# A Study on Some New Generalizations of Reversed Dynamic Inequalities of Hilbert-Type via Supermultiplicative Functions

Authors: M. Zakarya; Ahmed I. Saied; Ghada ALNemer; H. A. Abd El-Hamid; H. M. Rezk
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this article, we establish some new generalizations of reversed dynamic inequalities of Hilbert-type via supermultiplicative functions by applying reverse Hölder inequalities with Specht’s ratio on time scales. We will generalize the inequalities by using a supermultiplicative function which the identity map represents a special case of it. Also, we use some algebraic inequalities such as the Jensen inequality and chain rule to prove the essential results in this paper. Our results (whenT≪ℕ) are essentially new.


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## Body

## 1. Introduction

In [1], Hardy established that
(1)∑n=1∞∑m=1∞wmznm+n≤πsinπ/p∑m=1∞wmp1/p∑n=1∞znq1/q,where wm,zn≥0 with 0<∑m=1∞wmp<∞,0<∑n=1∞znq<∞ and p>1,1/p+1/q=1. In [2], Hardy showed that
(2)∫0∞∫0∞fϑgyϑ+ydϑdy≤πsinπ/p∫0∞fpϑdϑ1/p∫0∞gqydy1/q,where f and g are measurable nonnegative functions such that 0<∫0∞fpϑdϑ<∞ and 0<∫0∞gqydy<∞. The constant π/sinπ/p is the best possible.In [3], Hölder proved that
(3)∑k=1nζkyk≤∑k=1nζkp1/p∑k=1nykq1/q,where ζk and yk are positive sequences and p,q>1 such that 1/p+1/q=1. The integral form of (3) is
(4)∫wzψτϖτdτ≤∫wzψpτdτ1/p∫wzϖqτdτ1/q,where ψ,ϖ∈Cw,z,ℝ+.In [4], Zhao and Cheung showed that if ψζ and ϖζ are nonnegative continuous functions and ψ1/pζϖ1/qζ is integrable on w,z, then
(5)∫wzψpζdζ1/p∫wzϖqζζ1/q≤wzSYψpζϑϖqζψζϖζdζ,with
(6)ϑ=wzψpζdζ,Y=wzϖqζdζ,p>1and1p+1q=1,where S. is the Specht’s ratio function (see [5]) and defined by
(7)Sl≔l1/l−1elogl1/l−1,l≠1,S1=1.Also, they proved that ifψ,ϖ∈Cw,z,ℝ+ and γ>0, then
(8)∫wzψγ+1ζϖγζdζ≤∫wzSGψγ+1ζ/Fϖγ+1ζψζdζγ+1∫wzϖζdζγ,where
(9)G=∫wzϖζdζ,F=∫wzψγ+1ζϖγζdζ.In addition to this, they proved the discrete case of (8) as follows:
(10)∑i=1nwim+1zim≤∑i=1nSzwim+1/wzim+1wi∑i=1nzim,where z=∑i=1nzi and w=∑i=1nwim+1/zim.In [6], the authors proved the reverse Hilbert-type inequalities by using the Specht’s ratio as follows: if 0≤p,q≤1 and λm,ψn are nonnegative and decreasing sequences of real numbers for m=1,2,⋯,k and n=1,2,⋯,r with k, r∈ℕ, then
(11)∑m=1k∑n=1rSp,q,k,r,m,n∑s=1mλsp∑t=1nψtqmn1/2≥2Cp,q,k,r∑m=1kλm∑s=1mλsp−12k−m+11/2×∑n=1rψn∑t=1nψtq−12r−n+11/2,where
(12)Cp,q,r,s=12pqkr1/2,Sp,q,k,r,m,n=Sk∑s=1mλs∑τ=1sλτp−12∑s=1kk−s+1λs∑τ=1sλτp−12×Sr∑t=1nψt∑τ=1tψτq−12∑t=1rr−t+1ψt∑τ=1tψτq−12×Smλu∑τ=1uλτp−12∑s=1mλs∑τ=1sλτp−12×Snψv∑τ=1vψτq−12∑t=1nψt∑τ=1tψτq−12,where
(13)Smλu∑τ=1uλτp−12∑s=1mλs∑τ=1sλτp−12=maxSmλ1∑τ=11λτp−12∑s=1mλs∑τ=1sλτp−12;Smλm∑τ=1mλτp−12∑s=1mλs∑τ=1sλτp−12,Snψv∑τ=1vψτq−12∑t=1nψt∑τ=1tψτq−12=maxSnψ1∑τ=11ψτq−12∑t=1nψt∑τ=1tψτq−12;Snψn∑τ=1nψτq−12∑t=1nψt∑τ=1tψτq−12.In 1990, Arin˘o and Muckenhoupt [7] proved an inequality which maps from Lp to Lp,1≤p<∞ and contains one weighted function that they characterized such that the inequality
(14)∫0∞wt1t∫0tfxdxpdt≤C∫0∞wtfptdt,holds for all nonnegative nonincreasing measurable function f on 0,∞ with a constant C>0 independent on f (here 1≤p<∞). The characterization reduces to the condition that the function w satisfies
(15)∫t∞wxxpdx≤Btp∫0twxdx,forallt∈0,∞andB>0.In 1993, Sinnamon [8] generalized (14) to map from Lp to Lq,0<q<1<p which has two different weighted functions and characterized the weights such that the inequality
(16)∫abux∫axftdtqdx1/q≤C∫abvxfpxdx1/p,holds for all measurable f≥0 and the constant C is independent of f (here 0<q<1<p and 1/r=1/q−1/p). The characterization reduces to the condition that the nonnegative functions u and v satisfy
(17)∫ab∫xbutdtr/p∫axv1−p′tdtr/p′uxdx=K<∞,p′=pp−1.Also, many authors study the inequalities with weighted functions (see [9–16]).For the discrete case of (14), in 2006, Bennett and Gross-Erdmann [11] characterized the weights such that the inequality
(18)∑n=1∞wn1n∑k=1ngkp≤C∑n=1∞wngnp,1≤p<∞,holds for all nonnegative nonincreasing sequence gn and C>0. The characterization reduces to the condition that the nonnegative sequence wn satisfies
(19)∑k=n∞wkkp≤Bnp∑k=1nwk,foralln∈ℕandB>0.In the last decades, a time scale theory has been discovered to unify the continuous calculus and discrete calculus. A time scaleT is defined as an arbitrary nonempty closed subset of the real numbers ℝ. Many authors proved some dynamic inequalities on time scales which unify the inequalities in the continuous calculus and discrete calculus (see [17–26]). In particular, in 2021, Saker et al. [25] unified the inequalities (14) and (18) to be general inequality on time scales and proved that T is a time scale with a∈T and 1≤p<∞. Furthermore, assume that f is a nonincreasing function and ∫a∞wtfptΔt<∞. Suppose there is a constant B>0 with
(20)∫t∞wsσs−apΔs≤Bσt−ap∫aσtwsΔsforallt∈a,∞T.Then,(21)∫a∞wt1σt−a∫aσtfxΔxpΔt≤ppB+1p∫a∞wtfptΔt.As special cases of (20), when (T≪ℝ, σt=t, a=0), we obtain the inequality (14) and when (T≪ℕ, σt=t+1, a=1), we have the inequality (18).Also, we can get new inequalities, for example, when (T≪hℕ:h>0, a=h), where σt=t+h, we get a new inequality in a new calculus of (21) in the form
(22)∑n=h∞wn1n∑k=1nfkp≤ppB+1p∑n=h∞wnfnp,1≤p<∞,For some reversed dynamic inequalities of Hilbert-type, Hölder-type, and Hardy-type on time scale, see the papers ([27–32]).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 via supermultiplicative functions on time scales.The following is a breakdown of the paper’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 (11) proved by Zhao and Cheung [6].


## 2. Preliminaries and Fundamental Axioms

For completeness, we recall the following concepts related to the notion of time scales. For more details of time scale analysis, we refer the reader to the two books by Bohner and Peterson [33, 34] which summarize and organize much of the time scale calculus. A time scale T is an arbitrary nonempty closed subset of the real numbers ℝ.First, we defineστ≔infu∈T:u>τ, CrdT,ℝ introduces the set of all such rd-continuous functions, and for any function u:T⟶ℝ, the notation uστ denotes uστ.The derivative of the productuϖ and the quotient u/ϖ (where ϖϖσ≠0) of two differentiable functions u and ϖ are given by
(23)uϖΔ=uΔϖ+uσϖΔ=uϖΔ+uΔϖσ,uϖΔ=uΔϖ−uϖΔϖϖσ.The integration by parts formula onT is
(24)∫υ0υλτφΔτΔτ=λτφτυ0υ−∫υ0υλΔτφστΔτ.The time scale chain rule ([9], Theorem 1.87) is
(25)ϖ∘φΔτ=ϖ′φϰφΔτ,whereϰ∈τ,στ,where it is supposed that ϖ:ℝ⟶ℝ is continuously differentiable and φ:T⟶ℝ is Δ−differentiable.Definition 1 (see [35]).
A functionL:I⟶ℝ+ is supermultiplicative if
(26)Lϰζ≥LϰLζ,∀ϰ,ζ∈I⊂ℝ.
WhenL is the identity map (i.e., Lζ=ζ), the inequality (26) holds with equality. L is said to be a submultiplicative function if the last inequality has the opposite sign.Lemma 2.
Ifw∈T, λ∈CrdT,ℝ, and 0≤γ≤1, then
(27)∫wσtλτΔτγ≥γ∫wσt∫wσϑλτΔτγ−1λϑΔϑ.Proof.
Using (25) on the term ∫wϑλτΔτ, we get
(28)∫wϑλτΔτγΔ=γ∫wζλτΔτγ−1λϑ,ζ∈ϑ,σϑ.
Sinceζ≤σϑ, we have (note 0≤γ≤1) that
(29)∫wζλτΔτγ−1≥∫wσϑλτΔτγ−1.
Substituting (29) into (28), we see that
(30)∫wϑλτΔτγΔ≥γ∫wσϑλτΔτγ−1λϑ.
Integrating (30) over ϑ from w to σt, we have
(31)∫wσt∫wϑλτΔτγΔΔϑ≥γ∫wσt∫wσϑλτΔτγ−1λϑΔϑ.That is,
(32)∫wσtλτΔτγ≥γ∫wσt∫wσϑλτΔτγ−1λϑΔϑ,which is (27).Lemma 3 (Specht’s ratio [5]).
Letα,β be positive numbers, p>1, and 1/p+1/q=1.
Then,(33)Sαβα1/pβ1/q≥αp+βq,where
(34)St=t1/t−1elogt1/t−1,t≠1.
The functionSt is strictly decreasing for 0<t<1 and strictly increasing for t>1. In addition, the following equations are true:
(35)S1=1,St=S1t∀t>0.Lemma 4 ([16, whenα=1]).
Letf,g∈Cw,zT,ℝ+ such that fβ,gν be Δ−integrable on w,zT. If β>1 and 1/β+1/ν=1, then
(36)∫wzSYfβζXgνζfζgζΔζ≥∫wzfβζΔζ1/β∫wzgνζΔζ1/ν,where X=∫wzfβζΔζ and Y=∫wzgνζΔζ.Lemma 5 (Jensen’s inequality, alnamer).
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
(37)Ψ1∫ζ0ζλτΔτ∫ζ0ζλτφτΔτ≤1∫ζ0ζλτΔτ∫ζ0ζλτΨφτΔτ.The inequality (37) is reversed when Ψ is continuous and concave.Lemma 6.
Letw∈T,λ,ψ be positive and decreasing functions,f,g be positive and nondecreasing functions, and 0≤p,q≤1. In addition, assume that ϕ,φ are positive, increasing, concave, and supermultiplicative functions. If β>1,ν>1 with 1/β+1/ν=1, then
(38)S∫wσtfνϑΔϑϕβλζ∫wσζλτΔτp−1fνζ∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ=maxS∫wσtfνϑΔϑϕβλw∫wσwλτΔτp−1fνw∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ;S∫wσtfνϑΔϑϕβλt∫wσtλτΔτp−1fνt∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ,(39)S∫wσξgνyΔyφβψη∫wσηψτΔτq−1gνη∫wσξφβψy∫wσyψτΔτq−1Δy=maxS∫wσξgνyΔyφβψw∫wσwψτΔτq−1gνw∫wσξφβψy∫wσyψτΔτq−1Δy;S∫wσξgνyΔyφβψξ∫wσξψτΔτq−1gνξ∫wσξφβψy∫wσyψτΔτq−1Δy.Proof.
Forϑ≤y, we have
(40)∫wσϑλτΔτ≤∫wσyλτΔτ,and then (where 0≤p≤1)
(41)∫wσϑλτΔτp−1≥∫wσyλτΔτp−1.Sinceλ is decreasing, we have from (41) (where ϑ≤y) that
(42)λϑ∫wσϑλτΔτp−1≥λy∫wσyλτΔτp−1.Using the facts thatβ>1, ϕ is an increasing function and (42), we get
(43)ϕβλϑ∫wσϑλτΔτp−1≥ϕβλy∫wσyλτΔτp−1,and then, we obtain (where ϑ≤y and f is nondecreasing) that
(44)1fνϑϕβλϑ∫wσϑλτΔτp−1≥1fνyϕβλy∫wσyλτΔτp−1.Thus, the function1/fνϑϕβλϑ∫wσϑλτΔτp−1 is decreasing. Therefore, we have for w≤ϑ that
(45)1fνwϕβλw∫wσwλτΔτp−1≥1fνϑϕβλϑ∫wσϑλτΔτp−1,and then,
(46)fνϑϕβλw∫wσwλτΔτp−1≥fνwϕβλϑ∫wσϑλτΔτp−1.Integrating the last inequality overϑ from w to σt, we have
(47)ϕβλw∫wσwλτΔτp−1∫wσtfνϑΔϑ≥fνw∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ,and then,
(48)∫wσtfνϑΔϑϕβλw∫wσwλτΔτp−1fνw∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ≥1.Since the function1/fνϑϕβλϑ∫wσϑλτΔτp−1 is decreasing, we have for ϑ≤t that
(49)1fνϑϕβλϑ∫wσϑλτΔτp−1≥1fνtϕβλt∫wσtλτΔτp−1,and then,
(50)fνtϕβλϑ∫wσϑλτΔτp−1≥fνϑϕβλt∫wσtλτΔτp−1.By integrating (50) over ϑ from w to σt, we have
(51)fνt∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ≥ϕβλt∫wσtλτΔτp−1∫wσtfνϑΔϑ.Thus,
(52)∫wσtfνϑΔϑϕβλt∫wσtλτΔτp−1fνt∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ≤1.From (48) and (52), we observe that
(53)∫wσtfνϑΔϑϕβλw∫wσwλτΔτp−1fνw∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ≥⋯≥1≥⋯≥∫wσtfνϑΔϑϕβλt∫wσtλτΔτp−1fνt∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ.SinceS. is decreasing on 0,1 and increasing on 1,∞, we get that one of
(54)S∫wσtfνϑΔϑϕβλw∫wσwλτΔτp−1fνw∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ,S∫wσtfνϑΔϑϕβλt∫wσtλτΔτp−1fνt∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ,is maximum (where S1=1), and it is in the form
(55)S∫wσtfνϑΔϑϕβλζ∫wσζλτΔτp−1fνζ∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ=maxS∫wσtfνϑΔϑϕβλw∫wσwλτΔτp−1fνw∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ;S∫wσtfνϑΔϑϕβλt∫wσtλτΔτp−1fνt∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ,which is (38). Similarly, with respect to ψ when 0≤q≤1, we get
(56)S∫wσξgνyΔyφβψη∫wσηψτΔτq−1gνη∫wσξφβψy∫wσyψτΔτq−1Δy=maxS∫wσξgνyΔyφβψw∫wσwψτΔτq−1gνw∫wσξφβψy∫wσyψτΔτq−1Δy;S∫wσξgνyΔyφβψξ∫wσξψτΔτq−1gνξ∫wσξφβψy∫wσyψτΔτq−1Δy,which is (39).Throughout the article, we will assume that the functions are nonnegative rd-continuous functions onw,∞T≔w,∞∩T and the integrals considered are assumed to exist. Now, we will present and justify our main findings.


## 3. Main Results

Theorem 7.
Letw∈T, 0≤p,q≤1, and λ,ψ be positive and decreasing functions. Assume that ω,ϖ,ϕ,φ are positive functions such that ϕ,φ are increasing, concave, and supermultiplicative functions with
(57)Aϕ≤ω≤Bϕ,Aφ≤ϖ≤Bφ,where A,B are positive constants. If f,g are positive and nondecreasing functions and β>1,ν>1 with 1/β+1/ν=1, then
(58)∫wσs∫wσrΦt,ξPtQξ×ωS∫wσtfνϑΔϑϕβλζ∫wσζλτΔτp−1fνζ∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ×1Pt∫wσtλτΔτpft×ϖS∫wσξgνyΔyφβψη∫wσηψτΔτq−1gνη∫wσξφβψy∫wσyψτΔτq−1Δy×1Qξ∫wσξψτΔτqgξΔtΔξ≥νA2B2Cp,q,r,s,ν∫wσrωβλϑ∫wσϑλτΔτp−1σr−ϑΔϑ1/β×∫wσsϖβψy∫wσyψτΔτq−1σs−yΔy1/β,holds for all r,s∈w,∞T, where
(59)Φt,ξ=ftgξ∫wσtfνϑΔϑ1/ν∫wσξgνyΔy1/ν×S∫wσrfνϑΔϑ∫wσtϕβλζ∫wσζλτΔτp−1fνt∫wσrϕβλϑ∫wσϑλτΔτp−1σr−ϑΔϑ×S∫wσsgνϑΔϑ∫wσξφβψη∫wσηψτΔτq−1gνξ∫wσsφβψy∫wσyψτΔτq−1σs−yΔy,with
(60)Cp,q,r,s,ν=ϕpφqν∫wσrfνϑΔϑ1/ν∫wσsgνϑΔϑ1/ν,(61)Pt=∫wσtS∫wσtfνϑΔϑϕβλϑ∫wσϑλτΔτp−1fνϑ∫wσtϕβλϑ∫wσϑλτΔτp−1ΔϑfϑΔϑ,(62)Qξ=∫wσξS∫wσξgνyΔyφβψy∫wσyψτΔτq−1gνy∫wσξφβψy∫wσyψτΔτq−1ΔygyΔy,such that
(63)S∫wσtfνϑΔϑϕβλζ∫wσζλτΔτp−1fνζ∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ=maxS∫wσtfνϑΔϑϕβλw∫wσwλτΔτp−1fνw∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ;S∫wσtfνϑΔϑϕβλt∫wσtλτΔτp−1fνt∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ,S∫wσξgνyΔyφβψη∫wσηψτΔτq−1gνη∫wσξφβψy∫wσyψτΔτq−1Δy=maxS∫wσξgνyΔyφβψw∫wσwψτΔτq−1gνw∫wσξφβψy∫wσyψτΔτq−1Δy;S∫wσξgνyΔyφβψξ∫wσξψτΔτq−1gνξ∫wσξφβψy∫wσyψτΔτq−1Δy.Proof.
Applying (27) with γ=p, we have
(64)∫wσtλτΔτp≥p∫wσtλϑ∫wσϑλτΔτp−1Δϑ.Multiplying the last inequality by(65)S∫wσtfνϑΔϑϕβλζ∫wσζλτΔτp−1fνζ∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑft,we get (where f is nondecreasing) that
(66)S∫wσtfνϑΔϑϕβλζ∫wσζλτΔτp−1fνζ∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ∫wσtλτΔτpft≥p∫wσtS∫wσtfνϑΔϑϕβλζ∫wσζλτΔτp−1fνζ∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ×λϑ∫wσϑλτΔτp−1ftΔϑ≥p∫wσtS∫wσtfνϑΔϑϕβλζ∫wσζλτΔτp−1fνζ∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ×λϑ∫wσϑλτΔτp−1fϑΔϑ.From Lemma6, the last inequality becomes
(67)S∫wσtfνϑΔϑϕβλζ∫wσζλτΔτp−1fνζ∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ∫wσtλτΔτpft≥p∫wσtS∫wσtfνϑΔϑϕβλϑ∫wσϑλτΔτp−1fνϑ∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ×λϑ∫wσϑλτΔτp−1fϑΔϑ.Similarly, we have for the decreasing functionψ, the nondecreasing function g and 0≤q≤1, that
(68)∫wσξψτΔτqgξ×S∫wσξgνyΔyφβψη∫wσηψτΔτq−1gνη∫wσξφβψy∫wσyψτΔτq−1Δy≥q∫wσξS∫wσξgνyΔyφβψy∫wσyψτΔτq−1gνy∫wσξφβψy∫wσyψτΔτq−1Δy×ψy∫wσyψτΔτq−1gyΔy.From (57), (67), and (68), we get (note ϕ is a positive, increasing and supermultiplicative function) that
(69)ωS∫wσtfνϑΔϑϕβλζ∫wσζλτΔτp−1fνζ∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ×1Pt∫wσtλτΔτpft≥AϕS∫wσtfνϑΔϑϕβλζ∫wσζλτΔτp−1fνζ∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ×1Pt∫wσtλτΔτpft≥AϕpPt∫wσtS∫wσtfνϑΔϑϕβλϑ∫wσϑλτΔτp−1fνϑ∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ×λϑ∫wσϑλτΔτp−1fϑΔϑ≥Aϕpϕ1Pt∫wσtS∫wσtfνϑΔϑϕβλϑ∫wσϑλτΔτp−1fνϑ∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ×λϑ∫wσϑλτΔτp−1fϑΔϑ,and then, by applying the Jensen inequality on the right hand side of (69) (where ϕ is a concave function), we have that
(70)ωS∫wσtfνϑΔϑϕβλζ∫wσζλτΔτp−1fνζ∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ×1Pt∫wσtλτΔτpft≥AϕpPt∫wσtS∫wσtfνϑΔϑϕβλϑ∫wσϑλτΔτp−1fνϑ∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ×fϑϕλϑ∫wσϑλτΔτp−1Δϑ.Similarly, with respect to (57) and (68), we see (where φ is a positive, increasing, concave, and supermultiplicative function) that
(71)ϖS∫wσξgνyΔyφβψη∫wσηψτΔτq−1gνη∫wσξφβψy∫wσyψτΔτq−1Δy×1Qξ∫wσξψτΔτqgξ≥AφqQξ∫wσξS∫wσξgνyΔyφβψy∫wσyψτΔτq−1gνy∫wσξφβψy∫wσyψτΔτq−1Δy×gyφψy∫wσyψτΔτq−1Δy.Multiplying (70) and (71), we see that
(72)PtωS∫wσtfνϑΔϑϕβλζ∫wσζλτΔτp−1fνζ∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ×1Pt∫wσtλτΔτpft×QξϖS∫wσξgνyΔyφβψη∫wσηψτΔτq−1gνη∫wσξφβψy∫wσyψτΔτq−1Δy×1Qξ∫wσξψτΔτqgξ≥Aϕp∫wσtS∫wσtfνϑΔϑϕβλϑ∫wσϑλτΔτp−1fνϑ∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ×fϑϕλϑ∫wσϑλτΔτp−1Δϑ×Aφq∫wσξS∫wσξgνyΔyφβψy∫wσyψτΔτq−1gνy∫wσξφβψy∫wσyψτΔτq−1Δy×gyφψy∫wσyψτΔτq−1Δy.Applying (36) on the right hand side of (72), we observe that
(73)PtωS∫wσtfνϑΔϑϕβλζ∫wσζλτΔτp−1fνζ∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ×1Pt∫wσtλτΔτpft×QξϖS∫wσξgνyΔyφβψη∫wσηψτΔτq−1gνη∫wσξφβψy∫wσyψτΔτq−1Δy×1Qξ∫wσξψτΔτqgξ≥A2ϕp∫wσtfνϑΔϑ1/v∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ1/β×φq∫wσξgνyΔy1/v∫wσξφβψy∫wσyψτΔτq−1Δy1/β.Multiplying (73) by
(74)Φt,ξ=ftgξ∫wσtfνϑΔϑ1/ν∫wσξgνyΔy1/ν×S∫wσrfνϑΔϑ∫wσtϕβλζ∫wσζλτΔτp−1fνt∫wσrϕβλϑ∫wσϑλτΔτp−1σr−ϑΔϑ×S∫wσsgνϑΔϑ∫wσξφβψη∫wσηψτΔτq−1gνξ∫wσsφβψy∫wσyψτΔτq−1σs−yΔy,and then, taking the integration over t from w to σr and the integration over ξ from w to σs, we get to obtain
(75)∫wσs∫wσrΦt,ξPtQξ×ωS∫wσtfνϑΔϑϕβλζ∫wσζλτΔτp−1fνζ∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ×1Pt∫wσtλτΔτpft×ϖS∫wσξgνyΔyφβψη∫wσηψτΔτq−1gνη∫wσξφβψy∫wσyψτΔτq−1Δy,(76)1Qξ∫wσξψτΔτqgξΔtΔξ≥A2ϕpφq∫wσr∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ1/βft×S∫wσrfνϑΔϑ∫wσtϕβλζ∫wσζλτΔτp−1fνt∫wσrϕβλϑ∫wσϑλτΔτp−1σr−ϑΔϑΔt×∫wσs∫wσξφβψy∫wσyψτΔτq−1Δy1/βgξ×S∫wσsgνϑΔϑ∫wσξφβψη∫wσηψτΔτq−1gνξ∫wσsφβψy∫wσyψτΔτq−1σs−yΔyΔξ.Applying the integration by parts formula on the term(77)∫wσrϕβλϑ∫wσϑλτΔτp−1σr−ϑΔϑ,with uϑ=σr−ϑ and vΔϑ=ϕβλϑ∫wσϑλτΔτp−1, we get
(78)∫wσrϕβλϑ∫wσϑλτΔτp−1σr−ϑΔϑ=σr−ϑvϑwσr+∫wσrvσϑΔϑ,where vϑ=∫wϑϕβλθ∫wσθλτΔτp−1Δθ and then where vw=0),
(79)∫wσrϕβλϑ∫wσϑλτΔτp−1σr−ϑΔϑ=∫wσr∫wσϑϕβλθ∫wσθλτΔτp−1ΔθΔϑ.Similarly, we see that(80)∫wσsφβψy∫wσyψτΔτq−1σs−yΔy=∫wσs∫wσyφβψθ∫wσθψτΔτq−1ΔθΔy.Substituting (79) and (80) into (76), we observe that
(81)∫wσs∫wσrΦt,ξPtQξ×ωS∫wσtfνϑΔϑϕβλζ∫wσζλτΔτp−1fνζ∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ×1Pt∫wσtλτΔτpft×ϖS∫wσξgνyΔyφβψη∫wσηψτΔτq−1gνη∫wσξφβψy∫wσyψτΔτq−1Δy×1Qξ∫wσξψτΔτqgξΔtΔξ≥A2ϕpφq∫wσr∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ1/βft×S∫wσrfνϑΔϑ∫wσtϕβλζ∫wσζλτΔτp−1Δζfνt∫wσr∫wσϑϕβλθ∫wσθλτΔτp−1ΔθΔϑΔt×∫wσs∫wσξφβψy∫wσyψτΔτq−1Δy1/βgξ×S∫wσsgνϑΔϑ∫wσξφβψη∫wσηψτΔτq−1Δηgνξ∫wσs∫wσyφβψθ∫wσθψτΔτq−1ΔθΔyΔξ,and then, by applying (36), (79), and (80), we see that
(82)∫wσs∫wσrΦt,ξPtQξ×ωS∫wσtfνϑΔϑϕβλζ∫wσζλτΔτp−1fνζ∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ×1Pt∫wσtλτΔτpft×ϖS∫wσξgνyΔyφβψη∫wσηψτΔτq−1gνη∫wσξφβψy∫wσyψτΔτq−1Δy×1Qξ∫wσξψτΔτqgξΔtΔξ≥A2ϕpφq∫wσrfνϑΔϑ1/v∫wσsgνϑΔϑ1/v×∫wσr∫wσϑϕβλθ∫wσθλτΔτp−1ΔθΔϑ1/β×∫wσs∫wσyφβψθ∫wσθψτΔτq−1ΔθΔy1/β=A2ϕpφq∫wσrfνϑΔϑ1/v∫wσsgνϑΔϑ1/v×∫wσrϕβλϑ∫wσϑλτΔτp−1σr−ϑΔϑ1/β×∫wσsφβψy∫wσyψτΔτq−1σs−yΔy1/β,and thus, we have from (57) and (60) that
(83)∫wσs∫wσrΦt,ξPtQξ×ωS∫wσtfνϑΔϑϕβλζ∫wσζλτΔτp−1fνζ∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ×1Pt∫wσtλτΔτpft×ϖS∫wσξgνyΔyφβψη∫wσηψτΔτq−1gνη∫wσξφβψy∫wσyψτΔτq−1Δy×1Qξ∫wσξψτΔτqgξΔtΔξ≥νA2B2Cp,q,r,s,ν∫wσrωβλϑ∫wσϑλτΔτp−1σr−ϑΔϑ1/β×∫wσsϖβψy∫wσyψτΔτq−1σs−yΔy1/β,which is (58).Remark 8.
IfA=B=1, then ω=ϕ,ϖ=φ and then we have for w∈T, 0≤p,q≤1,β>1,ν>1 such that 1/β+1/ν=1 that
(84)∫wσs∫wσrΦt,ξPtQξ×ϕS∫wσtfνϑΔϑϕβλζ∫wσζλτΔτp−1fνζ∫wσtϕβλϑ∫wσϑλτΔτp−1Δϑ×1Pt∫wσtλτΔτpft×φS∫wσξgνyΔyφβψη∫wσηψτΔτq−1gνη∫wσξφβψy∫wσyψτΔτq−1Δy×1Qξ∫wσξψτΔτqgξΔtΔξ≥νCp,q,r,s,ν∫wσrϕβλϑ∫wσϑλτΔτp−1σr−ϑΔϑ1/β×∫wσsφβψy∫wσyψτΔτq−1σs−yΔy1/β.Remark 9.
IfT≪ℝ, σs=s, and w=0, the inequality (58) becomes
(85)∫0s∫0rΦt,ξPtQξ×ωS∫0tfνϑdϑϕβλζ∫0ζλτdτp−1fνζ∫0tϕβλϑ∫0ϑλτdτp−1dϑ×1Pt∫0tλτdτpft×ϖS∫0ξgνydyφβψη∫0ηψτdτq−1gνη∫0ξφβψy∫0yψτdτq−1dy×1Qξ∫0ξψτdτqgξdtdξ≥νA2B2Cp,q,r,s,ν∫0rωβλϑ∫0ϑλτdτp−1r−ϑdϑ1/β×∫0sϖβψy∫0yψτdτq−1s−ydy1/β.
This inequality is new in continuous calculus.Remark 10.
As a special case of (84) when T≪ℕ, φϑ=ϕϑ=ϑ, fϑ=gϑ=1, w=1, and β=ν=2, we obtain (11) demonstrated in [36].


## 4. Conclusion and Future Work

In this paper, we studied some new generalizations of reversed dynamic inequalities of Hilbert-type via supermultiplicative functions on time scales by applying reverse Hölder inequalities and the Specht’s ratio function. We generalized the inequality proved by Zhao and Cheung [6] by using a supermultiplicative function which the identity map represents a special case of it and also by using the power greater than 2 when we used different powers β,ν>1 with 1/β+1/ν=1, where we get the special case when β=ν=2. Also, we added some increasing functions for generalizing the results where we get the results when we take the identity function as a special case. In the future, we will continue to generalize more reversed dynamic inequalities of Hilbert-type 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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