Some New Reverse Hilbert’s Inequalities on Time Scales

: This paper is interested in establishing some new reverse Hilbert-type inequalities, by using chain rule on time scales, reverse Jensen’s, and reverse Hölder’s with Specht’s ratio and mean inequalities. To get the results, we used the Specht’s ratio function and its applications for reverse inequalities of Hilbert-type. Symmetrical properties play an essential role in determining the correct methods to solve inequalities. The new inequalities in special cases yield some recent relevance, which also provide new estimates on inequalities of these type.

In addition, they proved the discrete case of (7) as follows: In [12], Zhao and Cheung established the reverse Hilbert inequalities by using the Specht's ratio and proved that if 0 ≤ α, β ≤ 1, {λ i }, {ψ j } are nonnegative and decreasing sequences of real numbers for i = 1, 2, ..., k and j = 1, 2, ..., r with k, r ∈ N, then where In addition, they proved that, if {λ i }, ω j are nonnegative sequences for i = 1, 2, ..., k, and j = 1, 2, ..., r with k, r ∈ N and {α i }, β j are positive sequences. Let φ, ψ are nonnegative, concave and supermultiplicative functions. Then, with and where the function S(.) is the Specht's ratio. In [12], the authors proved that, if {λ i }, ω j are nonnegative sequences for i = 1, 2, ..., k and j = 1, 2, ..., r with k, r ∈ N, then with In the last few decades, a new theory has been discovered to unify the continuous calculus and discrete calculus. It is called a time scale theory. A time scale T is an arbitrary nonempty closed subset of the real numbers R. Many authors established dynamic inequalities and generalized them on time scales. For more details, see ( [13][14][15][16][17]).
In particular, El-Deeb, Elsennary, and Wing-Sum Cheung [15] proved the reverse Hölder inequality on time scales by using the Specht's ratio function and proved that, where is the Specht's ratio (see [11]). In addition, they proved (12) with weighted functions and proved that if ψ, , where where The aim of this paper is to establish some new reverse Hilbert-type inequalities on time scales by using the Specht's ratio function and applying reverse Hölder inequalities on time scales.
The organization of the paper is as follows: in Section 2, we show some basics of the time scale theory and some lemmas on time scales needed in Section 3 where we prove our results. Our main results (when T = R) give the inequalities (9)-(11) proved by Zhao and Cheung [12].

Preliminaries and Basic Lemmas
A forward jump operator on time scales is defined by: σ(τ) := inf{r ∈ T : r > τ}. The set of all such rd-continuous functions is ushered by C rd (T, R) and for any function To learn more about the time scale calculus, see ( [18,19]).
The derivative of the product Φ and the quotient Φ/ (where σ = 0) of two differentiable functions Φ and are given by The integration by parts formula on time scales is given by The time scales chain rule (see Theorem 1.87 [18]) is as follows: and The inequality (18) holds with equality when h is the identity map (i.e., h(χ) = χ). If the last inequality has a reversed sign, then h is said to be a submultiplicative function. Lemma 1. Let T be a time scale with a ∈ T, λ is nonnegative rd-continuous function and Proof. By applying (17) on the term Since Substituting (21) into (20), we see that Integrating the last inequality over χ from a to σ(t), we observe that i.e., which is (19). The proof is complete.

Lemma 3 ([10]
). Let S(.) be a Specht's ratio function which is defined in Lemma 2, then the function S(t) is strictly decreasing for 0 < t < 1 and strictly increasing for t > 1. Furthermore, the following equations hold: In [15] for α = 1, we get the following lemma.
Theorem 1 (Jensen's inequality). Assume that T is a time scale with ζ 0 , ζ ∈ T and r 0 , is rd-continuous and Ψ : (r 0 , r) → R is continuous and convex, then Lemma 5. Let a ∈ T, λ, ψ be nonnegative and decreasing functions and 0 < α, β ≤ 1. Then, and then (where 0 < α ≤ 1) Since λ is decreasing, we have that Integrate the last inequality over χ from a to σ(t), to get and then Since the function λ(χ) , and by integrating the last inequality over χ from a to σ(t), we get that , and then From (27) and (28), we observe that Since the function (Specht's ratio S(.)) is decreasing on (0, 1) and increasing on (1, ∞), we observe that one of which is (25). Similarly, with respect to the decreasing function ψ when 0 < β ≤ 1, we have which is (26).

Main Results
Throughout the paper, we will assume that the functions are nonnegative rd-continuous functions on [a, b] T and the integrals considered are assumed to exist. We define the time Now, we can present and prove the first result of this section.

Remark 1.
As a special case of Theorem 2, when T = N, we can get (9) proved by Zhao and Cheung [12].