Some Dynamic Hilbert-Type Inequalities on Time Scales

Throughout this article, we will demonstrate some new generalizations of dynamic Hilbert type inequalities, which are used in various problems involving symmetry. We develop a number of those symmetric inequalities to a general time scale. From these inequalities, as particular cases, we formulate some integral and discrete inequalities that have been demonstrated in the literature and also extend some of the dynamic inequalities that have been achieved in time scales.

The following useful relationships are often used between the time scale calculus T and the difference calculus R and the difference calculus Z. Please mind that (i) if T = R, then (ii) if T = Z, then Within the following, we display some basic lemmas and some algebraic inequalities that play a key role in inaugurating the major findings of this paper. [24]). Let k, l ∈ T and ζ, χ ∈ C rd (T, R). Then
If ζ ∈ CC 1 rd (R, (p, q)) and Θ : (p, q) → R be a convex function, then where R is a rectangle in T 1 × T 2 defined by Define Then, Ω is ∆-integrable on T 2 and Π is ∆-integrable on T 1 and Lemma 6 (Young's inequality [27]). Let r > 0, µ q > 0 and ∑ p q=1 µ q = Ω p . Then The symmetry index is the most important parameter when evaluating functional asymmetries in athletes of different disciplines. Hence, the first objective of this paper is to establish a new inequality symmetry to Hilbert's type inequality. Our findings provide new estimates on time-scale for this form of inequality. During that paper, we must assume that all functions found in the theorems statements are non-negative, right-dense continuous (rd-continuous) and that the integrals considered exist.

Main Results
In this section, we state and prove our main results. Namely, we set a time scale model for inequalities (5) and (6). To prove our next theorems, we will assume that λ, µ be any two real numbers such that λ, µ > 1 with 1/λ + 1/µ = 1.

The One Dimension Version
where Proof. From the hypotheses, we have the following two identities hold, Further, by using Hölder's integral inequality (9), we have By multiplying (19) and (20), we get Using the inequality (14), we note Now, by setting Substituting (23) into (21) yields Dividing both sides of (24) by the last factor µ( Integrating both sides of (25) and using (9), we find that Applying Fubini's theorem on (26) and by taking advantage of the fact that σ(δ) ≥ δ, we conclude that which is equivalent to (15).
Therefore, by applying (27) on the right-hand side of (15) in Theorem 1, we get Remark 2. Clearly, for T = Z and t 0 = 0, inequality (28) in Remark 1 reduces to It is merely a similar variant of the consequence disparity in [15] Remark 1, attributed to Young and Byung.
Remark 3. For T = R and t 0 = 0, Remark 1 coincides with Remark 2 in [15]. (15) is nothing more than a close version of the following inequality established in [28] Theorem 6,

Remark 7.
As a particular state of Theorem 1 if T = Z and t 0 = 0, then we have relations (8) and inequality (15) where ∆a s = a s+1 − a s , This is just a similar version of (5) that premised in the Introduction.

The Two Dimension Version
In the next theorems, we define the two independent variable versions of the inequalities given in Theorems 1 and 2. Throughout this paragraph, we are always assuming that T 1 and T 2 are two defined time scales with (i) t 0 , s, t, x, z ∈ T 1 ; (ii) t 0 , ϑ, r, y, w ∈ T 2 . We denote the partial delta derivatives of u(s, ϑ) with respect to s, ϑ and sϑ by respectively.
where R(λ, µ) = 1 Proof. From the hypotheses, we have the following two identities hold, Further, by using Hölder's integral inequality (10), we find that and By multiplying (50) and (51), we get Applying (22) Dividing both sides of (53) by µ[( Integrating both sides of (54) and using (10), we see that Applying Fubini's theorem on (55), we conclude that As a particular state of Theorem 3, if T 1 = T 2 = R and t 0 = 0, we have relations (7) and inequality (47) reduce to Thus, form (62) and (63), it can be acquired that