# Some Dynamic Inequalities of Hilbert’s Type

Authors: A. M. Ahmed; Ghada AlNemer; M. Zakarya; H. M. Rezk
Journal: Journal of Function Spaces
 (2020)

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## Abstract

This paper is concerned with deriving some new dynamic Hilbert-type inequalities on time scales. The basic idea in proving the results is using some algebraic inequalities, Hölder’s inequality and Jensen’s inequality, on time scales. As a special case of our results, we will obtain some integrals and their corresponding discrete inequalities of Hilbert’s type.


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## Body

## 1. Introduction

It is evident that the Hilbert-type inequalities outplay a major role in mathematics, for pattern complex analysis, numerical analysis, and qualitative theory of differential equations and their implementations. In recent years, there were a lot of various refinements, generalizations, extensions, and applications of Hilbert’s inequality which have seemed in the literature. Hilbert’s discrete inequality and its integral formula ([1], Theorem 316) have been generalized in many trends (for example, see [2–6]). Lately, Pachpatte [7] proved new inequalities similar to those of Hilbert’s inequality, namely, he proved that if h, l ≥ 1, Am=∑s=1mas≥0, and Bn=∑t=1nbt≥0, then(1)∑i=1k∑j=1rAihBjli+j≤C∗h,l,k,r∑i=1kk+1−iAih−1ai21/2×∑j=1rr+1−jBjl−1bj21/2,where(2)C∗h,l,k,r=12hlkr.An integral analogue of (1) is given in the following result. Let h, l ≥ 1, Fx=∫0xfτdτ≥0, and Gy=∫0ygνdν≥0, for x,τ∈0,a and y,ν∈0,b. Then,(3)∫0a∫0bFhxGlyx+ydxdy≤Dh,l,a,b∫0aa−xFh−1xfx2dx1/2×∫0bb−yGl−1ygy2dy1/2,where(4)Dh,l,a,b=12hlab.In 2001, Kim [8] gave some generalizations of (1) and (3) by introducing a parameter α > 0 as(5)∑i=1k∑j=1rAihBjliα+jα1/α≤Ch,l,k,r,α∑i=1kk−i+1Aih−1ai21/2×∑j=1rr−j+1Bjl−1bj21/2,where h, l ≥ 1, Am=∑p=1map≥0, Bn=∑q=1nbq≥0, and(6)Ch,l,k,r,α=121/αhlkr.An integral analogue of (5) is given in the following result. Let p, q ≥ 1, α > 0, Ft=∫0tfτdτ≥0, and Gs=∫0sgνdν≥0, for t,τ∈0,a, s,ν∈0,b. Then,(7)∫0a∫0bFhxGlyxα+yα1/αdxdy≤Dh,l,a,b∫0aa−xFh−1xfx2dx1/2×∫0bb−yGl−1ygy2dy1/2,where(8)Dh,l,a,b=121/αhlab.In 2009, Yang [9] gave another generalization of (1) and (3) by introducing parameter α > 1 and γ > 1 as follows. Let h, l ≥ 1, Am=∑p=1map≥0, and Bn=∑q=1nbq≥0. Then,(9)∑i=1k∑j=1rAihBjlγiα−1α+γ/αγ+αjα−1α+γ/αγ≤Ch,l,k,r,α,γ∑i=1kk−i+1Aih−1aiα1/α×∑j=1rr−j+1Bjl−1bjγ1/γ,where(10)Ch,l,k,r,α,γ=hlα+γkα−1/αrγ−1/γ.An integral analogue of (9) is given as follows. If h, l ≥ 1, α > 1, γ > 1, Fx=∫0xfτdτ≥0, and Gy=∫0ygνdν≥0, for x,τ∈0,a and y,ν∈0,b, then(11)∫a0∫b0FhxGlyγxα−1α+γ/αγ+αyα−1α+γ/αγdxdy≤Dh,l,a,b,α,γ∫0aa−xFh−1xfxαdx1/α×∫0bb−yGl−1ygyγdy1/γ,where(12)Dh,l,a,b,α,γ=pqα+γaα−1/αbγ−1/γ.In [10], the authors deduced several generalizations of inequalities (1) and (3) on time scales, namely, they proved that if h and l ≥ 1 are real numbers, Ax1=∫wx1aτ1Δτ1, By1=∫wy1bτ1Δτ1, and η > 1, η∗ > 1 with η−1+η∗−1=1, then(13)∫wx2∫wy2Ahx1Bly1η∗x1−wη−1+ηy1−wη∗−1Δy1Δx1≤Mη,η∗∫wx2σx2−x1ax1Ah−1σx1ηΔx1η−1×∫wy2σy2−y1by1Bl−1σy1η∗Δy1η∗−1,where(14)Mη,η∗=hlηη∗x2−wη−1/ηy2−wη∗−1/η∗.In [11], the authors gave some extensions of inequalities (5) and (7) on time scales. Minutely, they proved that if γ > 0 and h and l ≥ 1 are real numbers, As=∫t0saτΔτ, Bt=∫t0tbτΔτ, and η > 1, η∗ > 1 with η−1+η∗−1=1, then(15)∫t0x∫t0yAhσsBlσtσs−t0γ+σt−t0γ2/ηγΔsΔt≤Ch,l,η,γ∫t0xσx−σsasAh−1σsη∗Δsη∗−1×∫t0yσy−σtbtBl−1σtη∗Δtη∗−1,where(16)Ch,l,η,γ=hl122/ηγx−t01/ηy−t01/η.Following this trend and to develop the study of dynamic inequalities on time scales, we will prove some new inequalities of Hilbert’s type on time scales, namely, we prove time scale versions of inequalities (9) and (11) on time scale T. These inequalities can be considered as extensions and generalizations of some Hilbert-type inequalities proved in [10]. We also derive some inequalities on time scale as special cases.


## 2. Definitions and Basic Results

In this division, we will present some fundamental concepts and effects on time scales which will be beneficial for deducing our main results. The following definitions and theorems are referred from [12, 13].Time scaleT is defined as a nonempty arbitrary locked subplot of real numbers R. We define the forward jump operator σ:T⟶T as(17)στ=infθ∈T:θ>τand the backward jump operator ρ:T⟶T as(18)ρτ=supθ∈T:θ<τ.From the above two definitions, it can be stated that a pointτ∈T with infT<τ<supT is called right-scattered if σ(τ) > τ, right-dense if σ(τ) = τ, left-scattered if ρ(τ) < τ, and left-dense if ρ(τ) = τ. If T has left-scattered maximum sm, then Tk=T−sm; otherwise, Tk=T. Finally, the graininess function μ:T⟶0,∞ for any τ∈T is defined by(19)μτ=στ−τ.For a functionχ:T⟶ℝ, the delta derivative of χ at τ∈Tk is defined as for each ɛ > 0, there is a neighborhood U of τ such that(20)χστ−χθ−χΔτστ−θ≤εστ−θ,for everyθ∈U.Moreover,χ is called delta differentiable on Tk if it is delta differentiable at every τ∈Tk.A functionχ:T⟶ℝ is called right-dense continuous (rd−continuous) as long as it is continuous at all right-dense points in T, and its left-sided limits exist (finite) at all left-dense points in T. The classes of real rd−continuous functions on an interval I will be denoted by CrdT,ℝ. For θ, t∈T, the Cauchy integral of χΔ is defined as(21)∫θtχΔτΔτ=χt−χθ.Note that(a)
IfT=ℝ, then(22)στ=τ,μτ=0,χΔτ=χ′τ,∫θtχΔτΔ=∫θtχτdτ.(b)
IfT=ℤ, then(23)στ=τ+1,μτ=1,χΔτ=Δχτ,∫θtχΔτΔτ=∑τ=st−1χτ.In what follows, we will present Hölder’s inequality, Jensen’s inequality, and the power rules for integration on time scales.Theorem 1.
(Hölder’s inequality (see [14, 15])). Let u,v∈T. For ζ,χ∈CrdT,ℝ, we have(24)∫uvζθχθΔθ≤∫uvζθηΔθη−1∫uvχθη∗Δθη∗−1,where η > 1 and η∗ > 1 with η−1+η∗−1=1.Theorem 2.
(Jensen’s inequality (see [14, 16])). Suppose that ζ∈Crdu,vT,w,z and η∈Crdu,vT,ℝ are nonnegative with(25)∫uvηsΔs>0.
IfΦ∈Crdw,z,ℝ is convex, then(26)Φ∫uvηsζsΔs∫uvζsΔs≤∫uvζsΦηsΔs∫uvζsΔs.Lemma 3.
(see [17]). Let u, s∈T, and ζ∈Crdu,sT,ℝ be nonnegative. If α ≥ 1, then(27)∫usζτΔτα≤α∫usζη∫uvζτΔτα−1Δv.
Now, we will present the formula that will reduce double integrals to single integrals which is the desired in [18].Lemma 4.
Letχ:T⟶ℝ and u,s,θ∈T. Then,(28)∫us∫uθχτΔτΔθ=∫uss−σθχτΔt,fors∈T,assuming the integrals considered exist.Lemma 5.
(see [19]). Let r > 0, μq > 0, and ∑q=1mμq=Ωm. Then,(29)∏q=1msqμq1/Ωm≤1Ωm∑q=1mμqsqr1/r.


## 3. Main Results

In this division, we will prove our main results. Throughout this section, we will assume that all functions are nonnegative and the integrals considered are assumed to exist. Also, we will assume thath and l ≥ 1 be real numbers and η > 1 and η∗ > 1 with η−1+η∗−1=1.Theorem 6.
Lets, θ, and t0∈T and f∈Crdt0,xT,ℝ+ and g∈Crdt0,yT,ℝ+. Suppose that Fs and Gθ are defined as(30)Fs=∫t0sfξΔξ,Gθ=∫t0θgξΔξ.
Then, fors∈t0,xT and θ∈t0,yT, we have(31)∫t0x∫t0yFhsGlθη∗s−t0η−1η+η∗/ηη∗+ηθ−t0η∗−1η+η∗/ηη∗ΔsΔθ≤Ch,l,η,η∗∫t0xσx−sFh−1σsfsηΔs1/η×∫t0yσy−θGl−1σθgθη∗Δθ1/η∗,where(32)Ch,l,η,η∗=hlη+η∗x−t0η−1/ηy−t0η∗−1/η∗.Proof.
By using inequality (27) (see Lemma 3), we see that(33)Fhs≤h∫t0sFh−1ξfξΔξ,(34)Glθ≤l∫t0θGl−1ξgξΔξ.
Then, we have(35)FhsGlθ≤hl∫t0sFh−1ξfξΔξ∫t0θGξgξΔξl−1.
Applying Hölder’s inequality (1) on ∫t0sFh−1ξfξΔξ with indices η and η/(η − 1), we find that(36)∫t0sFh−1ξfξΔξ≤s−t0η−1/η∫t0sFh−1ξfξηΔξ1/η,and on the integral ∫t0θGl−1ξgξΔξ with indices and η∗/(η∗ − 1), we find that(37)∫t0θGl−1ξgξΔξ≤θ−t0η∗−1/η∗∫t0θGl−1ξgξη∗Δξ1/η∗.
From (36) and (37), we get(38)FhsGlθ≤hls−t0η−1/ηθ−t0η∗−1/η∗∫t0sFh−1ξfξηΔξ1/η×∫t0θGl−1ξgξη∗Δξ1/η∗.
Using inequality (29) of power means, we observe that(39)s1ω1s2ω2r/ω1+ω2≤1ω1+ω2ω1s1r+ω2s2r.
Now, by settings1=s−t0η−1, s2=θ−t0η∗−1, ω1 = 1/η, ω2 = 1/η∗, and r = ω1 + ω2 in (39), we get(40)s−t0η−1/ηθ−t0η∗−1/η∗≤ηη∗η+η∗s−t0η−1η+η∗/ηη∗η+θ−t0η∗−1η+η∗/ηη∗η∗.
Substituting (40) into (38) yields(41)FhsGlt≤hlηη∗η+η∗s−t0η−1η+η∗/ηη∗η+θ−t0η∗−1η+η∗/ηη∗η∗×∫t0sFh−1ξfξηΔξ1/η∫t0θGl−1ξgξη∗Δξ1/η∗.
Dividing both sides of (41) by the last factor η∗s−t0η−1η+η∗/ηη∗+ηθ−t0η∗−1η+η∗/ηη∗, we obtain(42)FhsGltη∗s−t0η−1η+η∗/ηη∗+ηθ−t0η∗−1η+η∗/ηη∗≤hlη+η∗∫t0sFh−1ξfξηΔξ1/η×∫t0θGl−1ξgξη∗Δξ1/η∗.
Integrating the above relation and applying Hölder’s inequality (1), we have(43)∫t0x∫t0yFhsGltη∗s−t0η−1η+η∗/ηη∗+ηθ−t0η∗−1η+η∗/ηη∗ΔsΔθ≤hlη+η∗x−t0η−1/ηy−t0η∗−1/η∗∫t0x∫t0sFh−1ξfξηΔξΔs1/η×∫t0y∫t0θGl−1ξgξη∗ΔξΔθ1/η∗.
Applying Lemma4 on (43) and using the fact that σ(n) ≥ n, we conclude that(44)∫t0x∫t0yFhsGltη∗s−t0η−1η+η∗/ηη∗+ηθ−t0η∗−1η+η∗/ηη∗ΔsΔθ≤hlη+η∗x−t0η−1/ηy−t0η∗−1/η∗∫t0xσx−sFh−1σsfsηΔs1/η×∫t0yσy−θGl−1σθgθη∗Δθ1/η∗=Ch,l,η,η∗∫t0xσx−sFh−1σsfsηΔs1/η×∫t0yσy−θGl−1σθgθη∗Δθ1/η∗,which proves (31). This completes the proof.Remark 1.
Letting 1/η + 1/η∗ = 1 in (31), we get Theorem 3.1 due to Saker et al. ([11], Theorem 3.1).
By using relations (22) and putting T=ℝ and t0 = 0 in Theorem 6, we get the following conclusion.Corollary 7.
Assume thatf(ξ) and g(ξ) are two nonnegative functions, and define(45)Fs=∫0sfξdξ,Gq=∫0θgξdξ.
Then, fors ∈ (0, x) and θ ∈ (0, y), we have(46)∫0x∫0yFhsGltη∗sη−1η+η∗/ηη∗+ηθη∗−1η+η∗/ηη∗dsdθ≤C∗h,l,η,η∗∫0xx−sFh−1sfsηds1/η×∫0yy−θGl−1θgθη∗dθ1/η∗,where(47)C∗h,l,η,η∗=hlη+η∗xη−1/ηyη∗−1/η∗,which was proved by Yang ([9], Theorem 3.1).
By using relations (23) and putting T=ℤ and t0 = 0 in Theorem 6, we get the following conclusion.Corollary 8.
Assume that {ai} and {bj} are two nonnegative sequences of real numbers, and define(48)Ai=∑s=1ias,Bj=∑t=1jbt.
Then,(49)∑i=1k∑j=1rAihBjlη∗iη−1η+η∗/ηη∗+ηjη∗−1η+η∗/ηη∗≤C∗h,l,η,η∗∑i=1kk−i+1Aih−1aiη1/η×∑j=1rr−j+1Bjl−1bjη∗1/η∗,where(50)C∗h,l,η,η∗=hlη+η∗kη−1/ηrη∗−1/η∗,which was proved by Yang ([9], Theorem 2.1).Remark 2.
In Theorem6, setting h = l = 1, we have(51)∫t0x∫t0yFsGtη∗s−t0η−1η+η∗/ηη∗+ηθ−t0η∗−1η+η∗/ηη∗ΔsΔθ≤C0η,η∗∫t0xσx−sfsηΔs1/η∫t0yσy−θgθη∗Δθ1/η∗,where(52)C0η,η∗=1η+η∗s−t0η−1/ηθ−t0η∗−1/η∗.Remark 3.
In Remark2, if T=ℤ, T=ℝ, and t0 = 0, then we get Remarks 2 and 5, respectively, due to Yang [9].
In the following theorems, we give a further generalization of (51) obtained in Remark 2. Before we give our result, we assume that there exist two functions Φ and Ψ which are real-valued, nonnegative, convex, and submultiplicative functions defined on 0,∞. A function χ is a submultiplicative if χ(st) ≤ χ(s) χ(t) for s, t ≥ 0.Theorem 9.
Lets, θ, and t0∈T, f∈Crdt0,xT,ℝ+, g∈Crdt0,yT,ℝ+, and h(τ) and l(ξ) be two positive functions defined for τ∈t0,xT and ξ∈t0,yT. Suppose that F(s) and G(θ) are as defined in Theorem 6, and let(53)Hs=∫t0shτΔτ,Lθ=∫t0θlξΔξ.
Then, fors∈t0,xT and θ∈t0,yT, we have(54)∫t0x∫t0yΦFsΨGθη∗s−t0η−1η+η∗/ηη∗+ηθ−t0η∗−1η+η∗/ηη∗ΔsΔθ≤Dη,η∗∫t0xσx−shsΦfshsηΔs1/η×∫t0yσy−θlθΨgθlθη∗Δθ1/η∗,where(55)Dη,η∗=1η+η∗∫t0xΦHsHsη/η−1Δsη−1/η∫t0yΨLθLθη∗/η∗−1Δθη∗−1/η∗.Proof.
According to Theorem2 and the definition of function Φ, it is clear that(56)ΦFs=ΦHs∫t0shτfτ/hτΔτ∫t0shτΔτ≤ΦHsΦ∫t0shτfτ/hτΔτ∫t0shτΔτ≤ΦHsHs∫t0shτΦfτhτΔτ.
By applying Hölder’s inequality (1) on (56), we find that(57)ΦFs≤ΦHsHss−t0η−1/η∫t0shτΦfτhτηΔτ1/η.
Analogously,(58)ΨGθ≤ΨLθLθθ−t0η∗−1/η∗∫t0tlξΨgξlξη∗Δξ1/η∗.
Thus, from (57) and (58), it can be acquired that(59)ΦFsΨGθ≤s−t0η−1/ηθ−t0η∗−1/η∗ΦHsHs∫t0shτΦfτhτηΔτ1/η×ΨLθLθ∫t0tlξΨgξlξη∗Δξ1/η∗.
Applying (39) on the term s−t0η−1/η×θ−t0η∗−1/η∗, we get(60)ΦFsΨGθ≤ηη∗η+η∗s−t0η−1η+η∗/ηη∗η+t−t0η∗−1η+η∗/ηη∗η∗×ΦHsHs∫t0shτΦfτhτηΔτ1/η×ΨLθLθ∫t0tlξΨgξlξη∗Δξ1/η∗.
From (60), we observe that(61)ΦFsΨGθη∗s−t0η−1η+η∗/ηη∗+ηt−t0η∗−1η+η∗/ηη∗≤1η+η∗ΦHsHs∫t0shτΦfτhτηΔτ1/η×ΨLθLθ∫t0tlξΨgξlξη∗Δξ1/η∗.
Integrating the above relation and using Hölder’s inequality (1) again with indices η, η/(η − 1) and η∗, η∗/(η∗ − 1), we find that(62)∫t0x∫t0yΦFsΨGθη∗s−t0η−1η+η∗/ηη∗+ηθ−t0η∗−1η+η∗/ηη∗ΔsΔθ≤1η+η∗∫t0xΦHsHsη/η−1Δsη−1/η∫t0x∫t0shτΦfτhτηΔτΔs1/η×∫t0yΨLθLθη∗/η∗−1Δθη∗−1/η∗∫t0y∫t0θlξΨgξlξη∗ΔξΔθ1/η∗.
Applying Lemma4 on (62) and using σ(n) ≥ n, we get(63)∫t0x∫t0yΦFsΨGθη∗s−t0η−1η+η∗/ηη∗+ηθ−t0η∗−1η+η∗/ηη∗ΔsΔθ≤1η+η∗∫t0xΦHsHsη/η−1Δsη−1/η∫t0yΨLθLθη∗/η∗−1Δθη∗−1/η∗×∫t0xx−σshsΦfshsηΔs1/η∫t0yy−σθlθΨgθlθη∗Δθ1/η∗≤Dη,η∗∫t0xσx−shsΦfshsηΔs1/η∫t0yσy−θlθΨgθlθη∗Δθ1/η∗,which is (54). This completes the proof.Remark 4.
Letting 1/η + 1/η∗ = 1 in (54), then we get Theorem 3.2 due to Saker et al. [11].
By using relations (22) and putting T=ℝ and t0 = 0 in Theorem 9, we get the following conclusion.Corollary 10.
Assume thatf(s) and g(θ) are two nonnegative functions and h(s) and l(θ) are two positive functions, and let(64)Fs=∫0sfτdτ,Gθ=∫0θgτdτ,Hs=∫0shτdτ,Lθ=∫0θlτdτ.
Then, fors ∈ (0, x) and θ ∈ (0, y), we have(65)∫0x∫0yΦFsΨGθη∗sη−1η+η∗/ηη∗+ηθη∗−1η+η∗/ηη∗dsdt≤D∗η,η∗∫0xx−shsΦfshsηds1/η×∫0yy−θlθΨgθlθη∗dθ1/η∗,where(66)D∗∗η,η∗=1η+η∗∫0xΦHsHsη/η−1dsη−1/η∫0yΨLθLθη∗/η∗−1dθη∗−1/η∗.
It is clear that we can have the same inequality in [9], Theorem 3.2.
By using relations (23) and putting T=ℤ and t0 = 0 in Theorem 9, we get the following conclusion.Corollary 11.
Assume that {ai} and {bj} are two nonnegative sequences of real numbers and {hi} and {lj} are positive sequences, and define(67)Ai=∑p=1iap,Bj=∑q=1jbq,Hi=∑p=1ihp,Lj=∑q=1jlq.
Then,(68)∑i=1k∑j=1rΦAiΨBjη∗iη−1η+η∗/ηη∗+ηjη∗−1η+η∗/ηη∗≤D∗∗η,η∗∑i=1kk−i+1hiΦaihiη1/η×∑j=1rr−j+1ljΨbjljη∗1/η∗,where(69)D∗∗η,η∗=1η+η∗∑i=1kΦHiHiη/η−1η−1/η∑j=1rΨLjLjη∗/η∗−1η∗−1/η∗.Remark 5.
From inequality (39), we obtain(70)s1ω1s2ω2≤1ω1+ω2ω1s1ω1+ω2+ω2s2ω1+ω2,forω1>0,ω2>0.
If we apply (70) on (31) in Theorem 6 and (54) in Theorem 9, then we get the following, respectively, inequalities:(71)∫t0x∫t0yFhsGlθη∗s−t0η−1η+η∗/ηη∗+ηθ−t0η∗−1η+η∗/ηη∗ΔsΔθ≤ηη∗D0h,l,η,η∗η+η∗1η∫t0xσx−sFh−1σsfsηΔsη+η∗/ηη∗+1η∗∫t0yσy−θGl−1σθgθη∗Δθη+η∗/ηη∗,where(72)D0h,l,η,η∗=hlη+η∗x−t0η−1/ηy−t0η∗−1/η∗.
Also,(73)∫t0x∫t0yΦFsΨGθη∗s−t0η−1η+η∗/ηη∗+ηθ−t0η∗−1η+η∗/ηη∗ΔsΔθ≤ηη∗D0∗h,l,η,η∗η+η∗1η∫t0xσx−shsΦfshsηΔsη+η∗/ηη∗+1η∗∫t0yσy−θlθΨgθlθη∗Δθη+η∗/ηη∗,where(74)D0∗h,l,η,η∗=1η+η∗∫t0xΦHsHsη/η−1Δsη−1/η∫t0yΨLθLθη∗/η∗−1Δθη∗−1/η∗.Remark 6.
In Remark5, if T=ℤ, T=ℝ, and t0 = 0, then we get Remarks 3 and 5, respectively, due to Yang [9].
The following theorems deal with slight variants of inequality (54) given in Theorem 9.Theorem 12.
Lets, θ, and t0∈T, f∈Crdt0,xT,ℝ+, and g∈Crdt0,yT,ℝ+. Define(75)Fs=1s−t0∫t0sfξΔξ,Gθ=1θ−t0∫t0θgξΔξ.
Then, fors∈t0,xT and θ∈t0,yT, we have(76)∫t0x∫t0yΦFsΨGθs−t0θ−t0η∗s−t0η−1η+η∗/ηη∗+ηθ−t0η∗−1η+η∗/ηη∗ΔsΔθ≤Eη,η∗∫t0xσx−sΦfsηΔs1/η×∫t0yσy−θΨgθη∗Δθ1/η∗,where(77)Eη,η∗=1η+η∗x−t0η−1/ηy−t0η∗−1/η∗.Proof.
By assumption and using Jensen’s inequality (26), we see that(78)ΦFs=Φ1s−t0∫t0sfξΔξ≤1s−t0∫t0sΦfξΔξ.
By applying inequality (1) on (78) with indices η, η/(η − 1), we have(79)ΦFs≤1s−t0s−t0η−1/η∫t0sΦfξηΔξ1/η.
This implies that(80)ΦFss−t0≤s−t0η−1/η∫t0sΦfξηΔξ1/η.
Analogously,(81)ΨGθθ−t0≤θ−t0η∗−1/η∗∫t0tΨgξη∗Δξ1/η∗.
From (80) and (81), we get(82)ΦFsΨGθs−t0θ−t0≤s−t0η−1/ηθ−t0η∗−1/η∗∫t0sΦfξηΔξ1/η×∫t0tΨgξη∗Δξ1/η∗.
Applying elementary inequality (39) on the term s−t0η−1/η×θ−t0η∗−1/η∗, where s1=s−t0η−1, s2=t−t0η∗−1, ω1 = 1/η, ω2 = 1/η∗, and r = ω1 + ω2, we get(83)ΦFsΨGθs−t0θ−t0≤ηη∗η+η∗s−t0η−1η+η∗/ηη∗η+θ−t0η∗−1η+η∗/ηη∗η∗×∫t0sΦfξηΔξ1/η∫t0tΨgξη∗Δξ1/η∗.
From (83), we have(84)ΦFsΨGθs−t0θ−t0η∗s−t0η−1η+η∗/ηη∗+ηθ−t0η∗−1η+η∗/ηη∗≤1η+η∗∫t0sΦfξηΔξ1/η∫t0tΨgξη∗Δξ1/η∗.
Taking delta integrating on both sides of (84), first over s from t0 to x and then over θ from t0 to y, we find that(85)∫t0x∫t0yΦFsΨGθs−t0θ−t0η∗s−t0η−1η+η∗/ηη∗+ηθ−t0η∗−1η+η∗/ηη∗ΔsΔθ≤1η+η∗∫t0x∫t0sΦfξηΔξ1/ηΔs×∫t0y∫t0tΨgξη∗Δξ1/η∗Δθ.
By applying inequality (1) on (85) with indices η, η/(η − 1) and η∗, η∗/(η∗ − 1), we get(86)∫t0x∫t0yΦFsΨGθs−t0θ−t0η∗s−t0η−1η+η∗/ηη∗+ηθ−t0η∗−1η+η∗/ηη∗ΔsΔθ≤1η+η∗x−t0η−1/ηy−t0η∗−1/η∗∫t0x∫t0sΦfξηΔξΔs1/η×∫t0y∫t0tΨgξη∗ΔξΔθ1/η∗.
Applying Lemma4 on (86) and using the fact σ(n) ≥ n, we find that(87)∫t0x∫t0yΦFsΨGθs−t0θ−t0η∗s−t0η−1η+η∗/ηη∗+ηθ−t0η∗−1η+η∗/ηη∗ΔsΔθ≤1η+η∗x−t0η−1/ηy−t0η∗−1/η∗∫t0xx−σsΦfsηΔs1/η×∫t0yy−σθΨgθη∗Δθ1/η∗,≤Eη,η∗∫t0xσx−sΦfsηΔs1/η∫t0yσy−tΨgθη∗Δθ1/η∗.
The proof is complete.Remark 7.
Letting 1/η + 1/η∗ = 1 in (76), then we get Theorem 3.3 due to Saker et al. [11].
By using relations (22) and putting T=ℝ and t0 = 0 in Theorem 12, we get the following conclusion.Corollary 13.
Assume thatf(s) and gt are nonnegative functions, and define(88)Fs=1s∫0sfξdξ,Gθ=1θ∫0θgξdξ.
Then, fors ∈ (0, x) and θ ∈ (0, y), we have(89)∫0x∫0ysθΦFsΨGtη∗sη−1η+η∗/ηη∗+ηθη∗−1η+η∗/ηη∗dsdθ≤E∗η,η∗∫0xx−sΦfsηds1/η×∫0yy−θΨgθη∗dθ1/η∗,where(90)E∗η,η∗=1η+η∗xη−1/ηyη∗−1/η∗,which is the same inequality in [9], Theorem 3.3.
By using relations (23) and putting T=ℤ and t0 = 0 in Theorem 12, we get the following conclusion.Corollary 14.
Assume that {ai} and {bj} are two nonnegative sequences of real numbers, and define(91)Ai=1i∑p=1iap,Bj=1j∑q=1jbq.
Then,(92)∑i=1k∑j=1rijΦAmΨBnη∗iη−1η+η∗/ηη∗+ηjη∗−1η+η∗/ηη∗≤E∗∗η,η∗∑i=1kk−i+1Φaiη1/η∑j=1rr−j+1Ψbjη∗1/η∗,where(93)E∗∗η,η∗=1η+η∗kη−1/ηrη∗−1/η∗,which is the same inequality in [9], Theorem 2.3.Theorem 15.
Lets, θ, and t0∈T, f∈Crdt0,xT,ℝ+, g∈Crdt0,yT,ℝ+, and h(ξ) and l(ξ) be two positive functions defined for ξ∈t0,xT and ξ∈t0,yT and H and L be as defined in Theorem 9, and let(94)Fs=1Hs∫t0shξfξΔξ,Gθ=1Lθ∫t0θlξgξΔξ.
Then, fors∈t0,yT and θ∈t0,xT, we have(95)∫t0x∫t0yΦFsΨGθHsLθη∗s−t0η−1η+η∗/ηη∗+ηθ−t0η∗−1η+η∗/ηη∗ΔsΔθ≤Wη,η∗∫t0xσx−shsΦfsηΔs1/η×∫t0yσy−θlθΨgθη∗Δθ1/η∗,where(96)Wη,η∗=1η+η∗x−t0η−1/ηy−t0η∗−1/η∗.Proof.
Using the hypotheses of Theorem15 and Jensen’s inequality, we find that(97)ΦFs=Φ1Hs∫t0shξfξΔξ≤1Hs∫t0shξΦfξΔξ.
By applying inequality (1) on (97) with indices η, η/(η − 1), we have(98)ΦFs≤1Hss−t0η−1/η∫t0shξΦfξηΔξ1/η.
From (98), we get(99)ΦFsHs≤s−t0η−1/η∫t0shξΦfξηΔξ1/η.
Analogously,(100)ΨGθLθ≤θ−t0η∗−1/η∗∫t0θlξΨgξη∗Δξ1/η∗.
From (99) and (100), we find that(101)ΦFsΨGθHsLθ≤s−t0η−1/ηθ−t0η∗−1/η∗∫t0shξΦfξηΔξ1/η×∫t0θlξΨgξη∗Δξ1/η∗.
Applying elementary inequality (39), we get(102)ΦFsΨGθHsLθ≤ηη∗η+η∗s−t0η−1η+η∗/ηη∗η+θ−t0η∗−1η+η∗/ηη∗η∗∫t0shξΦfξηΔξ1/η∫t0θlξΨgξη∗Δξ1/η∗.
This implies that(103)ΦFsΨGθHsLθη∗s−t0η−1η+η∗/ηη∗+ηθ−t0η∗−1η+η∗/ηη∗≤1η+η∗∫t0shξΦfξηΔξ1/η∫t0θlξΨgξη∗Δξ1/η∗.
Taking delta integrating on both sides of (103), first over s from t0 to x and then over θ from t0 to y, we obtain(104)∫t0x∫t0yΦFsΨGθHsLθη∗s−t0η−1η+η∗/ηη∗+ηθ−t0η∗−1η+η∗/ηη∗ΔsΔθ≤1η+η∗∫t0x∫t0shξΦfξηΔξ1/ηΔs×∫t0y∫t0θlξΨgξη∗Δξ1/η∗Δθ.
By applying inequality (1) on (104) with indices η, η/(η − 1) and η∗, η∗/(η∗ − 1), we get(105)∫t0x∫t0yΦFsΨGθHsLθη∗s−t0η−1η+η∗/ηη∗+ηθ−t0η∗−1η+η∗/ηη∗ΔsΔθ≤1η+η∗s−t0η−1/ηθ−t0η∗−1/η∗∫t0x∫t0shξΦfξηΔξΔs1/η×∫t0y∫t0θlξΨgξη∗ΔξΔθ1/η∗.
Applying Lemma4 and using σ(n) ≥ n, we get(106)∫t0x∫t0yΦFsΨGθHsLθη∗s−t0η−1η+η∗/ηη∗+ηθ−t0η∗−1η+η∗/ηη∗ΔsΔθ≤Wη,η∗∫t0xσx−shsΦfsηΔs1/η×∫t0yσy−θlθΨgθη∗Δθ1/η∗.
This completes the proof.Remark 8.
Letting 1/η + 1/η∗ = 1 in (95), then we get Theorem 3.4 due to Saker et al. [11].
By using relations (22) and putting T=ℝ and t0 = 0 in Theorem 15, we get the following conclusion.Corollary 16.
Assume thatf(s) and gθ are two nonnegative functions and h(s) and l(θ) are two positive functions, and define(107)Hs=∫0shξdξ,Lθ=∫0θlξdξ,Fs=1Hs∫0sfξhξdξ,Gθ=1Lθ∫0θgξlξdξ.
Then, fors ∈ (0, x) and θ ∈ (0, y), we have(108)∫0x∫0yΦFsΨGθHsLθη∗sη−1η+η∗/ηη∗+ηθη∗−1η+η∗/ηη∗dsdθ≤W∗η,η∗∫0xx−shsΦfsηds1/η×∫0yy−θlθΨgθη∗dθ1/η∗,where(109)W∗η,η∗=1η+η∗xη−1/ηyη∗−1/η∗.
It is clear that it is the same inequality in [9], Theorem 3.4.
By using relations (23) and putting T=ℤ and t0 = 0 in Theorem 15, we have the following conclusion.Corollary 17.
Assume that {ai} and {bj} are nonnegative sequences and {hi} and {lj} are positive sequences, and define(110)Hi=∑p=1ihp,Lj=∑q=1jlq,Ai=1Hi∑p=1ihpap,Bj=1Lj∑q=1jlqbq.
Then,(111)∑i=1k∑j=1rHiLjΦAiΨBjη∗iη−1η+η∗/ηη∗+ηjη∗−1η+η∗/ηη∗≤W∗∗η,η∗∑i=1kk−i+1hiΦaiη1/η×∑j=1rr−j+1ljΨbjη∗1/η∗,where(112)W∗∗η,η∗=1η+η∗kη−1/ηrη∗−1/η∗,which is the same inequality in [9], Theorem 2.4.



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