# A Study on Some New Reverse Hilbert-Type Inequalities and Its Generalizations on Time Scales

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

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## Abstract

This study develops the study of reverse Hilbert-type inequalities on time scales where we can establish some new generalizations of reverse Hilbert-type inequalities via supermultiplicative functions on time scales by applying reverse Hölder inequalities. The main results will be proved by using Specht’s ratio, chain rule, and Jensen’s inequality. Our results (whenT=ℕ) are essentially new.


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## Body

## 1. Introduction

In [1], Hardy proved that(1)∑n=1∞∑m=1∞ΩmFnm+n≤πsinπ/d∑m=1∞Ωmd1/d∑n=1∞Fnl1/l,where Ωm, Fn≥0 with 0<∑m=1∞Ωmd<∞, 0<∑n=1∞Fnl<∞, and d>1, 1/d+1/l=1. The continuous form ([2]) of (1) is(2)∫0∞∫0∞fηgπη+πdηdπ≤πsinπ/d∫0∞fdηdη1/d∫0∞glπdπ1/l,where d>1, 1/d+1/l=1, f,g≥0 are the measurable functions such that 0<∫0∞fdηdη<∞ and 0<∫0∞glπdπ<∞. The constant π/sinπ/d in both (2) and (1) is sharp. In [2], Hardy showed that if d>1, l>1, 1/d+1/l≥1, and 0<λ=2−1/d+1/l≤1, then(3)∑n=1∞∑m=1∞ΩmFnm+nλ≤Kd,l∑m=1∞Ωmd1/d∑n=1∞Fnl1/l,where 1/d+1/l=1, λ=2−1/d+1/l=1, and the constant factor Kd,l is sharp. In [3], the authors showed that if 0≤d,l≤1 and λmm=1k, ψnn=1r are nonnegative and decreasing sequences of real numbers with k,r∈ℕ, and then(4)∑m=1k∑n=1rSd,l,k,r,m,n∑s=1mλsd∑t=1nψtlmn1/2≥2Cd,l,k,r∑m=1kλm∑s=1mλsd−12k−m+11/2×∑n=1rψn∑t=1nψtl−12r−n+11/2,where(5)Cd,l,r,s=12dlkr1/2,Sd,l,k,r,m,n=Sk∑s=1mλs∑τ=1sλτd−12∑s=1kk−s+1λs∑τ=1sλτd−12Smλu∑τ=1uλτd−12∑s=1mλs∑τ=1sλτd−12×Sr∑t=1nψt∑τ=1tψτl−12∑t=1rr−t+1ψt∑τ=1tψτl−12Snψv∑τ=1vψτl−12∑t=1nψt∑τ=1tψτl−12,where(6)Smλu∑τ=1uλτd−12∑s=1mλs∑τ=1sλτd−12=maxSmλ1∑τ=11λτd−12∑s=1mλs∑τ=1sλτd−12;Smλm∑τ=1mλτd−12∑s=1mλs∑τ=1sλτd−12,Snψv∑τ=1vψτl−12∑t=1nψt∑τ=1tψτl−12=maxSnψ1∑τ=11ψτl−12∑t=1nψt∑τ=1tψτl−12;Snψn∑τ=1nψτl−12∑t=1nψt∑τ=1tψτl−12,where S⋅ is Specht’s ratio ([4]) which is defined as follows:(7)Sl=l1/l−1elogl1/l−1,l≠1,S1=1.Also, they proved that ifλmm=1k, ψnn=1r are nonnegative sequences with k,r∈ℕ and dm, ln are positive sequences. Let ϕ, ψ are nonnegative, concave, and supermultiplicative functions. Then,(8)∑m=1k∑n=1rSk,r,m,nϕΛmψΩnmn1/2≥2Mk,r∑s=1kdsϕλsds2k−s+11/2×∑t=1rltψωtlt2r−t+11/2,with(9)Mk,r=12∑m=1kϕDm/Dm21/2∑n=1rψLn/Ln21/2,Sk,r,m,n=S∑s=1kdsϕλs/ds2k−s+1ϕDm/Dm2∑m=1kϕDm/Dm2∑s=1mdsϕλs/ds2×S∑t=1rltψωt/lt2r−t+1ψLn/Ln2∑n=1rψLn/Ln2∑t=1nltψωt/lt2,Λm=∑s=1mSmdsϕλs/ds2∑s=1mdsϕλs/ds2λs,Ωn=∑t=1nSnltψωt/lt2∑t=1nltψωt/lt2ωt,Dm=∑s=1mSmdsϕλs/ds2∑s=1mdsϕλs/ds2ds,Ln=∑t=1nSnltψωt/lt2∑t=1nltψωt/lt2lt.In [3], the authors showed that if λmm=1k, ψnn=1r are nonnegative sequences with k,r∈ℕ, then(10)∑m=1k∑n=1rSk,r,m,nΛmΩnmn1/2≥kr1/2∑m=1kλm2k−m+11/2∑n=1rωn2r−n+11/2,with(11)Sk,r,m,n=S∑s=1kλs2k−s+1k∑s=1mλs2S∑t=1rωt2r−t+1r∑t=1nωt2,Λm=∑s=1mSmλs2∑s=1mλs2λsΩn=∑t=1nSnωt2∑t=1nωt2ωt.The theory of time scales, which has recently received a lot of attention, was initiated by Hilger in his PhD thesis in order to unify discrete and continuous analysis [5]. The general idea is to prove a result for a dynamic equation or a dynamic inequality where the domain of the unknown function is a so-called time scale T, which may be an arbitrary closed subset of the real numbers ℝ.During the past decade, a number of dynamic inequalities have been established by some authors which are motivated by practical problems ([6–9]).The goal of this manuscript is to use reverse Hölder inequalities with Specht’s ratio onT to develop some new generalizations of reverse Hilbert-type inequalities and its generalizations via supermultiplicative functions.The following is a breakdown of the study’s structure. In Section2, we cover some fundamentals of time scale theory as well as several time scale lemmas that will be useful in Section 3, where we prove our findings. As specific examples (when T=ℕ), our major results yield (4), (8), and (10) proven by Zhao and Cheung [3].


## 2. Definitions and Basic Lemmas

First, we defineστ:=infu∈T:u>τ, where the forward jump operator, CrdT,ℝ introduces the set of all such rd-continuous functions, and for any function U:T⟶ℝ, the notation Uστ denotes Uστ.The derivatives of the productUϖ and the quotient U/ϖ (where ϖϖσ≠0) of two differentiable functions U and ϖ are given by(12)UϖΔ=UΔϖ+UσϖΔ=UϖΔ+UΔϖσ,UϖΔ=UΔϖ−UϖΔϖϖσ.The integration by parts formula onT is(13)∫υ0υλτφΔτΔτ=λτφτυ0υ−∫υ0υλΔτφστΔτ.The time scales chain rule ([10], Theorem 1.87) is(14)ϖ∘φΔτ=ϖ′φχφΔτ,whereχ∈τ,στ,where it is supposed that ϖ:ℝ⟶ℝ is continuously differentiable and φ:T⟶ℝ is Δ−differentiable. For further information on time scale calculus, see [10, 11].Definition 1.
(See [12]). A function L:I⟶ℝ+ is supermultiplicative if(15)Lχζ≥LχLζ,∀χ,ζ∈I⊂ℝ.
WhenL is the identity map (i.e., Lζ=ζ), the inequality (15) holds with equality. L is said to be a submultiplicative function if the last inequality has the opposite sign.Lemma 1.
Letz∈T, λ∈CrdT,ℝ be nonnegative and 0≤γ≤1. Then,(16)∫zσtλτΔτγ≥γ∫zσt∫zσηλτΔτγ−1ληΔη.Proof.
By applying (14) on the term ∫zηλτΔτ, we get(17)∫zηλτΔτγΔ=γ∫zζλτΔτγ−1λη,ζ∈η,ση.
Sinceζ≤ση, then we have (note 0≤γ≤1)(18)∫zζλτΔτγ−1≥∫zσηλτΔτγ−1.
Substituting (18) into (17), we see that(19)∫zηλτΔτγΔ≥γ∫zσηλτΔτγ−1λη.
Integrating (19) over η from z to σt, we have(20)∫zσt∫zηλτΔτγΔΔη≥γ∫zσt∫zσηλτΔτγ−1ληΔη,i.e.,(21)∫zσtλτΔτγ≥γ∫zσt∫zσηλτΔτγ−1ληΔη,which is (16).Lemma 2 (specht’s ratio [4]).
Letc,d be positive numbers, γ>1 and 1/γ+1/δ=1. Then,(22)Scdc1/γd1/δ≥cγ+dδ,where(23)Sl=l1/l−1elogl1/l−1,l≠1,S1=1.Lemma 3 (see [4]).
LetS⋅ be defined as in Lemma 2. Then, Sl is strictly decreasing for 0<l<1 and strictly increasing for l>1. In addition, the following equations are true:(24)S1=1,Sl=S1l∀l>0.Lemma 4 (see [13], when α=1).
Letf,g∈Cz,wT,ℝ+ s.t. fβ,gν be Δ−integrable on z,wT. If β>1 and 1/β1/ν=1, then(25)∫ZwSπfβζηgνζfζgζΔζ≥∫ZwfβζΔζ1/β∫ZwgνζΔζ1/v,where η=∫zbfβζΔζ and π=∫zbgνζΔζ.Lemma 5 (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(26)Ψ1∫ζ0ζλτΔτ∫ζ0ζλτφτΔτ≤1∫ζ0ζλτΔτ∫ζ0ζλτΨφτΔτ.The inequality (26) is reversed when Ψ is continuous and concave.The authors used Jensen’s inequality and related inequalities [14–19].Lemma 6.
Letz∈T, λ,ψ be nonnegative and decreasing functions, f,g are positive and nondecreasing functions, and 0≤d,l≤1. If β>1, ν>1 with 1/β1/v=1, then(27)S∫zσtfνηΔηλζ∫zσζλτΔτd−1βfνζ∫zσtλη∫zσηλτΔτd−1βΔη=maxS∫zσtfνηΔηλz∫zσzλτΔτd−1βfνz∫zσtλη∫zσηλτΔτd−1βΔη;S∫zσtfνηΔηλt∫zσtλτΔτd−1βfνt∫zσtλη∫zσηλτΔτd−1βΔηS∫zσξgνπΔπψη∫zσηψτΔτl−1βgνη∫zσξψπ∫zσπψτΔτl−1βΔπmaxS∫zσξgνπΔπψz∫zσzψτΔτl−1βgνz∫zσξψπ∫zσπψτΔτl−1βΔπ;S∫zσξgνπΔπψξ∫zσξψτΔτl−1βgνξ∫zσξψπ∫zσπψτΔτl−1βΔπProof.
Forη≤π, we have(28)∫zσηλτΔτ≤∫zσπλτΔτ,and then (where 0≤d≤1),(29)∫zσηλτΔτd−1≥∫zσπλτΔτd−1.
Sinceλ is decreasing and f is nondecreasing, we get(30)1fνηλη∫zσηλτΔτd−1β≥1fνπλπ∫zσπλτΔτd−1β,thus, the function 1/fνηλη∫zσηλτΔτd−1β is decreasing. Therefore, we have for z≤η that(31)1fνzλz∫zσzλτΔτd−1β≥1fνηλη∫zσηλτΔτd−1β,and then(32)fνηλz∫zσzλτΔτd−1β≥fνzλη∫zσηλτΔτd−1β.
Integrating the last inequality overη from z to σt, to get(33)λz∫zσzλτΔτd−1β∫zσtfνηΔη≥fνz∫zσtλη∫zσηλτΔτd−1βΔη,and then(34)∫zσtfνηΔηλz∫zσzλτΔτd−1βfνz∫zσtλη∫zσηλτΔτd−1βΔη≥1.
Since the function1/fνηλη∫zσηλτΔτd−1β is decreasing, we have for η≤t that(35)1fνηλη∫zσηλτΔτd−1β≥1fνtλt∫zσtλτΔτd−1β,and then(36)fνtλη∫zσηλτΔτd−1β≥fνηλt∫zσtλτΔτd−1β.
By integrating (37) over η from z to σt, we have(37)fνt∫zσtλη∫zσηλτΔτd−1βΔη≥λt∫zσtλτΔτd−1β∫zσtfνηΔη,thus,(38)∫zσtfνηΔηλt∫zσtλτΔτd−1βfνt∫zσtλη∫zσηλτΔτd−1βΔη≤1.
From (34) and (39), we observe that(39)∫zσtfνηΔηλz∫zσzλτΔτd−1βfνz∫zσtλη∫zσηλτΔτd−1βΔη≥…≥1≥…≥∫zσtfνηΔηλt∫zσtλτΔτd−1βfνt∫zσtλη∫zσηλτΔτd−1βΔη.
SinceS⋅ is decreasing on 0,1 and increasing on 1,∞, so we obtain that one of(40)S∫zσtfνηΔηλz∫zσzλτΔτd−1βfνz∫zσtλη∫zσηλτΔτd−1βΔη,S∫zσtfνηΔηλt∫zσtλτΔτd−1βfνt∫zσtλη∫zσηλτΔτd−1βΔη,is maximum (where S1=1), and it is in the form(41)S∫zσtfνηΔηλζ∫zσζλτΔτd−1βfνζ∫zσtλη∫zσηλτΔτd−1βΔη=maxS∫zσtfνηΔηλz∫zσzλτΔτd−1βfνz∫zσtλη∫zσηλτΔτd−1βΔη;S∫zσtfνηΔηλt∫zσtλτΔτd−1βfνt∫zσtλη∫zσηλτΔτd−1βΔη,which is (27). Similarly, with respect to ψ when 0≤l≤1, we get(42)S∫zσξgνπΔπψη∫zσηψτΔτl−1βgνη∫zσξψπ∫zσπψτΔτl−1βΔπ=maxS∫zσξgνπΔπψz∫zσzψτΔτl−1βgνz∫zσξψπ∫zσπψτΔτl−1βΔπ;S∫zσξgνπΔπψξ∫zσξψτΔτl−1βgνξ∫zσξψπ∫zσπψτΔl−1βΔπ,which is (27).


## 3. Main Results

Throughout the study, we will assume that the functions are nonnegative rd-continuous functions onz,∞T:=z,∞T∩T and the integrals considered exist. Now, we can state and prove our results.Theorem 1.
Letz∈T, 0≤d,l≤1, and λ, ψ be nonnegative and decreasing functions. If f,g are positive and nondecreasing functions and β>1, ν>1 with 1/β+1/v=1, then(43)∫zσs∫zσrSd,l,t,ξ,r,s,β∫zσtλτΔτd∫zσξψτΔτlf2tg2ξ∫zσtfνηΔη1/v∫zσξgνπΔπ1/vΔtΔξ≥νCd,l,r,s,ν∫zσrλt∫zσtλτΔτd−1βσr−tΔt1/β×∫zσsψξ∫zσξψτΔτl−1βσs−ξΔξ1/β,holds for all r,s∈z,∞T, where(44)Cd,l,r,s,ν=dlν∫zσrfνηΔη1/v∫zσsgνηΔη1/v,Sd,l,t,ξ,r,s,β=S∫zσtfνηΔηλζ∫zσζλτΔτd−1βfνζ∫zσtλη∫zσηλτΔτd−1βΔη×S∫zσξgνπΔπψη∫zσηψτΔτl−1βgνη∫zσξψπ∫zσπψτΔτl−1βΔπ×S∫zσrfνηΔη∫zσtλζ∫zσζλτΔτd−1βfνt∫zσrλη∫zσηλτΔτd−1βσr−ηΔη×S∫zσsgνηΔη∫zσξψη∫zσηψτΔτl−1βgνξ∫zσsψπ∫zσπψτΔτl−1βσs−πΔπ,such that(45)S∫zσtfνηΔηλζ∫zσζλτΔτd−1βfνζ∫zσtλη∫zσηλτΔτd−1βΔη=maxS∫zσtfνηΔηλz∫zσzλτΔτd−1βfνz∫zσtλη∫zσηλτΔτd−1βΔη;S∫zσtfνηΔηλt∫zσtλτΔτd−1βfνt∫zσtλη∫zσηλτΔτd−1βΔηS∫zσξgνπΔπψη∫zσηψτΔτl−1βgνη∫zσξψπ∫zσπψτΔτl−1βΔπ=maxS∫zσξgνπΔπψz∫zσzψτΔτl−1βgνz∫zσξψπ∫zσπψτΔτl−1βΔπ;S∫zσξgνπΔπψξ∫zσξψτΔτl−1βgνξ∫zσξψπ∫zσπψτΔτl−1βΔπ.Proof.
.Applying (16) with γ=d, we have(46)∫zσtλτΔτd≥d∫zσtλη∫zσηλτΔτd−1Δη.
Multiplying the last inequality by(47)S∫zσtfνηΔηλζ∫zσζλτΔτd−1βfνζ∫zσtλη∫zσηλτΔτd−1βΔηft,we get (where f is nondecreasing)(48)S∫zσtfνηΔηλζ∫zσζλτΔτd−1βfνζ∫zσtλη∫zσηλτΔτd−1βΔη×∫zσtλτΔτdft≥d∫zσtS∫zσtfνηΔηλζ∫zσζλτΔτd−1βfνζ∫zσtλη∫zσηλτΔτd−1βΔη×λη∫zσηλτΔτd−1ftΔη≥d∫zσtS∫zσtfνηΔηλζ∫zσζλτΔτd−1βfνζ∫zσtλη∫zσηλτΔτd−1βΔη×λη∫zσηλτΔτd−1fηΔη.
From Lemma6, the last inequality becomes(49)S∫zσtfνηΔηλζ∫zσζλτΔτd−1βfνζ∫zσtλη∫zσηλτΔτd−1βΔη×∫zσtλτΔτdft≥d∫zσtS∫zσtfνηΔηλη∫zσηλτΔτd−1βfνη∫zσtλη∫zσηλτΔτd−1βΔη×λη∫zσηλτΔτd−1fηΔη.
Similarly, we have for the decreasing functionψ, the nondecreasing function g and 0≤l≤1, that(50)S∫zσξgνπΔπψη∫zσηψτΔτl−1βgνη∫zσξψπ∫zσπψτΔτl−1βΔπ∫zσξψτΔτlgξ≥l∫zσξS∫zσξgνπΔπψπ∫zσπψτΔτl−1βgνπ∫zσξψπ∫zσπψτΔτl−1βΔπ×ψπ∫zσπψτΔτl−1gπΔπ.
From (49) and (50), we get(51)S∫zσtfνηΔηλζ∫zσζλτΔτd−1βfνζ∫zσtλη∫zσηλτΔτd−1βΔη∫zσtλτΔτd×S∫zσξgνπΔπψη∫zσηψτΔτl−1βgνη∫zσξψπ∫zσπψτΔτl−1βΔπ×∫zσξψτΔτlftgt≥dl∫zσtS∫zσtfνηΔηλη∫zσηλτΔτd−1βfνη∫zσtλη∫zσηλτΔτd−1βΔη×λη∫zσηλτΔτd−1fηΔη×∫zσξS∫zσξgνπΔπψπ∫zσπψτΔτl−1βgνπ∫zσξψπ∫zσπψτΔτl−1βΔπ×ψπ∫zσπψτΔτl−1gπΔπ.
Applying (25) on the right-hand side of (51), we get(52)S∫zσtfνηΔηλζ∫zσζλτΔτd−1βfνζ∫zσtλη∫zσηλτΔτd−1βΔη×S∫zσξgνπΔπψη∫zσηψτΔτl−1βgνη∫zσξψπ∫zσπψτΔτl−1βΔπ×∫zσtλτΔτd∫zσξψτΔτlftgt≥dl∫zσtfνηΔη1/v∫zσtλη∫zσηλτΔτd−1βΔη1/β×∫zσξgνπΔπ1/v∫zσξψπ∫zσπψτΔτl−1βΔπ1/β.
Multiply (52) by(53)S∫zσrfνηΔη∫zσtλζ∫zσζλτΔτd−1βfνt∫zσrλη∫zσηλτΔτd−1βσr−ηΔη×S∫zσsgνηΔη∫zσξψη∫zσηψτΔτl−1βgνξ∫zσsψπ∫zσπψτΔτl−1βσs−πΔπ.
We see that(54)Sd,l,t,ξ,r,s,β∫zσtλτΔτd∫zσξψτΔτlftgξ=S∫zσtfνηΔηλζ∫zσζλτΔτd−1βfνζ∫zσtλη∫zσηλτΔτd−1βΔη×S∫zσξgνπΔπψη∫zσηψτΔτl−1βgνη∫zσξψπ∫zσπψτΔτl−1βΔπ×S∫zσrfνηΔη∫zσtλζ∫zσζλτΔτd−1βfνt∫zσrλη∫zσηλτΔτd−1βσr−ηΔη×S∫zσsfνηΔη∫zσξψη∫zσηψτΔτl−1βgνξ∫zσsψπ∫zσπψτΔτl−1βσs−πΔπ×∫zσtλτΔτd∫zσξψτΔτlftgt≥dl∫zσtfνηΔη1/v∫zσtλη∫zσηλτΔτd−1βΔη1/β×∫zσξgνπΔπ1/v∫zσξψπ∫zσπψτΔτl−1βΔπ1/β×S∫zσrfνηΔη∫zσtλζ∫zσζλτΔτd−1βfνt∫zσrλη∫zσηλτΔτd−1βσr−ηΔη×S∫zσsfvηΔη∫zσξψη∫zσηψτΔτl−1βgνξ∫zσsψπ∫zσπψτΔτl−1βσs−πΔπ.
Multiplying the two sides of (54) by ftgξ/∫zσtfνηΔη1/v∫zσξgνπΔπ1/v and then taking the integration over t from z to σr and the integration over ξ from z to σs, we have(55)∫zσs∫zσrSd,l,t,ξ,r,s,β∫zσtλτΔτd∫zσξψτΔτlf2tg2ξ∫zσtfνηΔη1/v∫zσξgνπΔπ1/vΔtΔξ≥dl∫zσrS∫zσrfνηΔη∫zσtλζ∫zσζλτΔτd−1βfνt∫zσrλη∫zσηλτΔτd−1βσr−ηΔη×∫zσtλη∫zσηλτΔτd−1βΔη1/βftΔt×∫zσsS∫zσsfνηΔη∫zσξψη∫zσηψτΔτl−1βgνξ∫zσsψπ∫zσπψτΔτl−1βσs−πΔπ×∫zσξψπ∫zσπψτΔτl−1βΔπ1/βgξΔξ.
Applying formula (13) on the term(56)∫zσrλη∫zσηλτΔτd−1βσr−ηΔη,with uη=σr−η and vΔη=λη∫zσηλτΔτd−1β, we get(57)∫zσrλη∫zσηλτΔτd−1βσr−ηΔη=σr−ηvη|zσr+∫zσrvσηΔη,where vη=∫zηλθ∫zσθλτΔτd−1βΔθ and then (where vz=0)(58)∫zσrλη∫zσηλτΔτd−1βσr−ηΔη=∫zσr∫zσηλθ∫zσθλτΔτd−1βΔθΔη.
Similarly, we see that(59)∫zσsψπ∫zσπψτΔτl−1βσs−πΔπ=∫zσs∫zσπψθ∫zσθψτΔτl−1βΔθΔπ.
Substituting (58) and (59) into (55) and then by applying (25), we observe that(60)∫zσs∫zσrSd,l,t,ξ,r,s,β∫zσtλτΔτd∫zσξψτΔτlf2tg2ξ∫zσtfνηΔη1/v∫zσξgνπΔπ1/vΔtΔξ≥dl∫zσrS∫zσrfνηΔη∫zσtλζ∫zσζλτΔτd−1βΔζfνt∫zσr∫zσηλθ∫zσθλτΔτd−1βΔθΔη×∫zσtλη∫zσηλτΔτd−1βΔη1/βftΔt×∫zσsS∫zσsgνηΔη∫zσξψη∫zσηψτΔτl−1βΔηgνξ∫zσs∫zσπψθ∫zσθψτΔτl−1βΔθΔπ×∫zσξψπ∫zσπψτΔτl−1βΔπ1/βgξΔξ≥dl∫zσrfνηΔη1/v∫zσr∫zσtλη∫zσηλτΔτd−1βΔηΔt1/β×∫zσsgνηΔη1/v∫zσs∫zσξψπ∫zσπψτΔτl−1βΔπΔξ1/β.
From (58), (59), and (60), the last inequality becomes(61)∫zσs∫zσrSd,l,t,ξ,r,s,β∫zσtλτΔτd∫zσξψτΔτlf2tg2ξ∫zσtfνηΔη1/v∫zσξgνπΔπ1/vΔtΔξ≥dl∫zσrfνηΔη1/v∫zσrλt∫zσtλτΔτd−1βσr−tΔt1/β×∫zσsgνηΔη1/v∫zσsψξ∫zσξψτΔτl−1βσs−ξΔξ1β=νCd,l,r,s,ν∫zσrλt∫zσtλτΔτd−1βσr−tΔt1/β×∫zσsψξ∫zσξψτΔτl−1βσs−ξΔξ1/β,which is (43).Remark 1.
IfT=ℕ, fη=gη=1 and β=ν=2, in Theorem 1, then we obtain (4) demonstrated in [3].Theorem 2.
Letz∈T and λ,ω be nonnegative functions. Also, we assume that d,l,f,g are positive functions. If ϕ,ψ are nonnegative, concave, and supermultiplicative functions and β>1, ν>1 with 1/β+1/ν=1, then(62)∫zσs∫zσrSt,r,s,ζ,ν,βϕΛtψΩζ∫zσtfνηΔη1/v∫zσζgνπΔπ1/vΔtΔζ≥νMr,s,ν∫zσrdηϕληdηβσr−ηΔη1/β×∫zσslπψωπlπβσs−πΔπ1/βholds for all r,s∈z,∞T, with(63)Mr,s,ν=1ν∫zσrϕDtDtνΔt1/v∫zσsψLζLζνΔζ1/vSt,r,s,ζ,ν,β=S∫zσrdηϕλη/dηβσr−ηΔηϕDt/Dtν∫zσrϕDt/DtνΔt∫zσtdηϕλη/dηβΔη×S∫zσslπψωπ/lπβσs−πΔπψLζ/Lζν∫zσsψLζ/LζνΔζ∫zσζlπψωπ/lπβΔπ,Λt=∫zσtS∫zσtfνηΔηdηϕλη/dηβfνη∫zσtdηϕλη/dηβΔηfηληΔη,Ωζ=∫zσζS∫zσζgνπΔπlπψωπ/lπβgνπ∫zσζlπψωπ/lπβΔπgπωπΔπ,Dt=∫zσtS∫zσtfνηΔηdηϕλη/dηβfνη∫zσtdηϕλη/dηβΔηdηfηΔη,Lζ=∫zσζS∫zσζgνπΔπlπψωπ/lπβgνπ∫zσζlπψωπ/lπβΔπlπgπΔπ.Proof.
Using the fact thatϕ is a supermultiplicative function, apply Jensen’s inequality and then apply (25). Then,(64)ϕΛt=ϕDt∫zσtS∫zσtfvηΔηdηϕλη/dηβ/fvη∫zσtdηϕλη/dηβΔηdηfηλη/dηΔη∫zσts∫zσtfvηΔηdηϕλη/dηβ/fvη∫zσtdηϕλη/dηβΔηdηfηΔη≥ϕDtϕ∫zσtS∫zσtfvηΔηdηϕλη/dηβ/fvη∫zσtdηϕλη/dηβΔηdηfηλη/dηΔη∫zσts∫zσtfvηΔηdηϕλη/dηβ/fvη∫zσtdηϕλη/dηβΔηdηfηΔη≥ϕDtDt∫zσtS∫zσtfvηΔηdηϕλη/dηβfvη∫zσtdηϕλη/dηβΔηdηϕληdηfηΔη≥ϕDtDt∫zσtfvηΔη1/v∫zσtdηϕληdηβΔη1/β.
Similarly, we can get(65)ψΩζ≥ψLζLζ∫zσζgνπΔπ1/v∫zσζlπψωπlπβΔπ1/β.
Multiplying the both sides of (64) and (65), respectively, by(66)S∫zσrdηϕλη/dηβσr−ηΔηϕDt/Dtν∫zσrϕDt/DtνΔt∫zσtdηϕλη/dηβΔη,S∫zσslπψωπ/lπβσs−πΔπψLζ/Lζν∫zσsψLζ/LζνΔζ∫zσζlπψωπ/lπβΔπ,and then multiplying these inequalities, we get(67)S∫zσrdηϕλη/dηβσr−ηΔηϕDt/Dtν∫zσrϕDt/DtνΔt∫zσtdηϕλη/dηβΔηϕΛt×S∫zσslπψωπ/lπβσs−πΔπψLζ/Lζν∫zσsψLζ/LζνΔζ∫zσζlπψωπ/lπβΔπψΩζ≥ϕDtDt∫zσtfνηΔη1/v∫zσtdηϕληdηβΔη1/β×S∫zσrdηϕλη/dηβσr−ηΔηϕDt/Dtν∫zσrϕDt/DtνΔt∫zσtdηϕλη/dηβΔη×ψLζLζ∫zσζgνπΔπ1/v∫zσζlπψωπ/lπβΔπ1/β×S∫zσslπψωπ/lπβσs−πΔπψLζ/Lζν∫zσsψLζ/LζνΔζ∫zσζlπψωπ/lπβΔπ.
Dividing the two sides of (67) on ∫zσtfνηΔη1/v∫zσζgνπΔπ1/v and then taking the integration over ζ from z to σs and then the integration over t from z to σr, we have(68)∫zσs∫zσrSt,r,s,ζ,ν,βϕΛtψΩζ∫zσtfνηΔη1/v∫zσζgνπΔπ1/vΔtΔζ≥∫zσrS∫zσrdηϕλη/dηβσr−ηΔηϕDt/Dtν∫zσrϕDt/DtνΔt∫zσtdηϕλη/dηβΔη×ϕDtDt∫zσtdηϕληdηβΔη1/βΔt×∫zσsS∫zσslπψωπ/lπβσs−πΔπψLζ/Lζν∫zσsψLζ/LζνΔζ∫zσζlπψωπ/lπβΔπ×ψLζLζ∫zσζlπψωπlπβΔπ1/βΔζ.
By using the integration by parts, we can see that(69)∫zσrdηϕληdηβσr−ηΔη=∫zσr∫zσηdθϕλθdθβΔθΔη.
Also, we can obtain that(70)∫zσslπψωπlπβσs−πΔπ=∫zσs∫zσπlθψωθlθβΔθΔπ.
Substituting (69) and (70) into (68), we have(71)∫zσs∫zσrSt,r,s,ζ,ν,βϕΛtψΩζ∫zσtfνηΔη1/v∫zσζgνπΔπ1/vΔtΔζ≥∫zσrS∫zσr∫zσηdθϕλθ/dθβΔθΔηϕDt/Dtν∫zσrϕDt/DtνΔt∫zσtdηϕλη/dηβΔη×ϕDtDt∫zσtdηϕληdηβΔη1/βΔt×∫zσsS∫zσs∫zσπlθψωθ/lθβΔθΔπψLζ/Lζν∫zσsψLζ/LζνΔζ∫zσζlπψωπ/lπβΔπ×ψLζLζ∫zσζlπψωπlπβΔπ1/βΔζ.
Applying (25) on the right-hand side of (71), we have(72)∫zσs∫zσrSt,r,s,ζ,ν,βϕΛtψΩζ∫zσtfνηΔη1/v∫zσζgνπΔπ1/vΔtΔζ≥∫zσrϕDtDtνΔt1/v∫zσr∫zσηdθϕλθdθβΔθΔη1/β×∫zσsψLζLζνΔζ1/v∫zσs∫zσπlθψωθlθβΔθΔπ1/β=νMr,s,ν∫zσr∫zσηdθϕλθdθβΔθΔη1/β×∫zσs∫zσπlθψωθlθβΔθΔπ1/β.
From (69) and (70), the inequality (72) becomes(73)∫zσs∫zσrSt,r,s,ζ,ν,βϕΛtψΩζ∫zσtfνηΔη1/v∫zσζgνπΔπ1/vΔtΔζ≥νMr,s,ν∫zσrdηϕληdηβσr−ηΔη1/β×∫zσslπψωπlπβσs−πΔπ1/β,which is (62).Remark 2.
IfT=ℕ, fη=gη=1, and β=ν=2, then we obtain (8) demonstrated in [3].Theorem 3.
Letz∈T and λ, ω be nonnegative functions. If f, g are the positive functions and β>1, ν>1 with 1/β+1/v=1, then(74)∫zσs∫zσrSt,r,s,ζ,ν,βΛtΩζ∫zσtfνηΔη1/v∫zσζgνπΔπ1/vΔtΔζ≥νMr,s,ν∫zσrλβησr−ηΔη1/β∫zσsωβπσs−πΔπ1/βholds for all r,s∈z,∞T, with(75)Mr,s,ν=1νσr−z1/vσs−z1/v,St,r,s,ζ,ν,β=S∫zσrλβησr−ηΔησr−z∫zσtλβηΔη×S∫zσsωβπσs−πΔπσs−z∫zσζωβπΔπ,Λt=∫zσtS∫zσtfνηΔηλβηfνη∫zσtλβηΔηfηληΔη,Ωζ=∫zσζS∫zσζgνπΔπωβπgνπ∫zσζωβπΔπgπωπΔπ.Proof.
Puttingϕη=η and ψπ=π in Theorem 2, we acquired (74).Remark 3.
As a special case of Theorem3, if T=ℕ, fη=gη=1 and β=ν=2, we can get (10) proved by Zhao and Cheung [3].


## 4. Conclusion and Future Work

In this study, we establish some new generalizations of reverse Hilbert-type inequalities on time scales by applying reverse Hölder inequalities on time scales and Specht’s ratio function. In the future, we will continue to generalize more fractional dynamic inequalities by using Kantorovich’s ratio and n-tuple fractional integral. In particular, such inequalities can be introduced by using fractional integrals and fractional derivatives of the Riemann–Liouville type on time scales. In addition to this, we may generalize these results to be with multidimensional Hilbert-type inequalities via supermultiplicative functions on time scales. It will also be very enjoyable to introduce such inequalities in quantum calculus.



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