# Some Hardy-Type Inequalities for Superquadratic Functions via Delta Fractional Integrals

Authors: Usama Hanif; Ammara Nosheen; Rabia Bibi; Khuram Ali Khan; Hamid Reza Moradi
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, Jensen and Hardy inequalities, including Pólya–Knopp type inequalities for superquadratic functions, are extended using Riemann–Liouville delta fractional integrals. Furthermore, some inequalities are proved by using special kernels. Particular cases of obtained inequalities give us the results on time scales calculus, fractional calculus, discrete fractional calculus, and quantum fractional calculus.


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## Body

## 1. Introduction

The study of Hardy inequalities has gained huge attention in the literature, and now it became a major field in applied and pure mathematics. The Hardy inequality has a long history and many variants. Together with the Sobolev inequalities, it is one of the most frequently used inequalities in the analysis. Firstly, Hardy inequality was discovered to simplify the proof of another inequality. It was then studied in its own right and acquired several useful variants, and it eventually turned out to be extremely useful in the theory of partial differential equations. Except for the direct application of Hardy’s inequality to the Schrodinger operator, other useful variants have been successfully developed for applications in other areas of physics [1].Hardy-type inequalities have crucial importance in the study of function spaces, especially of fractional regularity [2]. Another fundamental consequence is the trace theory of weighted Sobolev spaces; in turn, weighted Sobolev spaces are useful in the regularity theory of the superposition operators [2]. The familiar Hardy inequality (both in the discrete and continuous settings) as presented in [3] has been enormously studied and used as a model for the inquisition of more general integral inequalities [4, 5].In [6], Hardy proved the discrete inequality,(1)∑n=1∞1n∑i=1naip≤pp−1p∑n=1∞apn,p>1,where an≥0 for n≥1. Also, in [7], Hardy proved the continuous inequality, using the calculus of variation, which states that for nonnegative, integrable function over any finite interval 0,x and fp is integrable for p>1,(2)∫0∞1x∫0xftdtpdx≤pp−1p∫0∞fpxdx,where the constant p/p−1p is best possible.Hardy inequality (2) has been generalized by G. H. Hardy himself in [8]. There he showed that for any integral function fx>0 on 0,∞,p>1,(3)∫0∞1xm∫x∞ftdtpdx≤p1−mp∫0∞1xm−pfpxdx,m<1,(4)∫0∞1xm∫0xftdtpdx≤pm−1p∫0∞1xm−pfpxdx,m>1.Littlewood and Hardy [9] established the discrete version of (3) and (4).Since the discovery of these two inequalities, various papers which deal with new proofs, generalizations, and extensions have appeared in the literature, [6, 7, 10–12] are referred to readers. During the last decades, these inequalities were extended, a time scale version of weighted Hardy-type inequality is proved in [11], and some preliminary dynamic inequalities [13] are proved in time scales calculus. In [14, 15], some classical inequalities are proved for isotonic linear functionals and superquadratic functions.Recently, many new Hardy inequalities have been proved, including Hardy and Rellich inequalities for Bessel pairs [16], Hardy–Sobolev inequalities on hypersurfaces for Euclidean space [17], improved Hardy inequalities with exact remainder terms [18], Hardy inequalities with double singular weights [19], and Hardy inequalities for class functions in one-dimensional fractional Orlicz–Sobolev spaces [20]. A new approach for the fractional integral operator in time scales with variable exponent Lebesgue spaces is presented in [21] and n-dimensional integral-type inequalities via time scale calculus is studied in [22].


## 2. Preliminaries

### 2.1. Some Basics on Time Scales

A time scale is a closed subset of the real lineℝ, and its common notation is T. A time scale T may or may not be connected. Forward jump operator and backward jump operator are respectively defined as (5)σω=infs∈T:s>ω,ρω=sups∈T:s<ω.In general,σω≥ω and ρω≤ω. The forward and backward graininess functions μ,ν:T⟶0,+∞ are, respectively, defined by(6)μω=σω−ω,νω=ω−ρω.Ifσω>ω, then ω is said to be right scattered. If ρω<ω, then ω is said to be left scattered. Points that are left-scattered and right scattered at the same time are called isolated. If ω<supT and σω=ω, then ω is called the right-dense and if ω>infT and ρω=ω, then ω is called left-dense. Points that are left-dense and right-dense at the same time are called dense.Rd-continuous function: a functionℏ:T⟶ℝ is called rd-continuous if it is continuous at right-dense points in T and its left-sided limits exist at left-dense points in T.Notation: ifT is a time scale, then the set Tk is defined as(7)Tk=TρsupT,supT,supT<∞;T,supT=∞.Delta derivatives: take a functionℏ:T⟶ℝ. The delta derivative (Hilger derivative) ℏΔξ exists iff, for every ℑ>0, there exists a neighborhood ℵ of ξ such that(8)ℏσξ−ℏs−ℏΔξσξ−s≤ℑσξ−s,for all s∈ℵ, ξ∈T.


### 2.2. Some Principles of Fractional Calculus

#### 2.2.1. Riemann–Liouville Derivative

Riemann–Liouville derivative is the generalization of the usual derivative. It is based on Cauchy’s formula for calculating iterated integrals, if the first integral of a function, which must be equal to deriving it to−1, is as follows:(9)D−1ℏϑ=∫0ϑℏωdω.The calculation of the second can be simplified by interchanging the integration order,(10)D−2ℏϑ=∫0ϑ∫0ω2ℏω1dω1dω2=∫0ϑ∫ω1ϑℏω1dω2dω1=∫0ϑℏω1∫ω1ϑdω2dω1=∫0ϑℏωϑ−ωdω.This method can be applied repeatedly, resulting in the following formula for calculating iterated integrals,(11)D−nℏϑ=1n−1!∫0ϑℏωϑ−ωn−1dω.Now, this can be easily generalized to noninteger values, in what is the Riemann–Liouville derivative [23],(12)Dγℏϑ=1Γ−γ∫0ϑℏωϑ−ωγ+1dω.The Riemann–Liouville derivative with the lower integration limita would be [23](13)Daϑγℏϑ=1Γ−γ∫aϑℏωϑ−ωγ+1dω.


#### 2.2.2. Riemann–Liouville Integral

The Riemann–Liouville integral relates with a real functionℏ:ℝ⟶ℝ, another same kind of function Iγℏ for each value of the parameter γ>0. The integral is a generalization of the repeated antiderivative of ℏ of order real number γ.LetJ=a,b−∞<a<b<∞ be a finite interval of ℝ. The left and right Riemann–Liouville fractional integrals Daω−γℏ and Dtb−γℏ of order γ∈ℝ+ are defined in [23] by(14)Daω−γℏω=1Γγ∫aωω−sγ−1ℏsds,ω>a,γ>0,(15)Dtb−γℏω=1Γγ∫ωbs−ωγ−1ℏsds,ω<b,γ>0,respectively, provided the right-hand sides are pointwise defined on a,b. When γ=n∈N+, definitions (14) and (15) coincide with the nth integrals of the form,(16)Daω−nℏω=1n−1!∫aωω−sn−1ℏsds,Dtb−nℏω=1n−1!∫ωbs−ωn−1ℏsds.


### 2.3. Some Principles of Delta Fractional Calculus

Forμ,ν>0, one has that (see [24], p. 256)(17)∫uξξ−rℓ−1Γℓr−uJ−1ΓJdr=ξ−uℓ+J−1Γℓ+J,where Γ is the Gamma function. If time scale T is considered to be T=Tk, then the coordinate-wise rd-continuous functions are given by(18)hγ:T×T⟶ℝ,γ≥0,such that h0ξ,r=1,∀ξ,r∈T and(19)hγ+1ξ,r=∫rξhγy,rΔy,∀ξ,r,y∈T.Furthermore, forγ,β>1, it is assumed that(20)∫σrξhγ−1ξ,σshβ−1s,σrΔs=hγ+β−1ξ,σr,∀r<ξ;r,ξ∈T.In the caseT=ℝ and for σr=r, one obtains(21)hkr,ξ=r−ξκκ!,κ∈ℕ0=ℕ∪0,hγr,ξ=r−ξγΓγ+1,γ≥0,and also(22)∫rξξ−rγΓγ+1dξ=ξ−rγ+1Γγ+2=hγ+1ξ,r,satisfying (19). Furthermore, it is observed that for γ,β>1 and by using (17), one gets(23)∫rςhγ−1ς,ξhβ−1ξ,ϖdξ=∫rςς−ξγ−1Γγξ−ϖβ−1Γβdξ=ς−ϖγ+β−1Γγ+β=hγ+β−1ς,ϖ,which is satisfying (20).Ifg∈Crd and x∈Tk ([25], Chap: 1), then(24)∫xσxgςΔς=μxgx.


#### 2.3.1. Riemann–Liouville Delta Fractional Integral

Forγ≥1, the Δ-Riemann–Liouville type fractional integral is defined in [26] as(25)Kaγϕω=∫aωhγ−1ω,σνϕνΔν,for a,b∈T and ω∈a,b∩T. Notice that Ka0ϕ=ϕ, where ϕ∈L1a,b∩T (Lebesgue Δ-integrable functions on a,b∩T, [27]).


#### 2.3.2. Fubini’s Theorem on Time Scale

Considerr,l∈T and ϕ∈CrdT×T,ℝ. Then,(26)∫rl∫ηlϕη,ζΔζΔη=∫rl∫rσζϕη,ζΔηΔζ.It can be found in [28].


### 2.4. Superquadratic Function

Definition 1.
A functionΘ:0,∞⟶ℝ is called superquadratic 29 provided that, for all ω≥0, there exists a constant Cω∈ℝ such that(27)Θν−Θω−Θν−ω≥Cων−ω,for all ν≥0. We say that ϕ is subquadratic if −ϕ is a superquadratic function.The following lemma (see [29]) shows that every positive superquadratic function is also a convex function.Lemma 1.
LetΘ be a superquadratic function with Cω as in Definition 1. Then,(i)
Θ0≤0.(ii)
IfΘ0=Θ′0=0, then Cω=Θ′ω, where Θ is differentiable at ω>0.(iii)
IfΘ≥0, then Θ is convex and Θ0=Θ′0=0.


## 2.1. Some Basics on Time Scales

A time scale is a closed subset of the real lineℝ, and its common notation is T. A time scale T may or may not be connected. Forward jump operator and backward jump operator are respectively defined as (5)σω=infs∈T:s>ω,ρω=sups∈T:s<ω.In general,σω≥ω and ρω≤ω. The forward and backward graininess functions μ,ν:T⟶0,+∞ are, respectively, defined by(6)μω=σω−ω,νω=ω−ρω.Ifσω>ω, then ω is said to be right scattered. If ρω<ω, then ω is said to be left scattered. Points that are left-scattered and right scattered at the same time are called isolated. If ω<supT and σω=ω, then ω is called the right-dense and if ω>infT and ρω=ω, then ω is called left-dense. Points that are left-dense and right-dense at the same time are called dense.Rd-continuous function: a functionℏ:T⟶ℝ is called rd-continuous if it is continuous at right-dense points in T and its left-sided limits exist at left-dense points in T.Notation: ifT is a time scale, then the set Tk is defined as(7)Tk=TρsupT,supT,supT<∞;T,supT=∞.Delta derivatives: take a functionℏ:T⟶ℝ. The delta derivative (Hilger derivative) ℏΔξ exists iff, for every ℑ>0, there exists a neighborhood ℵ of ξ such that(8)ℏσξ−ℏs−ℏΔξσξ−s≤ℑσξ−s,for all s∈ℵ, ξ∈T.


## 2.2. Some Principles of Fractional Calculus

### 2.2.1. Riemann–Liouville Derivative

Riemann–Liouville derivative is the generalization of the usual derivative. It is based on Cauchy’s formula for calculating iterated integrals, if the first integral of a function, which must be equal to deriving it to−1, is as follows:(9)D−1ℏϑ=∫0ϑℏωdω.The calculation of the second can be simplified by interchanging the integration order,(10)D−2ℏϑ=∫0ϑ∫0ω2ℏω1dω1dω2=∫0ϑ∫ω1ϑℏω1dω2dω1=∫0ϑℏω1∫ω1ϑdω2dω1=∫0ϑℏωϑ−ωdω.This method can be applied repeatedly, resulting in the following formula for calculating iterated integrals,(11)D−nℏϑ=1n−1!∫0ϑℏωϑ−ωn−1dω.Now, this can be easily generalized to noninteger values, in what is the Riemann–Liouville derivative [23],(12)Dγℏϑ=1Γ−γ∫0ϑℏωϑ−ωγ+1dω.The Riemann–Liouville derivative with the lower integration limita would be [23](13)Daϑγℏϑ=1Γ−γ∫aϑℏωϑ−ωγ+1dω.


### 2.2.2. Riemann–Liouville Integral

The Riemann–Liouville integral relates with a real functionℏ:ℝ⟶ℝ, another same kind of function Iγℏ for each value of the parameter γ>0. The integral is a generalization of the repeated antiderivative of ℏ of order real number γ.LetJ=a,b−∞<a<b<∞ be a finite interval of ℝ. The left and right Riemann–Liouville fractional integrals Daω−γℏ and Dtb−γℏ of order γ∈ℝ+ are defined in [23] by(14)Daω−γℏω=1Γγ∫aωω−sγ−1ℏsds,ω>a,γ>0,(15)Dtb−γℏω=1Γγ∫ωbs−ωγ−1ℏsds,ω<b,γ>0,respectively, provided the right-hand sides are pointwise defined on a,b. When γ=n∈N+, definitions (14) and (15) coincide with the nth integrals of the form,(16)Daω−nℏω=1n−1!∫aωω−sn−1ℏsds,Dtb−nℏω=1n−1!∫ωbs−ωn−1ℏsds.


## 2.2.1. Riemann–Liouville Derivative

Riemann–Liouville derivative is the generalization of the usual derivative. It is based on Cauchy’s formula for calculating iterated integrals, if the first integral of a function, which must be equal to deriving it to−1, is as follows:(9)D−1ℏϑ=∫0ϑℏωdω.The calculation of the second can be simplified by interchanging the integration order,(10)D−2ℏϑ=∫0ϑ∫0ω2ℏω1dω1dω2=∫0ϑ∫ω1ϑℏω1dω2dω1=∫0ϑℏω1∫ω1ϑdω2dω1=∫0ϑℏωϑ−ωdω.This method can be applied repeatedly, resulting in the following formula for calculating iterated integrals,(11)D−nℏϑ=1n−1!∫0ϑℏωϑ−ωn−1dω.Now, this can be easily generalized to noninteger values, in what is the Riemann–Liouville derivative [23],(12)Dγℏϑ=1Γ−γ∫0ϑℏωϑ−ωγ+1dω.The Riemann–Liouville derivative with the lower integration limita would be [23](13)Daϑγℏϑ=1Γ−γ∫aϑℏωϑ−ωγ+1dω.


## 2.2.2. Riemann–Liouville Integral

The Riemann–Liouville integral relates with a real functionℏ:ℝ⟶ℝ, another same kind of function Iγℏ for each value of the parameter γ>0. The integral is a generalization of the repeated antiderivative of ℏ of order real number γ.LetJ=a,b−∞<a<b<∞ be a finite interval of ℝ. The left and right Riemann–Liouville fractional integrals Daω−γℏ and Dtb−γℏ of order γ∈ℝ+ are defined in [23] by(14)Daω−γℏω=1Γγ∫aωω−sγ−1ℏsds,ω>a,γ>0,(15)Dtb−γℏω=1Γγ∫ωbs−ωγ−1ℏsds,ω<b,γ>0,respectively, provided the right-hand sides are pointwise defined on a,b. When γ=n∈N+, definitions (14) and (15) coincide with the nth integrals of the form,(16)Daω−nℏω=1n−1!∫aωω−sn−1ℏsds,Dtb−nℏω=1n−1!∫ωbs−ωn−1ℏsds.


## 2.3. Some Principles of Delta Fractional Calculus

Forμ,ν>0, one has that (see [24], p. 256)(17)∫uξξ−rℓ−1Γℓr−uJ−1ΓJdr=ξ−uℓ+J−1Γℓ+J,where Γ is the Gamma function. If time scale T is considered to be T=Tk, then the coordinate-wise rd-continuous functions are given by(18)hγ:T×T⟶ℝ,γ≥0,such that h0ξ,r=1,∀ξ,r∈T and(19)hγ+1ξ,r=∫rξhγy,rΔy,∀ξ,r,y∈T.Furthermore, forγ,β>1, it is assumed that(20)∫σrξhγ−1ξ,σshβ−1s,σrΔs=hγ+β−1ξ,σr,∀r<ξ;r,ξ∈T.In the caseT=ℝ and for σr=r, one obtains(21)hkr,ξ=r−ξκκ!,κ∈ℕ0=ℕ∪0,hγr,ξ=r−ξγΓγ+1,γ≥0,and also(22)∫rξξ−rγΓγ+1dξ=ξ−rγ+1Γγ+2=hγ+1ξ,r,satisfying (19). Furthermore, it is observed that for γ,β>1 and by using (17), one gets(23)∫rςhγ−1ς,ξhβ−1ξ,ϖdξ=∫rςς−ξγ−1Γγξ−ϖβ−1Γβdξ=ς−ϖγ+β−1Γγ+β=hγ+β−1ς,ϖ,which is satisfying (20).Ifg∈Crd and x∈Tk ([25], Chap: 1), then(24)∫xσxgςΔς=μxgx.


### 2.3.1. Riemann–Liouville Delta Fractional Integral

Forγ≥1, the Δ-Riemann–Liouville type fractional integral is defined in [26] as(25)Kaγϕω=∫aωhγ−1ω,σνϕνΔν,for a,b∈T and ω∈a,b∩T. Notice that Ka0ϕ=ϕ, where ϕ∈L1a,b∩T (Lebesgue Δ-integrable functions on a,b∩T, [27]).


### 2.3.2. Fubini’s Theorem on Time Scale

Considerr,l∈T and ϕ∈CrdT×T,ℝ. Then,(26)∫rl∫ηlϕη,ζΔζΔη=∫rl∫rσζϕη,ζΔηΔζ.It can be found in [28].


## 2.3.1. Riemann–Liouville Delta Fractional Integral

Forγ≥1, the Δ-Riemann–Liouville type fractional integral is defined in [26] as(25)Kaγϕω=∫aωhγ−1ω,σνϕνΔν,for a,b∈T and ω∈a,b∩T. Notice that Ka0ϕ=ϕ, where ϕ∈L1a,b∩T (Lebesgue Δ-integrable functions on a,b∩T, [27]).


## 2.3.2. Fubini’s Theorem on Time Scale

Considerr,l∈T and ϕ∈CrdT×T,ℝ. Then,(26)∫rl∫ηlϕη,ζΔζΔη=∫rl∫rσζϕη,ζΔηΔζ.It can be found in [28].


## 2.4. Superquadratic Function

Definition 1.
A functionΘ:0,∞⟶ℝ is called superquadratic 29 provided that, for all ω≥0, there exists a constant Cω∈ℝ such that(27)Θν−Θω−Θν−ω≥Cων−ω,for all ν≥0. We say that ϕ is subquadratic if −ϕ is a superquadratic function.The following lemma (see [29]) shows that every positive superquadratic function is also a convex function.Lemma 1.
LetΘ be a superquadratic function with Cω as in Definition 1. Then,(i)
Θ0≤0.(ii)
IfΘ0=Θ′0=0, then Cω=Θ′ω, where Θ is differentiable at ω>0.(iii)
IfΘ≥0, then Θ is convex and Θ0=Θ′0=0.


## 3. Jensen Inequality for Superquadratic Functions via Delta Fractional Integrals

Theorem 1.
Lethγ−1:T×T⟶ℝ where γ≥1 and let ϑ1∈T. If ϒ1=λ1,λ2∩T with λ1,λ2∈T and ϕ:ϒ1⟶0,∞ is rd-continuous, then for every continuous superquadratic function Θ:0,∞⟶ℝ,(28)ΘKλ1γϕϑ1ϝϑ1≤1ϝϑ1∫λ1ϑ1Θϕs−Θϕs−Kλ1γϕϑ1ϝsΔs,where(29)ϝϑ1=∫λ1ϑ1hγ−1ϑ1,σϑ2Δϑ2.Proof.
LetΘ:0,∞⟶ℝ be a superquadratic function and ϑ0∈0,∞. According to (27), there is a constant Cϑ0 such that(30)Θϑ2≥Θϑ0+Cϑ0ϑ2−ϑ0+Θϑ2−ϑ0.
Sinceϕ is rd-continuous,(31)ϑ0=Kλ1γϕϑ1ϝϑ1,is well defined. The function Θ∘ϕ is also rd-continuous, so we may apply (30) with ϑ2=ϕs and (31) to obtain(32)Θϕs≥ΘKλ1γϕϑ1ϝϑ1+Cϑ0ϕs−Kλ1γϕϑ1ϝϑ1+Θϕs−Kλ1γϕϑ1ϝϑ1.
Integrating fromλ1 to ϑ1, we get(33)∫λ1ϑ1hγ−1ϑ1,σsΘϕsΔs−∫λ1ϑ1hγ−1ϑ1,σsΘϕs−Kλ1γϕϑ1ϝϑ1Δs−∫λ1ϑ1hγ−1ϑ1,σsΘKλ1γϕϑ1ϝϑ1Δs≥Cϑ0∫λ1ϑ1hγ−1ϑ1,σsϕs−Kλ1γϕϑ1ϝϑ1Δs=Cϑ0Kλ1γϕϑ1−∫λ1ϑ1hγ−1ϑ1,σsKλ1γϕϑ1ϝϑ1Δs=Cϑ0Kλ1γϕϑ1−ϝϑ1⋅Kλ1γϕϑ1ϝϑ1=Cϑ0Kλ1γϕϑ1−Kλ1γϕϑ1=0.Remark 1.
SinceΘ is nonnegative superquadratic function, therefore (by Lemma 1) a convex one too, the result of (28) refines the result given in [30] (Theorem 1) and by assuming γ=1, (28) becomes [31] (Theorem 2.5), and result of [31] (Theorem 2.5) refines the result given in [31] (Theorem 2.4). Furthermore, respective inequality is obtained in fractional calculus ([29], Theorem 2.3) by assuming γ=1 and T=ℝ.Example 1.
IfT=ℕ, then σϑ2=ϑ2+1, ∀ϑ2∈ℕ, and(34)hγ−1ϑ1,σϑ2=ϑ1−σϑ2γ−1γ−1!=ϑ1−ϑ2−1γ−1γ−1!.
By substituting (28) and (34), it becomes(35)Θ∑ϑ2=λ1ϑ1−1ϑ1−ϑ2−1γ−1ϕϑ2∑ϑ2=λ1ϑ1−1ϑ1−ϑ2−1γ−1≤γ−1!∑ϑ2=λ1ϑ1−1ϑ1−ϑ2−1γ−1∑s=λ1ϑ1−1Θϕs−Θϕs−∑ϑ2=λ1ϑ1−1ϑ1−ϑ2−1γ−1ϕϑ2∑ϑ2=λ1ϑ1−1ϑ1−ϑ2−1γ−1.Example 2.
IfT=qℤ, then σϑ2=qϑ2. In this case,(36)hγ−1ϑ1,σϑ2=∏υ=0γ−2ϑ1−qυσϑ2∑μ=0υqμ=∏υ=0γ−2ϑ1−qυ+1ϑ2∑μ=0υqμ.
By substituting (36) in (28), one gets(37)Θ∑ϑ2=λ1ϑ1−1∏υ=0γ−2ϑ1−qυ+1ϑ2/∑μ=0υqμϕϑ2∑ϑ2=λ1ϑ1−1∏υ=0γ−2ϑ1−qυ+1ϑ2/∑μ=0υqμ≤1∏υ=0γ−2ϑ1−qυ+1ϑ2/∑μ=0υqμ∑s=λ1ϑ1−1Θϕs−Θϕs−∑ϑ2=λ1ϑ1−1∏υ=0γ−2ϑ1−qυ+1ϑ2/∑μ=0υqμϕϑ2∑ϑ2=λ1ϑ1−1∏υ=0γ−2ϑ1−qυ+1ϑ2/∑μ=0υqμ.


## 4. Hardy Inequalities for Superquadratic Functions with General Kernels

Theorem 2.
Assumehγ−1:T×T⟶ℝ where γ≥1, β≥1, and ϒ1=λ1,λ2∩T and ϒ2=η1,η2∩T where λ1,λ2,η1,η2∈T, and ϑ1∈Tκ,s1,s2∈T such that λ1<s1<λ2 and η1<s2<η2. Define(38)Aκϕϑ1=1χϑ1∫λ1s1hγ−1s1,σϑ2κϑ1,ϑ2ϕϑ2Δϑ2−∫λ2s1hγ−1s1,σϑ2κϑ1,ϑ2ϕϑ2Δϑ2,where ϕ:ϒ1⟶ℝ is a λΔ-measurable function, κ:ϒ2×ϒ1⟶ℝ is a nonnegative kernel, and(39)χϑ1=∫λ1ϑ1hγ−1ϑ1,σϑ2κϑ1,ϑ2Δϑ2−∫λ2ϑ1hγ−1ϑ1,σϑ2κϑ1,ϑ2Δϑ2<∞,ϑ1∈ϒ2.
Letζ:ϒ2⟶ℝ+ be μΔ-integrable function. Denote(40)wϑ2=∫η1η2κϑ1,ϑ2ζϑ1χϑ1Δϑ1,ϑ2∈ϒ1.
Further assumeI=a,c, 0≤a<c≤∞, and Θ:I⟶R. If Θ∈CI,R is superquadratic, then(41)∫η1η2hβ−1s2,σϑ1ΘAκϕϑ1ζϑ1Δϑ1+∫λ1λ2hγ−1s1,σϑ2hβ−1s2,σϑ1∫η1η2Θϕϑ2−Aκϕϑ1ζϑ1κϑ1,ϑ2χϑ1Δϑ1Δϑ2≤∫λ1λ2hγ−1s1,σϑ2hβ−1s2,σϑ1Θϕϑ1wϑ2Δϑ2,holds for all λΔ-integrable ϕ:ϒ1⟶ℝ such that ϕϒ1⊂I. If Θ is subquadratic, then inequality (41) is reversed.Proof.
Consider(42)∫η1η2hβ−1s2,σϑ1ΘAκϕϑ1ζϑ1Δϑ1=∫η1η2hβ−1s2,σϑ1ζϑ1×Θ1χϑ1∫λ1s1hγ−1s1,σϑ2κϑ1,ϑ2ϕϑ2Δϑ2−∫λ2s1hγ−1s1,σϑ2κϑ1,ϑ2ϕϑ2Δϑ2Δϑ1=∫η1η2hβ−1s2,σϑ1ζϑ1Θ1χϑ1∫λ1s1hγ−1s1,σϑ2κϑ1,ϑ2ϕϑ2Δϑ2Δϑ1−∫η1η2hβ−1s2,σϑ1ζϑ1Θ1χϑ1∫λ2s1hγ−1s1,σϑ2κϑ1,ϑ2ϕϑ2Δϑ2Δϑ1.
By using Jensen’s inequality (28),(43)≤∫η1η2hβ−1s2,σϑ1ζϑ1χϑ1∫λ1s1hγ−1s1,σϑ2κϑ1,ϑ2Θϕϑ2Δϑ2Δϑ1−∫η1η2hβ−1s2,σϑ1ζϑ1χϑ1∫λ1s1hγ−1s1,σϑ2κϑ1,ϑ2Θϕϑ2−Aκϕϑ1Δϑ2Δϑ1−∫η1η2hβ−1s2,σϑ1ζϑ1χϑ1∫λ2s1hγ−1s1,σϑ2κϑ1,ϑ2Θϕϑ2Δϑ2Δϑ1+∫η1η2hβ−1s2,σϑ1ζϑ1χϑ1∫λ2s1hγ−1s1,σϑ2κϑ1,ϑ2Θϕϑ2−Aκϕϑ1Δϑ2Δϑ1≤∫η1η2hβ−1s2,σϑ1ζϑ1χϑ1∫λ1s1hγ−1s1,σϑ2κϑ1,ϑ2Θϕϑ2Δϑ2−∫λ2s1hγ−1s1,σϑ2κϑ1,ϑ2Θϕϑ2Δϑ2Δϑ1−∫η1η2hβ−1s2,σϑ1ζϑ1χϑ1∫λ1s1hγ−1s1,σϑ2κϑ1,ϑ2Θϕϑ2−Aκϕϑ1Δϑ2−∫λ2s1hγ−1s1,σϑ2κϑ1,ϑ2Θϕϑ2−Aκϕϑ1Δϑ2Δϑ1≤∫η1η2hβ−1s2,σϑ1ζϑ1χϑ1∫λ1λ2hγ−1s1,σϑ2κϑ1,ϑ2Θϕϑ2Δϑ2Δϑ1−∫η1η2hβ−1s2,σϑ1ζϑ1χϑ1∫λ1λ2hγ−1s1,σϑ2κϑ1,ϑ2Θϕϑ2−Aκϕϑ1Δϑ2Δϑ1.
By using Fubini’s theorem,(44)=∫λ1s1hγ−1s1,σϑ2hβ−1s2,σϑ1Θϕϑ2∫η1η2κϑ1,ϑ2ζϑ1χϑ1Δϑ1Δϑ2−∫λ1s1hγ−1s1,σϑ2hβ−1s2,σϑ1∫η1η2Θϕϑ2−Aκϕϑ1κϑ1,ϑ2ζϑ1χϑ1Δϑ1Δϑ2−∫λ1s2hγ−1s1,σϑ2hβ−1s2,σϑ1Θϕϑ2∫η1η2κϑ1,ϑ2ζϑ1χϑ1Δϑ1Δϑ2+∫λ1s2hγ−1s1,σϑ2hβ−1s2,σϑ1∫η1η2Θϕϑ2−Aκϕϑ1κϑ1,ϑ2ζϑ1χϑ1Δϑ1Δϑ2=∫λ1λ2hγ−1s1,σϑ2hβ−1s2,σϑ1Θϕϑ2∫η1η2κϑ1,ϑ2ζϑ1χϑ1Δϑ1Δϑ2−∫λ1λ2hγ−1s1,σϑ2hβ−1s2,σϑ1∫η1η2Θϕϑ2−Aκϕϑ1κϑ1,ϑ2ζϑ1χϑ1Δϑ1Δϑ2=∫λ1λ2hγ−1s1,σϑ2hβ−1s2,σϑ1Θϕϑ2wϑ2Δϑ2−∫λ1λ2hγ−1s1,σϑ2hβ−1s2,σϑ1∫η1η2Θϕϑ2−Aκϕϑ1κϑ1,ϑ2ζϑ1χϑ1Δϑ1Δϑ2.Remark 2.
If the functionΘ is subquadratic, then Jensen’s inequality for superquadratic functions on delta fractional integrals is reversed, which implies, according to the conclusions made above, that inequality sign in (41) is reversed.
In [30], authors prove the analogous results for convex functions; now, it is easy to see that when Θ in (41) is nonnegative, then inequality (41) gives the refinement of [30] (Theorem 2). By assuming γ,β=1, we obtain the inequality (41) in time scales ([32], Theorem 2.1), when ϕ is nonnegative in [32] (Theorem 2.1), then [32] (Theorem 2.1) gives the refinement of [27] (Theorem 3.2). Moreover, the inequality (41) is obtained in fractional calculus [33] by assuming T=ℝ and γ,β=1.Corollary 1.
Assume (38)–(40). If p≥2, then(45)∫η1η2hβ−1s2,σϑ1Aκpϕϑ2ζϑ1Δϑ1+∫λ1λ2hγ−1s1,σϑ2hβ−1s2,σϑ1∫η1η2ϕϑ2−Aκϕϑ1pκϑ1,ϑ2ζϑ1χϑ1Δϑ1Δϑ2≤∫λ1λ2hγ−1s1,σϑ2hβ−1s2,σϑ1ϕpϑ2wϑ2Δϑ2.holds for all λΔ-integrable ϕ:ϒ1⟶ℝ such that ϕΥ1⊂I.
If0≤p≤2, then (45) holds in reverse direction.Proof.
UseΘr=rp in Theorem 2.Remark 3.
We obtain the respective results in time scales ([32], Corollary 2.3) by assuming γ,β=1 in (45).Remark 4.
In particular, ifp=2 in Corollary 4.2, we get the following identity:(46)∫η1η2hβ−1s2,σϑ1Aκ2ϕϑ2ζϑ1Δϑ1+∫λ1λ2hγ−1s1,σϑ2hβ−1s2,σϑ1∫η1η2ϕϑ2−Aκϕϑ12κϑ1,ϑ2ζϑ1χϑ1Δϑ1Δϑ2=∫λ1λ2hγ−1s1,σϑ2hβ−1s2,σϑ1ϕ2ϑ2wϑ2Δϑ2.Corollary 2.
Assume (38)–(40), then(47)∫η1η2hβ−1s2,σϑ1eAκhϑ1ζϑ1Δϑ1+I≤∫λ1λ2hγ−1s1,σϑ2hβ−1s2,σϑ1hϑ2wϑ2Δϑ2,holds for all λΔ-integrable h:ϒ1⟶ℝ such that ϕϒ1⊂I with(48)Aκhϑ1=1χϑ1∫λ1s1hγ−1s1,σϑ2κϑ1,ϑ2lnhϑ2Δϑ2−∫λ2s1hγ−1s1,σϑ2κϑ1,ϑ2lnhϑ2Δϑ2,I=∫λ1λ2hγ−1s1,σϑ2hβ−1s2,σϑ1×∫η1η2elnhϑ2−Aκhϑ1−lnhϑ2−Aκhϑ1κϑ1,ϑ2ζϑ1χϑ1Δϑ1Δϑ2+∫λ1λ2hγ−1s1,σϑ2hβ−1s2,σϑ1lnhϑ2wϑ2Δϑ2−∫η1η2hβ−1s2,σϑ11+Aκhϑ1ζϑ1Δϑ1.Proof.
UseΘr=er−r−1 and ϕr=lnhr in Theorem 2.Remark 5.
By choosingγ,β=1 in (47), the respective result is obtained in [32] (Corollary 2.5).Corollary 3.
Assume (38)–(40) and further if T=ℝ, then(49)∫η1η2s2−ϑ1β−1β−1!Θ∫λ1λ2s1−ϑ2γ−1κϑ1,ϑ2ϕϑ2Δϑ2∫λ1λ2ϑ1−ϑ2γ−1κϑ1,ϑ2dϑ2ζϑ1dϑ1+∫λ1λ2s1−ϑ2γ−1s2−ϑ1β−1β−1!×∫η1η2Θϕϑ2−∫λ1λ2s1−ϑ2γ−1κϑ1,ϑ2ϕϑ2Δϑ2∫λ1λ2ϑ1−ϑ2γ−1κϑ1,ϑ2dϑ2κϑ1,ϑ2ζϑ1∫λ1λ2ϑ1−ϑ2γ−1κϑ1,ϑ2dϑ2dϑ1dϑ2≤∫λ1λ2s1−ϑ2γ−1s2−ϑ1β−1β−1!Θϕϑ1∫η1η2κϑ1,ϑ2ζϑ1∫λ1λ2ϑ1−ϑ2γ−1κϑ1,ϑ2dϑ2dϑ1dϑ2.Proof.
IfT=ℝ, then σϑ2=ϑ2, ∀ϑ2∈ℝ. In this case,(50)hγ−1s1,σϑ2=s1−σϑ2γ−1γ−1!=s1−ϑ2γ−1γ−1!,(51)hβ−1s2,σϑ1=s2−σϑ1β−1β−1!=s2−ϑ1β−1β−1!.
Substituting (50) and (51) in (41), one gets(52)∫η1η2s2−ϑ1β−1β−1!Θ∫λ1λ2s1−ϑ2γ−1κϑ1,ϑ2ϕϑ2Δϑ2∫λ1λ2ϑ1−ϑ2γ−1κϑ1,ϑ2dϑ2ζϑ1dϑ1+∫λ1λ2s1−ϑ2γ−1s2−ϑ1β−1β−1!×∫η1η2Θϕϑ2−∫λ1λ2s1−ϑ2γ−1κϑ1,ϑ2ϕϑ2Δϑ2∫λ1λ2ϑ1−ϑ2γ−1κϑ1,ϑ2dϑ2κϑ1,ϑ2ζϑ1∫λ1λ2ϑ1−ϑ2γ−1κϑ1,ϑ2dϑ2dϑ1dϑ2≤∫λ1λ2s1−ϑ2γ−1s2−ϑ1β−1β−1!Θϕϑ1∫η1η2κϑ1,ϑ2ζϑ1∫λ1λ2ϑ1−ϑ2γ−1κϑ1,ϑ2dϑ2dϑ1dϑ2.


## 5. Inequalities with Special Kernels

Corollary 4.
Assume (38)–(40) with kernel κ such that(53)κϑ1,ϑ2=1,ifλ1≤ϑ2<σϑ1≤λ2,0,otherwise,then(54)∫λ1λ2hβ−1s2,σϑ1Kλ1γϕσϑ1ϝσϑ1pζϑ1Δϑ1+∫λ1λ2hγ−1s1,σϑ2hβ−1s2,σϑ1∫ϑ2λ2ϕϑ2−Kλ1γϕσϑ1ϝσϑ1pζϑ1ϝσϑ1Δϑ1Δϑ2≤∫λ1λ2hγ−1s1,σϑ2hβ−1s2,σϑ1ϕpϑ2∫ϑ2λ2ζϑ1ϝσϑ1Δϑ1Δϑ2,holds for all λΔ-integrable h:ϒ1⟶ℝ such that ϕϒ1⊂I.Proof.
By using (53) in (38)–(40), respectively,(55)χϑ1=∫λ1σϑ1hγ−1σϑ1,σϑ2Δϑ2=ϝσϑ1,(56)wϑ2=∫ϑ2λ2ζϑ1ϝσϑ1Δϑ1,(57)Aκϕϑ1=∫λ1σϑ1hγ−1σϑ1,σϑ2ϕϑ2Δϑ2∫λ1σϑ1hγ−1σϑ1,σϑ2Δϑ2=Kλ1γϕσϑ1ϝσϑ1.
We useϒ1=ϒ2=λ1,λ2 and (55)–(57) in Corollary 4.2 to get(58)∫λ1λ2hβ−1s2,σϑ1Kλ1γϕσϑ1ϝσϑ1pζϑ1Δϑ1+∫λ1λ2hγ−1s1,σϑ2hβ−1s2,σϑ1∫ϑ2λ2ϕϑ2−Kλ1γϕσϑ1ϝσϑ1pζϑ1ϝσϑ1Δϑ1Δϑ2≤∫λ1λ2hγ−1s1,σϑ2hβ−1s2,σϑ1ϕpϑ2∫ϑ2λ2ζϑ1ϝσϑ1Δϑ1Δϑ2.Corollary 5.
Assumeϒ1=ϒ2=λ1,λ2 with λ2=∞ and (55)–(57) hold, then(59)∫λ1∞hβ−1s2,σϑ1Kλ1γϕσϑ1ϝσϑ1pζϑ1Δϑ1+∫λ1∞hγ−1s1,σϑ2hβ−1s2,σϑ1∫ϑ2∞ϕϑ2−Kλ1γϕσϑ1ϝσϑ1pζϑ1ϝσϑ1Δϑ1Δϑ2≤∫λ1∞hγ−1s1,σϑ2hβ−1s2,σϑ1ϕpϑ2∫ϑ2∞ζϑ1ϝσϑ1Δϑ1Δϑ2,holds for all λΔ-integrable h:ϒ1⟶ℝ such that ϕϒ1⊂I.Proof.
The statement follows from Corollary4 by using λ2=∞.Remark 6.
By assumingγ,β=1 in Corollaries 4 and 5, respectively, we obtain the results given in [32] (Corollaries 3.2 and 3.4) (for n=1).


## 6. Pólya–Knopp Type Inequalities

Example 3.
ForT=hℤ, σϑ2=ϑ2+h, ∀ϑ2∈ℤ. In this case,(60)hγ−1s1,σϑ2=s1−σϑ2γ−1γ−1!=s1−ϑ2−hγ−1γ−1!,(61)hβ−1s2,σϑ2=s2−σϑ1β−1β−1!=s2−ϑ1−hβ−1β−1!.
By using (60) and (61) in (41), one gets(62)∑ϑ1=η1η2−1s2−ϑ1−hβ−1β−1!Θ∑ϑ2=λ1λ2−1s1−ϑ2−hγ−1κϑ1,ϑ2ϕϑ2∑ϑ2=λ1ϑ1−1ϑ1−ϑ2−hγ−1κϑ1,ϑ2−∑ϑ2=λ2ϑ1−1ϑ1−ϑ2−hγ−1κϑ1,ϑ2ζϑ1+∑ϑ2=λ1λ2−1s1−ϑ2−hγ−1s2−ϑ1−hβ−1β−1!×∑ϑ1=η1η2−1Θϕϑ2−∑ϑ2=λ1λ2−1s1−ϑ2−hγ−1κϑ1,ϑ2ϕϑ2∑ϑ2=λ1ϑ1−1ϑ1−ϑ2−hγ−1κϑ1,ϑ2−∑ϑ2=λ2ϑ1−1ϑ1−ϑ2−hγ−1κϑ1,ϑ2×κϑ1,ϑ2ζϑ1∑ϑ2=λ1ϑ1−1ϑ1−ϑ2−hγ−1κϑ1,ϑ2−∑ϑ2=λ2ϑ1−1ϑ1−ϑ2−hγ−1κϑ1,ϑ2≤∑ϑ2=λ1λ2−1s1−ϑ2−hγ−1s2−ϑ1−hβ−1β−1!Θϕϑ1×∑ϑ1=η1η2−1κϑ1,ϑ2ζϑ1∑ϑ2=λ1ϑ1−1ϑ1−ϑ2−hγ−1κϑ1,ϑ2−∑ϑ2=λ2ϑ1−1ϑ1−ϑ2−hγ−1κϑ1,ϑ2.Corollary 6.
Suppose time scaleT consists of isolated points, ϒ1=ϒ2=λ1,∞∩T, λ1≥0, and ζ:ϒ2⟶ℝ+ is a μΔ-integrable function. For p≥2 inequality,(63)∑ϑ1∈λ1,∞Ts2−ϑ1−1β−1ζϑ1β−1!∑ϑ2∈λ1,σϑ1Tϑ1−ϑ2γ−1ϕϑ2μϑ2∑ϑ2∈λ1,σϑ1Tϑ1−ϑ2γ−1μϑ2pμϑ1+∑ϑ2∈λ1,∞Ts1−ϑ2−1γ−1s2−ϑ1−1β−1β−1!×∑ϑ1∈ϑ2,∞Tϕϑ2−∑ϑ2∈λ1,σϑ1Tϑ1−ϑ2γ−1ϕϑ2μϑ2∑ϑ2∈λ1,σϑ1Tϑ1−ϑ2γ−1μϑ2pζϑ1∑ϑ2∈λ1,σϑ1Tϑ1−ϑ2γ−1μϑ2μϑ1μϑ2≤∑ϑ2∈λ1,∞Ts1−ϑ2−1γ−1s2−ϑ1−1β−1ϕpϑ2β−1!∑ϑ1∈ϑ2,∞Tζϑ1∑ϑ2∈λ1,σϑ1Tϑ1−ϑ2γ−1μϑ2μϑ1μϑ2,holds for all λΔ-integrable ϕ:ϒ1⟶ℝ such that ϕϒ1⊂I, where μr=σr−r.Proof.
IfT consists of isolated points, then(64)hγ−1s1,σϑ2=s1−σϑ2γ−1γ−1!=s1−ϑ2−1γ−1γ−1!,(65)hγ−1σϑ1,σϑ2=σϑ1−σϑ2γ−1γ−1!=ϑ1−ϑ2γ−1γ−1!,(66)hβ−1s2,σϑ1=s2−σϑ1β−1β−1!=s2−ϑ1−1β−1β−1!.
By using (64)–(66) in Corollary 5, one gets(67)∑ϑ1∈λ1,∞Ts2−ϑ1−1β−1ζϑ1β−1!∑ϑ2∈λ1,σϑ1Tϑ1−ϑ2γ−1ϕϑ2μϑ2∑ϑ2∈λ1,σϑ1Tϑ1−ϑ2γ−1μϑ2μϑ1+∑ϑ2∈λ1,∞Ts1−ϑ2−1γ−1s2−ϑ1−1β−1β−1!×∑ϑ1∈ϑ2,∞Tϕϑ2−∑ϑ2∈λ1,σϑ1Tϑ1−ϑ2γ−1ϕϑ2μϑ2∑ϑ2∈λ1,σϑ1Tϑ1−ϑ2γ−1μϑ2pζϑ1∑ϑ2∈λ1,σϑ1Tϑ1−ϑ2γ−1μϑ2μϑ1μϑ2≤∑ϑ2∈λ1,∞Ts1−ϑ2−1γ−1s2−ϑ1−1β−1ϕpϑ2β−1!∑ϑ1∈ϑ2,∞Tζϑ1∑ϑ2∈λ1,σϑ1Tϑ1−ϑ2γ−1μϑ2μϑ1μϑ2.Remark 7.
By choosingγ,β=1 in (63), the respective Polya–Knopp type inequality is obtained in time scales ([32], Corollary 4.1).Example 4.
If we considerT=hℕ=hι:ι∈ℕ with fixed h>0, λ1=h and taking ζϑ1=1/σϑ1. (63) takes the form,(68)∑ι=1∞J−ι−1β−1β−1!ι+1∑κ=1ιι−κγ−1ϕκh∑κ=1ιι−κγ−1p+∑κ=1∞ıh−κh−hγ−1Jh−ιh−hβ−1β−1!×∑ι=1∞ϕϑ2−∑κ=1ιι−κγ−1ϕϑ2∑κ=1ιι−κγ−1p1ι+1∑κ=1ιιh−κhγ−1≤∑κ=1∞ıh−κh−hγ−1J−ι−1β−1ϕκhβ−1!∑ι=κ∞1ι+1∑κ=1ιιh−κhγ−1.Remark 8.
Notice that (68) becomes [32] Example 4.3 by choosing γ,β=1.Example 5.
If we consider time scaleT=ℕ2=ι2:ι∈ℕ, λ1=1 and(69)ζϑ1=2σϑ1−1σϑ1−ϑ12ϑ1+3=2ι+12−1ι+12−ι22ι+3=2ιι+22ι+12ι+3.
(63) takes the form,(70)∑ι=1∞J2−ι+12β−12ιι+2β−1!2ι+3∑κ=1ιι+12−κ+12γ−1ϕκ22κ+1∑κ=1ιι+12−κ+12γ−12κ+1p+∑κ=1∞ı2−κ+12γ−1J2−ι+12β−1β−1!×∑ι=κ∞ϕκ2−∑κ=1ιι+12−κ+12γ−1ϕκ22κ+1∑κ=1ιι+12−κ+12γ−12κ+1p×2ιι+22ι+3∑κ=1ιι+12−κ+12γ−12κ+12κ+1≤∑κ=1∞ı2−kh−hγ−1J2−nh−hβ−1ϕκ2β−1!×∑ι=κ∞2ιι+22ι+3∑κ=1ιι+12−κ+12γ−12κ+12κ+1.Example 6.
IfT=qℕ=qn:n∈ℕ with fixed q>1, then σϑ=qϑ=qn+1, and assume(71)ζϑ=2σϑ−1σϑ−ϑ2ϑ+3=2qn+1−1qnq−12qn+3.
In this case,(72)hγ−1s1,σϑ2=∏υ=0γ−2s1−qυσϑ2∑μ=0υqμ=∏υ=0γ−2s1−qυ+k+1∑μ=0υqμ,(73)hβ−1s2,σϑ1=∏υ=0β−2s2−qυσϑ1∑μ=0υqμ=∏υ=0β−2s2−qυ+n+1∑μ=0υqμ.
By using (63), (71)–(73) take the form,(74)∑n=1∞∏υ=0β−2s2−qυ+n+1∑μ=0υqμ2qn+1−1qn2qn+3∑k=1n∏υ=0γ−2s1−qυ+k+1/∑μ=0υqμϕqk∑k=1n∏υ=0γ−2qn−qυ+k+1/∑μ=0υqμϕqkp+∑k=1∞∏υ=0γ−2s1−qυ+k+1∑μ=0υqμ⋅∏υ=0β−2s2−qυ+n+1∑μ=0υqμ∑n=k∞ϕqk−∑k=1n∏υ=0γ−2s1−qυ+k+1/∑μ=0υqμϕqk∑k=1n∏υ=0γ−2qn−qυ+k+1/∑μ=0υqμp2qn+1−1/qn2qn+3∑k=1n∏υ=0γ−2qn−qυ+k+1/∑μ=0υqμ≤∑k=1∞∏υ=0γ−2s1−qυ+k+1∑μ=0υqμ⋅∏υ=0β−2s2−qυ+n+1∑μ=0υqμϕqkp∑n=k∞2qn+1−1/qn2qn+3∑k=1n∏υ=0γ−2qn−qυ+k+1/∑μ=0υqμ.Remark 9.
By choosingγ,β=1 in (70), the respective Polya–Knopp type inequality is obtained in time scales ([32], Example 4.4).


## 7. Conclusion

In the paper, some dynamic inequalities of Jensen and Hardy for superquadratic functions with Riemann–Liouville delta fractional integrals are generalized. Particular cases of obtained inequalities have been given in time scales [32] by assuming γ,β=1 and respective inequalities in fractional calculus [33] by choosing T=ℝ are also obtained. Furthermore, some discrete fractional inequalities for superquadratic functions are obtained by choosing T=hℤ and T=ℕ2. Inequalities in quantum discrete calculus are also obtained by choosing T=qℤ, which are new, up to the knowledge of authors.



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