# Time-Scale Integral Inequalities of Copson with Steklov Operator in High Dimension

Authors: Wedad Albalawi
Journal: Journal of Mathematics
 (2022)

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## Abstract

The paper derives some new time-scale (TS) dynamic inequalities for multiple integrals. The obtained inequalities are special cases of Copson integral using Steklov operator in (TS) version with high dimension. We prove the inequalities with several formulas for the operator and in different casesm>μ+1 and m<μ+1 for every μ≥1, using time-scales (TSs) setting for integral properties, chain rules, Fubini’s theorem, and Hölder’s inequality.


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## Body

## 1. Introduction

Equations and inequalities are the core of scientific study and have a great influence on a huge number of applications. A large number of physical phenomena and engineering studies have been analyzed and explained through equations and inequalities. For this reason, the study in this field developed rapidly and many types of inequalities and equations appeared. Dynamic inequalities on (TS) are some of the important inequalities that were extended by a lot of researchers and have interesting applications. Furthermore, dynamic inequalities are used to study the behaviour of dynamic equations.Mathematical analysis has been the most important study in mathematics for the past three decades. Integral inequalities are one of the main studies and the core of mathematical analysis. In the 20th century, a significant part of science was numerical inequalities as the first composition to be released in 1934, through the published study by P’olya et al. [1]. This framework of inequalities played a vital role in the improvement processes and various applications of mathematics.A large number of essential studies of integral inequalities appeared in the twentieth century, including pure and applied mathematics study. In 1920, Hardy produced the discrete Hardy inequality [2]. This inequality was also proved by himself in [3] (see also [4]), using the variations calculus to obtain the following inequality that is very valuable across both technological sciences and mathematics. If p>1 and h≥0 in 0,∞ andHx=∫0xhtdt<∞,then(1)∫0∞Hxpdx<Λp∫0∞hpxdx,where Λ=pp−1−1 is the best possible constant (BPC). Several important assessments and their implementation are done by inequality (1). Furthermore, the inequality is true in case 0<a<b<∞,(2)∫abHxpdx<Λp∫abhpxdx,where 0<∫0bhpxdx<∞. The classical inequality of Hardy declares that if p>1 and h is nonnegative and measurable on a,b, then (2) is valid except h≡0 a.e. in a,b, considering the (BPC).Integral inequalities (3) and (4) are established in 1928 by Hardy [5].Letf be a nonnegative measurable function on 0,∞:(3)Hhx≤∫0xhtdt,fora<p−1,∫x∞htdt,fora>p−1.Then,(4)∫0∞xa−pHhpxdx≤pp−a−1p∫0∞xahpxdx,forp>1.Later, in 1976, Copson studied the integral inequalities ([6], Theorem 1, Theorem 3) as follows.Leth and v be functions such that they are nonnegative measurable on 0,∞;Vx=∫0xvtdt,Chx≤∫0xhtvtdt,forc>1,∫x∞htvtdt,forc<1.Then,∫0∞V−cxvxChpxdx≤pc−1p∫0∞Vp−cxvxdx,forp≥1.Many papers included new extensions and generalizations for the inequalities above in more general settings. For instance, in 1979, some generalizations of Hardy-type inequality were proved by Chan [7]. Then, in 1992, Pachpatte [8] generalized the inequalities that were produced by Chan [7]. In 2005, P. Rehak used (TS) setting to extend Hardy’s inequalities [9]. In 2015, Pachpatte’s inequalities [8] were extended by Saker and O’Regan [10], with setting of (TSs). Later, some extensions of (TSs) Hardy inequalities were done for functions with high dimensions (see, for example, [11–14]).In 2021, Albalawi and Khan generalized the main integral of Hardy and Copson inequalities, using the Steklov operator. The operator is defined in the following formulas with considering conditions in two cases (for more details, see [15]).The aim of this paper is extending the study in [16] that was used for some new Hardy-type inequalities to obtain new special Copson inequalities with the Steklov operator (see [15]) in (TS) versions with high dimension. The results below are proved in two cases m>μ+1 and m<μ+1 by considering some general conditions that can be applied for any variable in the integral. To achieve this paper, we use (TSs) settings in integrals properties, chain rules, Hölder’s inequality, and Fubini’s theorem.The paper takes the following structure: After introduction, the main concepts of (TSs) are presented in Section2. Then, in Section 3, we generalized a class of Copson inequalities pertaining the Steklov operator with (TS) in high dimension. Lastly, conclusion of our results is presented.


## 2. Preliminaries and Lemmas on Time Scales

We state the main concepts of (TSs) that are used in this paper (for more details about (TS) calculus, see [17, 18]).(TS) calculus in continuous case and discrete analysis was introduced by Hilger [19] in 1988. We denote to a subset (TS) of the real numbers ℝ by T. Hence, the sets of numbers ℝ, ℤ, and ℕ can be considered as (TSs).Letσ:T⟶T be a forward jump operator, such that σt≔infs∈T:s>t, while ς:T⟶T is the backward jump operator, given by ςt=sups∈T:s<t for all t∈T.Ifσt>t, then t is right-scattered, and if ςt<t, t is left-scatted. In the case if points are right-scattered and left-scattered at the same time, then they will be isolated. The point t is right-dense if t<supT and σt=t, while t is left-dense if t>infT and ςt=t.Letg:T⟶ℝ be a continuous function and if it satisfied the continuity at all right-dense points in T and the limits of the left-sided exist (finite) at all left-dense points in T, then g is known rd-continuous. We use CrT,ℝ to denote the space of all rd-continuous.A functiong:T⟶ℝ is Δ-differentiable at t∈T, if there is a real number β=gΔt and for all ε>0, there exists a neighbor. U of t satisfiesgσt−gs−βσt−s≤εσt−s,for alls∈U.TheΔ-derivative of a function g in high order n∈N is given bygΔnt=gΔn−1tΔ.If theΔ-derivative of gΔn−1t exists, the following examples show that the delta derivative for every number set of (TSs).IfT=ℝ, thengΔt=g′=limΔt⟶0gt+Δt−gtΔt,for allt∈T.IfT=ℕ, thengΔt=gt+1−gt,for allt∈T.Letg:T⟶ℝ; if g is continuous at right-scattered t, then it is delta-derivative of the function g, given bygΔt=gσt−gtσt−t.In the case oft is not right-scattered, then the derivative of g is given bygΔt=lims⟶tgσt−gst−s=lims⟶∞gt−gst−s.Here, the limit exists. Note that ifT=ℝ, we haveσt=t,gΔt=g′t.IfT=ℤ, we haveσt=t+1,gΔt=Δgt,∫abgtΔt=∑t=ab−1gt.Lemma 1.
Leth;g:T⟶ℝ be delta-differentiable. Then,(5)hgΔ=hΔg+hσhΔ=fgΔ+fΔgσ,hgΔ=hΔg−hgΔggσ.The Cauchy integral of a delta-differential function ofggΔ is defined by∫adgΔtΔt=gd−ga,fora,d∈T.The time-scale integration by parts formula is given by(6)∫adhtgΔtΔt=htgtad−∫adhΔtgσΔt,a,d∈T.The infinite integrals are defined by∫a∞gtΔt=limd⟶∞∫adgtΔt.IfT=ℝ, we have∫adgtΔt=∫adgtdt.IfT=ℤ, we get∫adgtΔt=∑t=ad−1gt.Lemma 2 (chain rule [16]).
Assume a continuous function,w:ℝ⟶ℝ, a delta-differentiable, and w:T⟶ℝ, on Tc and a continuous differentiable h:ℝ⟶ℝ. Then, there exists c∈t,σt with(7)h∘wΔt=h′wcwΔt.Lemma 3 (dynamic Hölder inequality).
Leta,d∈T and h,w∈Crda,dT,0,∞. If p1,p2>1 with 1/p1+1/p2=1, then(8)∫adhtwtΔt≤∫adhp1tΔt1/p1∫adwp2tΔt1/p2.Theorem 4 (Fubini’s theorem [20]).
Letϒ,N,μΔ and Σ,L,γΔ be (TS) measure spaces with finite dimension. Consider ϒ×Σ,N×L,μΔ×γΔ as the measure space, where N×L is the σ−algebra product that is generated by E×F:E∈N,F∈L andμΔ×γΔE×F=μΔEγΔF.Then, Fubini’s theorem satisfied.
To be more accurate, ifξ:ϒ×Σ⟶ℝ is μΔ×γΔ—integrable,Ψγ=∫Σξγ,ΠΔΠ,exists  forΠ∈ϒ,andΨΠ=∫ϒξγ,ΠΔγ,exists forγ∈Σ.Then,∫ϒΔγ∫Σξγ,ΠΔΠ=∫ΣΔΠ∫ϒξγ,ΠΔγ.


## 3. Main Results

A new (TS) version of Copson-type inequality with Steklov operator for multiple integrals is obtained in this section. We consider the nonnegative rd-continuous functionswl, fl, gl, and vl are Δ-integrable and defined integrals. Throughout this paper, we set Kt1,…,tk as the Copson–Steklov-type operator considering the existence of the integral and also finite.Theorem 5.
LetTl be a (TS) and a∈0,∞Tl, for 1≤l≤k with l,k∈ℕ. In addition, let wl, fl, gl, and vl be nonnegative and rd-continuous functions on a,∞Tl. Furthermore, assume there exist μ,λ≥1 such thatwlΔltlwlσtl≤μVlΔltlVltl,andglΔltlglσtl≤λFΔlt1,…,tkFt1,…,tk,where Δl=∂/∂tl for every l,Vltl=∫atlvlslΔsl,withVl∞=∞,andwla=0,andFt1,…,tk≔∫at1…∫atk∏l=1k1glslvlslVlslfs1,…,skΔs1,…,Δsk.
Define the operator(9)Kt1,…,tk=∏l=1kgltlFt1,…,tk,Then(10)∫a∞…∫a∞∏l=1kwlσtlvltlVlmtlKσt1,…,tkpΔt1,…,Δtk≤pλ+1m−μ+1p∫a∞vktkVkmtkwkptkwkσtkp−1gkσtkgktkp∫a∞…∫a∞∏l=1k−1wlσtlvltlVlmtlfpt1,…,tkΔt1,…,Δtk−1Δtk,where p≥1 and m>μ+1.Proof.
We write the left side of (10) as follows:(11)∫a∞…∫a∞∏l=1k−1wlσtlvltlVlmtlΓkΔt1,…,Δtk−1,where Γk is the k-termΓk=∫a∞wkσtkvktkVkmtkKσt1,…,tkpΔtk.
Using formula (6) for integration by parts to compute Γk, we have(12)Γk=∫a∞wkσtkvktkVkmtkKσt1,…tkpΔtk=ztkutka∞−∫a∞utkztkΔkΔtk,where uΔktk=vktk/Vkmtk and then utk=−m+1Vk−m+1tk and zσtk=wkσtkKσt1,…,tkp, implying that ztk=wktkKt1,…,tkp, and hence,zΔktk=wkΔktkKσt1,…,tkp+pwktkKp−1t1,…,ckgkΔktkFt1,…,tk+gkσtkFΔkt1,…,tk.
Assumeλ≥1 such thatgkΔktkgkσtk≤λFΔkt1,…,tkFt1,…,tk,where FΔk=∂F/∂tk, and since cl∈tl,σtl, we havezΔktk≤wkΔktkKσt1,…,tkp+pλ+1wktkKσt1,…,tkp−1gkσtkFΔkt1,…tk.
Substituting the previous quantities in (12) and since Vl∞=∞ and wla=0, then we have∫a∞wkσtkvktkVkmtkKσt1,…,tkpΔtk=1m−1∫a∞1Vkm−1tkwkΔktkKσt1,…,tkpΔtk+pλ+1m−1∫a∞1Vkm−1tkwktkKσt1,…,tkp−1gkσtkFΔkt1,…,tkΔtk.
Assumeμ≥1 such thatwlΔltlwlσtl≤μVlΔltlVltl.Then, we obtain∫a∞wkσtkvktkVkmtkKσt1,…,tkpΔtk=μm−1∫a∞wkσtkvktkVkmtkKσt1,…,tkpΔtk+pλ+1m−1∫a∞1Vkm−1tkwktkKσt1,…,tkp−1gkσtkFΔkt1,…,tkΔtk.
SinceFΔkt1,…,tk=ft1,…,tk/gktkvktk/Vktk, then we have∫a∞wkσtkvktkVkmtkKσt1,…,tkpΔtk≤pλ+1m−1−μ∫a∞vktkVkmtkwktkKσt1,…,tkp−1gkσtkft1,…,tkgktkΔtk.Then, Hölder’s inequality (8) with indices p and p/p−1 can be applied:(13)Γk=∫a∞wkσtkvktkVkmtkKσt1,…,tkpΔtk≤pλ+1m−1−μp∫a∞vktkVkmtkwkptkwkσtkp−1fpt1,…,tkgkσtkgktkpΔtk.
SubstitutingΓk in (11) and applying Fubini’s Theorem 4, then we obtain the inequality∫a∞…∫a∞∏l=1kwlσtlvltlVlmtlKσt1,…tkpΔt1…Δtk≤pλ+1m−μ+1p∫a∞vktkVkmtkwkptkwkσtkp−1gkσtkgktkp∫a∞…∫a∞∏l=1k−1wlσtlvltlVlmtlfpt1,…,tkΔt1…Δtk−1Δtk.Corollary 6.
Ifl=1 in Theorem 5, inequality (10) becomes(14)∫a∞wσtvtVmtKσtpΔt≤pλ+1m−μ+1p∫a∞vtVmtwptwσtp−1gσtgtpfptΔt.Corollary 7.
IfT=ℝ in Corollary 6, we obtain(15)∫a∞wtvtVmtKptdt≤pλ+1m−μ+1p∫a∞vtVmtwtfptdt.Remark 8.
Assumeμ=0 and β>λ in Corollary 7; then, we have Corollary 3 in [15](16)∫a∞wtvtVmtKptdt≤βpm−1p∫a∞wtvtVmtfptdt.Theorem 9.
LetTl be (TS) and a∈0,∞Tl, for 1≤l≤k with l,k∈ℕ. In addition, let wl, fl, gl, and vl be nonnegative and rd-continuous functions on a,∞Tl. Furthermore, assume there exist λ,μ≥1 such thatwlΔltlwlσtl≥μVlΔltlVltl,andglΔltlglσtl≥λFΔlt1,…,tkFt1,…,tk,where FΔl=∂F/∂tl; for every l,Vltl=∫atlvlslΔsl,withVl∞=∞,andwla=0,andFt1,…,tk≔∫t1∞…∫tk∞∏l=1k1glslvlslVlslfs1,…skΔs1…Δsk.
Define the operatorKt1,…,tk=∏l=1kgltlFt1,…,tk.Then,(17)∫a∞…∫a∞∏l=1kwlσtlvltlVlmtlKσt1,…tkpΔt1…Δtk≤pλ+1μ+1−mp∫a∞vktkVkmtkwkptkwkσtkp−1gkσtkgktkp∫a∞…∫a∞∏l=1k−1wlσtlvltlVlmtlfpt1,…,tkΔt1…Δtk−1Δtk,where p≥1 and 0≤m<μ+1.Proof.
We write the left side of (17) as follows:(18)∫a∞…∫a∞∏l=1k−1wlσtlvltlVlmtlΓkΔt1…Δtk−1.
Use formula (6) to calculate the following k−term:(19)Γk=∫a∞wkσtkvktkVkmtkKσt1,…,tkpΔtk=utkztka∞+∫a∞utk−ztkΔkΔtk,where uΔktk=wkσtkvktk/Vktkm and zσtk=Kσt1,…,tkp.
Using (7) and the product rule (5), there exists ck∈sk,σsk such thatwkskVk1−mskΔk=wkΔkskVk1−msk+wkσ1−mVk−mckVkΔksk.
Assumeμ≥1 such thatwkΔktkwkσtk≥μVkΔktkVktk.
SinceVkΔksk=vksk≥0, sk≤ck≤σsk, and 0≤m<1, then(20)wkσVk−mskvksk≤11−m+μwkskVk1−mskΔk.
By integration, we haveutk=∫a∞wkσVk−mskvkskΔsk≤11−m+μ∫a∞wkskVk1−mskΔkΔsk.Now, we calculate −Kpt1,…,tkΔk and we obtainKpt1,…,tkΔk=pKp−1t1,…,ckgkΔktkFt1,…tk+gkσtkFΔkt1,…,tk.
Assumeλ≥1 such thatgkΔktkgkσtk≥λFΔkt1,…,tkFt1,…,tk,where(21)Ft1,…,tk≔gktk∫tk∞1gkskvkskVkskfs1,…,skΔsk,and since Vk∞=∞ and ck≥sk, then we haveKpt1,…,tkΔk≥pKp−1t1,…,tkλ+1gkσtkFΔkt1,…,tk≥pλ+1Kp−1t1,…,tk−gkσtkgktkvktkVktkft1,…,tk.Then,−Kpt1,…,tkΔk≤pλ+1Kp−1t1,…,tkgkσtkgktkvktkVktkft1,…,tk.
Hence, we have∫a∞wkσtkvktkVkmtkKσt1,…,tkpΔt≤pλ+1μ+1−m∫a∞wktkvktkVkmtkft1,…,tkgkσtkgktkKσt1,…,tkp−1Δtk.
Using Hölder’s inequality, wherep1=p and p2=p/p−1, we obtain(22)∫a∞wkσtkvktkVkmtkKσt1,…,tkpΔtk≤pλ+1μ+1−mp∫a∞vktkVkmtkwkptkwkσtkp−1fpt1,…,tkgkσtkgktkpΔtk.
Substituting (22) in (17), we have∫a∞…∫a∞∏l=1kwlσtlvltlVlmtlKσt1,…tkpΔt1…Δtk≤pλ+1μ+1−mp∫a∞vktkVkmtkwkptkwkσtkp−1gkσtkgktkp∫a∞…∫a∞∏l=1k−1wlσtlvltlVlmtlfpt1,…,tkΔt1…Δtk−1Δtk.Corollary 10.
Ifl=1 and T=ℝ in Theorem 9, we get(23)∫a∞wtvtVmtKptdt≤pλ+1μ+1−mp∫a∞vtVmtwtfptdt,whereKt=gt∫t∞1gsvsVsfsds.Remark 11.
Assumeμ=0 and θ>λ in Corollary 10; we have Corollary 5 in [15].Theorem 12.
LetTl be (TS) and a∈0,∞Tl, for 1≤l≤k with l,k∈ℕ. In addition, let wl, fl, gl, and vl be nonnegative rd-continuous functions on a,∞Tl. Furthermore, assume there exist λ,μ≥1 such thatwlΔltlwlσtl≥μVlΔltlVltl,andglΔltglσt≥λFΔlt1,…,tkFt1,…,tk,where for every l,Vltl=∫atlvlslΔsl,withVl∞=∞,andwla=0,andFt1,…,tk≔∫t1∞…∫tk∞∏l=1kglslvlslVlslfs1,…skΔs1…Δsk.
Define the operatorKt1,…,tk≔∏l=1k1gltlFt1,…,tk.Then,(24)∫a∞…∫a∞∏l=1kwlσtlvltlVlmtlKσt1,…tkpΔt1…Δtk≤pλ−1μ+1−mp∫a∞vktkVkmtkwkptkwkσtkp−1gktkgkσtkp∫a∞…∫a∞∏l=1k−1wlσtlvltlVlmtlfpt1,…,tkΔt1,…,Δtk−1Δtk,where p≥1 and 0≤m<μ+1.Proof.
We write the left side of (24) as follows:(25)∫a∞…∫a∞∏l=1k−1wlσtlvltlVlmtlΓkΔt1…Δtk−1.
Apply (6) to calculate the following k−term:(26)Γk=∫a∞wkσtkvktkVkmtkKσt1,…,tkpΔtk=utkztka∞+∫a∞utk−ztkΔkΔtk,where uΔktk=wkσtkvktk/Vkmtk. Using the chain rule on (TS) (7) and product rule (5), there exist ck∈sk,σsk such thatwkskVk1−mskΔk=wkΔkskVk1−msk+1−mwσskVk−mckVkΔksk.
Assumeμ≥1 such thatwkΔktkwkσtk≥μVkΔktkVktk.
SinceVkΔksk=vksk≥0, sk≤ck≤σsk, and 0≤m<μ+1, thenwkskVk1−mskΔk≥1−m+μwkσVk−mskvksk,implyingutk=∫atkwkσVk−mskvkskΔsk≤11−m+μ∫atkwkskVk1−mskΔkΔsk≤11−m+μwktkVk1−mtk.We calculate −Kpt1,…,tkΔk, and we obtainKpt1,…,tkΔk=pKp−1t1,…,ck−gkΔgkσtkgtkFt1,…,tk+1gkσtkFΔkt1,…,tk.
Assumeλ≥1 such thatgkΔtktkgktk≥λFΔkt1,…,tkFt1,…,tk,where(27)Ft1,…,tk≔1gktk∫atkgkskvkskVkskfs1,…,skΔsk,and since ck≥tk, then we haveKpt1,…,tkΔk≥pKp−1t1,…,tk1−λ1gkσtkFΔkt1,…,tk≥pλ−1Kp−1t1,…,tk1gkσtkgktkvktkVktkft1,…,tk.
SinceV∞=∞, then−Kpt1,…,tkΔk≤−pλ−1Kp−1t1,…,tkgktkgkσtkvktkVktkft1,…,tk≤pλ−1Kσp−1t1,…,tkgktkgkσtkvktkVktkft1,…,tk.
Hence, we have∫a∞wkσtkvktkVkmtkKσt1,…,tkpΔtk≤u∞z∞−uaza+pλ−1μ+1−m∫a∞wktkvktkVkmtkft1,…,tkgktkgkσtkKσt1,…,tkp−1Δtk≤pλ−1μ+1−m∫a∞wktkvktkVkmtkft1,…,tkgktkgkσtkKkσt1,…,tkp−1Δtk.Then, Hölder’s inequality (8) can be applied with indices p and p/p−1:(28)∫a∞wkσtkvktkVkmtkKσt1,…,tkpΔtk≤pλ−1μ+1−mp∫a∞vktkVkmtkwkptkwkσp−1fpt1,…,tkgktkgkσtkpΔtk.
Substituting (28) in (26), we have∫a∞…∫a∞∏l=1kwlσtlvltlVlmtlKσt1,…tkpΔt1…Δtk≤pλ−1μ+1−mp∫a∞vktkVkmtkwkptkwkσtkp−1gktkgkσtkp∫a∞…∫a∞∏l=1k−1wlσtlvltlVlmtlfpt1,…,tkΔt1…Δtk−1Δtk.Corollary 13.
Ifl=1 and T=ℝ in Theorem 12, we obtain(29)∫a∞wtvtVmtKptdt≤pλ−1μ+1−mp∫a∞vtVmtwtfptdt,whereKt=1gs∫atgtvsVsfsds.Example 14.
Chooseμ=m, λ=2, and p=1 in Theorem 12. Hence, we get(30)∫a∞…∫a∞∏l=1kwlσtlvltlVlmtlKσt1,…tkΔt1…Δtk≤∫a∞vktkVkmtkgktkgkσtkwktk∫a∞…∫a∞∏l=1k−1wlσtlvltlVlmtlft1,…,tkΔt1…Δtk−1Δtk.


## 4. Conclusions

(TSs) calculus is used in this paper to prove special cases of (TS) Copson–Steklov-type inequalities with several variables. The obtained inequalities would be interesting to apply in different fields of mathematics (functional spaces, partial differential equations, mathematical modeling). Furthermore, the inequalities can be discussed in calculus, discrete calculus, and quantum calculus. As a perspective, we propose to study these results for other kinds of operators and solve the singularity that appeared in Theorem12 with case m>μ+1.



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