# On Refinements of Multidimensional Inequalities of Hardy-Type via Superquadratic and Subquadratic Functions

Authors: M. Zakarya; Ghada AlNemer; H. A. Abd El-Hamid; Roqia Butush; H. M. Rezk
Journal: Journal of Mathematics
 (2022)

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## Abstract

By utilizing the peculiarities of superquadratic and subquadratic functions, we give the extensions for multidimensional inequalities of Hardy-type with general kernel. We use some algebraic inequalities such as the Minkowski inequality, the refined Jensen inequality, and the Bernoulli inequality to prove the essential results in this paper. The performance of the superquadratic functions is reliable and effective to obtain new dynamic inequalities on time scales. By utilizing special kernels, we also acquire numerous examples and implementations of the related inequalities.


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## Body

## 1. Introduction

In [1], Hardy proved that if λ>1, f≥0 over interval 0,∞ and ∫0∞fλsds<∞; then,(1)∫0∞1s∫0sftdtλds≤λλ−1λ∫0∞fλsds,where the constant λ/λ−1λ is sharp. By rewriting (1) with f1/λ rather than f and taking the limit as λ⟶∞, we acquire the limiting case of Hardy’s inequality known as the inequality of Pólya-Knopp ([2]), that is,(2)∫0∞exp1s∫0slnftdtds≤e∫0∞fsds.In [3], Kaijser et al. specified that both (1) and (2) are special cases of Hardy-Knopp’s inequality.(3)∫0∞Φ1s∫0sftdtdss≤∫0∞Φfsdss,where Φ∈CI,ℝ, I⊆ℝ is a convex function and f:ℝ+⟶ℝ+ is a locally integrable positive function.In [4], Kaijser et al. applied Fubini’s theorem and Jensen’s inequality to establish an invitingly popularization of (1). Particularly, it was evidenced that if ξ:0,β⟶ℝ≥0 and l:0,β×0,β⟶ℝ≥0, 0<β≤∞ such that(4)Ls=∫0sls,tdt>0,s∈0,β,and Φ∈CI,ℝ, I⊆ℝ is a convex function and υ is defined by(5)υt=t∫tβξsls,tLsdss<∞,t∈0,β.Then, the inequality,(6)∫0∞ξsΦAlfsdss≤∫0∞υsΦfsdss,holds for any non-negative integrable function f:0,β⟶ℝ such that fs⊆I where Alf is defined by(7)Alfs=1Ls∫0sls,tftdt,s∈0,β.As a popularization of (6), Krulic et al. [5] have demonstrated that if Ξ,Σ,μ and Ξ˜,Σ˜,μ˜ are measure spaces with positive σ-finite measures, ξ:Ξ⟶ℝ≥0 and l:Ξ×Ξ˜⟶ℝ≥0 are measurable functions such that ls,⋅ is a dμ˜-integrable function for s∈Ξ˜, L:Ξ⟶ℝ is defined as(8)Ls=∫Ξ˜ls,tdμ˜t<∞,s∈Ξ,and υ is defined by(9)υt=∫Ξξsls,tLsq/pdμsq/p<∞,t∈Ξ˜,where 0<p≤q<∞. If Φ is a convex function on I⊆ℝ, then the inequality,(10)∫ΞξsΦq/pAlfsdμs1/q≤∫Ξ˜υsΦfsdμ˜s1/p,holds for any non-negative dμ˜-integrable function f:Ξ˜⟶ℝ such that fΞ˜⊆I where Alf:Ξ⟶ℝ is defined by(11)Alfs=1Ls∫Ξ˜ls,tftdμ˜t,s∈Ξ.In [6], the authors showed that if r≥1, ξ:Ξ⟶ℝ≥0, l:Ξ×Ξ˜⟶ℝ≥0 are measurable functions such that ls,⋅ is a dμ˜-integrable function for s∈Ξ˜, L:Ξ⟶ℝ is defined by (8) and the function υ is defined as(12)υt=∫Ξξsls,tLsrdμs1/r<∞,t∈Ξ˜.Moreover, ifΦ is a non-negative superquadratic function, then(13)∫ΞξsΦrAlfsdμs+r∫Ξ∫Ξ˜ξsls,tLsΦr−1Alfs×Φft−Alfsdμsdμ˜t≤∫Ξ˜υtΦftdμ˜tr,holds for any non-negative dμ˜-integrable function f:Ξ˜⟶ℝ where Alf:Ξ⟶ℝ is defined by (11).In [7], the researchers demonstrated some Hardy-type inequalities with a general kernel. They have determined that if Ξ,Σ,μ and Ξ˜,Σ˜,μ˜ are two time scale measure spaces with positive σ-finite measures, l:Ξ×Ξ˜⟶ℝ is such that(14)Ls=∫Ξ˜ls,tΔμ˜t<∞,s∈Ξ,Φ is a convex function and ξ:Ξ⟶ℝ+≥0 such that(15)υt=∫Ξξsls,tlsΔμs<∞,t∈Ξ˜,then(16)∫ΞξtΦ1Ls∫Ξ˜ls,tftΔμ˜tΔμs≤∫Ξ˜υtΦftΔμ˜t,holds for all non-negative Δμ˜-integrable f:Ξ˜⟶ℝ such that fΞ˜⊂I.In [8], the authors improved the inequality (16) by replacing the function ft by an m-tuple of functions.(17)ft=f1t,f2t,…,fmt,such as f1t,f2t,…,fmt are Δμ˜-integrable on Ξ˜ as follows: let ξ:Ξ⟶ℝ, l:Ξ×Ξ˜⟶ℝ are non-negative such that ls,⋅ is a Δμ˜-integrable function for s∈Ξ˜, L:Ξ⟶ℝ and the function υ are defined by (14), (15), respectively. Then, for a convex function Φ over a convex set I⊂ℝm, the integral inequality,(18)∫ΞξtΦ1Ls∫Ξ˜ls,tftΔμ˜tΔμs≤∫Ξ˜υtΦftΔμ˜t,holds for all Δμ˜-integrable functions f:Ξ˜⟶ℝm such that fΞ˜⊂U⊂ℝm.In [9], the authors derived some inequalities of Hardy-type by utilizing the concept of superquadratic functions. Particularly, they proved that if ξ:Ξ⟶ℝ≥0 and l:Ξ×Ξ˜⟶ℝ≥0 are measurable functions such that ls,⋅ is a Δμ˜-integrable function for s∈Ξ˜, L:Ξ⟶ℝ and the function υ are defined by (14) and (15), respectively. Let Φ be a non-negative superquadratic function. Then, the inequality,(19)∫ΞξsΦAlfsΔμs+∫Ξ∫Ξ˜ξsls,tLsΦft−AlfsΔμsΔμ˜t≤∫Ξ˜υtΦftΔμ˜t,holds for all non-negative Δμ˜-integrable function f:Ξ˜⟶ℝ, where Alf:Ξ⟶ℝ is defined by(20)Alfs=1Ls∫Ξ˜ls,tftΔμ˜t,s∈Ξ.Moreover, in [10, 11], the authors generalized (19) and proved that if ξ:Ξ⟶ℝ≥0 and l:Ξ×Ξ˜⟶ℝ≥0 are measurable functions such that ls,⋅ is a Δμ˜-integrable function for s∈Ξ˜, L:Ξ⟶ℝ and the function υ are defined by (14) and (15), respectively. Let Φ be a non-negative superquadratic function. Then, the inequality(21)∫ΞξsΦAlfsΔμs+∫Ξ∫Ξ˜ξsls,tLsΦft−AlfsΔμsΔμ˜t≤∫Ξ˜vtΦftΔμ˜t,holds for all Δμ˜-integrable functions f:Ξ˜⟶ℝm such that fΞ˜⊂U⊂ℝm, where Alf:Ξ⟶ℝ is defined by(22)Alfs=1Ls∫Ξ˜ls,tftΔμ˜t,s∈Ξ.In [12], the authors deduced several generalizations of (19) on time scales. They proved that if r≥1, ξ:Ξ⟶ℝ≥0 and l:Ξ×Ξ˜⟶ℝ≥0 are measurable functions such that ls,⋅ is a Δμ˜-integrable function for s∈Ξ˜, L:Ξ⟶ℝ is defined by (14) and the function υ be defined by(23)υt=∫Ξξsls,tLsrΔμs1/r<∞,t∈Ξ˜.LetΦ be a non-negative superquadratic function. Then, the inequality,(24)∫ΞξsΦrAlfsΔμs+r∫Ξ∫Ξ˜ξsls,tLsΦr−1AlfsΦft−AlfsΔμsΔμ˜t≤∫Ξ˜υtΦftΔμ˜tr,holds for all non-negative Δμ˜-integrable function f:Ξ˜⟶ℝ where Alf:Ξ⟶ℝ is defined by (20).Another development of Hardy-type inequality (24) has been made by Saker et al. [13] as follows: let ξ:Ξ⟶ℝ and l:Ξ×Ξ˜⟶ℝ be non-negative functions such that ls,⋅ is a Δμ˜-integrable function for s∈Ξ˜, L:Ξ⟶ℝ is defined by(25)0<Ls=∫Ξ˜ls,tΔμ˜t1…Δμ˜tn<∞,s=s1,s2,…,sn∈Ξ,and υ is defined by(26)υt=∫Ξξsls,tLsrΔμs1…Δμsn1/r<∞,t=t1,t2,…,tm∈Ξ˜,where r≥1. If Φ is a non-negative superquadratic function, then the inequality,(27)∫ΞξsΦrAlfsΔμs1…Δμsn+r∫Ξ∫Ξ˜ξsls,tLsΦr−1AlfsΦft−Alfs×Δμs1…ΔμsnΔμ˜t1…Δμ˜tn≤∫Ξ˜υtΦftΔμ˜t1…Δμ˜tnr,holds for all non-negative Δμ˜-integrable function f:Ξ˜⟶ℝ where Alf:Ξ⟶ℝ is defined by(28)Alfs=1Ls∫Ξ˜ls,tftΔμ˜t1…Δμ˜tn,s∈Ξ.In order to develop dynamic time scale inequalities, we moved the reader to the articles [14–22].Motivated by the previous results, our major aim in this paper is to deduce several generalizations of general Hardy-type inequalities for multivariate superquadratic functions that involve more general kernels on arbitrary time scales.The paper is governed as follows: We remember some basic notions, definitions, and results of multivariate superquadratic functions on time scales in Section2. In Section 3, we prove some new refined dynamic inequalities of Hardy’s type with non-negative kernel by utilizing the peculiarities of superquadratic (or subquadratic) functions. In Section 4, we discuss several particular cases of Hardy-type inequality by choosing such special kernels. Eventually, in Section 5, we give more implementations of our obtained results on particular time scales.


## 2. Preliminaries

In this section, we will introduce some fundamental concepts and effects to integrals of time scales and for multivariate superquadratic functions which will be useful to deduce our major results.Before introducing the main results for multidimensional inequalities, it is necessary to present some further definitions. Firstly, supposem,n∈ℤ+, ℝm be the Euclidean space. Let(29)s=s1,s2,…,sm∈ℝm,t=t1,t2,…,tm∈ℝm,ft=f1t,f2t,…,fmt,be the function that is defined on t∈ℝm. We utilize the following notations:(30)s.t=s1t1,s2t2,…,smtm,st=s1t1,s2t2,…,smtm,st=s1t1s2t2…smtm,t=t1,t2,…,tm,s,t=∑i=1msiti,also, for s,t∈ℝm, we write s≤ts<t if component wise si≤tisi<ti, ∀1≤i≤m, the relations ≥, >rbin are defined analogously, 0=0,0,…,0∈ℝm is the null vector and 1=1,1,…,1∈ℝm. Correspondingly, for s,t∈ℝm, s<t, we define a,b=t∈ℝm:a<t<b and the n-cells a,b, a,b and a,b are defined similarly. Furthermore, the subsets Km and Km+ in ℝm are defined by(31)Km=0,∞m=t∈ℝm:0≤t,Km+=0,∞m=t∈ℝm:0<t.In particular,(32)t1=∏i=1mti,t2=∏i=1mti2andt−1=∏i=1mti−1,where m=m1,m2,…,mm. Correspondingly, a,b means the set a1,b1×a2,b2×⋯×am,bm, Δt=Δt1…Δtm and tp=t1…tmp.Now, we arraign the definition and some essential properties of superquadratic functions that are premised in [23].Definition 1.
(see [23]). A function Φ:Km⟶ℝ is called superquadratic if for all s∈Km, there exists a function cs∈ℝm such that(33)Φt−Φs−Φt−s≥cs,t−s,∀t∈Km.
If−Φ is superquadratic, then Φ is subquadratic and the reverse inequality of (22) is held.Definition 2.
(see [24]). A function Φ:0,∞⟶ℝ is superadditive provided Φx+y≥Φx+Φy for all x,y≥0. If the reverse inequality holds, then Φ is said to be subadditive.Lemma 1 (see [25]).
SupposeΦ:0,∞⟶ℝ is continuously differentiable and Φ0≤0. If Φ′ is superadditive or Φ′x/x is nondecreasing, then Φ is superquadratic.In the next, we recall a couple of beneficial examples of a superquadratic function.Example 1.
(see [23]). The power function Φ:Km⟶ℝ that is defined as Φt=tp is called superquadratic if p≥2 and subquadratic if 1<p≤2 (it is also readily seen that if 0<p≤1 then tp is a subquadratic function). Since the sum of superquadratic functions is also superquadratic, the function Φ:Km⟶ℝ is defined as follows:(34)Φt=∑i=1mtip,which is superquadratic on Km for each p≥2 (as shown in [23], Example 2) and the function Φ:Km⟶ℝ is defined by(35)Φt=−∑i=1mtip1/p,which is superquadratic on Km for each p≥1 (see [23], Example 3).Example 2 (see [23]).
Examples 4, 5, and 6 By utilizing the same argument as in Example 4, the functionsΦ1,Φ2,Φ3:Km⟶ℝ that define(36)Φ1t=∑i=1mticoshti−sinhti,Φ2t=ln1+∑i=1mti−∑i=1mti,Φ3t=∑i=1,i≠jmti2lnti,ifti>0,tj=0,0,ift=0,are superquadratic.
The following lemma shows that non-negative superquadratic functions are indeed convex functions:Lemma 2 (see [23]).
LetΦ:Km⟶ℝ be a superquadratic with ct=c1t,c2t,…,cmt that is defined as in Definition 1. Then,(i)
Φ0≤0 and ci0≤0∀1≤i≤m;(ii)
IfΦ0=0 and ∇Φ0=∂1Φ0,∂2Φ0,…,∂mΦ0=0, then cit=∂iΦt whenever ∂iΦt exists for some index 1≤i≤m at t∈Km; where ∇Φ means that the gradient of the function Φ and ∂iΦt denotes the partial derivative of Φ over the i-th variable;(iii)
IfΦ≥0, then Φ is convex and Φ0=∇Φ0=0.
The following definitions and theorems are referred from [26, 27]. Let Ti, 1≤i≤n be time scales, and(37)Λn=T1×T2×⋯×Tn=t=t1,t2,…,tn:ti∈Ti,1≤i≤n,which is called an n-dimensional time scale. Let E be Δ-measurable subset of Λn and f:E⟶ℝ be a Δ-measurable function. Then, the corresponding Δ-integral is called Lebesgue Δ-integral and is denoted by(38)∫Eft1,t2,…,tnΔ1t1…Δntn,∫EftΔ1t1…Δntn,∫EfdμΔor∫EftdμΔt,where μΔ is a σ-additive Lebesgue Δ-measure on Λn. Also, if(39)ft=f1t,f2t,…,fmt,which is an m-tuple of functions in n-variables such that f1,f2,…,fm are Lebesgue Δ-integrable on E, then ∫EfdμΔ denotes the m-tuple.(40)∫EfdμΔ1,…,∫EfmdμΔ,i.e., Δ-integral acts on each component of f.
Particularly, ifT is an arbitrary time scale and s,t⊂T includes only isolated points, then(41)∫stfθΔθ=∑θ∈s,tσθ−θfθ=∑θ∈s,tfθμθ,where μ is referred to as the graininess function on time scale.Lemma 3 (Minkowski’s inequality [13]).
LetΞ,Σ,μ and Ξ˜,Σ˜,μ˜ be two finite-dimensional time scale measures spaces. Suppose that ξ≥0, v≥0 and f≥0 that are defined on Ξ,Ξ˜ and Ξ×Ξ˜, respectively. If λ≥1, then(42)∫Ξ∫Ξ˜fs,tυtdμ˜tλξsdμs1/λ≤∫Ξ˜∫Ξfλs,tξsdμsυtdμ˜t,provided that all integrals in (23) exist. If 0<λ<1,(43)∫Ξ∫Ξ˜fυdμ˜λξdμ>0and∫Ξ˜fυdμ˜>0,then (42) is reversed. For λ<0, if in addition to (43) and(44)∫Ξfλξdμ>0,then (42) is again reversed.
In [25], (Corollary 5.1), Bibi get the following generalization of Jensen’s inequality and the converse of it for superquadratic functions.Theorem 1 (Jensen’s inequality).
LetΦ∈CKm,ℝ be a superquadratic function and w be Δ-integrable function for w≥0 and ∫EwdμΔ. Then, for every Δ-integrable functions f such that fE⊂Km and wf, wΦf are Δ-integrable, we have(45)Φ∫EwfdμΔ∫EwdμΔ≤∫EwΦfdμΔ−∫EwΦf−∫EwfdμΔ/∫EwdμΔ⋅1dμΔ∫EwdμΔ.IfΦ is subquadratic, then (45) is reversed.


## 3. Main Results

In this section, we prove multidimensional Hardy-type inequalities with general kernels on time scales. Before proceeding with results, we introduce the following notations:B1Ξ,Σ,μ and Ξ˜,Σ˜,μ˜ are two time scale delta measure spaces with positive σ-finite measures.B2l:Ξ×Ξ˜⟶ℝ be a non-negative so that ls,⋅ is Δμ˜-integrable function for s∈Ξ˜ and L:Ξ⟶ℝ be defined by(46)0<Ls=∫Ξ˜ls,tΔμ˜t1…Δμ˜tn<∞,s∈Ξ.B3ξ:Ξ⟶ℝ is Δμ-integrable and the function υ be defined by(47)υt=∫Ξξsls,tLsrΔμs1…Δμsn1/r<∞,t∈Ξ˜,wherer≥1.Theorem 2.
WeassumeB1−B3 are satisfied. If Φ∈CKm,ℝ is non-negative superquadratic function, then(48)∫ΞξsΦrAlfsΔμs1…Δμsn+r∫Ξ˜∫Ξξsls,tLsΦr−1AlfsΦft−Alfs×Δμs1…ΔμsnΔμ˜t1…Δμ˜tn≤∫Ξ˜υtΦftΔμ˜t1…Δμ˜tnr,holds for all Δμ˜-integrable functions f:Ξ˜⟶ℝm such that fΞ˜⊂Km⊂ℝm, where Alf:Ξ⟶ℝ is defined by(49)Alfs=1Ls∫Ξ˜ls,tftΔμ˜t1…Δμ˜tn,s∈Ξ.
If0<r<1 and Φ is subquadratic, then (48) is reversed.Proof.
We begin with the following identity:(50)ΦAlfs=Φ1Ls∫Ξ˜ls,tftΔμ˜t1…Δμ˜tn.
By applying (45) on (50), we find(51)ΦAlfs+1Ls∫Ξ˜ls,tΦft−AlfsΔμ˜t1…Δμ˜tn≤1Ls∫Ξ˜ls,tΦftΔμ˜t1…Δμ˜tn.
SinceΦ≥0 and r≥1, we have(52)ΦAlfs+1Ls∫Ξ˜ls,tΦft−AlfsΔμ˜t1…Δμ˜tnr≤1Ls∫Ξ˜ls,tΦftΔμ˜t1…Δμ˜tnr.
Furthermore, by employing the Bernoulli inequality [28],(53)1+xγ≤1+γx,for0<γ≤1andx>−1,γx+1−γ≤xγ,forγ≥1andx>0.
It follows that the L. H. S of (52) is not less than(54)ΦrAlfs+rΦr−1AlfsLs∫Ξ˜ls,tΦft−AlfsΔμ˜t1…Δμ˜tn,that is,(55)ΦrAlfs+rΦr−1AlfsLs∫Ξ˜ls,tΦft−AlfsΔμ˜t1…Δμ˜tn≤1Ls∫Ξ˜ls,tΦftΔμ˜t1…Δμ˜tnr.
By multiplying (55) with ξs and integrating it over Ξ with respect to Δμs1…Δμsn, we get(56)∫ΞξsΦrAlfsΔμs1…Δμsn+r∫ΞξsΦr−1AlfsLs∫Ξ˜ls,tΦft−AlfsΔμ˜t1⋯μ˜tn×Δμs1…Δμsn≤∫Ξξs1Ls∫Ξ˜ls,tΦftΔμ˜t1…Δμ˜tnrΔμs1…Δμsn.
Applying (42) on the R. H. S of (56), we get(57)∫Ξξs1Ls∫Ξ˜ls,tΦftΔμ˜t1…Δμ˜tnrΔμs1…Δμsn≤∫Ξ˜Φft∫Ξξsls,tLsrΔμs1…Δμsn1/rΔμ˜t1…Δμ˜tnr.
Finally, substituting (57) into (56) and using the definition (47) of the weight function υ, we have(58)∫ΞξsΦrAlfsΔμ1s1…Δμ1sn+r∫Ξ˜∫Ξξsls,tlsΦr−1AlfsΦft−Alfs×Δμs1…ΔμsnΔμ˜t1…Δμ˜tn≤∫Ξ˜υtΦftΔμ˜t1…Δμ˜tnr.
This proves (48). The proof of the case in which 0<r<1 and Φ is subquadratic is similar; the only difference is that the inequality sign in (48) is reversed. The proof is complete.Remark 1.
Form=1, inequality (48) in Theorem 2 reduces to (27).Remark 2.
Forn=1 and r=1, inequality (48) in Theorem 2 coincides with (21).Corollary 1.
Given thatξ and Alfs are as in Theorem 2 and ω≥0 is measurable function. Since Φ≥0 is superquadratic function, then the second term on the L. H. S of (48) is non-negative and the integral inequality(59)∫ΞξsΦrAlfsΔμs1…Δμsn≤∫Ξ˜υtΦftΔμ˜t1…Δμ˜tnr,holds.Remark 3.
By takingr=1 and n=1, inequality (59) in Corollary 1 reduces to (18).Remark 4.
Form=1 and n=1, inequality (59) in Corollary 1 coincides with inequality (2.2) which is [29], (Corollary 2.1.2).Remark 5.
If we rewrite (48) with r=λ/μ≥1, 0<μ≤λ<∞ or −∞<μ≤λ<0 and Φ≥0, then we get(60)∫ΞξsΦλ/μAlfsΔμs1…Δμsn+λμ∫Ξ∫Ξ˜ξsls,tLsΦλ/μ−1AlfsΦft−Alfs×Δμs1…ΔμsnΔμ˜t1…Δμ˜tn≤∫Ξ˜υtΦftΔμ˜t1…Δμ˜tnλ/μ.Remark 6.
Form=1 and n=1, inequality (60) coincides with inequality (3.13) which is [27], (Remark 3.5).Remark 7.
In Remark5, since Φ≥0, then the second term on the L. H. S. of (60) is non-negative and (60) reduces to the weighted Hardy-type inequality of the form(61)∫ΞξsΦλ/μAlfsΔμs1…Δμsn≤∫Ξ˜υtΦftΔμ˜t1…Δμ˜tnλ/μ,which is a refinement of general Hardy-type inequality established in [29] [Remark 2.1.4] and [5].
As a specific case of Theorem2 when Φt=tλ for λ≥2, we get the next result.Corollary 2.
Suppose that the assumptions of Theorem2 are satisfied and λ≥2. Then,(62)∫ΞξsAlfsλrΔμs1…Δμsn+r∫Ξ∫Ξ˜ξsls,tLsAlfsλr−1ft−Alfsλ×Δμs1…ΔμsnΔμ˜t1…Δμ˜tn≤∫Ξ˜υtfλtΔμ˜t1…Δμ˜tnr.
If0<r<1 and 1<λ≤2, then (62) is reversed.Remark 8.
Clearly, form=1, inequality (62) in Corollary 2 coincides with inequality (46) which is [13], (Corollary 2.1).
In fact, the functionΦs=es is not superquadratic but by working with the superquadratic function Φs=es−s−1 (see Lemma 3) and replacing fs by lnfs in Theorem 2, we obtain the next multidimensional version of the Pólya–Knopp type inequality.Corollary 3.
Suppose that the assumptions in Theorem2 are satisfied and assume that(63)Alfs=1Ls∫Ξ˜ls,tlnftΔμ˜t1…Δμ˜tn,s∈Ξ,then(64)∫ΞξsexpAlfs−Alfs−1rΔμs1…Δμsn+I≤∫Ξ˜υtft−lnft−1Δμ˜t1…Δμ˜tnr,where(65)I=r∫Ξ∫Ξ˜ξsls,tLsexpAlfs−Alfs−1r−1×explnft−Alfs−lnft−Alfs−1×Δμs1…ΔμsnΔμ˜t1…Δμ˜tn.
If0<r<1, then (64) is reversed.Remark 9.
Form=1, inequality (64) in Corollary 3 coincides with inequality (48) which is [13], (Corollary 2.2).
In the next results, we also suppose the following hypothesis:
B1′ Let Ξ=Ξ˜=a1,β1T×a2,β2T×⋯×an,βnT⊂ℝn, 0≤ai<βi≤∞, for every 1≤i≤n, where T is an arbitrary time scale and let Δμ1s=Δs, Δμ2t=Δt.Theorem 3.
WeassumeB1′ and B2 are satisfied. Suppose τ:Ξ⟶ℝ+ such that(66)ωt=s1…sn∫Ξ1σs1…σsnτsls,tLsrΔs1…Δsn1/r,t∈Ξ,where r≥1. If Φ∈CKm,ℝ is non-negative superquadratic function, then(67)∫a1β1…∫anβnτsΦrAl∗fsΔs1…Δsnσs1…σsn+r∫a1β1…∫anβn∫t1β1…∫tnβnτsls,tLsΦr−1Al∗fsΦft−Al∗fs×Δs1…Δsnσs1…σsnΔt1…Δtn≤∫a1β1…∫anβnωtΦftΔt1…Δtnr,holds for all Δμ˜-integrable functions f:Ξ⟶ℝm such that fΞ⊂Km⊂ℝm, where Al∗f:Ξ⟶ℝ is defined by(68)Al∗fs=1Ls∫a1β1…∫anβnls,tftΔt1…Δtn,s∈Ξ.
IfΦ is subquadratic function and 0<r<1, then (48) is reversed.Proof.
We get the result from Theorem2 by taking(69)ξs=τsσs1…σsn.Remark 10.
Forr=1, Theorem 3 coincides with [10], (Corollary 2.11).Remark 11.
In Theorem3, if we replace ωt by ωt/s1…sn and put m=1, then we get the result given in [13], (Corollary 2.3).


## 4. Inequalities with Special Kernels

In this section, we get some consequential inequalities of Hardy-type by selecting special kernels.Theorem 4.
SupposeB1′ and(70)ω2t=∫t1β1…∫tnβnξs1∏i=1nσsi−airΔs1…Δsn1/r<∞,such that ξ:Ξ⟶ℝ+ is Δμ-integrable function and r≥1. If Φ∈CKm,ℝ is non-negative superquadratic function, then(71)∫a1β1…∫anβnξsΦrAl∗∗fsΔs1…Δsn+r∫a1β1…∫anβn∫t1β1…∫tnβnξs∏i=1nσsi−aiΦr−1Al∗∗fsΦft−Al∗∗fs×Δs1…ΔsnΔt1…Δtn≤∫a1β1…∫anβnω2tΦftΔt1…Δtnr,holds for all Δμ˜-integrable functions f:Ξ⟶ℝm such that fΞ⊂Km⊂ℝm, where Al∗∗f:Ξ⟶ℝ is defined by(72)Al∗∗fs=1∏i=1nσsi−ai∫a1σs1…∫anσsnftΔt1…Δtn,s∈Ξ.
IfΦ is subquadratic function and 0<r<1, then (71) is reversed.Proof.
We get the result from Theorem2 by taking
We have the following in this case:(73)Ls=∫a1σs1…∫anσsnΔt1…Δtn=∏i=1nσsi−ai,Al=Al∗∗, ω=ω2.Remark 12.
Forr=1, Theorem 4 coincides with [10], (Corollary 3.1).Remark 13.
Form=1, Theorem 4 coincides with [13], (Corollary 2.7).
If we letai=0, i=1,…,n and ξs=1/s1…sn in Theorem 4, we have the next result.Corollary 4.
AssumingB1′, we define(74)ω3t=∫t1β1…∫tnβn1s1…sn∏i=1nσsirΔs1…Δsn1/r,t∈Ξ,such that ξ:Ξ⟶ℝ+ is Δμ-integrable function and r≥1. If Φ∈CKm,ℝ is non-negative superquadratic function, then(75)∫0β1…∫0βnΦrAl′fsΔs1…Δsn∏i=1nsi+r∫0β1…∫0βn∫t1β1…∫tnβnΦr−1Al′fsΦft−Al′fs×Δs1…Δsn∏i=1nsiσsiΔt1…Δtn≤∫0β1…∫0βnω3tΦftΔt1…Δtnr,holds for all Δμ˜-integrable functions f:Ξ⟶ℝm such that fΞ⊂Km⊂ℝm, where Al′f:Ξ⟶ℝ is defined by(76)Al′fs=1∏i=1nσsi∫0σs1…∫0σsnftΔt1…Δtn,s∈Ξ.
IfΦ is subquadratic function and 0<r<1, then (76) is reversed.Remark 14.
If we takeβi=∞i=1,…,n, then inequalities (75) and (76) reduces to(77)ω3t=∫t1∞…∫tn∞1s1…sn∏i=1nσsirΔs1…Δsn1/r,t∈Ξ,∫0∞…∫0∞ΦrAl′fsΔs1…Δsn∏i=1nsi+r∫0∞…∫0∞∫t1∞…∫tn∞Φr−1Al′fsΦft−AAl′fs×Δs1…Δsn∏i=1nsiσsiΔt1…Δtn≤∫0∞…∫0∞ω3tΦftΔt1…Δtnr.Remark 15.
Clearly, forr=1, inequalities (75) and (76), respectively, reduces to(78)ω3t=∫t1β1…∫tnβn1∏i=1nsiσsiΔs1…Δsn=∏i=1n1ti−1βi,and(79)∫0β1…∫0βn1∏i=1nsiΦAl′fsΔs1…Δsn+∫0β1…∫0βn∫t1β1…∫tnβn1∏i=1nsiσsiΦft−Al′fsΔs1…ΔsnΔt1…Δtn≤∫0β1…∫0βn∏i=1n1ti−1βiΦftΔt1…Δtn.
Furthermost, if we takeβi=∞i=1,…,n, then (80) becomes(80)∫0∞…∫0∞1∏i=1nsiΦAl′fsΔs1…Δsn+∫0∞…∫0∞∫t1∞…∫tn∞1∏i=1nsiσsiΦft−Al′fs×Δs1…ΔsnΔt1…Δtn≤∫0∞…∫0∞∏i=1nti,which is [10], [Remark 3.2].Theorem 5.
AssumeB1′ with βi=∞i=1,…,n. If Φ∈CKm,ℝ is non-negative superquadratic function, then(81)∫a1∞…∫an∞Φr∏i=1nsi∫s1∞…∫sn∞ft∏i=1ntiσtiΔt1…ΔtnΔs1…Δsn∏i=1nsi+r∫a1∞…∫an∞∫a1σt1…∫anσtnΦr−1∏i=1nsi∫s1∞…∫sn∞ft∏i=1ntiσtiΔt1…Δtn×Φft−∏i=1nsi∫s1∞…∫sn∞ft∏i=1ntiσtiΔt1…Δtn×Δs1…ΔsnΔt1…Δtn∏i=1ntiσti≤∫a1∞…∫an∞ωtΦftΔt1…Δtnr,holds for all Δμ˜-integrable functions f:Ξ⟶ℝm such that fΞ⊂Km⊂ℝm, where(82)ωt=∫a1∞…∫an∞1∏i=1nsi1/∏i=1ntiσti1/∏i=1nsirΔs1…Δsn1/r,t∈Ξ.IfΦ is subquadratic function and 0<r<1, then (82) is reversed.Proof.
We get the result from Theorem2 by taking(83)ls,t=∏i=1n1tiσti,ifsi≥tfor alli∈1,…,n,0,otherwise.and ξs=1/s1…sn.Corollary 5.
Clearly, forr=1, inequality (82) reduces to(84)∫a1∞…∫an∞Φ∏i=1nsi∫s1∞…∫sn∞ft∏i=1ntiσtiΔt1…ΔtnΔs1…ΔsnΠi=1nsi+∫a1∞…∫an∞∫a1σt1…∫anσtnΦft−∏i=1nsi∫s1∞…∫sn∞ft∏i=1ntiσtiΔt1…Δtn×Δs1…ΔsnΔt1…Δtn∏i=1ntiσti≤∫a1∞…∫an∞∏i=1n1−aiσtiΦftΔt1…Δtn∏i=1nti,which is [10], (Corollary 3.3).


## 5. Inequalities with Particular Time Scales

In this section, we get some consequential inequalities by selecting some specific time scales.Theorem 6.
AssumeT be an isolated time scale, ξ:Ξ⟶ℝ+ is Δμ-integrable function and Ξ=Ξ˜=a,∞T, a≥0. If λ≥1, then(85)∑s∈a,∞TξsAl″f1λs+Al″f2λsr/λμs+r∑t∈a,∞T∑s∈t,∞Tξsσs−aAl″f1λs+Al″f2λsr−1/λf1t−Al″f1sλ+f2t−Al′′f2sλ1/λμsμt≥∑t∈a,∞Tω4tf1λt+f2λt1/λμtr,holds for all Δμ˜-integrable functions f1,f2:Ξ˜⟶ℝ+, where(86)Al″fis=1σs−a∑t∈a,σsTfitμt,i=1,2,and(87)ω4t=∑s∈t,∞Tξsσs−arμs1/r.Proof.
We get the result from Theorem4 by taking(88)Φs=−∑i=1msiλ1/λ,with λ≥1, n=1 and m=2.Remark 16.
If we taker=1 in Theorem 6, then we get(89)∑s∈a,∞TξsAl′′f1λs+Al′′f2λs1/λμs+∑t∈a,∞T∑s∈t,∞Tξsσs−af1t−Al″f1sλ+f2t−Al″f2sλ1/λμsμt≥∑t∈a,∞Tω5tf1λt+f2λt1/λμt,where Al″fis, i=1, 2 is defined by (87) and(90)ω5t=∑s∈t,∞Tξsσs−aμs,which is [10], (Corollary 4.1).Theorem 7.
AssumingT be an isolated time scale, ξ:Ξ⟶ℝ+ is Δμ-integrable function and Ξ=Ξ˜=a,∞T, a≥0. Then,(91)∑s∈a,∞TξsμsA∧lf1scoshA∧lf1s−sinhA∧lf1s++A∧lf2scoshA∧lf2s−sinhA∧lf2sr+r∑t∈a,∞T∑s∈t,∞Tξsσs−aA∧lf1scoshA∧lf1s−sinhA∧lf1s+A∧lf2scoshA∧lf2s−sinhA∧lf2sr−1×f1t−A∧lf11scoshf1t−A∧lf1s−sinhf1t−A∧lf1s+f2t−A∧lf2scoshf2t−A∧lf2s−sinhf2t−A∧lf2sμsμt≤∑t∈a,∞Tω6tf1tcoshf1t−sinhf2t+f2tcoshf2t−sinhf2tμtr,holds for all Δμ˜-integrable functions f1,f2:Ξ˜⟶ℝ+, where(92)A^lfis=1σs−a∑t∈a,σsTfitμt,i=1,2,and(93)ω6t=∑s∈t,∞Tξsσs−arμs1/r.Proof.
We get the result from Theorem4 by taking(94)Φs=∑i=1msicoshsi−sinhsi,with n=1 and m=2.Remark 17.
If we taker=1 in Theorem 7, then we get(95)∑s∈a,∞TξsμsA^lf1scoshA^lf1s−sinhA^lf1s+A^lf2scoshA^lf2s−sinhA^lf2s+∑t∈a,∞T∑s∈t,∞Tξsσs−af1t−A^lf1scoshf1t−A^lf1s−sinhf1t−A^lf1s+f2t−A^lf2scoshf2t−A^lf2s−sinhf2t−A^lf2sμsμt≤∑t∈a,∞Tω7tf1tcoshf1t−sinhf1t+f2tcoshf2t−sinhf2tμt,where A^lfis, i=1,2 is defined by (93) and(96)ω7t=∑s∈t,∞Tξsσs−aμs,which is [10], (Corollary 4.3).Theorem 8.
AssumingT be an isolated time scale, ξ:Ξ⟶ℝ+ is Δμ-integrable function and Ξ=Ξ˜=a,∞T, a≥0. Then,(97)∑s∈a,∞TξsA¯lf12slnA¯lf1s+A¯lf22slnA¯lf2srμs+r∑t∈a,∞T∑s∈t,∞Tξsσs−aA¯lf12slnA¯lf1s+A¯lf22slnA¯lf2sr−1×f1t−A¯lf1s2lnf1t−A¯lf1s+f2t−A¯lf2s2lnf2t−A¯lf2sμsμt≤∑t∈a,∞Tω8tf12tlnf1t+f22tlnf2tμtr,holds for all Δμ˜-integrable functions f1,f2:Ξ˜⟶ℝ+, where(98)A¯lfis=1σs−a∑t∈a,σsTftμti,i=1,2,and(99)ω9t=∑s∈t,∞Tξsσs−arμs1/r.Proof.
We get the result from Theorem4 by taking(100)Φs=∑i=1msi2lnsi,with the assumption 0ln0=0.Remark 18.
If we taker=1 in Theorem 8, then we get(101)∑s∈a,∞TξsA¯lf12slnA¯lf1s+A¯lf22slnA¯lf2sμs+∑t∈a,∞T∑s∈t,∞Tξsσs−af1t−A¯lf1s2lnf1t−A¯lf1s+f2t−A¯lf2s2lnf2t−A¯lf2sμsμt≤∑t∈a,∞Tω9tf12tlnf1t+f22tlnf2tμt,where A¯lfis, i=1,2 is defined by (99) and(102)ω9t=∑s∈t,∞Tξsσs−aμs,which is [6], (Corollary 4.4).Theorem 9.
AssumingT be an isolated time scale, ξ:Ξ⟶ℝ+ is Δμ-integrable function and Ξ=Ξ˜=a,∞T, a≥0. Then,(103)∑s∈a,∞Tξsln1+A¨lf1s+A¨lf2s−A¨lf1s−A¨lf2srμs+r∑t∈a,∞T∑s∈t,∞Tξsσs−aln1+A¨lf1s+A¨lf2s−A¨lf1s−A¨lf2sr−1×ln1+f1t−A¨lf1s+f2t−A¨lf2s−f1t−A¨lf1s−f2t−A¨lf2sμsμt≤∑t∈a,∞Tω10tln1+f1t+f2t−f1t−f2tμtr,holds for all Δμ˜-integrable functions f1,f2:Ξ2⟶ℝ+, where(104)A¨lfis=1σs−a∑t∈a,σsTftiμt,i=1,2,and(105)ω10t=∑s∈t,∞Tξsσs−arμs1/r.Proof.
We get the result from Theorem4 by taking(106)Φs=ln1+∑i=1msi−∑i=1msi.Remark 19.
If we taker=1 in Theorem 9, then we get(107)∑s∈a,∞Tξsln1+A¨lf1s+A¨lf2s−A¨lf1s−A¨lf2sμs+∑t∈a,∞T∑s∈t,∞Tξsσs−aln1+f1t−A¨lf1s+f2t−A¨lf2s−f1t−A¨lf1s−f2t−A¨lf2sμsμt≤∑t∈a,∞Tω11tln1+f1t+f2t−f1t−f2tμt,where A¨lfis, i=1, 2 is defined by (105) and(108)ω11t=∑s∈t,∞Tξsσs−aμs,which is [10], (Corollary 4.5).


## 6. Conclusion and Future Work

This research article is dedicated for some general dynamic inequalities of Hardy’s type and their converses on time scales. These inequalities are considered rather in general terms and contain a number of special integral inequalities. In particular, our findings can be seen as refinements of some recent results closely linked to classical Hardy and Pólya–Knopp inequalities on time scale. We use some algebraic inequalities such as the Minkowski inequality, the refined Jensen inequality, and the Bernoulli inequality on time scales to prove the essential results in this paper. Our computed outcomes can be very useful as a starting point to get some continuous inequalities as special cases. In the future, such inequalities can be introduced by using fractional integrals and fractional derivatives of the Riemann−Liouville type on time scales. It will also be very enjoyable to introduce such inequalities in quantum calculus.



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