# New Modified Conformable Fractional Integral Inequalities of Hermite–Hadamard Type with Applications

Authors: Thabet Abdeljawad; Pshtiwan Othman Mohammed; Artion Kashuri
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this study, a few inequalities of Hermite–Hadamard type are constructed via the conformable fractional operators so that the normal version is recovered in its limit for the conformable fractional parameter. Finally, we present some examples to demonstrate the usefulness of conformable fractional inequalities in the context of special means of the positive numbers.


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## Body

## 1. Introduction

Recently, fractional differential equations have attracted more and more attention, which can be used to describe some biological, chemical, and physical phenomenon more accurately than the classical differential equations of integer order. Nowadays, these appear naturally in modeling long-term behaviors, particularly in the areas of viscous fluid dynamics, control systems, physics, engineering, and viscoelastic materials [1–3]. The fractional calculus has many interesting applications when it was applied on a new mathematical model in thermoelectricity theory. Also, in modern physical engineering, the fractional system order is used to modify the mathematical models that describe the governing equations of those models to be more economical than the classic uses [4, 5].There are many possible ways of defining fractional operators: Riemann–Liouville, Caputo, tempered, Marchaud, Atangana-Baleanu, and Hilfer. To find all these definitions, we advice our readers to visit the references [6–10]. Furthermore, all of these operators are considered special cases of a single, unifying, model of fractional operator by Fernandez et al. in [11].In 2014, Khalil et al. [12] defined a new well-behaved simple fractional derivative, namely, the conformable fractional derivative relying only on the basic limit definition of the classical first derivative. It was first introduced as a conformable fractional derivative, but it lacks some of the desired properties for fractional derivatives [13–16]. This operator and its properties and applications have been intensely studied in other works, of which we mention [13] in particular. In [13], the following conformable derivative was defined:
(1)Dρft=lim∈→0ft+∈t1−ρ−ft∈,where ρ∈0,1, t>0, and f:0,∞→R is a function. This operator relates to the large and thriving theory of conformable fractional derivatives, and it arises naturally in control theory.Now, iff is ρ-differentiable in some 0,ρ,ρ>0, limt→0+fρt exist, then we define
(2)fρ0=limt→0+fρt.Moreover, iff is differentiable, then we have
(3)Dρft=t1−ρf′t,where(4)f′t=lim∈→0ft+∈−ft∈.Briefly, we can writefρt for Dρft or dρ/dρtft to denote the conformable fractional derivatives of f of order ρ at t. In addition, if fρt exists, then we say that f is ρ-differentiable.Theorem 1 ([13]).
Letρ∈0,1 and f,g be ρ-differentiable at a point t>0. Then,
(i)
Dρx1f+x2g=x1Dρf+x2Dρg for all x1,x2∈R(ii)
Dρfg=fDρg+gDρf(iii)
Dρf/g=gDρf−fDρg/g2(iv)
Dρκ=0 for any constant κ∈R(v)
Dρ1/ρtρ=1Now, conformable fractional integral and its few basic properties are stated.Definition 2 ([13]).
For anyρ∈0,1 and 0≤x1≤x2, the function f:x1,x2→R is ρ-fractional integrable on x1,x2, if
(5)∫x1x2ftdρt=∫x1x2fttρ−1dtexists and is finite.Remark 3.
(a)
We indicate eachρ-fractional integrable functions by Lρ1x1,x2 on x1,x2(b)
The usual Riemann improper integral is defined by(6)Ix1ρft=Ix11tρ−1f=∫x1txρ−1fxdx,for each ρ∈0,1.Theorem 4 ([13, 16]).
For the differentiable functionf:x1,x2→R and ρ∈0,1, we have
(7)Ix1ρDρx1ft=ft−fx1,for all t>x1.Theorem 5 ([13]).
Letρ∈n,n+1 and fn be a continuous function. Then, for the function f:x1,∞→R and for all t>x1, we have
(8)Dρx1Ix1ρft=ft.
This property is often called the inverse property.Theorem 6 ([13, 16]).
For the two functionsf,g:x1,x2⟶R, we have
(9)∫x1x2fxD?x1gxdρx=fgx1x2−∫x1x2gxDρx1fxdρx,
such thatfg is differentiable.Theorem 7 ([13, 16]).
Letρ∈0,1 and f:x1,x2⟶R be a continuous function on x1,x2 with 0≤x1≤x2; then, we have
(10)Ix1ρftIx1ρft.Definition 8 ([17]).
A functionf:I⊆ℝ⟶ℝ is said to be convex on the interval I, if the inequality
(11)fηx1+1−ηx2≤ηfx1+1−ηfx2holds for all x1,x2∈I and η∈0,1.In view of recent results in theory of differential, integral, and fractional differential equations, it is becoming extremely worthless to ignore the existence of integral inequalities which are useful in determining bounds of unknown functions (see, e.g., [17–23]). Also, there are various integral inequalities in the literature and their numbers increase sharply every year. But the common inequality is the Hermite–Hadamard (HH) integral inequality, which was first found in 1893 by Hadamard in [24]. It has the following formula:
(12)fx1+x22≤1x2−x1∫x1x2fxdx≤fx1+fx22,where f:E⊆R⟶R is supposed to be convex on E with x1,x2∈E and x1<x2.The inequality of HH type (12) has been extended and applied in all models of fractional calculus, such as Riemann–Liouville [25–29], ψ-Riemann–Liouville [30], conformable fractional [31, 32], generalized fractional [33–35], time-scale fractional [36, 37], tempered fractional [38], and Atangana-Baleanu [39] models. Since our paper is about conformable fractional integral inequality, here, we give some inequalities involving conformable fractional integrals that are obtained in [32]:Lemma 9 ([32], Lemma 1).
Leth:x1,x2⊆ℝ⟶ℝ be an α-fractional differentiable mapping on x1,x2 with 0≤x1<x2. IfDαh∈Lα1x1,x2; then, the following identity for conformable fractional integral holds:
(13)Φαx1,x2=∑i=14δi,where
(14)δ1=x2−x14∫01x1ζ+1−ζ3x1+x242α−1−3x1+x24αx1ζ+1−ζ3x1+x24α−1×Dαhx1ζ+1−ζ3x2+x24dαζ,δ2=x2−x14∫013x1+x24ζ+1−ζx1+x222α−1−3x1+x24α3x1+x24ζ+1−ζx1+x22α−1×Dαh3x1+x24ζ+1−ζx1+x22dαζ,δ3=x2−x14∫01x1+x22ζ+1−ζx1+3x242α−1−x1+3x24αx1+x22ζ+1−ζx1+3x24α−1×Dαhx1+x22ζ+1−ζx1+3x24dαζ,δ4=x2−x14∫01x1+3x24ζ+1−ζx22α−1−x1+3x24αx1+3x24ζ+1−ζx2α−1×Dαhx1+3x24ζ+1−ζx2dαζ,Φαx1,x2=3x1+x24α−x1αhx1+x2α−x1+3x24αhx2+x1+3x24α−3x1+x24αhx1+x22−α∫x1x2hxdαx.Theorem 10 ([32], Theorem 2).
Leth:x1,x2⊆ℝ⟶ℝ be an α-fractional differentiable mapping on x1,x2 with 0≤x1<x2. IfDαh∈Lα1x1,x2 and h′ is convex on x1,x2, then the following inequality for conformable fractional integral holds:
(15)Φαx1,x2≤x2−x14d1α∣h′x1∣+d2α∣h′x2∣+d3αh′3x1+x24+d4αh′x1+x22+d5αh′x1+3x24,where
(16)d1α=1124x1α+3x1+x24x1α−1+3x1+x24α−1x1−53x1+x24α,d3α=112x1α+3x1+x24x1α−1+3x1+x24α−1x1+x1+x223x1+x24α−1+x1+x22α−13x1+x24+x1+x22α−53x1+x24α,d4α=1127x1+x22α+x1+x223x1+x24α−1+x1+x22α−13x1+x24+x1+x22x1+3x24α−1+x1+x22α−1x1+3x24−53x1+x24α−5x1+3x24α,d5α=112x2α+x1+x22α+x1+3x24x1+x22α−1+x1+3x24α−1x1+x22+vx1+3x24α−1+x2α−1x1+3x24−5x1+3x24α.Theorem 11 ([32], Theorem 3).
Leth:x1,x2⊆ℝ⟶ℝ be an α-fractionaldifferentiablemappingonx1,x2with0≤x1<x2. IfDαh∈Lα1x1,x2andh′q is convex on x1,x2, then the following inequality for conformable fractional integral holds:
(17)Φαx1,x2≤x2−x14A1α1−1/qA2αh′x1q+A3αh′3x1+x24q1/q+B1α1−1/qB2αh′3x1+x24q+B3αh′x1+x22q1/q+C1α1−1/qC2αh′x1+x22q+C3αh′x1+3x24q1/q+D1α1−1/qD2αh′x1+3x24q+D3αh′x2q1/q,where
(18)A1α=4u3x1+x2α−4x1α+14αx2−x1α+1−α3x1+x2α4α,A2α=3x1+x2α+2−4x1α+1αx2−x1+x1+x24αx2−x12α+1α+2−3x1+x2α22α+1,A3α=4x1α+2−3x1+x2α+25u−v−α3x1+x2α3u−vx1+x24αx2−x12α+1α+2−3x1+x2α22α+1,B1α=2x1+x2α+2−3x1+x2α+14αx2−x1α+1−3x1+x2α4α,B2α=2x1+x2α+2−3x1+x2α+1x1+3x2−α3x1+x2α+1x2−x14αx2−x12α+1α+2−3x1+x2α22α+1,B3α=3x1+x2α+2+42x1+x2α+1+α2x1+x2α+1x2−x14αx2−x12α+1α+2−3x1+x2α22α+1,C1α=4x1+3x2α+1−2x1+x2α+24αx2−x1α+1−x1+3x2α4α,C2α=α2x2−x12x1+3x2α+16bx1+x2x1+3x2α−3αx12+x22x1+3x2α22α+1x2−x12α+1α+2−8b2x1+x2α+1+2αx2−x12x1+x2α+122α+1x2−x12α+1α+2−x1+3x2α22α+1,C3α=2x1+x2α+2+x1+3x2α+2+αx1+3x2α+2−5x1+x2x1+3x2α+14αx2−x12α+1α+2,D1α=4x2α+1−x1+3x2α+14αα+1x2−x1−x1+3x2α22α+1,D2α=4x2α+2+x1+3x2α+1x1−5x2+αx1+3x2α+1x2−x14αx2−x12α+1α+2,D3α=α4x2α+2x2−x1−24x2α+1x1+x2+x1+3x2α+24αx2−x12α+1α+2.Theorem 12 ([32], Theorem 4).
Leth:x1,x2⊆ℝ⟶ℝ be an α-fractional differentiable mapping on x1,x2 with 0≤x1<x2. If Dαh∈Lα1x1,x2 and h′q is concave on x1,x2, then the following inequality for conformable fractional integral holds:
(19)Φαx1,x2≤x2−x14A1αh′P1αA1α+B1αh′P2αB1α+C1αh′P3αC1α+D1αh′P4αD1α,where A1α,B1α,C1α, and D1α are given in Theorem 11 and
(20)P1α=21−α3x1+x2α+2−22α+5−22α+17x1+6x2x2−x13x1+x2α−α7x1+x2x2−x13x1+x2α22α+3x2−x1α+2,P2α=22x1+x2α+2−8x1+x223x1+x2α−α5x1+3x2x2−x13x1+x2α22α+3x2−x1α+2,P3α=8x1+x22x1+3x2α−4αx1+x2αx1+3x2α−αx1+3x2α+2−2α+3x1+x2α+222α+1x2−x1α+2,P4α=22α−1αx1+3x2+4αx1+3x23+44x2α+2−4α+2x22x1+3x2−4x1+3x2α+2−22α+3αx22x1+3x24α+2x2−x1α+2.In view of the above indices, we establish new integral inequalities of HH type for the convex functions via the conformable fractional operators. By doing this, we will try to demonstrate the usefulness of conformable fractional inequalities in the context of special means of the positive numbers.


## 2. Main Results

The next result is necessary in the rest of the work.Lemma 13.
Letf:x1,x2⊆R⟶R be an ρ-differentiable function on x1,x2, 0≤x1<x2. If Dρf∈Lρ1x1,x2, then we have
(21)Δρx1,x2=x2−x14∫01x1t+1−t3x1+x242ρ−3x1+x241+ρx1t+1−t3x1+x24ρ−1×Dρfx1t+1−t3x1+x24dρt+∫013x1+x24t+1−tx1+x222ρ−x1+x221+ρ3x1+x24t+1−tx1+x22ρ−1×Dρf3x1+x24t+1−tx1+x22dρt+∫01x1+x22t+1−tx1+3x242ρ−x1+3x241+ρx1+x22t+1−tx1+3x24ρ−1×Dρfx1+x22t+1−tx1+3x24dρt+∫01x1+3x24t+1−tx22ρ−x21+ρx1+3x24t+1−tx2ρ−1×Dρfx1+3x24t+1−tx2dρt,where
(22)Δρx1,x2≔3x1+x241+ρ−x11+ρfx1+x21+ρ−x1+3x241+ρfx2+x1+3x241+ρ−3x1+x241+ρfx1+x22−1+ρIx1ρf3x1+x24+I3x1+x2/4ρfx1+x22+Ix1+x2/2ρfx1+3x24+Ix1+3x2/4ρfx2.Proof.
Making use of Definition 8, we have(23)ℏ1≔x2−x14∫01x1t+1−t3x1+x242ρ−3x1+x241+ρx1t+1−t3x1+x24ρ−1×Dρfx1t+1−t3x1+x24dρt=x2−x14∫01x1t+1−t3x1+x241+ρ−3x1+x241+ρf′x1t+1−t3x1+x24dt.
On integrating by parts, one can have(24)ℏ1≔3x1+x241+ρ−x1t+1−t3x1+x241+ρfx1t+1−t3x1+x2401−1+ρx2−x14∫01x1t+1−t3x1+x24ρfx1t+1−t3x1+x24d=3x1+x241+ρ−x11+ρfx1−1+ρ∫01fx1t+1−t3x1+x2/4x1t+1−t3x1+x2/4ρdt,making the change of variable x≔x1t+1−t3x1+x2/4,t∈0,1, to get
(25)ℏ1=3x1+x241+ρ−x11+ρfx1−1+ρ∫x13x1+x2/4fxxρdx=3x1+x241+ρ−x11+ρfx1−1+ρIx1ρf3x1+x24.
Similarly, we get(26)ℏ2≔x2−x14∫013x1+x24t+1−tx1+x222ρ−3x1+x241+ρ3x1+x24t+1−tx1+x22ρ−1Dρf3x1+x24t+1−tx1+x22dρt=x1+x221+ρ−3x1+x241+ρfx1+x22−1+ρI3x1+x2/4ρfx1+x22,ℏ3≔x2−x14∫01x1+x22t+1−tx1+3x242ρ−x1+3x241+ρx1+x22t+1−tx1+3x24ρ−1Dρfx1+x22t+1−tx1+3x24dρt=x1+3x241+ρ−x1+x221+ρfx1+x22−1+ρIx1+x2/2ρfx1+3x24,ℏ4≔x2−x14∫01x1+3x24t+1−tx22ρ−x1+3x241+ρx1+3x24t+1−tx2ρ−1Dρfx1+3x24t+1−tx2dρt=x2ρ−x1+3x24ρfx2−1+ρIx1+3x2/4ρfb.
Consideringℏ1,ℏ2,ℏ3, and ℏ4, we get (21) and thus, our proof is completed.Theorem 14.
Letf:x1,x2⊆R⟶R be an ρ-differentiable function on x1,x2, 0≤x1<x2. If Dρf∈Lρ1x1,x2 and f′ is a convex function on x1,x2, then we have
(27)Δρx1,x2≤x2−x124∑ℓ=14Dℓ,where
(28)D1=3x1+x241+ρ−x11+ρ2f′x1+f′3x1+x24,D2=x1+x221+ρ−3x1+x241+ρ2f′3x1+x24+f′x1+x22,D3=x1+3x241+ρ−x1+x221+ρ2f′x1+x22+f′x1+3x24,D4=x21+ρ−x1+3x241+ρ2f′x1+3x24+f′x2.Proof.
Making use of Lemma13 and the properties of Dρ, we have
(29)Δρx1,x2≤x2−x14∫01x1t+1−t3x1+x241+ρ−3x1+x241+ρ×f′x1t+1−t3x1+x24dt+x2−x14∫013x1+x24t+1−tx1+x221+ρ−x1+x221+ρ×f′3x1+x24t+1−tx1+x22dt+x2−x14∫01x1+x22t+1−tx1+3x241+ρ−x1+3x241+ρ×f′x1+x22t+1−tx1+3x24dt+x2−x14∫01x1+3x24t+1−tx21+ρ−x1+3x241+ρ×f′x1+3x24t+1−tx2dtx2−x14κ1+κ2+κ3+κ4.
Forρ∈0,1 and x>0, the function x1+ρ is convex. So, we have
(30)κ1=∫01x1t+1−t3x1+x241+ρ−3x1+x241+ρf′x1t+1−t3x1+x24dt≤∫013x1+x241+ρ−x11+ρtf′x1t+1−t3x1+x24dt.
Then, by the convexity off′t, we can deduce
(31)κ1=3x1+x241+ρ−x11+ρ∫01tf′x1t+1−t3x1+x24dt≤3x1+x241+ρ−x11+ρ∫01ttf′x1+1−tf′3x1+x24dt≤3x1+x241+ρ−x11+ρ2f′x1+f′3x1+x2/46.
Analogously, we can deduce(32)κ2=∫013x1+x24t+1−tx1+x221+ρ−x1+x221+ρf′3x1+x24t+1−tx1+x22dt≤x1+x221+ρ−3x1+x241+ρ2f′3x1+x2/4+f′x1+x2/26,κ3=∫01x1+x22t+1−tx1+3x241+ρ−x1+3x241+ρf′x1+x22t+1−tx1+3x24dt≤x1+3x241+ρ−x1+x221+ρ2f′x1+x2/2+f′x1+3x2/46,κ4=∫01x1+3x24t+1−tx21+ρ−x1+3x241+ρf′x1+3x24t+1−tx2dt≤x21+ρ−x1+3x241+ρ2f′x1+3x2/4+f′x26.
Substitutingκi’s intoΔ, we obtain the desired result.Corollary 15.
Theorem14 with ρ=1 becomes
(33)Δ1x1,x2≤x2−x124∑ℓ=14D¯ℓ,where
(34)D¯1=3x1+x242−x122f′x1+f′3x1+x24,D¯2=x1+x222−3x1+x2422f′3x1+x24+f′x1+x22,D¯3=x1+3x242−x1+x2222f′x1+x22+f′x1+3x24,D¯4=x22−x1+3x2422f′x1+3x24+f′x2.Theorem 16.
Letf:x1,x2⊆R⟶R be an ρ-differentiable function on x1,x2 such that 0≤x1<x2. If Dρf∈Lρ1x1,x2, f′q is a convex function on x1,x2, then we have
(35)Δρx1,x2≤x2−x14A1ρ1−1/q1x2−x12B1ρf′x1q+C1ρf′3x1+x24ρ1/ρ+A2ρ1−1/ρ1x2−x12B2ρf′3x1+x24ρ+C2ρf′x1+x22ρ1/ρ+A3ρ1−1/ρ1x2−x12B3ρf′x1+x22ρ+C3ρf′x1+3x24q1/q+A4ρ1−1/q1x2−x12B4ρf′x1+3x24q+C4ρf′x2q1/ρ,where
(36)A1ρ=22+ρx12+ρ−3x1+x242+ρ−3x1+x241+ρ,A2ρ=22+ρ3x1+x242+ρ−x1+x222+ρ−3x1+x241+ρ,A3ρ=22+ρx1+x222+ρ−x1+3x242+ρ−x1+3x241+ρ,A4ρ=22+ρx1+3x242+ρ−x22+ρ−3x1+x241+ρ,B1ρ=x12+ρ−3x1+x2/43+ρ3+ρ−3x1+x24x12+ρ−3x1+x2/42+ρ2+ρ+3x1+x241+ρ3x1+x2/42−x122+3x1+x242+ρx1−x24,B2ρ=3x1+x2/43+ρ−x1+x2/23+ρ3+ρ−x1+x223x1+x2/42+ρ−x1+x2/22+ρ2+ρ+3x1+x241+ρ3x1+x2/42−x1+x2/222−3x1+x242+ρx1−x24,B3ρ=x1+x2/23+ρ−x1+3x2/43+ρ3+ρ−x1+3x24x1+x2/23+ρ−x1+3x2/43+ρ2+ρ+x1+3x241+ρx1+x2/2−x1+3x2/422−x1+3x242+ρx1−x24,B4ρ=x12+ρ−3x1+x2/43+ρ3+ρ−x2x1+3x2/43+ρ−x23+ρ2+ρ+x1+3x241+ρx1+3x2/42−x222−x1+3x242+ρx1−x24,C1ρ=x12+ρ−3x1+x2/43+ρ3+ρ−x1x12+ρ−3x1+x2/42+ρ2+ρ+3x1+x241+ρ3x1+x2/42−x122+3x1+x242+ρx1−x24,C2ρ=3x1+x2/43+ρ−x1+x2/23+ρ3+ρ−3x1+x243x1+x2/42+ρ−x1+x2/22+ρ2+ρ+3x1+x241+ρ3x1+x2/42−x1+x2/222−3x1+x242+ρx1−x24,C3ρ=x1+x2/23+ρ−x1+3x2/43+ρ3+ρ−x1+x22x1+x2/23+ρ−x1+3x2/43+ρ2+ρ+x1+3x241+ρx1+x2/2−x1+3x2/422−x1+3x242+ρx1−x24,C4ρ=x12+ρ−3x1+x2/43+ρ3+ρ−x1+3x24x1+3x2/43+ρ−x23+ρ2+ρ+x1+3x241+ρx1+3x2/42−x222−x1+3x242+ρx1−x24.Proof.
From Lemma13, we have
(37)Δρx1,x2≤x2−x14I1+I2+I3+I4,where
(38)I1=∫01x1t+1−t3x1+x241+ρ−3x1+x241+ρf′x1t+1−t3x1+x24dt,I2=∫013x1+x24t+1−tx1+x221+ρ−3x1+x241+ρf′3x1+x24t+1−tx1+x22dt,I4=∫01x1+3x24t+1−tx21+ρ−x1+3x241+ρf′x1+3x24t+1−tx2dt,making use of the power-mean inequality to get
(39)I1≤∫01x1t+1−t3x1+x241+ρ−3x1+x241+ρdt1−1/q×∫01x1t+1−t3x1+x241+ρ−3x1+x241+ρf′x1t+1−t3x1+x24qdt1/q.
Sincef′q is convex on x1,x2, then for any t∈0,1, we have
(40)I1≤∫01x1t+1−t3x1+x241+ρ−3x1+x241+ρdt1−1/q×∫01x1t+1−t3x1+x241+ρ−3x1+x241+ρf′x1qt+f′3x1+x24q1−tdt1/q.
So we have(41)I1≤A1ρ1−1/qB1ρf′x1q+C1ρf′3x1+x24q1/q.
Analogously, we can deduce(42)I2≤A2ρ1−1/qB2ρf′3x1+x24q+C2ρf′x1+x22q1/q,I3≤A3ρ1−1/qB3ρf′x1+x22q+C3ρf′x1+3x24q1/q,I4≤A4ρ1−1/qB4ρf′x1+3x24q+C4ρf′x2q1/q,whereAiρ,Biρ, andCiρare as before. Then, by substituting the above inequalities into (37), we get the desired result.Remark 17.
Theorem16 with q=1 reduces to Theorem 14.Corollary 18.
Theorem16 with ρ=1 becomes
(43)Δ1x1,x2≤x2−x14A111−1/q1x2−x12B11f′x1q+C11f′3x1+x24q1/q+A211−1/q1x2−x12B21f′3x1+x24q+C21f′x1+x22q1/q+A311−1/q1x2−x12B31f′x1+x22q+C31f′x1+3x24q1/q+A411−1/q1x2−x12B41f′x1+3x24q+C41f′x2q1/q,where
(44)A11=23x13−3x1+x243−3x1+x242,A21=233x1+x243−x1+x223−3x1+x242,A31=23x1+x223−x1+3x243−x1+3x242,A41=23x1+3x243−x23−3x1+x242,B11=x13−3x1+x2/444−3x1+x24x13−3x1+x2/433+3x1+x2423x1+x2/42−x122+3x1+x243x1−x24,B21=3x1+x2/44−x1+x2/244−x1+x223x1+x2/43−x1+x2/233+3x1+x2423x1+x2/42−x1+x2/222−3x1+x243x1−x24,B31=x1+x2/24−x1+3x2/444−x1+3x24x1+x2/24−x1+3x2/443+x1+3x242x1+x2/2−x1+3x2/422−x1+3x243x1−x24,B41=x13−3x1+x2/444−x2x1+3x2/44−x243+x1+3x242x1+3x2/42−x222−x1+3x243x1−x24,C11=x13−3x1+x2/444−x1x13−3x1+x2/433+3x1+x2423x1+x2/42−x122+3x1+x243x1−x24,C21=3x1+x2/44−x1+x2/244−3x1+x243x1+x2/43−x1+x2/233+3x1+x2423x1+x2/42−x1+x2/222−3x1+x243x1−x24,C31=x1+x2/24−x1+3x2/444−x1+x22x1+x2/24−x1+3x2/443+x1+3x242x1+x2/2−x1+3x2/422−x1+3x243x1−x24,C41=x13−3x1+x2/444−x1+3x24x1+3x2/44−x243+x1+3x242x1+3x2/42−x222−x1+3x243x1−x24.Theorem 19.
Letf:x1,x2⊆R⟶R be an ρ-differentiable function on x1,x2 such that 0≤x1<x2. If Dρf∈Lρ1x1,x2, f′q is a convex function on x1,x2, then for q>1 and 1/p+1/q=1, we have
(45)Δρx1,x2≤x2−x142qE11/ρp,ρf′x1q+f′3x1+x24q1/q+E21/pp,ρf′3x1+x24q+f′x1+x22q1/q+E31/pp,ρf′x1+x22q+f′x1+3x24q1/q+E41/pp,ρf′x1+3x24q+f′x2q1/q,where
(46)E1p,ρ=∫01x1t+1−t3x1+x241+ρ−3x1+x241+ρpdt,E2p,ρ=∫013x1+x24t+1−tx1+x221+ρ−3x1+x241+ρpdt,E3p,ρ=∫01x1+x22t+1−tx1+3x241+ρ−x1+3x241+ρpdt,E4p,ρ=∫01x1+3x24t+1−tx21+ρ−x1+3x241+ρpdt.Proof.
From Lemma13, we have
(47)Δρx1,x2≤x2−x14I1+I2+I3+I4,where
(48)I1≤∫01x1t+1−t3x1+x241+ρ−3x1+x241+ρf′x1t+1−t3x1+x24dt,I2≤∫013x1+x24t+1−tx1+x221+ρ−3x1+x241+ρf′3x1+x24t+1−tx1+x22dt,I3≤∫01x1+x22t+1−tx1+3x241+ρ−x1+3x241+ρf′x1+x22t+1−tx1+3x24dt,I4≤∫01x1+3x24t+1−tx21+ρ−x1+3x241+ρf′x1+3x2x1+3x2/4t+1−tx2dt,making use of Hölder’s inequality to get
(49)I1≤∫01x1t+1−t3x1+x241+ρ−3x1+x241+ρpdt1/p×∫01f′x1t+1−t3x1+x24qdt1/q.
Sincef′q is convex on x1,x2, then for any t∈0,1, we have
(50)I1≤∫01x1t+1−t3x1+x241+ρ−3x1+x241+ρpdt1/p×∫01f′x1qt+f′3x1+x24q1−tdt1/q.
It follows that(51)I1≤E11/pp,ρf′x1q+f′3x1+x2/4q21/q.
Analogously, we can deduce(52)I2≤E21/pp,ρf′3x1+x2/4q+f′x1+x2/2q21/q,I3≤E31/pp,ρf′x1+x2/2q+f′x1+3x2/4q21/q,I4≤E41/pp,ρf′x1+3x2/4q+f′x2q21/q.
Then, by substituting∣Ii∣s (for i=1,2,3,4) into (47), we get the desired result.Corollary 20.
Theorem19 with ρ=1 becomes
(53)Δ1x1,x2≤x2−x142qE11/ppf′x1q+f′3x1+x24q1/q+E21/ppf′3x1+x24q+f′x1+x22q1/q+E31/ppf′x1+x22q+f′x1+3x24q1/q+E41/ppf′x1+3x24q+f′x2q1/q,where
(54)E1p=∫01x1t+1−t3x1+x242−3x1+x242pdt,E2p=∫013x1+x24t+1−tx1+x222−3x1+x242pdt,E3p=∫01x1+x22t+1−tx1+3x242−x1+3x242pdt,E4p=∫01x1+3x24t+1−tx22−x1+3x242pdt.Theorem 21.
Letf:x1,x2⊆R⟶R be an ρ-differentiable function on x1,x2 such that 0≤x1<x2. If Dρf∈Lρ1x1,x2 and f′q is a concave function on x1,x2, we have
(55)Δρx1,x2≤x2−x14A1ρf′J1ρA1ρ+A2ρf′J2ρA2ρ+A3ρf′J3ρA3ρ+A4ρf′J4ρA4ρ,where Aiρ,i=1,2,3,4 are given in Theorem 16 and
(56)J1ρ=−22+ρ3+ρx2−x13x1+x243+ρ−x13+ρ−3x1+x241+ρ7x1+x28,J2ρ=22+ρ3+ρx2−x1x1+x223+ρ−3x1+x243+ρ−3x1+x241+ρ5x1+3x28,J3ρ=−22+ρ3+ρx2−x1x1+3x243+ρ−x1+x223+ρ−x1+3x241+ρ3x1+5x28,J4ρ=22+ρ3+ρx2−x1x23+ρ−x1+3x243+ρ−x1+3x241+ρx1+7x28.Proof.
Sincef′q is concave on x1,x2, then for any t∈0,1, we have
(57)f′x1t+1−t3x1+x24q≥f′x1t+f′3x1+x241−tq,or we can say
(58)f′x1t+1−t3x1+x24≥f′x1t+f′3x1+x241−t.
This tells us that the function∣f′∣is also concave. Then, by making use of I1 and Jensen’s inequality with the inequality (58), we get
(59)∣I1∣≤∫01x1t+1−t3x1+x241+ρ−3x1+x241+ρdt×f′∫01x1t+1−t3x1+x2/41+ρ−3x1+x2/41+ρx1t+1−t3x1+x2/4dt∫01x1t+1−t3x1+x2/41+ρ−3x1+x2/41+ρdt≤22+ρx12+ρ−3x1+x242+ρ−3x1+x241+ρ×f′−22+ρ/3+ρx2−x13x1+x2/43+ρ−x13+ρ−3x1+x2/41+ρ7x1+x2/82/2+ρx12+ρ−3x1+x2/42+ρ−3x1+x2/41+ρ=A1ρf′J1ρA1ρ.
By the same way, can deduce∣Ii∣≤Aiρf′Jiρ/Aiρ,i=2,3,4, whereAiρare as before and
(60)J2ρ=22+ρ3+ρx2−x1x1+x223+ρ−3x1+x243+ρ−3x1+x241+ρ5x1+3x288,J3ρ=−22+ρ3+ρx2−x1x1+3x243+ρ−x1+x223+ρ−x1+3x241+ρ3x1+5x28,J4ρ=22+ρ3+ρx2−x1x23+ρ−x1+3x243+ρ−x1+3x241+ρx1+7x28.
These rearrange to the desired (55).Corollary 22.
Theorem21 with ρ=1 becomes
(61)Δ1x1,x2≤x2−x14A11∣f′J11A11∣+A21∣f′J21A21∣+A31∣f′J31A31∣+A41∣f′J41A41∣,where Ai1,i=1,2,3,4 are as in Corollary 18 and
(62)J11=−2x2−x13x1+x244−x14−3x1+x2427x1+x28,J21=2x2−x1x1+x224−3x1+x244−3x1+x2425x1+3x28,J31=−2x2−x1x1+3x244−x1+x224−x1+3x2423x1+5x28,J41=2x2−x1x24−x1+3x244−x1+3x242x1+7x28.


## 3. Applications to Special Means

Consider the two special means of positive real numbers (x1,x2∈R+):
(i)
The arithmetic mean:(63)Ax1,x2=x1+x22(ii)
Theρ,r-th generalized logarithmic mean:(64)Lρ,rx1,x2=ρx2r+ρ−x1r+1r+ρx2ρ−x1ρ1/r,r∈R\−ρ,0,ρ∈0,1.Proposition 23.
For any0<x1<x2,r>1 and ρ∈0,1, we have
(65)∣3x1+x241+ρ−x11+ρx1r+x21+ρ−x1+3x241+ρx2r+x1+3x241+ρ−3x1+x241+ρArx1,x2−1+ρρ3x1+x24ρ−x1ρ×Lρ,rrx1,3x1+x24+Aρx1,x2−3x1+x24ρLρ,rr3x1+x24,Ax1,x2+x1+3x24ρ−Aρx1,x2Lρ,rrAx1,x2,x1+3x24+x2ρ−x1+3x24ρLρ,rrx1+3x24,x2∣≤rx2−x124∑ℓ=14Uℓ,where
(66)U1=3x1+x24ρ+1−x1ρ+12x1r−1+3x1+x24r−1,U2=Aρ+1x1,x2−3x1+x24ρ+123x1+x24r−1+Ar−1x1,x2,U3=x1+3x24ρ+1−Aρ+1x1,x22Ar−1x1,x2+x1+3x24r−1,U4=x2ρ+1−x1+3x24ρ+12x1+3x24r−1+x2r−1.Proof.
Letfx=xr for x>0 and r>1. Then, by applying Theorem 14 for the convex function fx, we get the desired result.Proposition 24.
Let0<x1<x2,r>1 and ρ∈0,1; then, for any q≥1, we have
(67)3x1+x241+ρ−x11+ρx1r+x21+ρ−x1+3x241+ρx2r+x1+3x241+ρ−3x1+x241+ρArx1,x2−1+ρρ3x1+x24ρ−x1ρ×Lρ,rrx1,3x1+x24+Aρx1,x2−3x1+x24ρLρ,rr3x1+x24,Ax1,x2+x1+3x24ρ−Aρx1,x2Lρ,rrAx1,x2,x1+3x24+x2ρ−x1+3x24ρLρ,rrx1+3x24,x2≤r4x2−x1q−2/qA1ρ1−1/qB1ρx1r−1q+C1ρ3x1+x24r−1q1/q+A2ρ1−1/qB2ρ3x1+x24r−1q+C2ρAr−1qx1,x21/q+A3ρ1−1/qB3ρAr−1qx1,x2+C3ρx1+3x24r−1q1/q+A4ρ1−1/qB4ρx1+3x24r−1q+C4ρx2r−1q1/q,where Aiρ,Biρ, and Ciρ (for i=1,2,3,4) are as in Theorem 16.Proof.
Letfx=xr for x>0 and r>1. Then, by applying Theorem 16 for the convex function fx, we get the desired inequality.Proposition 25.
For any0<x1<x2,r>1 and ρ∈0,1, then for q>1 and 1/p+1/q=1, we have
(68)3x1+x241+ρ−x11+ρx1r+x21+ρ−x1+3x241+ρx2r+x1+3x241+ρ−3x1+x241+ρArx1,x2−1+ρρ3x1+x24ρ−x1ρ×Lρ,rrx1,3x1+x24+Aρx1,x2−3x1+x24ρLρ,rr3x1+x24,Ax1,x2+x1+3x24ρ−Aρx1,x2Lρ,rrAx1,x2,x1+3x24+x2ρ−x1+3x24ρLρ,rrx1+3x24,x2≤rx2−x142qE11/pp,ρx1r−1q+3x1+x24r−1q1/q+E21/pp,ρ3x1+x24r−1q+Ar−1qx1,x21/q+E31/pp,ρAr−1qx1,x2+x1+3x24r−1q1/q+E41/pp,ρx1+3x24r−1q+x2r−1q1/q,where Eip for i=1,2,3,4, are as in Theorem 19.Proof.
Letfx=xr for x>0 and r>1. Then, the result follows by applying Theorem 19 for the convex function fx.


## 4. Conclusion

In this article, we have considered new integral inequalities for conformable fractional operators through convexity of the functionf, its derivative Dρf, and the power of its derivative Dρfq. Finally, we see that Propositions 23–25 confirm the usefulness of conformable fractional inequalities in the context of special means of the positive numbers.



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