Dynamic Hilbert-Type Inequalities with Fenchel-Legendre Transform

: Our work is based on the multiple inequalities illustrated in 2020 by Hamiaz and Abuelela. With the help of a Fenchel-Legendre transform, which is used in various problems involving symmetry, we generalize a number of those inequalities to a general time scale. Besides that, in order to get new results as special cases, we will extend our results to continuous and discrete calculus.

We will need the following important relations between calculus on time scales T and either continuous calculus on R or discrete calculus on Z. Note that: (1) Next is Hölder's and Jensen's inequality: Lemma 1 ([19]). Let a, b ∈ T and f , g ∈ C rd ([a, b] T , [0, ∞)). If p, q > 1 with 1  . Now, we present the Fenchel-Legendre transform and refer, for example, to [11][12][13], for more details.
The domain of h * is the set of slopes of all the affine functions minorizing the function h over R n . An equivalent formula for (3) is introduced as follows: Corollary 1. Assuming h : R n −→ R is differentiable, strictly convex and 1-coercive function. Then for all x ∈ Dom(h), and y ∈ Dom(h * ).
In addition, we will use the following definition and lemma as we will see in the proof of our results: Lemma 4 ([20]). Assuming T is a time scale with x, a ∈ T such that x a. If f 0 andα 1, then Next, we write Fubini's theorem on time scales.
Lemma 5 (Fubini's Theorem, see [55]). Assume that (X, Σ 1 , µ ∆ ) and (Y, Σ 2 , ν ∆ ) are two finite-dimensional time scales measure spaces. Moreover, suppose that f : X × Y → R is a delta integrable function and define the functionsπ Thenπ 1 is delta integrable on Y andπ 2 is delta integrable on X and In this manuscript, by using Fubini's theorem and the Fenchel-Legendre transform, which is used in various problems involving symmetry, we extend the discrete results proved in [1] on time scales. We start from the inequalities treated in the Theorem 1. Our results can be applied to give more general forms of some previously proved inequalities through substituting h and h * by suitable functions as we will see in the following two sections.
The following section contains our main results.

Main Results
We start by establishing the following useful inequality: Lemma 6. Assume x and y ∈ R such that x + y 1, then for γ > 0, and α β 1 2 , we get Proof. For x + y 1 and α β 1, we have (x + y) From |x| + |y| 1 n |x| 1 n + |y| 1 n , for all n 1. Thus, from (9), and since 2β 1, we obtain: Now, since γ > 0, by taking the power 1 /γ for both sides of (10), we get: This proves our claim.
In Theorem 2, if we choose T = R, then we have relation (1) and the next results: In Theorem 2, if we chose T = Z, then we get (2), and the next result:

Corollary 4.
With the hypotheses of Theorem 2 we have: Proof. Using the Fenchel-Young inequality (5) in (11) and (12). This proves the claim.
then for σ(s) ∈ [t 0 , x] T and σ(t) ∈ [t 0 , y] T , we have that where Proof. From the properties ofΦ and using (2), we obtain Using (1) in (29), we see that In addition, from the convexity and submultiplicative property ofΨ, we get by using (2) and (1): From (30) and (31), we havȇ Applying Lemma 6 on the right hand side of (33), we see that From (34), we haveΦ From (35), we obtain From (36), by using (1), we have From (37), by using (5), we obtain By using the facts σ(x) x and σ(y) y, we obtain This completes the proof.
In Theorem 4, taking T = R, we have (1) and the result: In Theorem 4, taking T = Z, gives (2) and the result: In Corollary 6, if p = q = 2 we get the result due to Hamiaz and Abuelela ([1], Theorem 5).

Corollary 7.
Under the hypotheses of Theorem 4 the following inequality hold: Proof. Using (5) in (28). This proves our claim.

Corollary 10.
With the hypotheses of Theorem 5, we get: where M 5 defined as in (44).

Some Applications
We can apply our inequalities to obtain different formulas of Hilbert-type inequalities by suggesting h * (y) and h(x) by some functions: In (12), as a special case, if we take h(x) = x 2 2 , we have h * (x) = x 2 2 see [12], we get where C 2 (L, K, p) defined as in Theorem 2. Consequently, for α = β = 1, inequality (55) produces On the other hand if we take h(n) = n r r , r > 1, then h * (m) = m k k where 1 r + 1 k = 1 and n, m R + , then (12) gives

Conclusions
In this paper, with the help of a Fenchel-Legendre transform, which is used in various problems involving symmetry, we generalized a number of Hilbert-type inequalities to a general time scale. Besides that, in order to obtain some new inequalities as special cases, we also extended our inequalities to discrete and continuous calculus. In the future, we can generalize these inequalities in a different way by using other mathematical tools.