# Inequalities of Hardy Type via Superquadratic Functions with General Kernels and Measures for Several Variables on Time Scales

Authors: H. M. Rezk; H. A. Abd El-Hamid; A. M. Ahmed; Ghada AlNemer; M. Zakarya
Journal: Journal of Function Spaces
 (2020)

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## Abstract

We use the properties of superquadratic functions to produce various improvements and popularizations on time scales of the Hardy form inequalities and their converses. Also, we include various examples and interpretations of the disparities in the literature that exist. In particular, our findings can be seen as refinements of some recent results closely linked to the time-scale inequalities of the classical Hardy, Pólya-Knopp, and Hardy-Hilbert. Some continuous inequalities are derived from the main results as special cases. The essential results will be proved by making use of some algebraic inequalities such as the Minkowski inequality, the refined Jensen inequality, and the Bernoulli inequality on time scales.


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## Body

## 1. Introduction

In [1], Hardy claimed this fundamental inequality and proved it:
(1)∫0∞1θ∫0θgηdηqdθ≤qq−1q∫0∞gqθdθ,where 1<q<∞,g≥0, and q/q−1q are sharp. They have emerged in the literature since the discovery of (1) numerous papers concerned with new arguments, generalizations, and extensions. One of the most common generalizations for (1) is the disparity of Pólya-Knopp’s inequality (see [2]), which is
(2)∫0∞exp1θ∫0θlngηdηdθ≤e∫0∞gθdθ.In [3], Kaijser et al. signalized that both (1) and (2) are special states of the Hardy-Knopp’s inequality:
(3)∫0∞Θ1θ∫0θgηdηdθθ≤∫0∞Θgθdθdθθ,where Θ∈C0,∞,ℝ is a convex function.In [4], Cizmeija et al. proved that if ζ:0,α⟶ℝ≥0,Θ is a convex on β,γ where −∞≤β≤γ≤∞,g:0,α⟶ℝ with gθ∈β,γ,∀θ∈0,α as an integrable function and υ is defined by
(4)υη≔η∫ηαζθθ2dθ,forη∈0,α,then the integral inequality
(5)∫0∞ζθΘ1θ∫0θgηdηdθθ≤∫0∞υθΘgθdθθ,is valid.In [5], Kaijser et al. applied the inequality of Jensen for convex functions and the theorem of Fubini to establish an invitingly popularization (1). Particularly, it was proved that if ζ:0,α⟶ℝ≥0 and l:0,α×0,α⟶ℝ≥0,0<α≤∞ such that
(6)Lθ≔∫0θlθ,ηdη>0,θ∈0,α,and Θ∈CI,ℝ,I⊆ℝ is a convex function, g:0,α⟶ℝ such that gθ∈I,∀θ∈0,α be integrable function, and υ is defined by
(7)υη≔η∫ηαξθlθ,ηLθdθθ<∞,η∈0,α,then the integral inequality
(8)∫0∞ξθΘAlgθdθθ≤∫0∞υθΘgθdθθ,is valid, where Alg is defined by
(9)Algθ≔1Lθ∫0θlθ,ηgηdη,θ∈0,α.As a popularization of (8), Krulic et al. [6] have demonstrated that if Ω1,∑1,μ1 and Ω2,∑2,μ2 are two measure spaces with positive σ finite measures ζ:Ω1⟶ℝ≥0 and l:Ω1×Ω2⟶ℝ≥0 such that
(10)Lθ≔∫Ω2lθ,ηdμ2η>0,θ∈Ω1,and Θ is a convex function on an interval I⊆ℝ,g:Ω2⟶ℝ≥0 with gΩ2⊆I be measurable function and υ is defined by
(11)υη≔∫Ω1ξθlθ,ηLθq/pdμ1θq/p<∞,η∈Ω2,then the integral inequality
(12)∫Ω1ξθΘq/pAlgθdμ1θ1/q≤∫Ω2υηΘgηdμ2η1/q,is valid, where 0<p≤q<∞ and Alg:Ω1⟶ℝ are defined by
(13)Algθ≔1Lθ∫Ω2lθ,ηgηdμ2η,θ∈Ω1.Observe that inequality (12) is a generalization of Hardy inequality (1). Namely, let Ω1=Ω2=ℝ+=0,∞,dμ1θ=dθ,dμ2η=dη and uθ=1/θ, and if 1<p=q<∞ and Θ:0,∞⟶ℝ are defined by Θθ=θp, then (1) is followed directly from (12), which can be rewritten with gηp/p−1η1/p−1 instead of gη and
(14)lθ,η≔1θχ0<η≤θ<∞θ,η.In the same setting, except withgηη1/p instead of gη and with
(15)lθ,η≔θη1/qθ+η−1,relation (12) becomes the Hardy-Hilbert integral inequality (see [7]).
(16)∫0∞∫0∞gηθ+ηdηpdθ≤πsinπ/pp∫0∞gpθdθ.In [8], Abramovich et al. considered a superquadratic function Θ instead of a convex function Θ and obtained the following refinement of inequality (12) in the particular case p=q, as
(17)∫Ω1ζθΘAlgθdμ1θ+∫Ω1∫Ω2ζθlθ,ηLθΘgη−Algθdμ1θdμ2η≤∫Ω2υηΘgηdμ2η.In [9], Aleksandra et al. proved that, if λ≤1,Ω1,Σ1,μ1 and Ω2,Σ2,μ2 are two measure spaces with positive σ-finite measures, ζ:Ω1⟶ℝ≥0,l:Ω1×Ω2⟶ℝ≥0 such that L:Ω1⟶ℝ is defined as in (10), Θ∈CI,ℝ,I⊆ℝ is a convex function, g:Ω2⟶ℝ≥0 such that gΩ2⊆I be measurable function and is defined by
(18)υη≔∫Ω1ζθlθ,ηLθλdμ1θ1/λ<∞,η∈Ω2,then the integral inequality
(19)∫Ω1ζθΘλAlgθdμ1θ+λ∫Ω1∫Ω2ζθlθ,ηLθΘλ−1gη−Algθdμ1θdμ2η≤∫Ω2υηΘgηdμ2ηλ,is valid, where Alg:Ω1⟶ℝ is defined by (13).In the past few years, several researchers have suggested the study of dynamic time-scale inequalities. In [10], the authors showed a number of Hardy-type inequalities with a general kernel on time scale. Namely, they have determined that if Ω1,Σ1,μ1 and Ω2,Σ2,μ2 are two time-scale measure spaces, l:Ω1×Ω2⟶ℝ≥0 such that
(20)Lθ≔∫Ω2lθ,ηΔη<∞,θ∈Ω1,and ζ:Ω1⟶ℝ+≥0 such that
(21)υη≔∫Ω1lθ,ηζθLθΔθ<∞,η∈Ω2,then the integral inequality
(22)∫Ω1ζθΘ1Lθ∫Λlθ,ηgηΔηΔθ≤∫Ω2υηΘgηΔη,is available for all Δμ2-integrable g:Ω2⟶ℝ such that gΩ2⊂I and Θ∈CI,ℝ,I⊂ℝ are a convex function.Moreover, Donchev et al. [11] improved the inequality (22) by replacing the function gη by an m-tuple of functions gη=g1η,g2η,⋯,gmη such that g1η,g2η,⋯,gmη are Δμ2-integrable on Ω2 in the following way. If Ω1,Σ1,μ1 and Ω2,Σ2,μ2 are two time-scale measure spaces, U⊂ℝm a convex set and l:Ω1×Ω2⟶ℝ+ such that
(23)Lθ≔∫Ω2lθ,ηΔη<∞,θ∈Ω1,and ζ:Ω1⟶ℝ such that
(24)υη≔∫Ω1lθ,ηζθLθΔθ<∞,η∈Ω2,then for every a convex function Θ, the integral inequality
(25)∫Ω1ζθΘ1Lθ∫Ω2lθ,ηgηΔηΔθ≤∫Ω2υηΘgηΔη,is available for all Δμ2-integrable functions g:Ω2⟶ℝm such that gΩ2⊂U⊂ℝm.In [12], the authors have specified the time scale version of (17). That is, they proved it if Ω1,Σ1,μ1 and Ω2,Σ2,μ2 are two time-scale measure spaces with positive σ-finite measures, ζ:Ω1⟶R≥0 and l:Ω1×Ω2⟶ℝ≥0 such that lθ,. is a Δμ2-integrable function for θ∈Ω2, and L:Ω1⟶ℝ is defined as
(26)Lθ≔∫Ω2lθ,ηΔμ2η>0,θ∈Ω1,(27)υη≔∫Ω1ζθlθ,ηLθΔμ1θ<∞,η∈Ω2.IfΘ:α,∞⟶ℝ≥0,α≥0 and a superquadratic function, then
(28)∫Ω1ζθΘAlgθΔμ1θ+∫Ω1∫Ω2ζθlθ,ηLθΘgη−AlgθΔμ1θΔμ2η≤∫Ω2υηΘgηΔμ2η,is available for all Δμ2-integrable function g:Ω2⟶ℝ≥0, and Alg is defined by
(29)Algθ≔1Lθ∫Ω2lθ,ηgηΔμ2η,θ∈Ω1.In [13], Saker et al. obtained the following refined Jensen’s inequality for superquadratic
(30)Θ∫Ω2lθ,ηgηΔμ2η∫Ω2lθ,ηΔμ2η≤∫Ω2lθ,η∫Ω2lθ,ηΔμ2ηΘgη−Θgη−AlgθΔμ2η,and in the same paper, he employed the above result to derive the following inequality of Hardy type:
(31)∫Ω1ζθΘλAlgθΔμ1θ+λ∫Ω1∫Ω2ζθlθ,ηLθΘλ−1AlgθΘgη−AlgθΔμ1η≤∫Ω2υηΘgηΔμ2ηλ,where(32)υη≔∫Ω1ζθlθ,ηLθλΔμ1θ1/λ<∞,μ∈Ω2,λ≥1, ζ:Ω1⟶ℝ≥0, and l:Ω1×Ω2⟶ℝ≥0 such that lθ,. is a Δμ2-integrable function for θ∈Ω2 and L:Ω1⟶ℝ is defined by (26), Θ:0,∞⟶ℝ≥0 is a superquadratic function, and Alg is defined by (29).Another development of Hardy-type inequality (28) has been made by Bibi [14] and Fabelurin [15] as follows. If Ω1,Σ1,μ1 and Ω2,Σ2,μ2 are two time-scale measure spaces, ζ:Ω1⟶ℝ≥0 and l:Ω1×Ω2⟶ℝ≥0 such that lθ,. are a Δμ2-integrable function for θ∈Ω2,L:Ω1⟶ℝ is defined by (26) and Θ∈CKm,ℝ is a superquadratic function, then
(33)∫Ω1ξθΘAlgθΔμ1θ+∫Ω1∫Ω2ξθlθ,ηLθΘgη−AlgθΔμ2η≤∫Ω2υηΘgηΔμ2η,is available for all Δμ2-integrable functions g:Ω2⟶ℝm such that gΩ2⊂Km, where Alg:Ω1⟶ℝ is defined by
(34)Algθ≔1Lθ∫Ω2lθ,ηgηΔμ2η,θ∈Ω1.For developing of dynamic inequalities on time scale calculus, we refer the reader to the articles [16–26].Motivated by the above results, our major aim in this paper is to deduce few nouveau 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 Preliminaries. In Inequalities with General Kernel, we obtain the extensions to the general kernel of Hardy-type inequality. In Inequalities with Specific Time Scales, we extend the latest results from Inequalities with General Kernel to several specific time scales. In Inequalities with Specific Time Scales, we discuss several particular cases of Hardy-type inequality by choosing such special kernels. In Inequalities with Specific Kernels, we derive enhanced forms of certain well-known Hardy-Hilbert-type inequalities.


## 2. Preliminaries

In this section, we will present some fundamental concepts and effects to integrals of time scales and for multivariate superquadratic functions which will be useful to deduce our major results. Letℝm be the Euclidean space, θ≔θ1,θ2,⋯,θm∈ℝm, η≔η1,η2,⋯,ηm∈ℝm, and gt≔g1t,g2t,⋯,gmt be the function defined on θ⊂ℝm. Throughout this supplement, we utilize the following notations:
(35)θ.η≔θ1η1,θ2η2,⋯,θmηm,θ≔θ1,θ2,⋯,θmandθ,η≔∑i=1mθiηi.Also,θ≤ηθ<η means that θi≤ηiθi<ηi, ∀1≤i≤m, and 0≔0,0,⋯,0 is the null vector. The subsets Km and Km+ in ℝm are defined by
(36)Km≔0,∞m≔θ∈ℝm:0≤θ,Km+≔0,∞m≔θ∈ℝm:0<θ.Now, we arraign the definition and few essential properties of superquadratic functions that premised in [27].Definition 1.
A functionΘ:Km⟶ℝ is named a superquadratic function if ∀θ∈Km,∃cθ∈ℝm such that
(37)Θη−Θθ−Θη−θ≥cθ,η−θ,∀η∈Km.
If−Θ is a superquadratic, then Θ is a subquadratic, and the reverse inequality of (37) is available.In the following, we recall a couple of beneficial examples of a superquadratic function.Example 1.
By [2], Example 1, the power function Θ:0,∞⟶ℝ, defined by Θθ≔θp, is called a superquadratic if p≥2 and a subquadratic if 1<p≤2 (it is also readily seen that if 0<p≤1 then θp is a subquadratic function). Since the sum of superquadratic functions is also superquadratic, then
(38)Θθ≔∑i=1mθip,
is a superquadratic onKm for each p≥2.Example 2 ([2], Examples 4, 5, and 6,).
By utilizing the same argument as in Example 1, the functionsΘ1,Θ2,Θ3:Km⟶ℝ defined as
(39)Θ1θ≔∑i=1mθicoshθi−sinhθi,Θ2θ≔ln1+∑i=1mθi−∑i=1mθi,Θ3θ≔∑i=1,i≠jmθi2lnθi,ifθi>0,θj=0,0,ifθ=0,are superquadratic.The following lemma shows that nonnegative superquadratic functions are indeed convex functions.Lemma 2.
Suppose thatΘ is a superquadratic with cθ≔c1θ,c2θ,⋯,cnθ as in Definition 1. Then
(i)
Θ0≤0 and ci0≤0∀1≤i≤m(ii)
IfΘ0≔0 and ∇Θ0≔0, then ciθ≔∂igθ, whenever ∂igθ exists for some index 1≤i≤m at θ∈Km(iii)
IfΘ≥0, then Θ is convex and Θ0≔0 and ∇Θ0≔0.In the following, we recall the inequality of Minkowski and the inequality of Jensen for superquadratic functions on time scales which are utilized in the proof of the essential results. The following definitions and theorems are referred from [28, 29]. Let Ti,1≤i≤m be time scales, and
(40)Λm≔T1×T2×⋯×Tm≔t=t1,t2,⋯,tm:ti∈Ti,1≤i≤m,is called an m-dimensional time scale. Consider E to be Δ-measurable subplot of Λm and g:E⟶ℝ a Δ-measurable function; then, the corresponding Δ-integral named Lebesgue Δ-integral is denoted by
(41)∫Egt1,t2,⋯,tmΔ1t1⋯Δmtm,∫EgtΔt,∫EgdμΔor∫EgtdμΔt,where μΔ is a σ-additive Lebesgue Δ-measure on Λm. Also, if gt≔g1t,g2t,⋯,gmt is an m-tuple of functions such that g1,g2,⋯,gm are Lebesgue Δ-integrable on E, then ∫EgdμΔ denotes the m-tuple:
(42)∫Eg1dμΔ,⋯,∫EgmdμΔ,i.e.,Δ-integral acts on each component of g.Lemma 3.
AssumeΩ1,Σ1,μ1 and Ω2,Σ2,μ2 are two time-scale measure spaces, and suppose that u≥0,υ≥0 and g≥0 on Ω1,Ω2 and Ω1×Ω2, respectively. If q≥1, then
(43)∫Ω1∫Ω2gθ,ηυηdμ2ηquθdμ1θ1/q≤∫Ω2∫Ω1gqθ,ηuθdμ1θυηdμ2η,is available provided all integrals in (43) exist. If 0<q<1 and
(44)∫Ω1∫Ω2gυdμ2qudμ1>0,∫Ω2gυdμ2>0,is available, then (43) is reversed. For q<0, in addition with (44), if
(45)∫Ω1gqudμ1>0,is available, then the sign of (43) is reversed.Theorem 4 ([14], Theorem 3.1).
AssumeΩ1,Σ1,μ1 and Ω2,Σ2,μ2 are two finite-dimensional time-scale measure spaces. Let Θ∈CKm,ℝ≥0 be continuous and superquadratic, l:Ω1×Ω2⟶ℝ≥0 such that lθ,. is Δμ2-integrable for θ∈Ω2. Then, the inequality
(46)Θ∫Ω2lθ,ηgηΔμ2η∫Ω2lθ,ηΔμ2η≤∫Ω2lθ,ηΘgη−Θgη−1/∫Ω2lθ,ηΔμ2η∫Ω2lθ,ηgηΔμ2ηΔμ2η∫Ω2lθ,ηΔμ2η,holds for all functions g such that gE⊂Km. If Θ is a subquadratic, then (46) is reversed.


## 3. Inequalities with General Kernel

In this section, we get the Hardy inequality for several variables via multivariate superquadratic functions. Before presenting the results, we labeled the following hypothesis.(A1)Ω1,Σ1,μ1 and Ω2,Σ2,μ2 are two time-scale measure spaces with positive σ-finite measures(A2)l:Ω1×Ω2⟶ℝ≥0 such that
(47)Lθ≔∫Ω2lθ,ηΔμ2η<∞,θ∈Ω1.(A3)ξ:Ω1⟶ℝ is Δμ1-integrable, and the function ω is defined by
(48)ωη≔∫Ω1ξθlθ,ηLθλΔμ1θ1/λ<∞,η∈Ω2,where λ≥1.Theorem 5.
Assume (A1)–(A3) are satisfied. IfΘ∈CKm,ℝ≥0 and is superquadratic, then
(49)∫Ω1ξθΘλAlgθΔμ1θ+λ∫Ω1∫Ω2ξθlθ,ηLθΘλ−1AkgθΘgη−A1gθΔμ2η≤∫Ω2ωηΘgηΔμ2ηλ,is available for g:Ω2⟶ℝm that is a nonnegative Δμ2-integrable function such that gΩ2⊂Km and Alg:Ω1⟶ℝ defined by
(50)Algθ≔1Lθ∫Ω2lθ,ηgηΔμ2η,θ∈Ω1.
IfΘ is subquadratic and 0<λ<1, then (49) is reversed.Proof.
We begin with an explicit identity(51)ΘAlgθ≔Θ1Lθ∫Ω2lθ,ηgηΔμ2η.
By applying the refined Jensen inequality (46) on (51), we find
(52)ΘAlgθ+1Lθ∫Ω2lθ,ηΘgη−AlgθΔμ2η≤1Lθ∫Ω2lθ,ηΘgηΔμ2η.
Then, sinceλ≥1 and Θ≥0, we get
(53)ΘAlgθ+1Lθ∫Ω2lθ,ηΘgη−AlgθΔμ2ηλ≤1Lθ∫Ω2lθ,ηΘgηΔμ2ηλ.
Furthermost, by utilizing the famous inequality of Bernoulli, it ensues that the L. H. S. of (53) became
(54)ΘAlgθ+1Lθ∫Ω2lθ,ηΘgη−AlgθΔμ2ηλ≤ΘλAlgθ+λΘλ−1AlgθLθ∫Ω2lθ,ηΘgη−AlgθΔμ2η,that is, we get
(55)ΘλAlgθ+λΘλ−1AlgθLθ∫Ω2lθ,ηΘgη−AlgθΔμ2η≤1Lθ∫Ω2lθ,ηΘgηΔμ2ηλ.
Multiplying (55) by ξθ and integrating it over Ω1 with respect to Δμ1θ, we have
(56)∫Ω1ξθΘλAlgθΔμ1θ+λ∫Ω1ξθ1Lθ∫Ω1lθ,ηΘgη−AlgθΔμ2ηΔμ1θ≤∫Ω1ξθ1Lθ∫Ω1lθ,ηΘgηΔμ2ηλΔμ1θ.
Applying the inequality of Minkowski on the R. H. S. of (56), we get
(57)∫Ω1ξθΘ1Lθ∫Ω2lθ,ηΘgηΔμ2ηλΔμ1θ≤∫Ω2Θgη∫Ω2ξθlθ,ηLθλΔμ1θ1/λΔμ2ηλ.
Finally, substituting (57) into (56) and utilizing the definition (48) of the weight function ω, we get
(58)∫Ω1ξθΘλAlgθΔμ1θ+λ∫Ω1∫Ω2ξθlθ,ηLθΘλ−1AlgθΘgη−AlgθΔμ1θΔμ2η≤∫Ω2ωηΘgηΔμ2ηλ,which is (49). If Θ is subquadratic and 0<λ<1, the corresponding results can be obtained similarly.Remark 6.
Ifλ=1 and m=1 in Theorem 5, then (49) reduces to (28) premised in Introduction.Remark 7.
For the Lebesgue scale measuresΔμ1θ=Δθ,Δμ2η and m=1, Theorem 5 coincides with Theorem 2.1.1 in [30].Remark 8.
As a special case of Theorem5 when T=ℝ and m=1, we have the inequality (19).Corollary 9.
Given thatξ and Algθ are as in Theorem 5 and ω≥0, then, since Θ≥0 and superquadratic, the second term on the L. H. S. of (49) is nonnegative and the integral inequality
(59)∫Ω1ξθΘλAlgθΔμ1θ≤∫Ω2ωηΘgηΔμ2ηλ,is valid.Remark 10.
By takingλ=1 in Corollary 9, inequality (59) reduces to (25).Remark 11.
For the Lebesgue scale measuresΔμ1θ=Δθ,Δμ2η=Δη and m=1, Corollary 9 coincides with Corollary 2.1.2 in [30].Remark 12.
Rewrite (49) with λ=qp−1≥1 such that 0<p≤q<∞ or −∞<p≤q<0; then
(60)∫Ω1ξθΘq/pAlgθΔμ1θ+qp∫Ω1∫Ω2ξθlθ,ηLθΘq/p−1AlgθΘgη−AlgθΔμ1θΔμ2η≤∫Ω2ωηΘgηΔμ2ηq/p.Remark 13.
Form=1, inequality (60) coincides with inequality (3.13) in ([28], Remark 3.5).Remark 14.
In Remark12, since Θ≥0, then the second term on the L. H. S. of (60) is nonnegative. Hence, (60) reduces to
(61)∫Ω1ξθΘq/pAlgθΔμ1θ≤∫Ω2ωηΘgηΔμ2ηq/p,which is a refinement of the Hardy-type inequality in ([27], Remark 2.1.4) and [6].In the following, we labeled some specific superquadratic functions starting with power functions.Theorem 15.
Assume (A1)–(A3) are satisfied. Ifgi:Ω2⟶ℝ1≤i≤m are Δμ2-integrable functions such that giΩ2⊂0,∞, then the inequality
(62)∫Ω1ξθ∑i=1mAlgipθrΔμ1θ+λ∫Ω1∫Ω2ξθlθ,ηLθ∑i=1mAkgipθλ−1∑i=1mgiη−AlgiθpΔμ1θΔμ2η≤∫Ω2ωη∑i=1mgiηpΔμ2ηλ,is valid, where p≥2 and
(63)Algiθ≔1Lθ∫Ω2lθ,ηgiηΔμ2η,θ∈Ω1.
If0<λ<1 and 1<p≤2, then (62) is reversed.Proof.
We get the result from Theorem5 by putting
(64)Θθ≔∑i=1mθip,
in (49).Remark 16.
Form=1, Theorem 15 reduces to Corollary 3.1 in [13]. In particular, for p=1 and λ=1, Theorem 15 reduces to Remark 3.11 in [13].Remark 17.
For the Lebesgue scale measuresΔμ1θ=Δθ,Δμ2η=Δη and m=1. Theorem 15 coincides with Corollary 2.1.5 in [30].Theorem 18.
Assume (A1)–(A3) are satisfied. Ifgi:Ω2⟶ℝ1≤i≤m are Δμ2-integrable functions such that giΩ2⊂0,∞, then the inequality
(65)∫Ω1ξθ∑i=1mexpAigiθ−Aigiθ−1λΔμ1θ+I≤∫Ω2ωη∑i=1mgiη−loggiη−1Δμ2ηλ,is valid, where
(66)I≔λ∫Ω1∫Ω2ξθlθ,ηLθ∑i=1mexploggiη−Algiθ−loggiη−Algiθ−1×∑i=1mexpAlgiθ−Algiθ−1λ−1Δμ1θΔμ2η,and
(67)Algiθ≔1Lθ∫Ω2lθ,ηloggiηΔμ2η,θ∈Ω1.
If0<λ<1, then (65) is reversed.Proof.
We get the result from Theorem5 by putting
(68)Θθ≔∑i=1mexpθi−θi−1,in (49) and with loggη instead of gη.Remark 19.
By takingm=1 in Theorem 18, inequality (65) reduces to inequality 3.16 in [28], Corollary 3.2.Remark 20.
Form=1 and λ=1, the relation (65) that is regarded as a generalization and a refinement of the Pólya-Knopp’s inequality which coincided with Remark 3.12 in [13].Theorem 21.
Assume (A1)–(A3) are satisfied. Ifgi:Ω2⟶ℝ1≤i≤m are Δμ2-integrable functions such that giΩ2⊂0,∞, then the inequality
(69)∫Ω1∑i=1mAigiθcoshAigiθ−sinhAigiθλξθΔμ1θ+λ∫Ω1∫Ω2ξθlθ,ηLθ∑i=1mAigiθcoshAigiθ−sinhAigiθλ−1×∑i=1mgiη−Aigiθcoshgiη−Aigiθ−sinhgiη−AigiθΔμ1θΔμ2η≤∫Ω2ωη∑i=1mgiη−coshgiη−sinhgiηΔμ2ηλ,is valid, where Algi is defined as in (63). If 0<λ<1, then (69) is reversed.Proof.
We get the result from Theorem5 by putting
(70)Θθ≔∑i=1mθicoshθi−sinhθi,in (49).Remark 22.
Forλ=1, Theorem 21 reduces to Theorem 2.5 in [14]. In particular, for m=1 and λ=1, Theorem 21 coincides with Corollary 2.6 in [14].Theorem 23.
Assume (A1)–(A3) are satisfied. Ifgi:Ω2⟶ℝ1≤i≤m are Δμ2-integrable functions such that giΩ2⊂0,∞, then the inequality.
(71)∫Ω1ξθln1+∑i=1mAigiθ−∑i=1mAigiθλΔμ1θ+λ∫Ω1∫Ω1ξθlθ,ηLθln1+∑i=1mAigiθ−∑i=1mAigiθλ−1×ln1+∑i=1mgiηAigiθ−∑i=1mgiηAigiθλΔμ1θΔμ2η≤∫Ω2ωηln1+∑i=1mgiη−∑i=1mgiηΔμ2ηλ,is valid, where Algi is defined as in (63). If 0<λ<1, then (71) is reversed.Proof.
We get the result from Theorem5 by putting
(72)Θθ≔∑i=1mθi2lnθi,in (49) with the assumption 0ln0=0.Remark 24.
Forλ=1, Theorem 23 reduces to Theorem 2.7 in [14]. In particular, for m=1 and λ=1, Theorem 23 coincides with Corollary 2.8 in [14].Theorem 25.
Assume (A1)–(A3) are satisfied. Ifgi:Ω2⟶ℝ1≤i≤m are Δμ2-integrable functions such that giΩ2⊂0,∞, then the inequality
(73)∫Ω1ξθln1+∑i=1mAigiθ−∑i=1mAigiθλΔμ1θ+λ∫Ω1∫Ω1ξθlθ,ηLθln1+∑i=1mAigiθ−∑i=1mAigiθλ−1×ln1+∑i=1mgiηAigiθ−∑i=1mgiηAigiθλΔμ1θΔμ2η≤∫Ω2ωηln1+∑i=1mgiη−∑i=1mgiηΔμ2ηλ,is valid, where Algi is defined as in (63). If 0<λ<1, then (73) is reversed.Proof.
We get the result from Theorem5 by taking
(74)Θθ≔ln1+∑i=1mθi−∑i=1mθi,in (49).Remark 26.
Forλ=1, Theorem 25 reduces to Theorem 2.9 in [14]. In particular, for m=1 and λ=1, Theorem 25 coincides with Corollary 2.10 in [14].Now, to wrap up this section, we consider yet another implementation of Theorem5 rigged with finite measure spaces.Corollary 27.
Let the supposition of Theorem5 be satisfied and denote ∫Ω1Δμ1θ=Ω1 and∫Ω2Δμ2θ=Ω2 such that Ω1,Ω2<∞: setting lθ,η and ξθ=1. Then, Lθ=∫Ω2Δμ2θ=Ω2 and
(75)ωη≔∫Ω11Ω2λΔμ1θ1/λ=1Ω2λ∫Ω1Δμ1θ1/λ=Ω11/λΩ2.
Hence, the following inequality(76)∫Ω1Θ1Ω2∫Ω2gηΔμ2ηλΔμ1θ+λΩ2∫Ω1∫Ω2Θ1Ω2∫Ω2gηΔμ2ηλ−1×Θgη−1Ω2∫Ω2gηΔμ2ηΔμ1θΔμ2η≤Ω1Ω2∫Ω2ΘgηΔμ2ηλ,is valid. If Θ is subquadratic and 0<λ<1, then (76) is reversed.Remark 28.
By takingm=1 in Corollary 27, inequality (76) reduces to inequality 3.19 in [28], Corollary 3.2.Remark 29.
For the Lebesgue scale measuresΔμ1θ=Δθ,Δμ2η=Δη and m=1, Corollary 27 coincides with Corollary 2.1.6 in [30].Remark 30.
ForT=ℝ,m=1, and λ=1, Corollary 27 reduces to Corollary 3.3 in [8].


## 4. Inequalities with Specific Time Scales

In this section, by selecting few different time scales, we get some consequential inequalities. More precisely, assume0≤α<β≤∞ are points in T and S1≔θ,η∈T:0≤α<η≤θ<β. Applying Theorem 5 to Ω2=Ω2=α,βT,Δμ1θ=Δθ, and Δμ2η=Δη, we get the following conclusion.Theorem 31.
Assume0≤α<β≤∞ and l:α,βT×α,βT⟶ℝ≥0 such as Lθ≔∫αθkθ,ηΔη<∞,θ∈α,βT
Suppose thatξθ:α,βT⟶ℝ and
(77)ωη≔∫ηβξθlθ,ηLθλΔθ1/λ<∞,η∈α,βT,where λ≥1. If Θ∈CKm,ℝ≥0 and is superquadratic, then
(78)∫αβξθΘλAlgθΔθ+λ∫αβ∫αθξθlθ,ηLθΘλ−1AlgθΘgη−AlgθΔθΔη≤∫αβωηΘgηΔηλ,is available for all nonnegative integrable functions g:α,βT⟶ℝm and for Alg:α,βT⟶ℝ defined as
(79)Algθ≔1Lθ∫αθlθ,ηgηΔη,θ∈α,βT.
If0<λ<1 and Θ are subquadratic, then (78) is reversed.Remark 32.
By takingm=1 and replacingξθ, ωη, and lθ,η, respectively, ξθ/θ−α,ωη/η−α, andlχS1θ,η where χS1 denotes the characteristic function over S1 in Theorem 31, inequality (78) reduces to inequality 4.1 in [28], Theorem 4.1.On the other hand, for0≤α<β≤∞, consider the set
(80)S2≔θ,η∈T:β<θ≤η<∞.Then, puttingΩ1=Ω2=β,∞T where T is a time scale, Δμ1θ=Δθ and Δμ2η=Δη. We obtain a dual form of Theorem 31 as follows.Theorem 33.
Suppose that0≤β<∞ξ~θ:β,∞T⟶ℝ≥0 and l~:β,∞T×β,∞T×β,∞T⟶ℝ≥0 such that
(81)L~θ≔∫θ∞l~θ,ηΔη<∞,θ∈β,∞T,ω~η≔∫βηξ~θl~θ,ηLθλΔθ1/λ<∞,η∈β,∞T,where λ≥1. If Θ∈CKmℝ≥0 and superquadratic, then
(82)∫β∞ξ~θΘλAlgθΔθ+λ∫β∞∫θ∞ξ~θl~θ,ηLθΘλ−1A~lgθΘgη−AlgθΔθΔη≤∫β∞ω~ηΘgηΔη,is available for all nonnegative Δη-integrable functions g:β,∞T⟶ℝm and for the operator A~lg:β,∞T⟶ℝ defined by
(83)A~lgθ≔1L~θ∫θ∞l~θ,ηgηΔη,θ∈β,∞T.
IfΘ is subquadratic and 0<λ<1, then (82) is reversed.Remark 34.
By takingm=1 and replacing ξ~θ,ω~η, and l~θ,η, respectively, by ξ~θ/θ−α,ω~η/η−α, and l~χS2θ,η where χS2 denotes the characteristic function over S2 in Theorem 33; inequality (82) reduces to inequality 4.7 in [28], Theorem 4.2.


## 5. Inequalities with Specific Kernels

In this section, we find some consequential inequalities of the Hardy type by selecting specific kernels and weight functions.Corollary 35.
Suppose that the assumptions of Theorem31 are satisfied only with
(84)lθ,η≔0,ifα≤η≤σθ≤β.
Define(85)Lθ≔∫ασθlθ,ηΔη>0,θ∈α,βT.
IfΘ∈CKm,ℝ≥0 and is superquadratic, then (78) is available for all nonnegative ∆η-integrable functions g:α,βT⟶ℝm defined as
(86)Algθ≔1Lθ∫ασlθ,ηgηΔη,θ∈α,βT.
IfΘ is subquadratic and 0<λ<1, then (78) is reversed.Corollary 36.
Assume that the assumptions of Theorem31 is satisfied only with
(87)lθ,η≔0,ifα≤σθ≤η≤β.
Define(88)Lθ≔∫σθβlθ,ηΔη>0,θ∈α,βT.
IfΘ∈CKm,ℝ≥0 and is superquadratic, then (78) is available for all nonnegative integrable functions g:α,βT⟶ℝm(89)Algθ≔1Lθ∫σθβlθ,ηgηΔη,θ∈α,βT.
IfΘ is subquadratic and 0<λ<1, then (78) is reversed.Corollary 37.
Assume that the assumptions of Theorem31 is satisfied only with l:α,βT×α,βT⟶ℝ defined as
(90)lθ,η≔1,if0≤α≤η<σθ≤β,0,otherwise,and ξθ:α,βT⟶ℝ; then Lθ≔∫ασθlθ,ηΔη=σθ−α,θ∈α,βT, and Algθ in this case is the classical Hardy and denoted by
(91)Hgθ≔1σθ−α∫ασθgηΔη,θ∈α,βT.
If we let(92)ωη≔∫ηβξθ1σθ−αλΔθ<∞,η∈α,βT,where λ≥1, then (78) became
(93)∫αβξθΘλ1σθ−α∫ασθgηΔηΔθ+λ∫αβ∫ηβΘλ−11σθ−α∫ασθgηΔηΘgη−1σθ−α∫ασθgηΔηξθσθ−αΔθΔη≤∫αβωηΘgηΔη.
IfΘ is subquadratic and 0<λ<1, then (93) is reversed.Remark 38.
Form=1 and replacing ξθ,ωη by ξθ/θ−α and ωη/η−α in (93), Corollary 37 coincides with Example 4.1 in [13].Remark 39.
By takingT=ℝ,α=0, and replacing ξθ,ωη by ξθ/θ andωη/η in (93), we have
(94)∫0βξθΘλθ−1∫0θgηdηdθθ+λ∫0β∫ηβΘλ−11θ∫0θgηdηΘgη−1θ∫0θgηdηξθθ2dθdη≤∫0βωηΘgηdηηλ,where
(95)ωη≔η∫ηβξθ1θλdθθ1/λ,η∈0,β.
IfΘ is subquadratic and 0<λ<1, then (94) is reversed, which is a refinement of 4.6 in [28], Remark 4.2.Corollary 40.
In Corollary37, if α=0 and ξθ=1/θ, then (93) reduces to
(96)∫0βΘλ1σθ∫0θgηdηdθθ+λ∫0β∫ηβΘλ−11σθ∫0σθgηΔηΘgη−1σθ∫0σθgηΔη1θσθΔθΔη≤∫0βωηΘgηΔηλ,where
(97)ωη≔∫ηβ1σθλΔθθ1/λ<∞,η∈α,βT.
Furthermore, ifβ=∞, then (96) becomes
(98)∫0∞Θλ1σθ∫0σθgηΔηΔθθ+λ∫0∞∫η∞Θλ−11σθ∫0σθgηΔηΘgη−1σθ∫0σθgηΔη1θσθΔθΔη≤∫0∞ωηΘgηΔηλ,where
(99)ωη≔∫η∞1σθλΔθθ1/λ<∞,η∈α,∞T.Remark 41.
Forλ=1, inequality (96) reduces to
(100)∫0βΘ1σθ∫0θgηΔηΔθθ+∫0β∫ηβΘgη−1σθ∫0σθgηΔη1θσθΔθΔη≤∫0βωηΘgηΔη,where
(101)ωη≔∫ηβΔθθσθ=1η−1β,η∈α,βT,while inequality (98) reduces to
(102)∫0∞Θ1σθ∫0σθgηΔηΔθθ+∫0∞∫η∞Θgη−1σθ∫0σθgηΔη1θσθΔθΔη≤∫0∞ΘgηΔηη.Example 3.
Considering Theorem33 with l:β,∞T×β,∞T⟶ℝ defined by
(103)lθ,η≔1/ησηifη≥θ0,otherwise,and ξθ:β,∞T⟶ℝ≥0, then
(104)Lθ≔∫θ∞lθ,ηΔη=∫θ∞1ηση=−∫θ∞1ηΔΔη=1θ,θ∈β,∞T.The operator Algθ is defined as
(105)Algθ≔θ∫θ∞1ησηgηΔη,θ∈β,∞T,and if we let
(106)ωη≔∫ηβθ−1θησηλΔθ1/λ<∞,η∈β,∞T,where λ≥1, then (82) became
(107)∫0∞Θλθ∫θ∞1ησηgηΔηΔθθ+λ∫β∞∫θ∞Θλ−1θ∫θ∞1ησηgηΔηΘgη−θ∫0∞1ησηgηΔη1ησηΔθΔη≤∫β∞ωηΘgηΔηλ.
IfΘ is subquadratic and 0<λ<1, then (107) is reversed.Remark 42.
Forλ=1, inequality (107) reduces to
(108)∫β∞Θθ∫β∞1ησηgηΔηΔθθ+λ∫β∞∫θ∞Θgη−θ∫0∞1ησηgηΔη1ησηΔθΔη≤∫β∞ωηΘgηΔη,where
(109)ωη≔1ηση∫ηβΔθ=1ησηβ−η.


## 6. Some Particular Cases

In this section, we obtain a popularization and a refinement of the classical inequality of the Hardy-Hilbert type (16) for numerous variables on time scales. It is clarified in the result below.Theorem 43.
Assume that the assumptions of Theorem31 are satisfied only with Ω1=Ω2=0,∞T,p>1,λ>0 and replace Δμ1θ and Δμ2η by the Lebesgue scale measure Δθ and Δη.
Furthermore, define(110)L1θ≔∫0∞θ/η−1/pθ+ηΔηandL2η≔∫0∞θ/η1−1/pθ+ηλΔθ1/λ.
Ifλ≥1 and p≥2, then
(111)∫0∞L1θλ1−p∫0∞gηθ+ηΔηλpΔθ+λ∫0∞∫0∞η−1/pL1λ−11−pθ∫0∞gηθ+ηΔηpλ−1×gηη1/p−1L1θ∫0∞gηθ+ηΔηpθp1−1θ+ηΔθΔη≤∫0∞L2ηgηΔηλ,is available for all nonnegative integrable Δη-integrable functions g:α,βT⟶ℝm. If 0<λ<1, then (111) is reversed.Proof.
Utilizingξθ≔L1θ/θλ and
(112)lθ,η≔θη−1/pθ+η,ifθ≠0,η≠0,θ+η≠00,otherwise,
in Theorem15, we obtain
(113)Lθ≔∫0∞θη−1/pθ+ηΔη=L1θ,ωη≔∫0∞ξθlθ,ηLθλΔθ≔∫0∞L1θθλlθ,ηLθλΔθλ1≔∫0∞lθ,ηθλΔθλ1≔η−1∫0∞θη1−1/pθ+ηλΔθλ1≔1η∫0∞θη1−1/pθ+ηλΔθλ1L2ηη,and the operator Algθ in this case is defined as
(114)Algθ≔1L1θ∫0∞θη−1/pθ+ηgηΔη.
UtilizingAlgθ in (62), we obtain
(115)∫0∞L1θθλ1L1θ∫0∞θη−1/pθ+ηgηΔηλpΔθ+λ∫0∞∫0∞L1θθλθη−1/pθ+ηL1θ1L1θ∫0∞θη−1/pθ+ηgηΔηpλ−1×gη−1L1θ∫0∞θη−1/pθ+ηgηΔηpΔθΔη≤∫0∞K2ηηgpηΔηλ.
Hence,(116)∫0∞L1θλ1−p∫0∞gηη−1/pθ+ηΔηλpΔθ+λ∫0∞∫0∞L1λ−11−pθ∫0∞gηη−p1θ+ηΔηpλ−1×gη−1L1θ1θ∫0∞gηη−p−1θ+ηΔηpη−p−1θ+ηθ−11−p−1ΔθΔη≤∫0∞K2ηηgpηΔηλ.
Finally, replacinggη by gηη1/p in (116), we get (111). The cases 0<λ<1 and 1<p≤2 are proved in the same way.Remark 44.
Form=1, Theorem 43 reduces to Theorem 5.1 in [13]. In particular, for λ=1, Theorem 43 is a refinement of Theorem 5.5 in [10].Remark 45.
By takingT=ℝ,λ=1, and p≥2, in Theorem 43 and utilizing the known fact that
(117)∫0∞θη−1/pθ+ηdη=∫0∞θη1−1/pθ+ηdθ=πsinpπ,then (111) becomes
(118)∫0∞∫0∞gηθ+ηdηpdθ+πsinπp−1p−1∫0∞∫0∞η−p−1gηηp−1−sinπp−1πθp−1∫0∞gηθ+ηdηpθp−1−1θ+ηdθdη≤πsinπp−1p∫0∞gpηdη,which is a refinement of (16). For m=1, (118) has been established in [3], Corollary 3.2.In the following theorem, we introduce a generalized form of (111) on time scales.Theorem 46.
Suppose thatλ>0,p>1 and s,δ∈ℝ. Furthermore, assume
(119)∫0∞θδη/θs−2/p+1θ+ηsλΔθ1/λandL1θ≔∫0∞ηθ−1s−2/pθ+ηsΔη,
whereλ≥1 and p≥2; then
(120)∫0∞L1θλ1−pθλδ−s+1∫0∞gηθ+ηsΔηpλ+λ∫0∞∫0∞ηp2−sL1λ−11−pθ∫0∞gηθ+ηsΔηpλ−1×gηηp2−s−1L1θθp2−s∫0∞gηθ+ηsΔηpθpλ+s−21+pλ−p/pθ+ηsΔθΔη≤∫0∞L2ηgpηΔηλ,is available for all nonnegative integrable functions g:α,βT⟶ℝm. If 0<λ<1 and 1<p≤2, then (120) is reversed.Proof.
Rewrite (62) in Theorem 15 with Ω1=Ω2=0,∞T,Δμ1θ=Δθ, and Δμ2η=Δη. Let us define ξθ≔L1θθδ−1λ and
(121)lθ,η≔θηs−2/pθ+ηs,ifθ≠0,η≠0,θ+η≠00,otherwise.
We have(122)Lθ≔∫0∞θηs−2/pθ+ηsΔη=L1θ,ωη≔∫0∞ξθlθ,ηLθΔθλ−1≔∫0∞L1θθδθlθ,ηL1θΔθλ−1≔∫0∞L1θθδθlθ,ηL1θΔθλ−1≔1η∫0∞θδθηPs−2θθ+ηsgηΔθλ−1≔1η∫0∞θδθηPs−2+1θ+ηsλΔθλ−1≔L2ηη,and the operator Algθ in this case is defined as
(123)Algθ≔1L1θ∫θ∞θηs−2/pθ+ηsgηΔη.
Now, substitutingL,ω and Algθ in (62), we get
(124)∫0∞L1θθδ−1λ1L1θs∫0∞θηs−2/pθ+ηsgηΔθpλΔθ+λ∫0∞∫0∞L1θθδ−1λθηs−2/pθ+ηsL1θ1L1θ∫0∞θηs−2/pθ+ηsgηΔθpλ−1×gη−1L1θ∫0∞θηs−2/pθ+ηsgηΔηpΔθΔη≤∫0∞L2ηηgpηΔηλ.
Hence,(125)∫0∞L1θλ1−pθδλ1θλs−1∫0∞gηs−2/pθ+ηsΔηpλ+λ∫0∞∫0∞L1λ−11−pθ∫0∞gηs−2/pθ+ηsΔηpλ−1×gη−1L1θθps−2∫0∞ηs−2/pgηθ+ηsΔηpηps−2θ+ηs1θθpλ+s−21+pλ−p/pθ+ηsΔθΔη≤∫0∞L2ηgpηΔηλ.
Finally, considering (125) with gηη2−s/p instead of gη, we obtain (120). The cases 0<λ<1 and 1<p≤2 are proved in the same way.Remark 47.
Form=1, Theorem 46 coincides with Theorem 5.2 in [13].Remark 48.
Clearly, forp>1,δ=0, and s=1, Theorem 46 reduces to Theorem 43.


## 7. Conclusion and Future Work

The study of dynamic inequalities on time scales has a lot of scope. This research article is devoted to some general Hardy-type dynamic inequalities and their converses on time scales. Inequalities are considered in rather general forms and contain several special integral inequalities. In particular, our findings can be seen as refinements of some recent results closely linked to the time-scale inequalities of the classical Hardy, Pólya-Knopp, and Hardy-Hilbert. 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. The performance of the superquadratic method for functions is reliable and effective to obtain new dynamic inequalities on time scales. This method has more advantages: it is direct and concise. Thus, the proposed method can be extended to some forms for Hardy’s and related dynamic inequalities in mathematical and physical sciences. Our computed outcomes can be very useful as a starting point to get some continuous inequalities, especially from the obtained dynamic inequalities. In the future, we will get some discrete inequalities from the main results. Also, we will suppose thatgt≔g1t,⋯,gt is an m-tuple of functions and t=t1,t2,⋯,tn is n-tuple of variables to get the general forms of Hardy’s and related inequalities on time scales. Similarly, in the future, we can present such inequalities by using Riemann-Liouville-type fractional integrals and fractional derivatives on time scales. It will also be very interesting to present such inequalities on quantum calculus.



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