Some Dynamic Inequalities of Hilbert’s Type

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.

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, , and , thenwhere

An integral analogue of (1) is given in the following result. Let h, l ≥ 1, , and , for and . Then,where

In 2001, Kim [8] gave some generalizations of (1) and (3) by introducing a parameter α > 0 aswhere h, l ≥ 1, , , and

An integral analogue of (5) is given in the following result. Let p, q ≥ 1, α > 0, , and , for , . Then,where

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, , and . Then,where

An integral analogue of (9) is given as follows. If h, l ≥ 1, α > 1, γ > 1, , and , for and , thenwhere

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, , , and η > 1,  > 1 with , thenwhere

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, , , and η > 1,  > 1 with , thenwhere

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 . 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 scale is defined as a nonempty arbitrary locked subplot of real numbers . We define the forward jump operator asand the backward jump operator as

From the above two definitions, it can be stated that a point with 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 has left-scattered maximum s_m, then ; otherwise, . Finally, the graininess function for any is defined by

For a function , the delta derivative of χ at is defined as for each ɛ > 0, there is a neighborhood U of τ such that

Moreover, χ is called delta differentiable on if it is delta differentiable at every .

A function is called right-dense continuous (rd−continuous) as long as it is continuous at all right-dense points in , and its left-sided limits exist (finite) at all left-dense points in . The classes of real rd−continuous functions on an interval I will be denoted by . For θ, , the Cauchy integral of χ^Δ is defined as

Note that(a)If , then(b)If , then

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 . For , we havewhere η > 1 and  > 1 with .

Theorem 2. (Jensen’s inequality (see [14, 16])). Suppose that and are nonnegative withIf is convex, then

Lemma 3. (see [17]). Let u, , and be nonnegative. If α ≥ 1, thenNow, we will present the formula that will reduce double integrals to single integrals which is the desired in [18].

Lemma 4. Let and . Then,assuming the integrals considered exist.

Lemma 5. (see [19]). Let r > 0, μ_q > 0, and Then,

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 that h and l ≥ 1 be real numbers and η > 1 and  > 1 with .

Theorem 6. Let s, θ, and and and . Suppose that and are defined asThen, for and , we havewhere

Proof. By using inequality (27) (see Lemma 3), we see thatThen, we haveApplying Hölder’s inequality (1) on with indices η and η/(η − 1), we find thatand on the integral with indices and /( − 1), we find thatFrom (36) and (37), we getUsing inequality (29) of power means, we observe thatNow, by setting , , ω₁ = 1/η, ω₂ = 1/, and r = ω₁ + ω₂ in (39), we getSubstituting (40) into (38) yieldsDividing both sides of (41) by the last factor , we obtainIntegrating the above relation and applying Hölder’s inequality (1), we haveApplying Lemma 4 on (43) and using the fact that σ(n) ≥ n, we conclude thatwhich proves (31). This completes the proof.

Remark 1. Letting 1/η + 1/ = 1 in (31), we get Theorem 3.1 due to Saker et al. ([11], Theorem 3.1).
By using relations (22) and putting and t₀ = 0 in Theorem 6, we get the following conclusion.

Corollary 7. Assume that f(ξ) and (ξ) are two nonnegative functions, and defineThen, for s ∈ (0, x) and θ ∈ (0, y), we havewherewhich was proved by Yang ([9], Theorem 3.1).
By using relations (23) and putting and t₀ = 0 in Theorem 6, we get the following conclusion.

Corollary 8. Assume that {a_i} and {b_j} are two nonnegative sequences of real numbers, and defineThen,wherewhich was proved by Yang ([9], Theorem 2.1).

Remark 2. In Theorem 6, setting h = l = 1, we havewhere

Remark 3. In Remark 2, if , , 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 . A function χ is a submultiplicative if χ(st) ≤ χ(s) χ(t) for s, t ≥ 0.

Theorem 9. Let s, θ, and , , , and h(τ) and l(ξ) be two positive functions defined for and . Suppose that F(s) and G(θ) are as defined in Theorem 6, and letThen, for and , we havewhere

Proof. According to Theorem 2 and the definition of function Φ, it is clear thatBy applying Hölder’s inequality (1) on (56), we find thatAnalogously,Thus, from (57) and (58), it can be acquired thatApplying (39) on the term , we getFrom (60), we observe thatIntegrating the above relation and using Hölder’s inequality (1) again with indices η, η/(η − 1) and , /( − 1), we find thatApplying Lemma 4 on (62) and using σ(n) ≥ n, we getwhich is (54). This completes the proof.

Remark 4. Letting 1/η + 1/ = 1 in (54), then we get Theorem 3.2 due to Saker et al. [11].
By using relations (22) and putting and t₀ = 0 in Theorem 9, we get the following conclusion.

Corollary 10. Assume that f(s) and (θ) are two nonnegative functions and h(s) and l(θ) are two positive functions, and letThen, for s ∈ (0, x) and θ ∈ (0, y), we havewhereIt is clear that we can have the same inequality in [9], Theorem 3.2.
By using relations (23) and putting and t₀ = 0 in Theorem 9, we get the following conclusion.

Corollary 11. Assume that {a_i} and {b_j} are two nonnegative sequences of real numbers and {h_i} and {l_j} are positive sequences, and defineThen,where

Remark 5. From inequality (39), we obtainIf we apply (70) on (31) in Theorem 6 and (54) in Theorem 9, then we get the following, respectively, inequalities:whereAlso,where

Remark 6. In Remark 5, if , , 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. Let s, θ, and , , and . DefineThen, for and , we havewhere

Proof. By assumption and using Jensen’s inequality (26), we see thatBy applying inequality (1) on (78) with indices η, η/(η − 1), we haveThis implies thatAnalogously,From (80) and (81), we getApplying elementary inequality (39) on the term , where , , ω₁ = 1/η, ω₂ = 1/, and r = ω₁ + ω₂, we getFrom (83), we haveTaking delta integrating on both sides of (84), first over s from t₀ to x and then over θ from t₀ to y, we find thatBy applying inequality (1) on (85) with indices η, η/(η − 1) and , /( − 1), we getApplying Lemma 4 on (86) and using the fact σ(n) ≥ n, we find thatThe proof is complete.

Remark 7. Letting 1/η + 1/ = 1 in (76), then we get Theorem 3.3 due to Saker et al. [11].
By using relations (22) and putting and t₀ = 0 in Theorem 12, we get the following conclusion.

Corollary 13. Assume that f(s) and are nonnegative functions, and defineThen, for s ∈ (0, x) and θ ∈ (0, y), we havewherewhich is the same inequality in [9], Theorem 3.3.
By using relations (23) and putting and t₀ = 0 in Theorem 12, we get the following conclusion.

Corollary 14. Assume that {a_i} and {b_j} are two nonnegative sequences of real numbers, and defineThen,wherewhich is the same inequality in [9], Theorem 2.3.

Theorem 15. Let s, θ, and , , , and h(ξ) and l(ξ) be two positive functions defined for and and H and L be as defined in Theorem 9, and letThen, for and , we havewhere

Proof. Using the hypotheses of Theorem 15 and Jensen’s inequality, we find thatBy applying inequality (1) on (97) with indices η, η/(η − 1), we haveFrom (98), we getAnalogously,From (99) and (100), we find thatApplying elementary inequality (39), we getThis implies thatTaking delta integrating on both sides of (103), first over s from t₀ to x and then over θ from t₀ to y, we obtainBy applying inequality (1) on (104) with indices η, η/(η − 1) and , /( − 1), we getApplying Lemma 4 and using σ(n) ≥ n, we getThis completes the proof.

Remark 8. Letting 1/η + 1/ = 1 in (95), then we get Theorem 3.4 due to Saker et al. [11].
By using relations (22) and putting and t₀ = 0 in Theorem 15, we get the following conclusion.

Corollary 16. Assume that f(s) and are two nonnegative functions and h(s) and l(θ) are two positive functions, and defineThen, for s ∈ (0, x) and θ ∈ (0, y), we havewhereIt is clear that it is the same inequality in [9], Theorem 3.4.
By using relations (23) and putting and t₀ = 0 in Theorem 15, we have the following conclusion.

Corollary 17. Assume that {a_i} and {b_j} are nonnegative sequences and {h_i} and {l_j} are positive sequences, and defineThen,wherewhich is the same inequality in [9], Theorem 2.4.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was funded by the Deanship of Scientific Research at Princess Nourah Bint Abdulrahman University through the Fast-track Research Funding Program.