# On Hardy–Knopp Type Inequalities with Kernels via Time Scale Calculus

Authors: H. M. Rezk; A. I. Saied; Ghada AlNemer; M. Zakarya
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we study the inequalities of Hardy–Knopp type with kernel functions which have two nonnegative different weighted functions in two different spaces, in a general domain called a time scale calculus. A time scale calculusT is considered as a unification of the continuous calculus and the discrete calculus. We will prove these inequalities in a time scale calculus to avoid proving them twice once in the continuous case and the second in the discrete case. Also, as special cases of the time scale calculus, we can prove some new inequalities in new different domains. Our results will be proved by using the definition of a general Hardy operator on time scale. These inequalities (when T=ℕ) are essentially new.


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## Body

## 1. Introduction

In [1],Hardy showed that(1)∑h=1∞1h∑j=1hbjη≤ηη−1η∑h=1∞bηh,η>1,where bh≥0 for h≥1 and ∑h=1∞bηh<∞.In [2], Hardy established the continuous case of (1), which says that for w≥0 and integrable over 0,s where s∈0,∞, then(2)∫0∞1s∫0swτdτηds≤ηη−1η∫0∞wηsds,η>1.The constantη/η−1η in (1) and (2) is the finest possible.In [3], Kaijser et al. generalized (2) with a convex function Ψ≥0 and demonstrated the general the Hardy–Knopp inequality:(3)∫0∞Ψ1s∫0swτdτdss≤∫0∞Ψwsdss.In [4], Čižmešija et al. established a generalization of (3) with two different weighted functions. Specifically, it was demonstrated that if 0<x≤∞, p:0,x⟶ℝ is a nonnegative function such that the function s⟶ps/s2 is locally integrable on 0,x and Ψ is convex on y,z, where −∞≤y<z≤∞, the following inequality:(4)∫0xpsΨ1s∫0swτdτdss≤∫0xqsΨwsdss,holds for all integrable functions w:0,x⟶ℝ s.t. ws∈y,z∀s∈0,x and(5)qτ≔τ∫τxpss2ds,forτ∈0,x.In [5], Kaijser et al. generalized (2) as follows: if 0<x≤∞, p:0,x⟶ℝ, h:0,x×0,x⟶ℝ are nonnegative functions, and(6)qs≔s∫sxpτhτ,sHτdττ<∞,s∈0,x,then(7)∫0xpsΨAhwsdss≤∫0xqsΨwsdss,where Ψ is convex on I⊆ℝ, w:0,x⟶ℝ is a function with values in I, and(8)Ahws≔1Hs∫0shs,ζwζdζ,Hs≔∫0shs,ζdζ,s∈0,x.Also, in [5], it is established that if 1<η≤t<∞,λ∈1,η and 0<x<∞. Furthermore, assume that Ψ is a convex and strictly monotone function on y,z, −∞<y<z<∞, and assume that the general Hardy operator Ah defined by (8) and ps and qs are nonnegative weighted functions. Then,(9)∫0xΨAhwstpsdss1/t≤C∫0xΨηwsqsdss1/n,holds for the nonnegative function ws, y<ws<z, s∈0,x, and C>0 if(10)Bs≔sup0<ζ<xQζλ−1/η∫ζxhs,ζHstQstη−λ/ηpsdss1/t<∞,where(11)Qζ=∫0ζqτ−1/η−1τ1/η−1dτ.For the reader, we introduce some related papers like [6–9].In the last decades, a new theory has been discovered to unify the continuous calculus and discrete calculus. It is called a time scale theory. Many authors established dynamic inequalities and generalized them on time scales. For example, see papers [10–17]. In particular, Saker et al. [18] proved that(12)∫xyps∫xσswτΔτηΔs1/η≤C∫xyqswλsΔs1/λ,1<λ≤η<∞,holds for w∈Crdx,yT,ℝ iff(13)D=supx<s<y∫sypτΔτ1/η∫xσsq1−λ∗τΔτ1/λ∗<∞,λ∗=λλ−1,holds. Moreover, the estimate for the constant C>0 in (12) is(14)D≤C≤1+ηλ∗1/η1+λ∗η1/λ∗D,where the functions p and q which are nonnegative rd-continuous functions are called the weighted functions and condition (13) gives the characterization of these two functions which leads to the validation of inequality (12).In [19], Özkan et al. showed that if 0≤x<y≤∞, p∈Crdx,yT,ℝ is a nonnegative function such that ∫τyps/s−xσs−xΔs exists and(15)qτ≔τ−x∫τypss−xσs−xΔs,τ∈x,y.Furthermore, ifΨ:c,d⟶ℝ, c,d∈ℝ is continuous and convex, then(16)∫xypsΨ1σs−x∫xσswτΔτΔss−x≤∫xyqsΨwsΔss−x,holds for w∈Crdx,yT,ℝ s.t. ws∈c,d. Also, they proved that if p∈Crdy,∞,ℝ is a nonnegative function and(17)qτ≔1τ∫yτpsΔs,τ∈y,∞,then the inequality(18)∫xypsΨ1σs−x∫xσswτΔτΔss−x≤∫xyqsΨwsΔss−x,holds for w∈Crdy,∞T,ℝ s.t. ws∈c,d∀s∈y,∞.In [19], the authors proved inequality (2) for several functions on time scales and showed that if x∈T and wh,h=1,2,…,n, are nonnegative delta integrable functions, then(19)∫x∞w1στw2στ…wnστστ−xnrΔτ<rnr−nr∫x∞∑h=1nwhτrΔτ,where(20)whτ≔∫xτwhsΔs.Also, they proved that ifx,y∈T, wτ∈Crdx,yT,ℝ s.t. wτs∈c,d, c,d∈ℝ. Define whs≔1/s∫xswhsΔs, for h=1,2,…,n, and assume that p∈Crdx,yT,ℝ and(21)qτ≔τ−x∫τypss−xσs−xΔs,τ∈x,y.Furthermore, ifΨ:c,d⟶ℝ is continuous and convex, then(22)∫xypsΨ∏h=1nwhσs1/nΔss−x≤∫xyqsΨ∑h=1nwhsnΔss−x.In [20], Özkan and Yildirim generalized (16) with kernels and proved that if Ψ:c,d⟶ℝ s.t. c,d∈ℝ is continuous and convex, hs,ζ∈Crdx,y×x,y,ℝ and p∈Crdx,y,ℝ are nonnegative functions and(23)qζ=ζ−x∫ζyhs,ζHσspsΔss−x,ζ∈x,y.Then, the inequality(24)∫xypsΨAhwσs,sΔss−x≤∫xyqsΨwsΔss−x.satisfies for all delta integrable functions w∈Crdx,y,ℝ s.t. ws∈c,d, where(25)Ahwτ,s≔1Hτ,s∫xτhs,ζwζΔζ,Hs≔∫xshs,ζΔζ.The aim of this manuscript is to establish some new characterizations for dynamic inequalities of Hardy–Knopp type with kernels in different spaces to prove the following inequality:(26)∫x0xϕAhwσsqusΔsσs−x01/q≤C∫x0xϕηwζvζσζ−x0Δζ1/η,where 1<η≤q<∞.The paper is coordinated as follows: in Section2, we show some basics on T calculus and some theorems needed in Section 3, where we prove the main outcomes. Our key conclusions (when T=ℝ) give the characterizations of inequality (9) proved by Kaijser, Nikolova, Persson, and Wedestig. Also, we give some new characterizations of weights for new general theorems.


## 2. Preliminaries

A time scaleT is an arbitrary nonempty closed subset of the real numbers ℝ. The forward jump operator defined by στ≔infη∈T:η>τ. For any function Φ:T⟶ℝ, the notation Φστ denotes Φστ. The set of all such rd-continuous functions is denoted by CrdT,ℝ. To learn more about the time scale calculus, see [21, 22].The derivative ofΦϖ and Φ/ϖ (where ϖϖσ≠0) of two differentiable functions Φ and ϖ is(27)ΦϖΔ=ΦΔϖ+ΦσϖΔ=ΦϖΔ+ΦΔϖσ,ΦϖΔ=ΦΔϖ−ΦϖΔϖϖσ.The integration by parts formula onT is(28)∫υ0υλτφΔτΔτ=λτφτυ0υ−∫υ0υλΔτφστΔτ.The time scales chain rule is(29)ϖ°φΔτ=ϖ′φϰφΔτ,whereϰ∈τ,στ,such that ϖ:ℝ⟶ℝ is continuously differentiable and φ:T⟶ℝ is delta differentiable. The Hölder inequality on T is(30)∫ζ0ζλτφτΔτ≤∫ζ0ζλτγΔτ1/γ∫ζ0ζφτνΔτ1/υ,where ζ0,ζ∈T, λ,φ∈CrdI,ℝ, γ>1, and 1/γ+1/ν=1.Theorem 1.
(Jensen’s inequality). Assume thatζ0,ζ∈T and r0,r∈ℝ. If λ∈Crdζ0,ζT,ℝ, φ∈Crdζ0,ζT,r0,r and Ψ:r0,r⟶ℝ is continuous and convex, then(31)Ψ1∫ζ0ζλτΔτ∫ζ0ζλτφτΔτ≤1∫ζ0ζλτΔτ∫ζ0ζλτΨφτΔτ.
Direction of (31) will be reversed if Ψ is a concave function.
AssumeΞ,ℛ,μΔ and Π,S,λΔ are finite dimensional time scale measure spaces. We define Ξ×Π,ℛ×S,μΔ×λΔ such that ℛ×S is the product σ−algebra by G×H:G∈ℛ,H∈S and μΔ×λΔG×H=μΔGλΔH.Theorem 2.
(Minkowski’s inequality [11]). Assume that p,q, and w are nonnegative functions on Ξ,Π and Ξ×Π, respectively. If α≥1, then(32)∫Ξps∫Πws,ζqζΔζαΔs1/α≤∫Πqζ∫Ξwαs,ζpsΔs1/αΔζ.


## 3. Main Results

In what follows, we define the general Hardy operatorAh as follows:(33)Ahwr,s≔1Hr,s∫x0rhs,ζwζΔζ,Hr,s≔∫x0shs,ζΔζ,where r,s>x0 and w∈Crdx0,xT,ℝ is a delta integrable and hs,ζ∈Crdx0,xT×x0,xT,ℝ is a delta integrable and nonnegative function. Throughout this section, we will assume that the functions are nonnegative rd-continuous and the right-hand sides of the inequalities converge if the left sides converge. Now, we are ready to state and prove our main results.Theorem 3.
Assume thatx0,x∈T, 1<η≤q<∞, and r∈1,η. Let ϕ be a nonnegative and convex function on c,d, −∞<c<d<∞, and uandv are nonnegative weighted functions. Then,(34)∫x0xϕAhwσs,squsΔsσs−x01/q≤C∫x0xϕηwζvζσζ−x0Δζ1/η,holds for w≥0 and C>0, if(35)Ar=supζ∈x0,xTVσr−1/ηζ∫ζxhs,ζHσs,sqVσsqη−r/ηusΔsσs−x01/q<∞,where(36)Vζ=∫x0ζvτ−1/η−1στ−x01/η−1Δτ.Proof.
By applying (31), we have(37)∫x0xϕAhwσs,squsΔsσs−x0=∫x0xϕ1Hσs,s∫x0σshs,ζwζΔζqusσs−x0Δs≤∫x0x1Hσs,s∫x0σshs,ζϕwζΔζqusσs−x0Δs=∫x0x1Hqσs,s∫x0σshs,ζϕwζΔζqusσs−x0Δs.
Define a functiong such that(38)ϕηwζvζσζ−x0=ϕgζ,and then we have(39)∫x0σshs,ζϕwζΔζ=∫x0σshs,ζϕ1/ηgζVσζr−1/ηVσζ1−r/ηvζ−1/ησζ−x01/ηΔζ.
Applying (30) with η>1 and η/η−1 on the term,(40)∫x0σshs,ζϕ1/ηgζVσζr−1/ηVσζ1−r/ηvζ−1/ησζ−x01/ηΔζ,we get(41)∫x0σshs,ζϕ1/ηgζVσζr−1/ηVσζ1−r/ηvζ−1/ησζ−x01/ηΔζ≤∫x0σshηs,ζϕgζVσζr−1Δζ1/η×∫x0σsVσζ1−r/η−1vζ−1/η−1σζ−x01/η−1Δζη−1/η.
Substituting (41) into (39), we obtain(42)∫x0σshs,ζϕwζΔζ≤∫x0σshηs,ζϕgζVσζr−1Δζ1/η×∫x0σsVσζ1−r/η−1vζ−1/η−1σζ−x01/η−1Δζη−1/η.
And then we have from (37) that(43)∫x0xϕAhwσs,squsΔsσs−x0≤∫x0x∫x0σshηs,ζϕgζVσζr−1Δζq/η×∫x0σsVσζ1−r/η−1vζ−1/η−1σζ−x01/η−1Δζqη−1/ηusHqσs,sσs−x0Δs.
Since(44)Vζ=∫x0ζvτ−1/η−1στ−x01/η−1Δτ,then(45)VΔζ=vζ−1/η−1σζ−x01/η−1>0.
Therefore, the functionV is increasing. By applying (29) on the term V1−r−1/η−1ζ, we obtain(46)V1−r−1/η−1ζΔ=Vη−r/η−1ζΔ=η−rη−1V−r−1/η−1λVΔζ,where λ∈ζ,σζ. Thus, by substituting (45) into (46), we get(47)Vη−r/η−1ζΔ=η−rη−1V−r−1/η−1λvζ−1/η−1σζ−x01/η−1.
Sinceλ≤σζ and V is increasing, we have(48)Vλ≤Vσζ.
Using the relation1<r<η, we see(49)V−r−1/η−1λ≥Vσζ−r−1/η−1.
Substituting (49) into (47), we get (note η−r>0 and η−1>0) that(50)Vη−r/η−1ζΔ≥η−rη−1Vσζ−r−1/η−1vζ−1/η−1σζ−x01/η−1,and then(51)∫x0σsVη−r/η−1ζΔΔζ≥η−rη−1∫x0σsVσζ−r−1/η−1vζ−1/η−1σζ−x01/η−1Δζ.
Thus,(52)∫x0σsVσζ−r−1/η−1vζ−1/η−1σζ−x01/η−1Δζ≤η−1η−r∫x0σsVη−r/η−1ζΔΔζ=η−1η−rVσsη−r/η−1,and then we have from (52) that(53)∫x0σsVσζ1−r/η−1vζ−1/η−1σζ−x01/η−1Δζ≤η−1η−rVσsη−r/η−1.
Substituting (53) into (43), we get(54)∫x0xϕAhwσs,squsΔsσs−x0≤η−1η−rqη−1/η∫x0x∫x0σshηs,ζϕgζVσζr−1Δζq/η×Vσsη−r/ηHσs,squsσs−x0Δs.
From (30) and the definition of g in (38), we obtain(55)∫x0xϕAhwσs,squsΔsσs−x0≤η−1η−rqη−1/η∫x0x∫x0σshηs,ζϕηwζVσζr−1vζσζ−x0Δζq/η×Vσsη−r/ηHσs,squsσs−x0Δs.
Applying (32) on the term(56)∫x0x∫x0σshηs,ζϕηwζVσζr−1vζσζ−x0Δζq/η×Vσsη−r/ηHσs,squsσs−x0Δs,with q/η>1, we observe(57)∫x0x∫x0σshηs,ζϕηwζVσζr−1vζσζ−x0Δζq/η×Vσsη−r/ηHσs,squsσs−x0Δsη/q≤∫x0xϕηwζVσζr−1vζσζ−x0×∫ζxhqs,ζVσsη−r/ηHσs,squsσs−x0Δsη/qΔζ=∫x0xϕηwζVσζr−1vζσζ−x0×∫ζxVσsqη−r/ηhs,ζHσs,squsσs−x0Δsη/qΔζ.
Substituting (57) into (55), we get(58)∫x0xϕAhwσs,squsΔsσs−x0≤η−1η−rqη−1/η∫x0xϕηwζVσζr−1vζσζ−x0×∫ζxVσsqη−r/ηhs,ζHσs,squsσs−x0ΔsηqΔζq/η≤η−1η−rqη−1/ηAqr∫x0xϕηwζvζσζ−x0Δζq/η,and then(59)∫x0xϕAhwσs,squsΔsσs−x01/q≤η−1η−rη−1/ηAr∫x0xϕηwζvζσζ−x0Δζ1/η,
which is (34) with C=η−1/η−rη−1/ηAr.Remark 1.
IfT=ℝ, σs=s, and x0=0, then (34) reduces to (9), established in [5].Corollary 1.
IfT=ℕ, στ=τ+1, and x0=1, then (34) reduces to(60)∑n=1Nϕ1∑m=1nhn,m∑m=1nhn,mwmqunn1/q≤C∑m=1Nϕηwmvmm1/η,N∈ℕ,with C=η−1/η−rη−1/ηAr, such that(61)Ar=supn∈1,NVnr−1/η∑m=nNhm,nHm,mqVmqη−r/ηumm1/q<∞,where(62)Vm=∑τ=1mvτ−1/η−1τ1/η−1,Hm,n=∑τ=1mhn,τ.Theorem 4.
Letx0,x∈T, 1<η≤q<∞, and r∈1,η. Furthermore, assume that ϕ,ψ are nonnegative functions on c,d, −∞<c<d<∞, and ψ is a convex function such that(63)Aψ≤ϕ≤Dψ,where AandD are constants and uandv are nonnegative weighted functions. Then, the inequality(64)∫x0xϕAhwσs,squsΔsσs−x01/q≤C∫x0xϕηwζvζσζ−x0Δζ1/η,holds for the nonnegative function w and C>0, if(65)Ar=supζ∈x0,xTVσζr−1/η∫ζxusVσsqη−r/ηhs,ζHσs,sqΔsσs−x01/q<∞.Proof.
From (33) and by applying (31), we have(66)∫x0xϕAhwσs,squsΔsσs−x0≤Dq∫x0xψAhwσs,squsΔsσs−x0=Dq∫x0xψ1Hσs,s∫x0σshs,ζwζΔζqusσs−x0Δs≤Dq∫x0x1Hσs,s∫x0σshs,ζψwζΔζqusσs−x0Δs=Dq∫x0xusHqσs,sJqsΔsσs−x0,where(67)Js=∫x0σshs,ζψwζΔζ.
Define a functiong such that(68)ψηwζvζσζ−x0=ψgζ.
Substituting (68) into (67), we obtain(69)Js=∫x0σshs,ζψ1/ηgζvζ−1/ησζ−x01/ηΔζ.
Note that(70)Js=∫x0σshs,ζψ1/ηgζVσζr−1/ηVσζ1−r/ηvζ−1/ησζ−x01/ηΔζ.
Applying (30) with η>1 and η/η−1 on the term(71)∫x0σshs,ζψ1/ηgζVσζr−1/ηVσζ1−r/ηvζ−1/ησζ−x01/ηΔζ,we get(72)∫x0σshs,ζψ1/ηgζVσζr−1/ηVσζ1−r/ηvζ−1/ησζ−x01/ηΔζ≤∫x0σshηs,ζψgζVσζr−1Δζ1/η×∫x0σsVσζ1−r/η−1vζ−1/η−1σζ−x01/η−1Δζη−1/η.
Substituting (72) into (70), we observe(73)Js≤∫x0σshηs,ζψgζVσζr−1Δζ1/η×∫x0σsVσζ1−r/η−1vζ−1/η−1σζ−x01/η−1Δζη−1/η.
Substituting (73) into (66), we have(74)∫x0xϕAhwσs,squsΔsσs−x0≤Dq∫x0xusHqσs,sσs−x0∫x0σshηs,ζψgζVσζr−1Δζq/η×∫x0σsVσζ1−r/η−1vζ−1/η−1σζ−x01/η−1Δζqη−1/ηΔs.
Since(75)Vζ=∫x0ζvτ−1/η−1στ−x01/η−1Δτ,then(76)VΔζ=vζ−1/η−1σζ−x01/η−1>0.
Therefore, the functionV is increasing. Applying (29) on the term V1−r−1/η−1ζ, we obtain(77)V1−r−1/η−1ζΔ=Vη−r/η−1ζΔ=η−rη−1V−r−1/η−1λVΔζ,where λ∈ζ,σζ. Using the relation 1<r<η and substituting (76) into (77), we see that(78)Vη−r/η−1ζΔ=η−rη−1V−r−1/η−1λvζ−1/η−1σζ−x01/η−1.
Sinceλ≤σζ and V is increasing, we have(79)Vλ≤Vσζ.
Using the relation1<r<η, we get(80)V−r−1/η−1λ≥Vσζ−r−1/η−1.
Substituting (80) into (78), we have(81)Vη−r/η−1ζΔ≥η−rη−1Vσζ−r−1/η−1vζ−1/η−1σζ−x01/η−1,and then(82)∫x0σsVη−r/η−1ζΔΔζ≥η−rη−1∫x0σsVσζ−r−1/η−1vζ−1/η−1σζ−x01/η−1Δζ.
Thus,(83)∫x0σsVσζ−r−1/η−1vζ−1/η−1σζ−x01/η−1Δζ≤η−1η−r∫x0σsVη−r/η−1ζΔΔζ=η−1η−rVσsη−r/η−1,and then we have(84)∫x0σsVσζ1−r/η−1vζ−1/η−1σζ−x01/η−1Δζ≤η−1η−rVσsη−r/η−1.
Substituting (84) into (74), we get(85)∫x0xϕAhwσs,squsΔsσs−x0≤Dqη−1η−rqη−1/η∫x0x∫x0σshηs,ζψgζVσζr−1Δζq/η×Vσsqη−r/ηusΔsHqσs,sσs−x0.
Applying (32) on the term(86)∫x0x∫x0σshηs,ζψgζVσζr−1Δζq/ηVσsqη−r/ηusΔsHqσs,sσs−x0,with q/η>1, we observe(87)∫x0x∫x0σshηs,ζψgζVσζr−1Δζq/ηVσsqη−r/ηusΔsHqσs,sσs−x0η/q≤∫x0xψgζVσζr−1∫ζxVσsqη−r/ηushqs,ζΔsHqσs,sσs−x0η/qΔζ.
Substituting (87) into (85), we obtain(88)∫x0xϕAhwσs,squsΔsσs−x0≤Dqη−1η−rqη−1/η∫x0xψgζVσζr−1×∫ζxhqs,ζVσsqη−r/ηusHqσs,s.Δsσs−x0η/qΔζq/η.
Using assumption (65) and the definition of g in (68), we have from (63) and (88) that(89)∫x0xϕAhwσs,squsΔsσs−x0≤DqAqrη−1η−rqη−1/η∫x0xψgζΔζq/η=Dqη−1η−rqη−1/ηAqr∫x0xψηwζvζσζ−x0Δζq/η≤DqAqη−1η−rqη−1/ηAqr∫x0xϕηwζvζσζ−x0Δζq/η,and then(90)∫x0xϕAhwσs,squsΔsσs−x01/q≤DAη−1η−rη−1/ηAr∫x0xϕηwζvζσζ−x0Δζ1/η,which is (64) with the constant C=D/Aη−1/η−rη−1/ηAr.Corollary 2.
WhenT=ℝ, then σs=s, and inequality (64) in Theorem 3 reduces to the following inequality:(91)∫x0xϕAhwσs,squsdsσs−x01/q≤C∫x0xϕηwζvζσζ−x0dζ1/η,which holds for the nonnegative function w and C=D/Aη−1/η−rη−1/ηAr>0, if(92)Ar=supζ∈x0,xVζr−1/η∫ζxusVsqη−r/ηhs,ζHs,sqdss−x01/q<∞,where 1<η≤q<∞, r∈1,η, uandv are nonnegative weighted functions, ϕandψ are nonnegative functions on c,d, −∞<c<d<∞, and ψ is a convex function such that(93)Aψ≤ϕ≤Dψ,A,D are constants.Remark 2.
As a special case of Corollary3, if A=D=1 and x0=0, then we have (9) established in [5].Corollary 3.
IfT=ℕ, σn=n+1, and x0=1, we obtain a new discrete inequality:(94)∑n=1Nϕ1∑m=1nhn,m∑m=1nhn,mwmqunn1/q≤C∑m=1Nϕηwmvmm1/η,N∈ℕ,with C=D/Aη−1/η−rη−1/ηAr, such that(95)Ar=supn∈1,NVnr−1/η∑m=nNhm,nHm,mqVmqη−r/ηumm1/q<∞,where(96)Vm=∑τ=1mvτ−1/η−1τ1/η−1,Hm,n=∑τ=1mhn,τ.


## 4. Conclusion and Future Work

In this paper, we present some new inequalities of Hardy–Knopp type with kernel functions which have two nonnegative different weighted functions in two different spaceslη and lq for 1<η≤q<∞. Also, we established some inequalities using convex functions and applying general Hardy operator on time scale. In the future, we may generalize these results to be with multidimensional time scale. Also, some new dynamic inequalities of Hardy–Knopp type with kernel functions in two different weighted functions in two different spaces lη and lq for η≤q<∞ and 0<η<1. We hope general studies about a generalized Hardy operator on time scale and its application for some dynamic inequalities on time scales.



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*Source: 7997299-2022-06-06.xml*