Some New Generalizations of Reverse Hilbert-Type Inequalities via Supermultiplicative Functions

: Our work in this paper is based on the reverse Hölder-type dynamic inequalities illustrated by El-Deeb in 2018 and the reverse Hilbert-type dynamic inequalities illustrated by Rezk in 2021 and 2022. With the help of Specht’s ratio, the concept of supermultiplicative functions, chain rule, and Jensen’s inequality on time scales, we can establish some comprehensive and generalize a number of classical reverse Hilbert-type inequalities to a general time scale space. In time scale calculus, results are uniﬁed and extended. At the same time, the theory of time scale calculus is applied to unify discrete and continuous analysis and to combine them in one comprehensive form. This hybrid theory is also widely applied on symmetrical properties which play an essential role in determining the correct methods to solve inequalities. As a special case of our results when the supermultiplicative function represents the identity map, we obtain some results that have been recently published.

In [4], the researchers proved that, if φ(x) and (x) are nonnegative continuous functions on [z, r], then with where S(.) is the Specht's ratio function ( [5]) and defined by In [4], the researchers established that, if ψ, ∈ C((z, r), R + ) and q > 0, then where In addition, they established the discrete form of (8) as follows: where In [6], the researchers proved that, if 0 < p, δ ≤ 1, and {λ i } k i=1 , ω j r j=1 are nonnegative and decreasing sequences of real numbers with k, r ∈ N, then where In addition, they proved that where are nonnegative sequences with k, r ∈ N, {p i }, δ j are positive sequences φ, ψ are nonnegative, concave and supermultiplicative functions.
The primary objective of this article is to develop some new generalisations of reverse Hilbert-type inequalities via supermultiplicative functions by using reverse Hölder inequalities with Specht's ratio on T (a time scale T is defined as an arbitrary nonempty closed subset of the real numbers R).
The structure of the paper is summarised below. Section 2 covers some of the fundamentals of time scale theory as well as several time scale lemmas that will be useful in Section 3, where we prove our findings. As particular examples (when T = N), our major findings are (10), as demonstrated by Zhao and Cheung [6].

Preliminaries
The forward jump operator is defined as The set of all such rd-continuous functions is denoted by the space C rd (T, R), and for any function Z : T → R, the notation Z σ (c) denotes Z(σ(c)).
The derivatives of ZΩ and Z/Ω of two differentiable functions Z and Ω are given by The integration by parts formula on T is The time scales chain rule ([10] (Theorem 1.87)) is where Ω : R → R is continuously differentiable, and Z : T → R is ∆−differentiable. More information on time scale calculus can be found at ( [10,11]). Now, we will give some properties of multiplicative and supermultiplicative functions.
where L is the identity map (i.e., L(ζ) = ζ) and represents the multiplicative function. L is said to be a submultiplicative function if the last inequality has the opposite sign.

Lemma 6.
Let z ∈ T, λ, ψ be positive and decreasing functions, f , g are positive and nondecreasing functions and 0 ≤ p, δ ≤ 1. Furthermore, assume that φ, ϕ are positive, increasing, concave and supermultiplicative functions. If β > 1, Proof. For ϑ ≤ y, we have and then (where 0 ≤ p ≤ 1) Because λ is decreasing and ϑ ≤ y, we can deduce from (26) that Based on the knowledge β > 1, φ is an increasing function and (27), we can conclude that Then, we obtain (where ϑ ≤ y and f is nondecreasing) that is decreasing. Therefore, we have for and then Integrating (28) over ϑ from z to σ(t), we obtain and then Since the function 1 , and then Integrating (30) over ϑ from z to σ(t), we obtain Based on (29) and (31), we can see that Because S(.) is decreasing on (0, 1) and increasing on (1, ∞), we have that one of is maximum (where S(1) = 1), and it takes the shape and that is (24). In a similar manner, for ψ and 0 ≤ δ ≤ 1, we obtain which is (25).
Throughout the article, we will assume that the functions are nonnegative rd-continuous functions on [z, ∞) T := [z, ∞) ∩ T.

Principal Findings
Theorem 1. Let z ∈ T, 0 ≤ p, δ ≤ 1, λ, ψ be positive and decreasing functions and φ, ϕ are positive, increasing, concave and supermultiplicative functions. If f , g are positive and nondecreasing functions and β > 1, ν > 1 with 1/β + 1/ν = 1, then holds for all r, s ∈ [z, ∞) T , where functions. In the future work, we can generalize dynamic inequalities of this article using a fractional Riemann-Liouville integral on time scale calculus, and we can present some of these dynamic inequalities on quantum calculus. It will be interesting to present dynamic inequalities in two or more dimensions.