# Dynamic Inequalities in Quotients with General Kernels and Measures

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

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## Abstract

By utilizing the precepts related to the punctuation of time scalesT, we present some nouveau forms in quotients for Hardy’s and related inequalities on time scales. In particular, some recent results for the Pólya-Knopp, Hardy-Hilbert, and Hardy-Littlewood-Pólya-type inequalities are presented.


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## Body

## 1. Introduction

In [1], Hardy stated and proved the following integral inequality:
(1)∫0∞θ−1∫0θfηdηξdθ≤ξξ−1ξ∫0∞fξθdθ,forξ>1,where f is a positive function and the constant ξ/ξ−1ξ is the best ever. We rewrite (1) with the function fξ−1 rather than f, and by assuming limit ξ→∞, we acquired the limiting instance of the inequality of Hardy known as the inequality of Pólya-Knopp (see [2]), that is,
(2)∫0∞expθ−1∫0θlnfηdηdθ≤e∫0∞fθdθ.Lately, Kaijser et al. in [3] pointed out that both (1) and (2) are just special states of the much more general inequality of Hardy-Knopp for positive function f:
(3)∫0∞Φθ−1∫0θfηdηdθθ≤∫0∞Φfθdθθ,where Φ∈C0,∞,ℝ is a convex function. A popularization of inequality (3) with two weight functions is proved in [4]. Particularly, it was proved that if 0<b≤∞, λ:0,b→ℝ is a nonnegative function and Φ is a convex on a,c, then the inequality
(4)∫0bλθΦ1θ∫0θfηdηdθθ≤∫0bνθΦfθdθθ,is available for all integrable functions f:0,b→ℝ, such that fθ∈a,c and υ is defined by
(5)υη≔η∫ηbuθθ2dθ,forη∈0,b.Using the inequality of Jensen for convex functions and the theorem of Fubini, Kaijser et al. [5] established an inviting popularization of (1). Particularly, they proved that if λ:0,b→ℝ and k:0,b×0,b→ℝ are nonnegative functions such that 0<Kθ=∫0θkθ,ηdη<∞, for θ∈0,b and
(6)υη≔η∫ηbλθkθ,ηKθdθθ<∞,forη∈0,b,then
(7)∫0∞λθΦAkfθdθθ≤∫0∞νθΦfθdθθ,where Φ∈CI,ℝ is a convex function and
(8)Akfθ≔1Kθ∫0θkθ,ηfηdη,θ∈0,b.For further popularization of (7), Krulić et al. [6] proved that if Ω1,∑1,μ1 and Ω2,∑2,μ2 are two measure spaces with positive σ-finite measures λ:Ω1→ℝ and k:Ω1×Ω2→ℝ which are nonnegative measurable functions such that 0<Kθ=∫Ω2kθ,ηdμ2η<∞, θ∈Ω1, 0<ξ≤ξ∗<∞, and v is defined by
(9)νη≔∫Ω1λθkθ,ηKθξ∗/ξdμ1θξ/ξ∗<∞,η∈Ω2,then the inequality
(10)∫Ω1λθΦξ∗/ξAkfθdμ1θ1/ξ∗≤∫Ω2υηΦfηdμ2η1/ξ,is available for all measurable functions f:Ω2→ℝ such that fΩ2⊆I, where Φ∈CI,ℝ is a convex function and Ak is defined by
(11)Akfθ≔1Kθ∫Ω2kθ,ηfηdμ2η,θ∈Ω1.In [7], Iqbal et al. checked some new weighted Hardy-type inequalities on Ω1,∑1,μ1 and Ω2,∑2,μ2 measure spaces with positive σ-finite measures by replacing kθ,η by kθ,ηf2θ and f by f1/f2, where fi:Ω2→ℝi=1,2 are measurable functions in (10) as follows:
(12)∫Ω1λθΦg1θg2θξ∗/ξdμ1θ1/ξ∗≤∫Ω2υηΦf1ηf2ηdμ2η1/ξ,where 0<ξ≤ξ∗<∞, Φ∈CI,ℝ is a convex function, k:Ω1×Ω2→ℝ is a nonnegative measurable function, and v,giθi=1,2 are defined by
(13)νη≔f2η∫Ω1λθkθ,ηg2θξ∗/ξdμ1θξ/ξ∗<∞,g2θ≠0,η∈Ω2,giθ≔∫Ω2kθ,ηfiηdμ2ηi=1,2.In [8], the author founded the time scale version of (1). Particularly, he proved that if ξ>1, f∈Crda1,∞T,ℝ is a nonnegative function and ∫a1∞fξtΔt exists, then
(14)∫a1∞1σt−a1∫a1σtfθΔθξΔη≤ξξ−1ξ∫a1∞fξηΔη.In [9], the authors proved the time scale version of (4), which is given by
(15)∫a1b1λθΦ1σθ−a1∫a1σθfηΔηΔθθ−a1≤∫a1b1υθΦfθΔθθ−a1,where Φ∈Cc1,d1,ℝ is a convex function, λ∈Crda1,b1T,ℝ is a nonnegative function, and υ is defined by
(16)υt≔η−a1∫tb1ληθ−a1σθ−a1Δθ,forη∈a1,b1T.In [10], the authors outstretched a number of Hardy-type inequalities with certain kernels on time scale. Namely, they proved that if Ω1,∑1,μ1 and Ω2,∑2,μ2 are two time scale measure spaces, ζ:Ω1→ℝ+ and k:Ω1×Ω2→ℝ+, which are nonnegative measurable functions such that
(17)Kθ≔∫Ω2kθ,ηΔη<∞,θ∈Ω1,and υ is defined by
(18)υη≔∫Ω1kθ,ηζθKθΔθ<∞,η∈Ω2,then the inequality
(19)∫Ω1ζθΦ1Kθ∫Ω2kθ,ηfηΔηΔθ≤∫Ω2υηΦfηΔη,is available for all Δμ2-integrable f:Ω2→ℝ such that fΩ2⊂I and Φ∈CI,ℝ is a convex function. For development of dynamic inequalities on time scale calculus, we refer the reader to articles [11–18].The article is regimented as follows. In Section2, we recall the precepts related to the punctuation of time scales. In Section 3, we prove our results and give some remarks. Particularly, we prove a general dynamic weighted Hardy-type inequality with a nonnegative kernel. In Section 4, we critique a few particular states of the obtained inequalities, related to power and exponential functions and to the most simplest shapes of kernels.


## 2. Preliminaries

In this section, we will premise some fundamental precepts and effects on time scales which will be beneficial for deducing our major results. The following definitions and theorems are referred from [19, 20].A nonempty arbitrary locked subplot of the real numbersℝ is called a time scale which is denoted by T. For τ∈T, if infΦ=supT and supΦ=infT, then the forward jump operator σ:T→T and the backward jump operator ρ:T→T are defined as
(20)στ≔infθ∈T:θ>τ,ρτ≔supθ∈T:θ<τ,respectively. The Δ-derivative of χ:T→ℝ at τ∈Tk=T/ρsupT,supT is the number that enjoys the property that for all ε>0, there exists a neighborhood U⊂T of τ∈Tk such that
(21)χστ−χθ−χΔτστ−θ≤εστ−θ,forallθ∈U.Furthermore,χ is called a delta differentiable on Tk if it is delta differentiable at every τ∈Tk. Similarly, for τ∈Tk, we define the ∇-derivative of χ:T→ℝ at τ∈Tk=T/infT,σinfT as the number that enjoys the property that for all ε>0, there exists a neighborhood V⊂T of τ∈Tk such that
(22)χρτ−χθ−χ∇τρτ−θ≤ερτ−θ,forallθ∈V.Moreover,χ is called a nabla differentiable on Tk if it is nabla differentiable at every τ∈Tk. For θ,t∈Tk, the delta integral of χΔ is defined as
(23)∫θtχΔτΔτ≔χt−χθ.Similarly, forθ,t∈Tk, the nabla integral of χ∇ is defined as
(24)∫θtχ∇τ∇τ≔χt−χθ.Now, letχτ be differentiable on T in the Δ and ∇ senses. For τ∈Tkk, where Tkk=Tk∩Tk, the diamond-α dynamic derivative χ◊ατ is defined by
(25)χ◊ατ≔αχΔτ+1−αχ∇τ,0≤α≤1.The diamond-α derivative debases to the standard Δ-derivative for α=1 or the standard ∇-derivative for α=0.Next, we recall the inequality of Minkowski and the inequality of Jensen on time scales which are utilized in the proof of the major results.Theorem 1.
SupposeΩ,M,μΔ and Λ,L,λΔ are two finite-dimensional time scale measure spaces, and let u, v, and g be positive functions on Ω, Λ, and Ω×Λ, respectively. If ξ≥1, then the inequality
(26)∫Ω∫Λgθ,ηυηdυΔηξuθdμΔθ1/ξ≤∫Λ∫Ωgξθ,ηuθdμΔθυηdυΔη,is available for all integrals in (26). If 0<ξ<1 and
(27)∫Ω∫ΛgυdυΔξudμΔ>0,∫ΛgυdυΔ>0,are available, then (26) is reversed. For ξ<0, in addition with (27), if
(28)∫ΩgξudμΔ>0,is available, then again (26) is reversed.Theorem 2.
Lete,l∈T and u,v∈ℝ. Suppose that g1∈Crde,lT,u,vT and h1∈Crde,lT,ℝ are nonnegative with ∫elh1sΔs>0. If Φ∈Cu,v,ℝ is convex, then
(29)Φ∫elh1sg1sΔs∫elh1sΔs≤∫elΦh1sg1sΔs∫elh1sΔs.


## 3. Inequalities with General Kernels

In this section, we state and prove our major results. Before presenting the results, we labeled the following hypotheses.(H1)Ω1,Σ1,μ1 and Ω2,Σ2,μ2 are two time scale measure spaces(H2)k:Ω1×Ω2→ℝ is a positive measurable kernel and(30)Kθ≔∫Ω2kθ,ηΔμ2η<∞,θ∈Ω1.(H3)u:Ω1→ℝ is a Δμ2-integrable and(31)νη≔∫Ω1uθkθ,ηKθξ∗/ξΔμ1θξ/ξ∗<∞,where 0<ξ≤ξ∗<∞.In what follows, we will prove the foundation theorem that will be the decisive step in establishing our major result.Theorem 3.
Assume (H1)–(H3).
IfΦ∈CI,ℝ is a positive convex, then the following inequality
(32)∫Ω1uθΦAkfθξ∗/ξΔμ1θ1/ξ∗≤∫Ω2υηΦfηΔμ2η1/ξ,is available for all nonnegative Δμ2-integrable function f:Ω2→ℝ such that Imf⊆I, where Akf:Ω1→ℝ is defined by
(33)Akfθ≔1Kθ∫Ω2kθ,ηfηΔμ2η,θ∈Ω1.Proof.
We begin with an evident identity(34)∫Ω1uθΦAkfθξ∗/ξΔμ1θ=∫Ω1uθΦ1Kθ∫Ω2kθ,ηfηΔμ2ηξ∗/ξΔμ1θ.
Utilizing the inequality of Jensen (29) and the theorem of Minkowski (2) on (34), we find that
(35)∫Ω1uθΦAkfθξ∗/ξΔμ1θ≤∫Ω1uθ1Kθ∫Ω2kθ,ηΦfηΔμ2ηξ∗/ξΔμ1θ=∫Ω1uθ1Kθ∫Ω2kθ,ηΦfηΔμ2ηξ∗/ξΔμ1θξ/ξ∗ξ∗/ξ≤∫Ω2Φfη∫Ω1uθkθ,ηKθξ∗/ξΔμ1θξ/ξ∗Δμ2ηξ∗/ξ.
Taking into computation definition (31) of υ, it follows that
(36)∫Ω1uθΦAkfθξ∗/ξΔμ1θ≤∫Ω2υηΦfηΔμ2ηξ∗/ξ.
Finally, elevating (36) to the ξ∗th power, we acquired (33).Remark 4.
For the Lebesgue scale measuresΔμ1θ=Δθ, Δμ2η=Δη, and ξ=ξ∗=1, inequality (32) reduces to inequality (19) premised in Section 1. Choosing T=ℝ in Theorem 3, then inequality (32) reduces to inequality (10).In what follows, we labeled few particular convex functions starting with power functions.Corollary 5.
Suppose the supposition of Theorem3 be gratified only with Φx=xξ for ξ>1. Then, inequality (32) yields the following result:
(37)∫Ω1uθAkfθξ∗Δμ1θ1/ξ∗≤∫Ω2υηfηξΔμ2η1/ξ.Remark 6.
For the Lebesgue scale measuresΔμ1θ=Δθ, Δμ2η=Δη, and ξ=ξ∗, Corollary 5 coincides with Corollary 3.3 in [10].Now, considering Theorem3 with Φx=ex and f=lngξ, we get the following inviting result.Corollary 7.
Suppose the supposition of Theorem3 is gratified only with Φx=ex. If 0<ξ≤ξ∗<∞, then inequality (32) yields the following result:
(38)∫Ω1uθeξ/Kθ∫Ω2kθ,ηlngηΔμ2ηξ∗/ξΔμ1θ1/ξ∗≤∫Ω2υηgηξΔμ2η1/ξ.Remark 8.
For the Lebesgue scale measuresΔμ1θ=Δθ, Δμ2η=Δη, and ξ=ξ∗, Corollary 7 coincides with Corollary 3.4 in [10]. Also, in the special case ξ=1, Remark 8 coincides with Corollary 3.5 in [10].Now, by using a special substitution, we obtain our central result; that is, if we replacekθ,η by kθ,ηf2η and f by f1/f2, where fj:Ω2→ℝj=1,2 are measurable functions, we obtain these results.Theorem 9.
Assume (H1) and (H2) andυ be defined on Ω2 by
(39)νη≔f2η∫Ω1uθkθ,ηg2θξ∗/ξΔμ1θξ/ξ∗<∞,g2θ≠0,where 0<ξ≤ξ∗<∞. If Φ∈CI,ℝ is a positive convex, then the following inequality
(40)∫Ω1uθΦg1θg2θξ∗/ξΔμ1θ1/ξ∗≤∫Ω2υηΦf1ηf2ηΔμ2η1/ξ,is available for all measurable functions fj:Ω2→ℝj=1,2 and
(41)gjθ≔∫Ω2kθ,ηfjηΔμ2ηj=1,2.Remark 10.
ChoosingT=ℝ in Theorem 9, then inequality (40) reduces to inequality (12) established by Iqbal et al. (Theorem 2.1, [7]).As a special case of Theorem9 for ξ=ξ∗, we get the next corollary. Also, we note that the function Φ need not to be positive.Corollary 11.
Assume (H1) and (H2) andυ be defined on Ω2 by
(42)νη≔f2η∫Ω1uθkθ,ηg2θΔμ1θ<∞,g2θ≠0.
IfΦ∈CI,ℝ is a positive convex, then the following inequality
(43)∫Ω1uθΦg1θg2θΔμ1θ≤∫Ω2υηΦf1ηf2ηΔμ2η,is available for all measurable functions fj:Ω2→ℝj=1,2 and gj is defined by (41).Corollary 12.
Suppose the supposition of Theorem9 is gratified only with Φx=xξ for ξ>1. Then, inequality (40) yields the following result:
(44)∫Ω1uθg1θg2θξ∗Δμ1θ1/ξ∗≤∫Ω2υηf1ηf2ηξΔμ2η1/ξ.Corollary 13.
Suppose the supposition of Theorem9 is gratified only with Φx=ex. If 0<ξ≤ξ∗<∞, then inequality (40) yields the following result:
(45)∫Ω1uθeξ∗g1θ/ξg2θΔμ1θ1/ξ∗≤∫Ω2υηef1η/f2ηΔμ2η1/ξ.Remark 14.
ChoosingT=ℝ in Corollary 11, then (43) takes the form
(46)∫Ω1uθΦg1θg2θdμ1θ≤∫Ω2υηΦf1ηf2ηdμ2η,where
(47)νη≔f2η∫Ω1uθkθ,ηg2θdμ1θ<∞,g2θ≠0,which is the same result due to Iqbal et al. (Corollary 2.2, [7]).Corollary 15.
For the Lebesgue diamond-α scale measures Δμ1θ=◊αθ, Δμ2η=◊αη, Ω1=a1,b1T, and Ω2=c1,d1T, inequality (40) takes the form
(48)∫a1b1uθΦg1θg2θξ∗/ξ◊αθ1/ξ∗≤∫c1d1υηΦf1ηf2η◊αη1/ξ,where 0<ξ≤ξ∗<∞, Φ∈CI,ℝ is a positive convex, and
(49)νη=f2η∫a1b1uθkθ,ηg2θξ∗/ξ◊αθξ/ξ∗<∞,g2θ≠0,gjθ≔∫c1d1kθ,ηfjη◊αηj=1,2.Remark 16.
If we setξ=ξ∗=1, then inequality (48) takes the form
(50)∫a1b1uθΦg1θg2θ◊αθ≤∫c1d1υηΦf1ηf2η◊αη,which coincided with inequality (7) in [21].Remark 17.
If we setΦθ=−ln∣θ∣, then inequality (50) takes the form
(51)∫a1b1uθlng1θg2θ◊αθ≥∫c1d1υηlnf1ηf2η◊αη,which coincided with Corollary 4 in [21].Remark 18.
In Remark16, if we set f2η=1, g1θ=gθ, f1η=fη, and Φ∈C0,∞,ℝ be a convex function, then
(52)g2θ≔∫c1d1kθ,η◊αη=Kθ,∀θ∈a1,b1T.
We assume thatKθ>0 and define
(53)νη≔∫a1b1uθkθ,ηKθ◊αθ,∀θ∈c1,d1T.
Then, inequality (50) takes the form
(54)∫a1b1uθΦgθKθ◊αθ≤∫c1d1υηΦfη◊αη,which coincided with inequality (2.7) in [22] (Theorem 2.2).Remark 19.
Apply Remark8 for Φθ=−ln∣θ∣, which is convex with domain 0,∞. Then, inequality (54) takes the form
(55)∫a1b1uθlngθKθ◊αθ≥∫c1d1υηlnfη◊αη,which coincided with inequality (2.10) in [22] (Corollary 2.3).Remark 20.
Apply Remark18 for Φθ=eθ,θ≥0. Then, inequality (54) takes the form
(56)∫a1b1uθegθ/Kθ◊αθ≤∫c1d1υηefη◊αη,which coincided with inequality (2.11) in [22] (Corollary 2.4).Remark 21.
If we setα=1, then the delta version form of inequality (50) takes the form
(57)∫a1b1uθΦg1θg2θΔθ≤∫c1d1υηΦf1ηf2ηΔη,which coincided with inequality (8) in [21].Remark 22.
If we setα=0, then the nabla version form of inequality (50) takes the form
(58)∫a1b1uθΦg1θg2θ∇θ≤∫c1d1υηΦf1ηf2η∇η,which coincided with inequality (9) in [21].Remark 23.
In particular, ifT=ℝ and a1,b1T=c1,d1T, then inequality (50) takes the form
(59)∫a1b1uθΦg1θg2θdθ≤∫c1d1υηΦf1ηf2ηdη,where
(60)νη≔f2η∫a1b1uθkθ,ηg2θdθ,∀η∈a1,b1T,g2θ≠0,gjθ≔∫a1b1kθ,ηfjηdη,∀θ∈a1,b1T,j=1,2.
This result is given in [23, 24].


## 4. Inequalities with Special Kernels

The next theorem states the general result for Hardy’s inequality in quotient.Theorem 24.
Suppose0≤ξ≤ξ∗<∞ and u be a weight function. Define υ on e,lT by
(61)υη≔f2η∫ηl∫eσθf2ηΔη−ξ∗/ξuθΔθξ/ξ∗<∞.
IfΦ∈CI,ℝ is a positive convex, then the following inequality
(62)∫eluθΦ∫eσθf1ηΔη∫eσθf2ηΔηξ∗/ξΔθ1/ξ∗≤∫elυηΦf1ηf2ηΔη1/ξ,is available for all measurable functions fj:e,lT→ℝj=1,2.Proof.
Rewrite inequality (40) with Ω1=Ω2=e,lT, Δμ1θ=Δθ, and Δμ2η=Δη. Define the kernel k:e,lT×e,lT→ℝ by
(63)kθ,η≔1,if0≤e≤η≤σθ≤l,0,ifη>σθ.
Then,gj in (41) takes the form
(64)gjθ≔∫eσθfjηΔη.
Substitutinggjθj=1,2 in (40), we acquired (47).Corollary 25.
If we apply Theorem24 for Φθ=θξ, ξ≥1, then (62) takes the form
(65)∫eluθ∫eσθf1ηΔη∫eσθf2ηΔηξ∗Δθ1/ξ∗≤∫elυηf1ηf2ηξΔη1/ξ,where
(66)υη≔f2η∫ηl∫eσθf2ηΔη−ξ∗/ξuθΔθξ/ξ∗<∞.Remark 26.
If we put in Corollary25f2η=1, ξ=ξ∗, and l=∞, then we acquired Hardy’s inequality on time scale
(67)∫e∞uθσθ−e∫eσθf1ηΔηξΔθ≤∫e∞υηf1ξηΔη,where
(68)υη≔∫η∞uθσθ−eΔθ.Remark 27.
If we put in Theorem24T=ℝ, e=0, and l=∞, we see that σθ=θ and then (62) takes the form
(69)∫0∞uθΦ∫0θf1ηdη∫0θf2ηdηξ∗/ξdθ1/ξ∗≤∫0∞υηΦf1ηf2ηdη1/ξ,where
(70)υη≔f2η∫η∞∫0θf2ηdη−ξ∗/ξuθdθξ/ξ∗<∞,which is the same result due to Iqbal et al. (Theorem 2.4, [7]).Example 1.
If we putΦθ=θξ, ξ≥1, and a particular weight function
(71)uθ≔1θ2∫eσθf2ηΔηξ∗/ξ,θ∈e,lT,in (62), we obtain
(72)υη≔f2η∫ηl1θ2Δθξ/ξ∗,and inequality (62) becomes
(73)∫el∫eσθf1ηΔηξ∗∫eσθf2ηΔηξ∗1/ξ−1Δθθ21/ξ∗≤∫el∫ηl1θ2Δθξ/ξ∗f1ξηf21−ξηΔη1/ξ.
Forξ∗=ξ, inequality (73) reduces to
(74)∫el∫eσθf1ηΔηξ∫eσθf2ηΔη1−ξΔθθ2≤∫el∫ηl1θ2Δθf1ξηf21−ξηΔη.Remark 28.
WhenT=ℝ, e=0, l=∞, and f2η=1, we see that σθ=θ, and then (74) takes the form
(75)∫0∞1θ∫0θf1ηdηξdθθ≤∫0∞f1ξηdηη.Now, for the convex functionΦ∈CI,ℝ defined by Φθ=expθ, we can give the general form of Pólya-Knopp’s inequality in quotient.Theorem 29.
Suppose0≤ξ≤ξ∗<∞ and u be a weight function defined on e,lT. Define υ on e,lT by
(76)υη≔lnf2η∫ηl∫eσθlnf2ηΔη−ξ∗/ξuθΔθξ/ξ∗<∞.
Then, the following inequality:(77)∫eluθexp∫eσθlnf1ηΔη∫eσθlnf2ηΔηξ/ξ∗Δθ1/ξ∗≤∫elυηexplnf1ηlnf2ηΔη1/ξ,is available for all positive measurable functions fj:e,lT→ℝj=1,2.Proof.
We rewrite inequality (40) with Ω1=Ω2=e,lT, Δμ1θ=Δθ, Δμ2η=Δη, and Φ:ℝ+→ℝ defined by Φθ=expθ; we obtain
(78)∫eluθexpg1θg2θξ∗/ξΔθ1/ξ∗≤∫elυηexpf1ηf2ηΔη1/ξ.
Definekθ,η such as Theorem 24. Substituting gjθj=1,2 defined by (64) in (78), we acquired
(79)∫eluθexp∫eσθf1ηΔη∫eσθf2ηiηξ∗/ξΔθ1/ξ∗≤∫elυηexpf1ηf2ηΔη1/ξ.
Replacingfj by lnfjj=1,2 in (79), we acquired (77).Remark 30.
If we put in Theorem29T=ℝ, e=0, and l=∞, we see that σθ=θ and then (77) takes the form
(80)∫0∞uθexp∫0θlnf1ηdη∫0θlnf2ηdηξ∗/ξdθ1/ξ∗≤∫0∞υηexplnf1ηlnf2ηdη1/ξ,where
(81)υη≔lnf2η∫η∞∫0θlnf2ηdη−ξ∗/ξuθdθξ/ξ∗<∞,which is the same result due to Iqbal et al. (Corollary 2.6, [7]).Remark 31.
If we putf2η=exp., ξ=ξ∗, and l=∞ in Theorem 29, we have
(82)∫e∞uθexp∫eσθlnf1ηΔησθ−eΔθ≤∫e∞υηf1ηΔη,where
(83)υη≔∫η∞uθσθ−eΔθ<∞.Remark 32.
In a private case of Remark31, when T=ℝ and e=0, we see that σθ=θ, and inequality (82) reduces to
(84)∫0∞uθexp∫0θlnf1ηdηθdθ≤∫0∞υηf1ηdη,where
(85)υη≔∫η∞uθθdθ<∞.Example 2.
Particularly, for the weight function(86)uθ≔1θ2∫eσθlnf2ηΔηξ∗/ξ,θ∈e,lT,in Theorem 29, we obtain
(87)υη≔∫ηl1θ2Δθξ/ξ∗lnf2η,and inequality (77) becomes
(88)∫el1θ2∫eσθlnf2ηΔηξ∗/ξexp∫eσθlnf1ηΔη∫eσθlnf2ηΔηξ∗/ξΔθ1/ξ∗≤∫el∫ηl1θ2Δθξ/ξ∗lnf2ηexplnf1ηlnf2ηΔη1/ξ.Remark 33.
If we putT=ℝ, ξ=ξ∗, f2η=exp., e=0, and l=∞ in (88), we see that σθ=θ, and inequality (88) reduces to
(89)∫0∞exp1θ∫0θlnf1ηdηdθθ≤∫0∞f1ηdηη.The next general result is for Hardy-Hilbert’s inequality.Theorem 34.
Suppose0≤ξ≤ξ∗<∞ and u be a weight function defined on 0,∞T. Define υ on 0,∞T by
(90)υη≔f2η∫0∞uθθ+ηsξ∗/ξ∫0∞f2ηθ+ηsΔη−ξ∗/ξΔθξ/ξ∗<∞.
IfΦ∈CI,ℝ is a positive convex, then the following inequality
(91)∫0∞uθΦ∫0∞f1η/θ+ηsΔη∫0∞f2η/θ+ηsΔηξ∗/ξΔθ1/ξ∗≤∫0∞υηΦf1ηf2ηΔη1/ξ,is available for all measurable functions fj:0,∞T→ℝj=1,2.Proof.
We rewrite (40) with Ω1=Ω2=0,∞T, Δμ1θ=Δθ, and Δμ2η=Δη. Define the kernel k:0,∞T×0,∞T→ℝ by
(92)kθ,η≔η/θs−2/ξθ+ηs,ξ>1.
Then,gj defined in (64) becomes
(93)gjθ≔∫0∞ηθs−2/ξθ+η−sfjηΔη≔θ2−s/ξ∫0∞ηs−2/ξfjηθ+ηsΔη.
Substitutinggjθj=1,2 in (40), we acquired
(94)∫0∞uθΦ∫0∞ηs−2/ξf1η/θ+ηsΔη∫0∞ηs−2/ξf2η/θ+ηsΔηξ∗/ξΔθ1/ξ∗≤∫0∞υηΦf1ηf2ηΔη1/ξ.
Writingfjη instead of ηs−2/ξfjη in (94), we obtain (91).Remark 35.
If we put in Theorem34T=ℝ, then (91) takes the form
(95)∫0∞uθΦ∫0∞f1η/θ+ηsdη∫0∞f2η/θ+ηsdηξ∗/ξdθ1/ξ∗≤∫0∞υηΦf1ηf2ηdη1/ξ,where
(96)υη≔f2η∫0∞uθθ+ηsξ∗/ξ∫0∞f2ηθ+ηsdη−ξ∗/ξdθξ/ξ∗<∞,which is the same result due to Iqbal et al. (Theorem 2.8, [7]).Example 3.
For0<α<sξ∗/ξ and taking the particular weight function
(97)uθ≔θα−1∫0∞f2ηθ+ηsΔηξ∗/ξ,θ∈0,∞T,we obtain,
(98)vη≔f2η∫0∞θα−1θ+ηsξ∗/ξΔθξ/ξ∗.
Let the functionΦ:0,∞→ℝ be defined by Φθ=θξ; then, inequality (91) becomes
(99)∫0∞θα−1∫0∞f1ηθ+ηsΔηξ∗∫0∞f2ηθ+ηsΔηξ∗1/ξ−1Δθ1/ξ∗≤∫0∞∫0∞θα−1θ+ηsξ∗/ξΔθξ∗/ξf1ηξf2η1−ξΔη1/ξ.Remark 36.
WhenT=ℝ in Example 3, we see that
(100)vη≔ηαξ/ξ∗−sf2ηBα,sξ∗ξ−αξ/ξ∗,and then (99) takes the form
(101)∫0∞θα−1∫0∞f1ηθ+ηsdηξ∗∫0∞f2ηθ+ηsdηξ∗1/ξ−1dθ1/ξ∗≤Bα,sξ∗ξ−α1/ξ∗∫0∞ηαξ/ξ∗−sf1ηξf2η1−ξdη1/ξ,where B is the beta function, which is the same result due to Iqbal et al. (Example 2.9, [7]).In the forthcoming theorem, we give the Hardy-Littlewood-Pólya inequality in quotient.Theorem 37.
Suppose0≤ξ≤ξ∗<∞ and u be a weight function defined on 0,∞T. Define υ on 0,∞T by
(102)υη≔f2η∫0∞uθmaxθ,ηsξ∗/ξ∫0∞f2ηmaxθ,ηsΔη−ξ∗/ξΔθξ/ξ∗<∞.
IfΦ∈CI,ℝ is a positive convex, then the following inequality
(103)∫0∞uθΦ∫0∞f1η/maxθ,ηsΔη∫0∞f2η/maxθ,ηsΔηξ∗/ξΔθ1/ξ∗≤∫0∞υηΦf1ηf2ηΔη1/ξ,is available for all measurable functions fj:0,∞T→ℝj=1,2.Proof.
Rewrite inequality (40) with Ω1=Ω2=0,∞T, Δμ1θ=Δθ, and Δμ2η=Δη. Define the kernel Φ:0,∞T×0,∞T→ℝ by
(104)kθ,η≔ηθs−2/ξmaxθ,η−s.
Then,gj defined in (64) becomes
(105)gjθ≔∫0∞η/θs−2/ξmaxθ,ηsfjηΔη≔θ2−s/ξ∫0∞ηs−2/ξfjηmaxθ,ηsΔη.
Substitutinggjθj=1,2 in (40), we acquired
(106)∫0∞uθΦ∫0∞ηs−2/ξf1η/maxθ,ηsΔη∫0∞ηs−2/ξf2η/maxθ,ηsΔηξ∗/ξΔθ1/ξ∗≤∫0∞υηΦf1ηf2ηΔη1/ξ.
Writingfjη instead of ηs−2/ξfjη in (106), we obtain (103).Remark 38.
WhenT=ℝ in Theorem 37, then (103) takes the form
(107)∫0∞uθΦ∫0∞f1η/maxθ,ηsdη∫0∞f2η/maxθ,ηsdηξ∗/ξdθ1/ξ∗≤∫0∞υηΦf1ηf2ηdη1/ξ,where
(108)υη≔f2η∫0∞uθmaxθ,ηsξ∗/ξ∫0∞f2ηmaxθ,ηsdη−ξ∗/ξdθξ/ξ∗<∞,which is the same result due to Iqbal et al. (Theorem 2.10, [7]).Example 4.
For0<α<sξ∗/ξ and taking the particular weight function
(109)uθ≔θα−1∫0∞f2ηmaxθ,ηsΔηξ/ξ∗,θ∈0,∞T,we obtain
(110)vη≔f2η∫0∞θα−1maxθ,ηsξ∗/ξΔθξ∗/ξ.Let the functionΦ:0,∞→ℝ be defined by Φθ=θξ, then inequality (103) becomes
(111)∫0∞θα−1∫0∞f1ηmaxθ,ηsΔηξ∗∫0∞f2ηmaxθ,ηsΔηξ∗1/ξ−1Δθ1/ξ∗≤∫0∞∫0∞θα−1maxθ,ηsξ∗/ξΔθξ/ξ∗f1ηξf2η1−ξΔη1/ξ.Remark 39.
WhenT=ℝ in Example 4, we see that
(112)vη≔ηαξ/ξ∗−sf2ηsξ∗αsξ∗−αξξ/ξ∗,and then inequality (111) becomes
(113)∫0∞θα−1∫0∞f1ηmaxθ,ηsdηξ∗∫0∞f2ηmaxθ,ηsdηξ∗1/ξ−1dθ1/ξ∗≤sξ∗αsξ∗−αξ1/ξ∗∫0∞ηαξ/ξ∗−sf1ξηf21−ξηdη1/ξ,which is the same result due to Iqbal et al. (Example 2.11, [7]).Now, we give the result for the Hardy-Hilbert-type inequality in quotient.Theorem 40.
Let0≤ξ≤ξ∗<∞ and u be a weight function defined on 0,∞T. Define υ on 0,∞T by
(114)υη≔f2η∫0∞uθlnη−lnθη−θξ∗/ξ∫0∞lnη−lnθη−θf2ηΔη−ξ∗/ξΔθξ/ξ∗<∞.
IfΦ∈CI,ℝ is a positive convex, then the following inequality
(115)∫0∞uθΦ∫0∞lnη−lnθ/η−θf1ηΔη∫0∞lnη−lnθ/η−θf2ηΔηξ∗/ξΔθ1/ξ∗≤∫0∞υηΦf1ηf2ηΔη1/ξ,is available for all measurable functions fj:0,∞T→ℝj=1,2.Proof.
Rewrite inequality (40) with Ω1=Ω2=0,∞T, Δμ1θ=Δθ, and Δμ2η=Δη. For α∈0,1, we define the kernel k:0,∞T×0,∞T→ℝ by
(116)kθ,η≔lnη/θη−θθηα.
Then,gj defined in (64) takes the form
(117)gjθ≔∫0∞lnη−lnθη−θθηαfjηΔη≔θα∫0∞lnη−lnθη−θη−αfjηΔη.
Substitutinggjθj=1,2 in (40), we acquired
(118)∫0∞uθΦ∫0∞lnη−lnθ/η−θη−αf1ηΔη∫0∞lnη−lnθ/η−θη−αf2ηiηξ∗/ξΔθ1/ξ∗≤∫0∞υηΦf1ηf2ηΔη1/ξ.
Writingfjη instead of η−αfjη in (118), we obtain (115).Remark 41.
If we putT=ℝ in Theorem 34, we acquired the Hardy-Hilbert-type inequality in [7] (Theorem 2.12):
(119)∫0∞uθΦ∫0∞lnη−lnθ/η−θf1ηdη∫0∞lnη−lnθ/η−θf2ηdηξ∗/ξdθ1/ξ∗≤∫0∞υηΦf1ηf2ηdη1/ξ,where
(120)υη≔f2η∫0∞uθlnη−lnθη−θξ∗/ξ∫0∞lnη−lnθη−θf2ηdη−ξ∗/ξdθξ/ξ∗<∞.Example 5.
Forα∈0,1 and for
(121)uθ≔θ−αlnη/θη−θf2ηΔηξ∗/ξ,θ∈0,∞T,in (114), we obtain
(122)υη≔η1−αξ/ξ∗−1f2ηC,where
(123)C≔∫0∞z−αlnzz−1ξ∗/ξΔzξ/ξ∗.
Letξ≥1 and the function Φ:0,∞→ℝ be defined by Φθ=θξ. Then, inequality (115) becomes
(124)∫0∞θ−α∫0∞lnη/θη−θf1ηΔηξ∗∫0∞lnη/θη−θf2ηΔηξ∗1/ξ−1Δθ1/ξ∗≤C∫0∞η1−αξ/ξ∗−1f1ξηf21−ξηΔη1/ξ.Remark 42.
WhenT=ℝ in Example 5, we see that
(125)uθ≔θ−αlnη/θη−θf2ηdηξ∗/ξ,θ∈0,∞,υη≔η1−αξ/ξ∗−1f2ηC,where
(126)C≔∫0∞z−αlnzz−1ξ∗/ξdzξ/ξ∗,and then inequality (124) becomes
(127)∫0∞θ−α∫0∞lnη/θη−θf1ηdηξ∗∫0∞lnη/θη−θf2ηdηξ∗ξ−1−1dθ1/ξ∗≤C∫0∞η1−αξ/ξ∗−1f1ξηf21−ξηdη1/ξ,which is the same result due to Iqbal et al. (Example 2.13, [7]).


## 5. Conclusions

In this paper, with the help of special kernels, we obtained some nouveau forms for Hardy’s and related inequalities on time scales in quotients and also we presented some recent results for the Pólya-Knopp-, Hardy-Hilbert-, Hardy-Littlewood-Pólya-type inequalities. Furthermore, we got some continuous inequalities as special cases of the obtained dynamic inequalities.



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