12,s+1,s+3,121−♭2♭1+2♭1−2s+22F12,s+2,s+3,1−♭2♭1−♭1−2s+12F12,s+1,s+2,1−♭2♭1,(56)s7=♭1−222F12,1,3,121−♭2♭1−♭1−22ss+1s+22F12,s+1,s+3,121−♭2♭1+2♭1−2s+1s+22F12,2,s+3,1−♭2♭1−♭1−2s+12F12,1,s+2,1−♭2♭1,(57)s8=♭1−22s1+s2+s2F12,s+1,s+3,121−♭2♭1−♭1−221+ss+2s+32F12,s+2,s+4,121−♭2♭1+2♭1−2s+3s+22F12,s+2,s+4,1−♭2♭1−♭1−2s+1s+22F12,s+2,s+3,1−♭2♭1.Remark 27.
In Corollary26, if we take
(1)
ψ=δi2, then we can obtain the result for a strongly harmonic s-convex function(2)
ψ=−δ∣i∣, then we can obtain the result for an approximate harmonic s-convex function(3)
ψ=−δip, then we can obtain the result for an approximate harmonic s-convex function of order p>0Theorem 28.
IfY:♭1,♭2⟶ℝ is a uniform harmonic χ-convex function with ♭1<♭2 and i∈0,1, then we have
(58)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭121p+11/p∫0111−i♭1+i♭22χiY′♭1q+χ1−iY′♭2q−1−iχiψ1♭2−1♭11/qdi.Proof.
Using Lemma21, modulus property, and uniform harmonic χ-convexity of Y′q, then we have
(59)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭12∫011−2i1−i♭1+i♭22Y′♭1♭21−i♭1+i♭2di≤♭1♭2♭2−♭12∫011−2ip1/p∫0111−i♭1+i♭22qχiY′♭1q+χ1−iY′♭2q−1−iχiψ1♭2−1♭11/qdi.
This completes the proof.Now, we discuss some special cases of Theorem28:
(i)
If we takeχi=1 in Theorem 28, then we have the result for uniform harmonic P-convex functions(ii)
If we takeχi=i in Theorem 22, then we have the result for uniform harmonic convex functions(iii)
If we takeχi=is in Theorem 22, then we have the result for uniform harmonic s-convex functionsCorollary 29.
Under the assumptions of Theorem28, if ∣Y′∣ is a uniform harmonic P-convex function, then
(60)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭121p+11/p♭21−2q−♭11−2qY′♭1q+Y′♭2q♭2−♭11−2q−♭122F12q,1,3,1−♭2♭1ψ1♭2−1♭11/q.Corollary 30.
Under the assumptions of Theorem22, if ∣Y′∣ is a uniform harmonic convex function, then
(61)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭121p+11/p∫0111−i♭1+i♭22qiY′♭1+1−iY′♭2−1−iiψ1♭2−1♭1di≤♭1♭2♭2−♭121p+11/ps9Y′♭1q+s10Y′♭2q−s11ψ1♭2−1♭11/q,where
(62)s9=♭1−2q22F12q,2,3,1−♭2♭1,(63)s10=♭1−2q22F12q,1,3,1−♭2♭1,(64)s11=♭1−2q22F12q,2,4,1−♭2♭1.Remark 31.
In Corollary30, if we take
(1)
ψ=δi2, then we can obtain the result for a strongly harmonic convex function(2)
ψ=−δi, then we can obtain the result for an approximate harmonic convex function(3)
ψ=−δip, then we can obtain the result for an approximate harmonic convex function of order p>0Corollary 32.
Under the assumptions of Theorem22, if ∣Y′∣ is a uniform harmonic s-convex function, then
(65)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭121p+11/p∫0111−i♭1+i♭22qisY′♭1+1−isY′♭2−1−iisψ1♭2−1♭1di≤♭1♭2♭2−♭121p+11/ps12Y′♭1q+s13Y′♭2q−s14ψ1♭2−1♭11/q,where
(66)s12=♭1−2qs+12F12q,s+1,s+2,1−♭2♭1,s13=♭1−2qs+12F12q,1,s+2,1−♭2♭1,s14=♭1−2qs+2s+12F12q,s+1,s+3,1−♭2♭1.Remark 33.
In Corollary32, if we take
(1)
ψ=δi2, then we can obtain the result for a strongly harmonic s-convex function(2)
ψ=−δ∣i∣, then we can obtain the result for an approximate harmonic s-convex function(3)
ψ=−δip, then we can obtain the result for an approximate harmonic s-convex function of order p>0Theorem 34.
IfY:♭1,♭2⟶ℝ is a uniform harmonic χ-convex function with ♭1<♭2, 1/p+1/q=1, and i∈0,1, then
(67)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭12s151/p∫011−2i1−i♭1+i♭22χiY′♭1q+χ1−iY′♭2q−1−iχiψ1♭2−1♭11/qdi,where
(68)s15=♭1−222F12,1,3,121−♭2♭1+♭1−22F12,2,3,1−♭2♭1−1♭1♭2.Proof.
Using Lemma21, modulus property, power mean inequality, and uniform harmonic χ-convexity of Y′ and 1/p+1/q=1, then we have
(69)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭12∫011−2i1−i♭1+i♭221/p∫011−2i1−i♭1+i♭22Y′♭1♭21−i♭1+i♭2qdi1/q≤♭1♭2♭2−♭12∫011−2i1−i♭1+i♭221/p∫011−2i1−i♭1+i♭22χiY′♭1q+χ1−iY′♭2q−1−iχiψ1♭2−1♭1di1/q.
This completes the proof.Now, we discuss some special cases of Theorem34:
(i)
If we takeχi=1 in Theorem 34 and the proceedings are similar to Corollary 23, then we have the result for uniform harmonic P-convex functions(ii)
If we takeχi=i in Theorem 34, then we have the result for uniform harmonic convex functions(iii)
If we takeχi=is in Theorem 34, then we have the result for uniform harmonic s-convex functionsCorollary 35.
Under the assumptions of Theorem34, if ∣Y′∣ is a uniform harmonic P-convex function, then
(70)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭12s151/ps1Y′♭1q+Y′♭2q−s2ψ1♭2−1♭11/qdi,where s1 and s2 are given by (48) and (49), respectively.Corollary 36.
Under the assumptions of Theorem34, if Y′ is a uniform harmonic convex function, then
(71)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭12s151/ps3Y′♭1q+s4Y′♭2q−s5ψ1♭2−1♭11/q,where s3, s4, and s5 are given by (51), (52), and (53), respectively.Corollary 37.
Under the assumptions of Theorem34, if ∣Y′∣ is a uniform harmonic s-convex function, then
(72)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭12s151/ps6Y′♭1q+s7Y′♭2q−s8ψ1♭2−1♭11/q,where s6, s7, and s8 are given by (55), (56), and (57), respectively.Theorem 38.
IfY:♭1,♭2⟶ℝ is a uniform harmonic χ-convex function with ♭1<♭2 and i∈0,1, then we have
(73)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭12s161/p∫01∣1−2ipχiY′♭1∣q+χ1−iY′♭2q−1−iχiψ1♭2−1♭1di1/q,where
(74)s16=♭21−2p−♭11−2p♭2−♭11−2p.Proof.
Using Lemma21, modulus property, and uniform harmonic χ-convexity of Y′q, then we have
(75)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭12∫011−2i1−i♭1+i♭22Y′♭1♭21−i♭1+i♭2di≤♭1♭2♭2−♭12∫0111−i♭1+i♭22p1/p∫011−2iqχiY′♭1q+χ1−iY′♭2q−1−iχiψ1♭2−1♭11/qdi.
This completes the proof.Now, we discuss some special cases of Theorem38:
(i)
If we takeχi=1 in Theorem 38, then we have the result for uniform harmonic P-convex functions(ii)
If we takeχi=i in Theorem 34 and use a similar technique as in Corollary 39, then we have the result for uniform harmonic convex functionsCorollary 39.
Under the assumptions of Theorem38, if ∣Y′∣ is a uniform harmonic P-convex function, then
(76)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭12s161/p∫01/21−2ipY′♭1q+Y′♭2q−1−iψ1♭2−1♭1di+∫1/212i−1pY′♭1q+Y′♭2q−1−iψ1♭2−1♭1di1/q≤♭1♭2♭2−♭12s161/p1q+1Y′♭1q+Y′♭2q−12q+1ψ1♭2−1♭11/q,where s16 is given by (74).Corollary 40.
Under the assumptions of Theorem34, if ∣Y′∣ is a uniform harmonic convex function, then
(77)Y♭1+Y♭22+2♭1♭2α♭2−♭1Y∘Λk−♭1♭2Bαα♭2−♭1I1/♭2CFαY∘Λk+I1/♭1CFαY∘Λk≤♭1♭2♭2−♭12s161/pY′♭1q+Y′♭2q2q+1−12q+1q+3ψ1♭2−1♭11/q,where s16 is given by (74).


## 3. Applications

In this section, we present some applications to special means. We now recall some special means for two nonnegative numbers:(1) The Arithmetic Mean. A♭1,♭2≔♭1+♭2/2(2) The Geometric Mean. G♭1,♭2≔♭1♭2(3) The Harmonic Mean. H♭1,♭2≔2♭1♭2/♭1+♭2(4) The Logarithmic Mean. L♭1,♭2≔♭2−♭1/ln♭2−ln♭1(5) Thep-Logarithmic Mean. Lp♭1,♭2≔♭2p+1−♭1p+1/p+1♭2−♭11/p,p∈ℝ−−1,0Proposition 41.
For♭1,♭2>0, we have
(78)A♭1n+2,♭2n+2+♭1+♭2♭2−♭1kn+2H♭1,♭2−G2♭1,♭2Łnn♭1,♭2≤G2♭1,♭2♭2−♭1211+p1/pn+2qs9♭1qn+1+n+2qs10♭2qn+11/q,where s9 and s10 are given by (62) and (63), respectively.Proof.
Forα=1, ψ=0, and Yi=in+2 in Corollary 30, we obtain our required result.Proposition 42.
For♭1,♭2>0, we have
(79)μλ+2μA♭1λ/μ+2,♭2λ/μ+2+λ+2μ♭1+♭2μ♭2−♭1kλ/μ+2H♭1,♭2−λ+2μμG2♭1,♭2Łλ/μλ/μ♭1,♭2≤G2♭1,♭2♭2−♭1211+p1/ps9♭1qλ+qμ/μ+s10♭2qλ+qμ/μ1/q.Proof.
Forα=1, ψ=0, and Yi=μ/λ+2μiλ/μ+2 in Corollary 30, we obtain our required result.Proposition 43.
For♭1,♭2>0, we have
(80)A♭1n+2,♭2n+2+♭1+♭2♭2−♭1kn+2H♭1,♭2−G2♭1,♭2Łnn♭1,♭2≤G2♭1,♭2♭2−♭12s111/pn+2qq+1A♭1qn+1,♭2qn+11/q,where s11 is given by (74).Proof.
Forα=1, ψ=0, and Yi=in+2 in Corollary 40, we obtain our required result.Proposition 44.
For♭1,♭2>0, we have
(81)μλ+2μA♭1λ/μ+2,♭2λ/μ+2+λ+2μ♭1+♭2μ♭2−♭1kλ/μ+2H♭1,♭2−λ+2μμG2♭1,♭2Łλ/μλ/μ♭1,♭2≤G2♭1,♭2♭2−♭12s111/pA♭1qλ+qμ/μ,♭2qλ+qμ/μq+11/q.Proof.
Forα=1, ψ=0, and Yi=μ/λ+2μiλ/μ+2 in Corollary 40, we obtain our required result.



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