Hardy-Type Inequalities on Time Scale via Convexity in Several Variables

Abstract

We extend some Hardy-type inequalities with general kernels to arbitrary time scales using multivariable convex functions. Some classical and new inequalities are deduced seeking applications.

1. Introduction

The significant Hardy inequality is published in [1] (1952) (both in the continuous and discrete settings). More general Hardy integral inequalities have been studied in continuous cases. We notice only [2–6] and the references therein.

Recently, this inequality is studied in discrete case, and some variants of it are proved in case of time scales [7–9].

In [10], the authors study Hardy-type inequalities using convex functions of one variable with general kernels to arbitrary time scales.

The aim of this paper is to provide Hardy-type inequalities using multivariable convex function with general kernels to arbitrary time scales. Notice that time scales include continuous and discrete time cases under unified approach.

Firstly, we recall necessary preliminary facts needed afterward. The main results are given in Section 3. Section 4 is devoted to some inequalities with certain kernels. In the last section, we discuss some particular cases of Hardy-type inequalities.

2. Preliminaries

First, we recall the basic concepts used in the paper and refer the interested reader to [11] for the theory of time scales. A time scale is any nonempty closed subset of the real line 𝕋⊂ℝ. On nondensity points we define forward, respectively, backward jump operators 𝜎,𝜌∶𝕋→𝕋 as 𝜎(𝑡){},𝜌(𝑡){}.=inf𝑠∈𝕋∶𝑠>𝑡=sup𝑠∈𝕋∶𝑠<𝑡 (1) The point 𝑡 is said to be right-scattered if 𝜎(𝑡)>𝑡 and left-scattered if 𝜌(𝑡)<𝑡, respectively. Clearly, 𝑡 is right-dense if 𝜎(𝑡)=𝑡 and left dense if 𝜌(𝑡)=𝑡, respectively.

Let 𝑛∈ℕ; we define 𝑛-dimensional time scale by the Cartesian product of given time scales 𝕋_𝑖, 𝑖∈{1,…,𝑛}, as Ω^𝑛=𝑎=𝑎₁,𝑎₂,…,𝑎_𝑛∶𝑎_𝑖∈𝕋_𝑖,𝑖∈{1,…,𝑛}. (2) Evidently, Ω^𝑛 is a complete metric space with distance as follows: 𝑑𝑎,𝑏=^𝑛_𝑖=1𝑏_𝑖−𝑎_𝑖^21/2,𝑎,𝑏∈Ω^𝑛. (3) Now we are going to describe the construction of Lebesgue measure in Ω^𝑛. We refer to [12–14] for the theory of measure spaces and measurable functions on time scales.

Let ℱ be the family of all 𝑛-dimensional time scale intervals in Ω^𝑛; that is, 𝑉=𝑎,𝑏=𝑎₁,𝑏₁×𝑎₂,𝑏₂×⋯×𝑎_𝑛,𝑏_𝑛, (4) with 𝑎=(𝑎₁,𝑎₂,…,𝑎_𝑛), 𝑏=(𝑏₁,𝑏₂,…,𝑏_𝑛)∈Ω^𝑛, and 𝑎_𝑖≤𝑏_𝑖 for all 𝑖∈{1,…,𝑛}. Let 𝑚∶ℱ→[0,∞) be the set function that assigns to each 𝑛-dimensional time scale interval 𝑉=[𝑎,𝑏) its volume as follows: 𝑚(𝑉)=^𝑛_𝑖=1𝑏_𝑖−𝑎_𝑖. (5) Let ∅≠𝐸⊂Ω^𝑛. If there exists finite or countable system of pairwise disjoint 𝑛-dimensional time scale intervals 𝑉_𝑘=[𝑎_𝑘,𝑏_𝑘) with 𝐸⊂⋃_𝑘≥1𝑉_𝑘, then the outer measure 𝑚^∗ of 𝐸 is defined by 𝑚^∗(𝐸)=inf_𝑘≥1𝑚𝑉_𝑘,𝐸⊂_𝑘≥1𝑉_𝑘. (6) If there is no such covering of 𝐸, then 𝑚^∗(𝐸)=∞.

A subset 𝐴 of Ω^𝑛 is said to be measurable (or Δ-measurable) if 𝑚^∗(𝐸)=𝑚^∗(𝐸∩𝐴)+𝑚^∗𝐸∩𝐴^𝑐 (7) holds for all 𝐸⊂Ω^𝑛, where 𝐴^𝑐=Ω^𝑛⧵𝐴. The family ℳ of all 𝑚^∗-measurable subsets of Ω^𝑛 is a 𝜎-algebra generated by ℱ. The restriction of 𝑚^∗ to ℳ, which we denote by 𝜇_Δ, is a 𝜎-additive measure on ℳ. Clearly, ℱ⊂ℳ and 𝜇_Δ(𝑉)=𝑚(𝑉) for each 𝑉∈ℱ. The measure 𝜇_Δ (called the Lebesgue Δ-measure on Ω^𝑛) is Carathéodory extension of the original measure 𝑚 defined on ℱ. We call (Ω^𝑛,ℳ,𝜇_Δ) an 𝑛-dimensional time scale measure space.

Denote 𝑎_𝑖=inf_{𝑡∈𝕋𝑖}𝑡. Similarly, 𝑏̃_𝑖=sup_{𝑡∈𝕋𝑖}𝑡. Let Ω^𝑛_∞ be the set of all points 𝑏=(𝑏₁,𝑏₂,…,𝑏_𝑛)∈Ω^𝑛 for which there exists at least one 𝑏_𝑖 such that 𝑏_𝑖=𝑏̃_𝑖. From [11, Theorem 3.1], we know that if 𝑡⃗=(𝑡₁,𝑡₂,…,𝑡_𝑛)∈Ω^𝑛⧵Ω^𝑛_∞, then the single-point set {𝑡} is Δ-measurable and 𝜇_Δ({𝑡})=^𝑛_𝑖=1𝜎_𝑖𝑡_𝑖−𝑡_𝑖. (8) Obviously, for all 𝑖∈{1,…,𝑛}, the set Ω^𝑛₀=Ω^𝑛⧵Ω^𝑛_∞ can be represented as a finite or countable union of intervals of the family ℱ; hence it is Δ-measurable. Furthermore, the set Ω^𝑛_∞=Ω^𝑛⧵Ω^𝑛₀ is Δ-measurable being the difference of two Δ-measurable sets Ω^𝑛 and Ω^𝑛₀, but Ω^𝑛_∞ does not have a finite or countable covering intervals of ℱ, that is, for any Δ-measurable subset 𝐴 of Ω^𝑛 such that 𝐴∩Ω_∞≠∅ has Δ-measure ∞. In particular, if 𝑎∈𝕋, where 𝕋 is an arbitrary time scale, then the set [𝑎,∞)={𝑡∈𝕋∶𝑎≤𝑡} is Δ-measurable.

The function 𝑓∶Ω^𝑛→ℝ∶=[−∞,∞] is said to be Δ-measurable if for every 𝛼∈ℝ, the set 𝑓⁻¹([−∞,𝛼))={𝑡=(𝑡₁,…,𝑡_𝑛)∈Ω^𝑛∶𝑓(𝑡)<𝛼} is Δ-measurable. It is easy to see that 𝑓 is Δ-measurable if and only if for each open set 𝐺⊂ℝ, the set 𝑓⁻¹(𝐺)={𝑡∈Ω^𝑛∶𝑓(𝑡)∈𝐺} is Δ-measurable. Moreover, if 𝑓∶Ω^𝑛→ℝ is Δ-measurable and Φ∶𝐼⊂ℝ→ℝ is a continuous function, then Φ∘𝑓∶Ω^𝑛→ℝ is Δ-measurable.

Having the 𝜎-additive measure 𝜇_Δ on Ω^𝑛, we possess the corresponding integration theory for functions 𝑓∶𝐸⊂Ω^𝑛→ℝ, according to the general Lebesgue integration theory (see, e.g., [15]). The Lebesgue integral associated with the measure 𝜇_Δ on Ω^𝑛 is called the Lebesgue Δ-integral. For a Δ-measurable set 𝐸⊂Ω^𝑛 and a Δ-measurable function 𝑓∶𝐸→ℝ, the corresponding Δ-integral of 𝑓 over 𝐸 will be denoted by _𝐸𝑓𝑡₁,…,𝑡_𝑛Δ𝑡₁⋯Δ𝑡_𝑛or_𝐸𝑓(𝑡)Δ𝑡. (9) Notice that all theorems of the general Lebesgue integration theory, including the Lebesgue dominated convergence theorem, hold also for Lebesgue Δ-integrals on Ω^𝑛.

If 𝕋 is a time scale and the interval [𝑎,𝑏)⊂𝕋 contains only isolated points, then ^𝑏_𝑎𝑓(𝑡)Δ𝑡=_{𝑡∈[𝑎,𝑏)}(𝜎(𝑡)−𝑡)𝑓(𝑡). (10)

Let (Ω,ℳ,𝜇_Δ) and (Λ,ℒ,𝜆_Δ) be two finite dimensional time scale measure spaces. We define the product measure space (Ω×Λ,ℳ×ℒ,𝜇_Δ×𝜆_Δ), where ℳ×ℒ is the product 𝜎-algebra generated by {𝐸×𝐹∶𝐸∈ℳ,𝐹∈ℒ} and 𝜇_Δ×𝜆_Δ(𝐸×𝐹)=𝜇_Δ(𝐸)𝜆_Δ(𝐹). (11)Fubini's Theorem in Time Scales see ([16, Theorem 1.1]). If 𝑓∶Ω×Λ→ℝ is a 𝜇_Δ×𝜆_Δ-integrable function and if we define the function 𝜑(𝑦)=∫_Ω𝑓(𝑥,𝑦)Δ𝑥 for a.e. 𝑦∈Λ and 𝜓(𝑥)=∫_Λ𝑓(𝑥,𝑦)Δ𝑦 for a.e. 𝑥∈Ω, then 𝜑 is 𝜆_Δ-integrable on Λ, 𝜓 is 𝜇_Δ-integrable on Ω, and _ΩΔ𝑥_Λ𝑓𝑥,𝑦Δ𝑦=_ΛΔ𝑦_Ω𝑓𝑥,𝑦Δ𝑥. (12)

3. Inequalities with General Kernels

Let 𝐟(𝑦)=(𝑓₁(𝑦),…,𝑓_𝑚(𝑦)) be 𝑚-tuple of functions such that 𝑓_𝑗(𝑦) are 𝜆_Δ-integrable for all 𝑗∈{1,…,𝑚}. Then ∫_Λ𝐟(𝑦)Δ(𝑦) denotes the 𝑚-tuple (∫_Λ𝑓₁(𝑦)Δ(𝑦),…,∫_Λ𝑓_𝑚(𝑦)Δ(𝑦)); that is, Δ-integral acts on each component of 𝐟(𝑦).

The following Jensen's inequality on time scales is given in [17, Theorem 2.1].

Theorem 1. Let (Ω,ℳ,𝜇_Δ) and (Λ,ℒ,𝜆_Δ) be two time scale measure spaces. Suppose 𝑈⊂ℝ^𝑚 is a closed convex set, Φ∈𝐶(𝑈,ℝ) is convex, and 𝐟(Λ)⊂𝑈. Moreover, let 𝑘∶Ω×Λ→ℝ be nonnegative such that 𝑘(𝑥,⋅) is 𝜆_Δ-integrable. Then one has Φ∫_Λ𝑘𝑥,𝑦𝐟𝑦Δ𝑦∫_Λ𝑘≤∫𝑥,𝑦Δ𝑦^Λ𝑘𝑥,𝑦Φ𝐟𝑦Δ𝑦∫_Λ𝑘𝑥,𝑦Δ𝑦. (13)

Throughout this section, we assume that the following hypotheses hold.H1(Ω,ℳ,𝜇_Δ) and (Λ,ℒ,𝜆_Δ) are two time scale measure spaces.H2𝑘∶Ω×Λ→ℝ₊ is such that 𝐾(𝑥)∶=∫_Λ𝑘(𝑥,𝑦)Δ𝑦<∞,𝑥∈Ω.H3𝜉∶Ω→ℝ₊ is such that 𝑤(𝑦)∶=∫_Ω(𝑘(𝑥,𝑦)𝜉(𝑥)/𝐾(𝑥))Δ𝑥,𝑦∈Λ.

Theorem 2. If 𝑈⊂ℝ^𝑚 is a closed convex set such that Φ∶𝑈→ℝ is convex and continuous, then _Ω𝜉(𝑥)Φ1𝐾(𝑥)_Λ𝑘𝐟𝑦≤𝑥,𝑦Δ𝑦Δ𝑥_Λ𝑤𝑦Φ𝐟𝑦Δ𝑦 (14) holds for all 𝜆_Δ-integrable functions 𝐟∶Λ→ℝ^𝑚 such that 𝐟(Λ)⊂𝑈.

Proof. Using Jensen's inequality (13) for several variables and the Fubini theorem on time scales, we find that _Ω𝜉(𝑥)Φ1𝐾(𝑥)_Λ𝑘𝐟𝑦=𝑥,𝑦Δ𝑦Δ𝑥_Ω𝜉(𝑥)Φ1𝐾(𝑥)_Λ𝑘𝑥,𝑦𝑓₁𝑦1Δ𝑦,…,𝐾(𝑥)×_Λ𝑘𝑥,𝑦𝑓_𝑚𝑦≤Δ𝑦Δ𝑥_Ω𝜉(𝑥)𝐾(𝑥)_Λ𝑘Φ𝐟𝑦=𝑥,𝑦Δ𝑦Δ𝑥_ΛΦ𝐟𝑦_Ω𝑘𝑥,𝑦𝜉(𝑥)𝐾(𝑥)=Δ𝑥Δ𝑦_Λ𝑤𝑦Φ𝐟𝑦Δ𝑦. (15) The proof is therefore complete.

Remark 3. If Φ is concave, then (14) holds in reverse direction.

Corollary 4. Let Ψ∶[𝑙₁,𝑙₁́)×⋯×[𝑙_𝑚,𝑙_𝑚́)→ℝ₊ be continuous function and define 𝐿_𝑗𝑓_𝑗,Λ=𝐿_𝑗⁻¹∫_Λ𝑘𝑥,𝑦𝐿_𝑗𝑓_𝑗𝑦Δ𝑦𝐾(𝑥) (16) for all 𝑗∈{1,…,𝑚}. If Φ(𝑠₁,…,𝑠_𝑚)=Ψ(𝐿₁⁻¹(𝑠₁),…,𝐿_𝑚⁻¹(𝑠_𝑚)) is convex, then _Ω𝜉(𝑥)Ψ𝐿₁𝑓₁,Λ,…,𝐿_𝑚𝑓_𝑚≤,ΛΔ𝑥_Λ𝑤𝑦Ψ𝑓₁𝑦,…,𝑓_𝑚𝑦Δ𝑦 (17) holds for all 𝑓_𝑗(Λ)⊂[𝑙_𝑗,𝑙_𝑗́) and continuous monotone functions 𝐿_𝑗∶[𝑙_𝑗,𝑙_𝑗́)→ℝ such that 𝐿_𝑗∘𝑓_𝑗 are 𝜆_Δ-integrable for all 𝑗∈{1,…,𝑚}.

Proof. Replace in Theorem 2  𝑓_𝑗(𝑦) with 𝐿_𝑗(𝑓_𝑗(𝑦)) for all 𝑗∈{1,…,𝑚} and Φ(𝑠₁,…,𝑠_𝑚) with Ψ(𝐿₁⁻¹(𝑠₁),…,𝐿_𝑚⁻¹(𝑠_𝑚)).

Remark 5. In case that 𝕋=ℕ and 𝑚=2, Corollary 4 is as [18, Corollary 1.2].

Remark 6. In case that 𝑚=2, we can use the results of Beck [19] (see also [20], page 194) as applications of Corollary 4, which corresponds to the generalizations of Hölder’s and Minkoski's inequalities. In classical case, many authors have studied these types of generalizations; see, for example, [21–23].

Furthermore, in the paper, we use 𝑝>1 with (1/𝑝)+(1/𝑝)́=1.

Corollary 7. If 𝐟(𝑥)=(𝑓₁(𝑥),𝑓₂(𝑥)), then _Ω𝜉(𝑥)1𝐾(𝑥)_Λ𝑘(𝑥,𝑦)(𝑓₁(𝑦))^𝑝Δ𝑦^1/𝑝×1𝐾(𝑥)_Λ𝑘(𝑥,𝑦)(𝑓₂(𝑦))^́𝑝Δ𝑦^́𝑝1/≥Δ𝑥_Λ𝑤𝑦𝑓₁𝑦𝑓₂𝑦Δ𝑦 (18) holds for all 𝜆_Δ-integrable 𝑓_𝑗∶Λ→ℝ₊, where 𝑗∈{1,2}.

Proof. Use 𝑚=2,Ψ(𝑠₁,𝑠₂)=𝑠₁𝑠₂,𝐿₁(𝑡₁)=(𝑡₁)^𝑝,𝐿₂(𝑡₂)=(𝑡₂)^́𝑝 in Corollary 4; then Φ(𝑠₁,𝑠₂)=𝑠₁^1/𝑝𝑠^́𝑝₂^1/ is concave in Theorem 2.

Corollary 8. If 𝐟(𝑥)=(𝑓₁(𝑥),𝑓₂(𝑥)), then _Ω𝜉(𝑥)1𝐾(𝑥)_Λ𝑘(𝑥,𝑦)(𝑓₁(𝑦))^𝑝Δ𝑦^1/𝑝+1𝐾(𝑥)_Λ𝑘(𝑥,𝑦)(𝑓₂(𝑦))^𝑝Δ𝑦^1/𝑝𝑝≥Δ𝑥_Λ𝑤𝑦(𝑓₁(𝑦)+𝑓₂(𝑦))^𝑝Δ𝑦 (19) holds for all 𝜆_Δ-integrable 𝑓_𝑗∶Λ→ℝ₊, where 𝑗∈{1,2}.

Proof. Use 𝑚=2, Ψ(𝑠₁,𝑠₂)=(𝑠₁+𝑠₂)^𝑝, 𝐿₁(𝑡₁)=(𝑡₁)^𝑝, 𝐿₂(𝑡₂)=(𝑡₂)^𝑝 in Corollary 4; then Φ(𝑠₁,𝑠₂)=(𝑠₁^1/𝑝+𝑠₂^1/𝑝)^𝑝 is concave in Theorem 2.

Remark 9. If 𝑝<1, then reverse inequalities hold in Corollaries 4, 7, and 8.

4. Inequalities with Special Kernels

Throughout this section and in the next section, we assume that the following hypothesis holds.H4Ω=Λ=[𝑎₁,𝑏₁)_𝕋×[𝑎₂,𝑏₂)_𝕋×⋯×[𝑎_𝑛,𝑏_𝑛)_𝕋, 0≤𝑎_𝑖<𝑏_𝑖≤∞ for all 𝑖∈{1,…,𝑛}, where 𝕋 is an arbitrary time scale.

Corollary 10. Assume that 𝜉∶Ω→ℝ₊𝑤𝑦=issuchthat_𝑏1𝑦1⋯^𝑏𝑛_𝑦𝑛𝑘𝑥₁,…,𝑥_𝑛,𝑦𝜉𝑥₁,…,𝑥_𝑛𝐾𝑥₁,…,𝑥_𝑛×Δ𝑥₁⋯Δ𝑥_𝑛,𝑦∈Λ. (20) If 𝑈⊂ℝ^𝑚 is a closed convex set such that Φ∶𝑈→ℝ is convex and continuous; then ^𝑏1_𝑎1⋯^𝑏𝑛_𝑎𝑛𝜉𝑥₁,…,𝑥_𝑛Φ𝐴_𝑘𝐟𝑥₁,…,𝑥_𝑛Δ𝑥₁⋯Δ𝑥_𝑛≤^𝑏1_𝑎1⋯^𝑏𝑛_𝑎𝑛𝑤𝑦₁,…,𝑦_𝑛Φ𝐟𝑦₁,…,𝑦_𝑛×Δ𝑦₁⋯Δ𝑦_𝑛 (21) holds for all 𝜆_Δ-integrable 𝐟∶Λ→ℝ^𝑚 such that 𝐟(Λ)⊂𝑈, where 𝐴_𝑘𝐟(𝑥)=1𝐾(𝑥)^𝜎(𝑥1)_𝑎1⋯^{𝜎(𝑥𝑛)}_𝑎𝑛𝑘𝑥,𝑦₁,…,𝑦_𝑛×𝐟𝑦₁,…,𝑦_𝑛Δ𝑦₁⋯Δ𝑦_𝑛. (22)

Proof. Statement follows from Theorem 2 using 𝑘𝑥₁,…,𝑥_𝑛,𝑦₁,…,𝑦_𝑛=0,if𝑎_𝑖≤𝑦_𝑖≤𝜎𝑥_𝑖forall𝑖∈{1,…,𝑛}, (23) since in this case 𝐾(𝑥)=^𝜎(𝑥1)_𝑎1⋯^{𝜎(𝑥𝑛)}_𝑎𝑛𝑘𝑥,𝑦₁,…,𝑦_𝑛Δ𝑦₁⋯Δ𝑦_𝑛. (24)

Corollary 11. Assume that 𝜉∶Ω⟶ℝ₊𝑤𝑦=issuchthat_𝑦1𝑎1⋯^𝑦𝑛_𝑎𝑛𝑘𝑥₁,…,𝑥_𝑛,𝑦𝜉𝑥₁,…,𝑥_𝑛𝐾𝑥₁,…,𝑥_𝑛×Δ𝑥₁⋯Δ𝑥_𝑛,𝑦∈Λ. (25) If 𝑈⊂ℝ^𝑚 is a closed convex set such that Φ∶𝑈→ℝ is convex and continuous, then (21) holds for all 𝜆_Δ-integrable 𝐟∶Λ→ℝ^𝑚 such that 𝐟(Λ)⊂𝑈, and 𝐴_𝑘𝐟(𝑥)=1𝐾(𝑥)^𝑏1_𝜎𝑥1⋯^𝑏𝑛_𝜎𝑥𝑛𝑘𝑥,𝑦₁,…,𝑦_𝑛×𝐟𝑦₁,…,𝑦_𝑛Δ𝑦₁⋯Δ𝑦_𝑛. (26)

Proof. Statement follows from Theorem 2 using 𝑘𝑥₁,…,𝑥_𝑛,𝑦₁,…,𝑦_𝑛=0,if𝑎_𝑖≤𝜎𝑥_𝑖≤𝑦_𝑖forall𝑖∈{1,…,𝑛}, (27) since in this case 𝐾(𝑥)=^𝑏1_𝜎(𝑥1)⋯^𝑏𝑛_{𝜎(𝑥𝑛)}𝑘𝑥,𝑦₁,…,𝑦_𝑛Δ𝑦₁⋯Δ𝑦_𝑛. (28)

Theorem 12. Assume that 𝜉∶Ω→ℝ₊𝑤𝑦=issuchthat_𝑏1𝑦1⋯^𝑏𝑛_𝑦𝑛𝜉𝑥₁,…,𝑥_𝑛∏^𝑛_𝑖=1𝜎𝑥_𝑖−𝑎_𝑖Δ𝑥₁⋯Δ𝑥_𝑛,𝑦∈Λ. (29) If 𝑈⊂ℝ^𝑚 is a closed convex set such that Φ∶𝑈→ℝ is convex and continuous, then ^𝑏1_𝑎1⋯^𝑏𝑛_𝑎𝑛𝜉𝑥₁,…,𝑥_𝑛Φ𝐴_𝑘𝐟𝑥₁,…,𝑥_𝑛Δ𝑥₁⋯Δ𝑥_𝑛≤^𝑏1_𝑎1⋯^𝑏𝑛_𝑎𝑛𝑤𝑦₁,…,𝑦_𝑛Φ𝐟𝑦₁,…,𝑦_𝑛×Δ𝑦₁⋯Δ𝑦_𝑛 (30) holds for all 𝜆_Δ-integrable 𝐟∶Λ→ℝ^𝑚 such that 𝐟(Λ)⊂𝑈, where 𝐴_𝑘𝐟𝑥₁,…,𝑥_𝑛=1∏^𝑛_𝑖=1𝜎𝑥_𝑖−𝑎_𝑖×^𝜎(𝑥1)_𝑎1⋯^{𝜎(𝑥𝑛)}_𝑎𝑛𝐟̃𝑦₁,…,𝑦_𝑛×Δ𝑦₁⋯Δ𝑦_𝑛. (31)

Proof. Statement follows from Theorem 2 using 𝑘𝑥₁,…,𝑥_𝑛,𝑦₁,…,𝑦_𝑛=1if𝑎_𝑖≤𝑦_𝑖<𝜎𝑥_𝑖≤𝑏_𝑖{},,𝑖∈1,…,𝑛0otherwise, (32) since in this case 𝐾𝑥₁,…,𝑥_𝑛=^𝜎(𝑥1)_𝑎1⋯^{𝜎(𝑥𝑛)}_𝑎𝑛Δ𝑦₁⋯Δ𝑦_𝑛=^𝑛_𝑖=1𝜎𝑥_𝑖−𝑎_𝑖. (33) Thus 𝐴_𝑘=𝐴_𝑘,𝑤=𝑤.

Corollary 13. If 𝑎_𝑖=0 for all 𝑖∈{1,…,𝑛} in 𝐇𝟒 and 𝑈⊂ℝ^𝑚 is a closed convex set such that Φ∶𝑈→ℝ is convex and continuous, then ^𝑏1₀⋯^𝑏𝑛₀Φ𝐴_𝑘𝐟𝑥₁,…,𝑥_𝑛Δ𝑥₁⋯Δ𝑥_𝑛𝑥₁⋯𝑥_𝑛≤^𝑏1₀⋯^𝑏𝑛₀^𝑛_𝑖=11𝑦_𝑖−1𝑏_𝑖Φ𝐟𝑦₁,…,𝑦_𝑛Δ𝑦₁⋯Δ𝑦_𝑛 (34) holds for all 𝜆_Δ-integrable 𝐟∶Λ→ℝ^𝑚 such that 𝐟(Λ)⊂𝑈, where 𝐴_𝑘𝐟𝑥₁,…,𝑥_𝑛1∶=∏^𝑛_𝑖=1𝜎𝑥_𝑖×^𝜎(𝑥1)₀⋯^{𝜎(𝑥𝑛)}₀𝐟𝑦₁,…,𝑦_𝑛×Δ𝑦₁⋯Δ𝑦_𝑛. (35)

Proof. The statement follows from Theorem 12 using 𝜉(𝑥₁,…,𝑥_𝑛)=1/𝑥₁⋯𝑥_𝑛, since in this case 𝑤𝑦₁,…,𝑦_𝑛=^𝑏1_𝑦1⋯^𝑏𝑛_𝑦𝑛1∏^𝑛_𝑖=1𝑥_𝑖𝜎𝑥_𝑖Δ𝑥₁⋯Δ𝑥_𝑛=^𝑛_𝑖=11𝑦_𝑖−1𝑏_𝑖. (36)

Example 14. If 𝑏_𝑖=∞ for all 𝑖∈{1,…,𝑛} in addition to the assumptions of Corollary 13, then (34) takes the form ^∞₀⋯^∞₀Φ𝐴_𝑘𝐟𝑥₁,…,𝑥_𝑛Δ𝑥₁⋯Δ𝑥_𝑛𝑥₁⋯𝑥_𝑛≤^∞₀⋯^∞₀Φ𝐟𝑦₁,…,𝑦_𝑛Δ𝑦₁⋯Δ𝑦_𝑛𝑦₁⋯𝑦_𝑛. (37)

Remark 15. Clearly, if the left-hand side is ∞ in (34), then right-hand side is also ∞.

Remark 16. For 𝕋=ℝ and 𝑚=1, inequality (37) is proved in [4, 24].

5. Some Particular Cases

In this section, firstly, we give Hilbert-type inequality on time scales.

Theorem 17. If 𝑛=1, 𝑎₁=0,𝑎𝑛𝑑𝑏₁=∞ in 𝐇𝟒 and for 𝑞>1 one defines 𝐾₁(𝑥)=^∞₀(𝑦/𝑥)^−1/𝑞𝐾𝑥+𝑦Δ𝑦,₂𝑦=^∞₀(𝑦/𝑥)^1−1/𝑞𝑥+𝑦Δ𝑥, (38) then ^∞₀^∞₀𝑦/𝑥^1−1/𝑞𝑔₁𝑦^𝑝𝑥+𝑦Δ𝑦^1/𝑝×^∞₀(𝑦/𝑥)^1−1/𝑞(𝑔₂(𝑦))^𝑝′𝑥+𝑦Δ𝑦^1/𝑝′≥Δ𝑥_∞0𝐾₂𝑦𝑔₁𝑦𝑔₂𝑦Δ𝑦 (39) holds for all 𝜆_Δ-integrable 𝑔_𝑗∶Λ→ℝ₊, where 𝑗∈{1,2}.

Proof. Use 𝜉(𝑥)=𝐾₁(𝑥)/𝑥 and 𝑘(𝑥,𝑦)=^{(𝑦/𝑥)−1/𝑞/(𝑥+𝑦),𝑥≠0,𝑦≠0,𝑥+𝑦≠0,}_0,otherwise in Corollary 7 to obtain ^∞₀𝐾₁(𝑥)𝑥1𝐾₁(𝑥)^∞₀(𝑦/𝑥)^−1/𝑞(𝑓₁(𝑦))^𝑝𝑥+𝑦Δ𝑦^1/𝑝×1𝐾₁(𝑥)^∞₀(𝑦/𝑥)^−1/𝑞(𝑓₂(𝑦))^𝑝′𝑥+𝑦Δ𝑦^1/𝑝′≥Δ𝑥_∞0𝑤𝑦𝑓₁𝑦𝑓₂𝑦Δ𝑦, (40) where 𝑤𝑦=^∞₀𝑘𝑥,𝑦𝜉(𝑥)𝐾₁(𝑥)Δ𝑥=^∞₀𝑘𝑥,𝑦𝑥=1Δ𝑥𝑦^∞₀(𝑦/𝑥)^1−1/𝑞𝐾𝑥+𝑦Δ𝑥=²𝑦𝑦. (41) Using this value in (40), we obtain ^∞₀^∞₀(𝑦/𝑥)^−1/𝑞(𝑓₁(𝑦))^𝑝𝑥+𝑦Δ𝑦^1/𝑝×^∞₀(𝑦/𝑥)^−1/𝑞(𝑓₂(𝑦))^𝑝′𝑥+𝑦Δ𝑦^1/𝑝′Δ𝑥𝑥≥^∞₀𝐾₂𝑦𝑓₁𝑦𝑓₂𝑦Δ𝑦𝑦. (42) Now, if we replace 𝑓₁(𝑦) with 𝑔₁(𝑦)𝑦^1/𝑝 and 𝑓₂(𝑦) with 𝑔₂(𝑦)y^1/𝑝′, we obtain (39).

Another Proof of (42). Consider L.H.S of (40) and applying Hölder inequality on time scale [25, Theorem 6.2] and Fubini theorem on time scale [16, Theorem 1.1], we have ^∞₀𝐾₁(𝑥)𝑥1𝐾₁(𝑥)^∞₀(𝑦/𝑥)^−1/𝑞(𝑓₁(𝑦))^𝑝𝑥+𝑦Δ𝑦^1/𝑝×1𝐾₁(𝑥)^∞₀(𝑦/𝑥)^−1/𝑞(𝑓₂(𝑦))^𝑝′𝑥+𝑦Δ𝑦^1/𝑝′≥Δ𝑥_∞0∞0(𝑦/𝑥)^−1/𝑞𝑓₁𝑦𝑓₂𝑦𝑥+𝑦Δ𝑦Δ𝑥𝑥=^∞₀𝑓₁𝑦𝑓₂𝑦^∞₀(𝑦/𝑥)^−1/𝑞𝑥+𝑦Δ𝑥𝑥=Δ𝑦_∞0𝑓₁𝑦𝑓₂𝑦𝑦^∞₀(𝑦/𝑥)^1−1/𝑞=𝑥+𝑦Δ𝑥Δ𝑦_∞0𝐾₂𝑦𝑓₁𝑦𝑓₂𝑦Δ𝑦𝑦. (43)

Example 18. It is known that ^∞₀(𝑦/𝑥)^−1/𝑞𝑥+𝑦𝑑𝑦=^∞₀(𝑦/𝑥)^1−1/𝑞𝜋𝑥+𝑦𝑑𝑥=,sin𝜋/𝑞 (44) for all 𝑥,𝑦∈ℝ₊=(0,∞) with 𝑞>1. If 𝕋=ℝ; then from (39), we obtain ^∞₀^∞₀(𝑦/𝑥)^1−1/𝑞(𝑓₁(𝑦))^𝑝𝑥+𝑦𝑑𝑦^1/𝑝×^∞₀(𝑦/𝑥)^1−1/𝑞(𝑓₂(𝑦))^𝑝′𝑥+𝑦𝑑𝑦^1/𝑝′≥𝜋𝑑𝑥sin𝜋/𝑞^∞₀𝑓₁𝑦𝑓₂𝑦𝑑𝑦. (45)
In the rest of paper, we take 𝑛=1, 𝑎≥0, 𝑏=∞ in 𝐇𝟒.

Theorem 19. If (38) is satisfied, then ^∞₀^∞₀(𝑦/𝑥)^1−1/𝑞(𝑓₁(𝑦))^𝑝𝑥+𝑦Δ𝑦^1/𝑝+^∞₀(𝑦/𝑥)^1−1/𝑞(𝑓₂(𝑦))^𝑝𝑥+𝑦Δ𝑦^1/𝑝𝑝≥Δ𝑥_∞0𝐾₂𝑦(𝑓₁(𝑦)+𝑓₂(𝑦))^𝑝Δ𝑦 (46) holds for all 𝜆_Δ-integrable 𝑓_𝑗∶Λ→ℝ₊, where 𝑗∈{1,2}.

Proof. Use 𝜉(𝑥)=𝐾₁(𝑥)/𝑥 and 𝑘(𝑥,𝑦)=^{(𝑦/𝑥)−1/𝑞/(𝑥+𝑦),𝑥≠0,𝑦≠0,𝑥+𝑦≠0,}_0,otherwise in Corollary 7 to obtain ^∞₀^∞₀(𝑦/𝑥)^−1/𝑞(𝑓₁(𝑦))^𝑝𝑥+𝑦Δ𝑦^1/𝑝+^∞₀(𝑦/𝑥)^−1/𝑞(𝑓₂(𝑦))^𝑝𝑥+𝑦Δ𝑦^1/𝑝𝑝Δ𝑥𝑥≥^∞₀𝐾₂𝑦(𝑓₁(𝑦)+𝑓₂(𝑦))^𝑝Δ𝑦𝑦. (47) Now, if we replace 𝑓₁(𝑦) with 𝑓₁(𝑦)𝑦^1/𝑝 and 𝑓₂(𝑦) with 𝑓₂(𝑦)𝑦^1/𝑝, we obtain (46).

Remark 20. (a) We can give another proof of (47) using Minkowski's inequality on time scale [25, Theorem  7.2].
(b) If 𝑝<1, then we have reverse inequalities.
Now we consider some generalizations of the Pólya-Knopp type inequalities.

Corollary 21. If (29) holds, and furthermore, if 𝑈⊂ℝ^𝑚 is a closed convex set such that Φ∶𝑈→ℝ is convex and continuous, then ^∞_𝑎𝜉(𝑥)Φ1𝜎(𝑥)−𝑎_𝑎^𝜎(𝑥)𝐟𝑦Δ𝑦Δ𝑥≤^∞_𝑎𝑤𝑦Φ𝐟𝑦Δ𝑦 (48) holds for all 𝜆_Δ-integrable 𝐟∶Λ→ℝ^𝑚 such that 𝐟(Λ)⊂𝑈.

Proof. The statement follows from Theorem 12 using 𝑛=1.

Corollary 22. Assume (29). Then ^∞_𝑎𝜉(𝑥)1𝜎(𝑥)−𝑎_𝑎^𝜎(𝑥)(𝑓₁(𝑦))^𝑝Δ𝑦^1/𝑝×1𝜎(𝑥)−𝑎_𝑎^𝜎(𝑥)(𝑓₁(𝑦))^𝑝′Δ𝑦^1/𝑝′≥Δ𝑥_∞𝑎𝑤𝑦𝑓₁𝑦𝑓₂𝑦Δ𝑦 (49) holds for all 𝜆_Δ-integrable 𝑓_𝑗∶Λ→ℝ₊, where 𝑗∈{1,2}.

Proof. Statement follows from Corollary 7 using 𝑚=2.

Corollary 23. Assume (29). Then ^∞_𝑎𝜉(𝑥)1𝜎(𝑥)−𝑎_𝑎^𝜎(𝑥)(𝑓₁(𝑦))^𝑝Δ𝑦^1/𝑝+1𝜎(𝑥)−𝑎_𝑎^𝜎(𝑥)(𝑓₂(𝑦))^𝑝Δ𝑦^1/𝑝𝑝≥Δ𝑥_∞𝑎𝑤𝑦(𝑓₁(𝑦)+𝑓₂(𝑦))^𝑝Δ𝑦 (50) holds for all 𝜆_Δ-integrable 𝑓_𝑗∶Λ→ℝ₊, where 𝑗∈{1,2}.

Proof. Statement follows from Corollary 8 using 𝑚=2.

Example 24. When 𝕋 consists of isolated points, then from Corollary 22, we have _{[)𝑥∈𝑎,∞}𝜉(𝑥)(𝜎(𝑥)−𝑥)𝜎(𝑥)×−𝑎_{[(𝑥𝑦∈𝑎,𝜎))}(𝑓₁(𝑦))^𝑝𝜎𝑦−𝑦^1/𝑝×_{[(𝑥𝑦∈𝑎,𝜎))}(𝑓₂(𝑦))^𝑝′𝜎𝑦−𝑦^1/𝑝′≥_{[)𝑦∈𝑎,∞}𝑤𝑦𝑓₁𝑦𝑓₂𝑦𝜎𝑦−𝑦, (51) where 𝑤(𝑦)=∑_{𝑥∈[𝑦,∞)}(𝜉(𝑥)(𝜎(𝑥)−𝑥)/(𝜎(𝑥)−𝑎)).

Example 25. When 𝕋 consists of isolated points, then from Corollary 23, we have _{[)𝑥∈𝑎,∞}𝜉(𝑥)(𝜎(𝑥)−𝑥)𝜎(𝑥)×−𝑎_{[(𝑥𝑦∈𝑎,𝜎))}(𝑓₁(𝑦))^𝑝𝜎𝑦−𝑦^1/𝑝+_{[(𝑥𝑦∈𝑎,𝜎))}(𝑓₂(𝑦))^𝑝𝜎𝑦−𝑦^1/𝑝𝑝≥_{[)𝑦∈𝑎,∞}𝑤𝑦𝑓₁(𝑦)+𝑓₂(𝑦)^𝑝𝜎𝑦−𝑦, (52) where 𝑤(𝑦) is the same as in Example 24.

Example 26. For 𝕋=ℎℕ={ℎ𝑛∶𝑛∈ℕ} with ℎ>0, 𝑎=1, and 𝜉(𝑥)=1/𝜎(𝑥), (51) takes the form ^∞_𝑛=11𝑛(𝑛+1)^𝑛_𝑘=1(𝑓₁(𝑘ℎ))^𝑝1/𝑝×^𝑛_𝑘=1(𝑓₂(𝑘ℎ))^{𝑝′1/𝑝′}≥^∞_𝑛=11𝑛𝑓₁(𝑛ℎ)𝑓₂(𝑛ℎ). (53) Inequality (52) takes the form ^∞_𝑛=11𝑛(𝑛+1)^𝑛_𝑘=1(𝑓₁(𝑘ℎ))^𝑝1/𝑝+^𝑛_𝑘=1(𝑓₂(𝑘ℎ))^{𝑝1/𝑝𝑝}≥^∞_𝑛=11𝑛𝑓₁(𝑛ℎ)+𝑓₂(𝑛ℎ)^𝑝. (54)

Example 27. For 𝕋=ℕ²={𝑛²∶𝑛∈ℕ},𝑎=1, and 𝜉(𝑥)=2(𝜎(𝑥)−1)(𝜎(𝑥)−𝑥)²2√,𝑥+3 (55) inequality (51) takes the form ^∞_𝑛=12()×2𝑛+1)(2𝑛+3_𝑛𝑘=1(2𝑘+1)(𝑓₁(𝑘²))^𝑝1/𝑝×^𝑛_𝑘=1(2𝑘+1)(𝑓₂(𝑘²))^{𝑝′1/𝑝′}≥^∞_𝑘=1𝑓₁𝑘²𝑓₂𝑘². (56)
Inequality (52) takes the form ^∞_𝑛=12()×2𝑛+1)(2𝑛+3_𝑛𝑘=1(2𝑘+1)(𝑓₁(𝑘²))^𝑝1/𝑝+^𝑛_𝑘=1(2𝑘+1)(𝑓₂(𝑘²))^{𝑝1/𝑝𝑝}≥^∞_𝑘=1𝑓₁(𝑘²)+𝑓₂(𝑘²)^𝑝. (57)
If we take 𝜉(𝑥)=𝜎(𝑥)−1(𝑥),𝑥𝜎 (58) Inequality (51) takes the form ^∞_𝑛=1(2𝑛+1)𝑛²(𝑛+1)^2𝑛_𝑘=1(2𝑘+1)(𝑓₁(𝑘²))^𝑝1/𝑝×^𝑛_𝑘=1(2𝑘+1)(𝑓₂(𝑘²))^{𝑝′1/𝑝′}≥^∞_𝑘=1(2𝑘+1)𝑘²𝑓₁𝑘²𝑓₂𝑘². (59) Inequality (52) takes the form ^∞_𝑛=1(2𝑛+1)𝑛²(𝑛+1)²×^𝑛_𝑘=1(2𝑘+1)(𝑓₁(𝑘²))^𝑝1/𝑝+^𝑛_𝑘=1(2𝑘+1)(𝑓₂(𝑘²))^{𝑝1/𝑝𝑝}≥^∞_𝑘=1(2𝑘+1)𝑘²𝑓₁𝑘²+𝑓₂𝑘^2𝑝. (60)
By replacing 𝑓₁(𝑘²)=(𝑘²/(2𝑘+1))^1/𝑝𝑓₁(𝑘²),𝑓₂(𝑘²)=(𝑘²/(2𝑘+1))^́𝑝1/𝑓₂(𝑘²) in (59) and in (60), we have ^∞_𝑛=1(2𝑛+1)𝑛²(𝑛+1)^2𝑛_𝑘=1𝑘²(𝑓₁(𝑘²))^𝑝1/𝑝×^𝑛_𝑘=1𝑘²(𝑓₂(𝑘²))^{𝑝′1/𝑝′}≥^∞_𝑘=1𝑓₁𝑘²𝑓₂𝑘²,_∞𝑛=1(2𝑛+1)𝑛²(𝑛+1)^2𝑛_𝑘=1𝑘²(𝑓₁(𝑘²))^𝑝1/𝑝+^𝑛_𝑘=1𝑘²(𝑓₂(𝑘²))^{𝑝1/𝑝𝑝}≥^∞_𝑘=1𝑓₁(𝑘²)+𝑓₂(𝑘²)^𝑝, (61) respectively.

Example 28. For 𝕋=𝑞^ℕ={𝑞^𝑛∶𝑛∈{ℕ}, 𝑞>1}, 𝑎=𝑞 and 𝜉(𝑥)=𝜎(𝑥)−𝑎𝜎(𝑥𝜎(𝑥)),)(−𝑥 (62) The inequality (51) takes the form ^∞_𝑛=1𝑞−1𝑞^𝑛𝑛_𝑘=1𝑞^𝑘−1(𝑓₁(𝑞^𝑘))^𝑝1/𝑝×^𝑛_𝑘=1𝑞^𝑘−1(𝑓₂(𝑞^𝑘))^{𝑝′1/𝑝′}≥^∞_𝑛=1𝑓₁𝑞^𝑛𝑓₂𝑞^𝑛. (63) Inequality (52) takes the form ^∞_𝑛=1𝑞−1𝑞^𝑛×^𝑛_𝑘=1𝑞^𝑘−1(𝑓₁(𝑞^𝑘))^𝑝1/𝑝+^𝑛_𝑘=1𝑞^𝑘−1(𝑓₂(𝑞^𝑘))^{𝑝1/𝑝𝑝}≥^∞_𝑛=1(𝑓₁(𝑞^𝑛)+𝑓₂(𝑞^𝑛))^𝑝. (64)

Remark 29. (a) In classical case for ℎ=1, inequalities (53) and (54) are the same as (1.7) and (1.9), also (63) and (64) are the same as (1.6) and (1.8) in [18, Corollary 1.3], respectively, while according to authors knowledge, (56), (57), (59), (60), and (61) are not existing in the literature.
(b) For 𝑝<1, we get the reverse inequalities.

Remark 30. The results given in Section 5 can analogously be proved for Ω=Λ=[𝑎₁,∞)_𝕋×⋯×[𝑎_𝑛,∞)_𝕋.

Remark 31. The results given in Corollaries 7 and 8 and in their given applications can also be obtained for finite value of 𝑚>2 analogously.

Acknowledgments

The research of the first author is supported by a grant of the Romanian National Authority for Scientific Research, CNCS UEFISCDI, project no. PN-II-ID-PCE-2011-3-0154. The research of the second author is partially supported by the Higher Education Commission, Pakistan, and the research of the third author is supported by the Croatian Ministry of Science, Education and Sports under the research Grant 117-1170889-0888.