A Study on Some New Generalizations of Reversed Dynamic Inequalities of Hilbert-Type via Supermultiplicative Functions

In this article, we establish some new generalizations of reversed dynamic inequalities of Hilbert-type via supermultiplicative functions by applying reverse Hölder inequalities with Specht ’ s ratio on time scales. We will generalize the inequalities by using a supermultiplicative function which the identity map represents a special case of it. Also, we use some algebraic inequalities such as the Jensen inequality and chain rule to prove the essential results in this paper. Our results (when T ≪ ℕ ) are essentially new.

In [3], Hölder proved that where ðζ k Þ and ðy k Þ are positive sequences and p, q > 1 such that 1/p + 1/q = 1: The integral form of (3) is where ψ,ϖ ∈ Cððw, zÞ, ℝ + Þ: In [4], Zhao and Cheung showed that if ψðζÞ and ϖðζÞ are nonnegative continuous functions and ψ 1/p ðζÞϖ 1/q ðζÞ is integrable on ½w, z, then with ϑ= z w ψ p ζ ð Þdζ, Y= z w ϖ q ζ ð Þdζ, p > 1 and 1 p where Sð:Þ is the Specht's ratio function (see [5]) and defined by Also, they proved that if ψ,ϖ ∈ Cððw, zÞ, ℝ + Þ and γ > 0, then where In addition to this, they proved the discrete case of (8) as follows: where z = ∑ n i=1 z i and w = ∑ n i=1 w m+1 i /z m i : In [6], the authors proved the reverse Hilbert-type inequalities by using the Specht's ratio as follows: if 0 ≤ p, q ≤ 1 and fλ m g,fψ n g are nonnegative and decreasing sequences of real numbers for m = 1, 2, ⋯, k and n = 1, 2, ⋯, r with k, r ∈ ℕ, then where 2 Journal of Function Spaces In 1990, Arino and Muckenhoupt [7] proved an inequality which maps from L p to L p ,1 ≤ p<∞ and contains one weighted function that they characterized such that the inequality holds for all nonnegative nonincreasing measurable function f on ð0, ∞Þ with a constant C > 0 independent on f (here 1 ≤ p<∞). The characterization reduces to the condition that the function w satisfies In 1993, Sinnamon [8] generalized (14) to map from L p to L q ,0 < q < 1 < p which has two different weighted functions and characterized the weights such that the inequality holds for all measurable f ≥ 0 and the constant C is independent of f (here 0 < q < 1 < p and 1/r = 1/q − 1/p). The characterization reduces to the condition that the nonnegative functions u and v satisfy Also, many authors study the inequalities with weighted functions (see [9][10][11][12][13][14][15][16]).
For the discrete case of (14), in 2006, Bennett and Gross-Erdmann [11] characterized the weights such that the inequality holds for all nonnegative nonincreasing sequence g n and C > 0: The characterization reduces to the condition that the nonnegative sequence w n satisfies w k , for all n ∈ ℕ and B > 0: In the last decades, a time scale theory has been discovered to unify the continuous calculus and discrete calculus. A time scale T is defined as an arbitrary nonempty closed subset of the real numbers ℝ. Many authors proved some dynamic inequalities on time scales which unify the inequalities in the continuous calculus and discrete calculus (see [17][18][19][20][21][22][23][24][25][26]). In particular, in 2021, Saker et al. [25] unified the inequalities (14) and (18) to be general inequality on time scales and proved that T is a time scale with a ∈ T and 1 ≤ p < ∞: Furthermore, assume that f is a nonincreasing function and Then, As special cases of (20), when (T ≪ ℝ, σðtÞ = t, a = 0), we obtain the inequality (14) and when (T ≪ ℕ, σðtÞ = t + 1, a = 1), we have the inequality (18).
The goal of this manuscript is to use reverse Hölder inequalities with Specht's ratio on T to develop some new generalizations of reverse Hilbert-type inequalities via supermultiplicative functions on time scales.
The following is a breakdown of the paper's structure. In Section 2, we cover some 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 specific examples (when T = ℕ), our major results yield (11) proved by Zhao and Cheung [6].

Preliminaries and Fundamental Axioms
For completeness, we recall the following concepts related to the notion of time scales. For more details of time scale analysis, we refer the reader to the two books by Bohner and Peterson [33,34] which summarize and organize much of the time scale calculus. A time scale T is an arbitrary nonempty closed subset of the real numbers ℝ.
First, we define σðτÞ ≔ inf fu ∈ T : u > τg, C rd ðT , ℝÞ introduces the set of all such rd-continuous functions, and for any function u : T ⟶ ℝ, the notation u σ ðτÞ denotes uðσðτÞÞ: The derivative of the product uϖ and the quotient u/ϖ (where ϖϖ σ ≠ 0) of two differentiable functions u and ϖ 3 Journal of Function Spaces are given by The integration by parts formula on T is The time scale chain rule ( [9], Theorem 1.87) is where it is supposed that ϖ : ℝ ⟶ ℝ is continuously differentiable and φ : Definition 1 (see [35]). A function L : When L is the identity map (i.e., LðζÞ = ζ), the inequality (26) holds with equality. L is said to be a submultiplicative function if the last inequality has the opposite sign.
Journal of Function Spaces Proof. For ϑ ≤ y, we have and then (where 0 ≤ p ≤ 1) Since λ is decreasing, we have from (41) (where ϑ ≤ y) that Using the facts that β > 1, ϕ is an increasing function and (42), we get and then, we obtain (where ϑ ≤ y and f is nondecreasing) that Thus, the function 1/f ν ðϑÞϕ β ½λðϑÞð Ð σðϑÞ w λðτÞΔτÞ p−1 is decreasing. Therefore, we have for w ≤ ϑ that and then, Integrating the last inequality over ϑ from w to σðtÞ, we have and then, Since the function and then,

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By integrating (50) over ϑ from w to σðtÞ, we have Thus, From (48) and (52), we observe that Since Sð:Þ is decreasing on ð0, 1Þ and increasing on ð 1, ∞Þ, we get that one of is maximum (where Sð1Þ = 1), and it is in the form which is (38). Similarly, with respect to ψ when 0 ≤ q ≤ 1, which is (39). Throughout the article, we will assume that the functions are nonnegative rd-continuous functions on ½w,∞Þ T ≔ ½w, ∞Þ ∩ T and the integrals considered are assumed to exist. Now, we will present and justify our main findings.

Main Results
Theorem 7. Let w ∈ T , 0 ≤ p, q ≤ 1, and λ, ψ be positive and decreasing functions. Assume that ω,ϖ,ϕ,φ are positive functions such that ϕ, φ are increasing, concave, and supermultiplicative functions with where A, B are positive constants. If f , g are positive and nondecreasing functions and β > 1, ν > 1 with 1/β + 1/ν = 1, then Journal of Function Spaces holds for all r, s ∈ w,∞Þ T , where with such that Proof. Applying (27) with γ = p, we have Multiplying the last inequality by we get (where f is nondecreasing) that 7 Journal of Function Spaces From Lemma 6, the last inequality becomes Similarly, we have for the decreasing function ψ, the nondecreasing function g and 0 ≤ q ≤ 1, that From (57), (67), and (68), we get (note ϕ is a positive, increasing and supermultiplicative function) that ω S and then, by applying the Jensen inequality on the right hand side of (69) (where ϕ is a concave function), we have that Similarly, with respect to (57) and (68), we see (where φ is a positive, increasing, concave, and supermultiplicative function) that Multiplying (70) and (71), we see that Journal of Function Spaces Applying (36) on the right hand side of (72), we observe that Multiplying (73) by and then, taking the integration over t from w to σðrÞ and the integration over ξ from w to σðsÞ, we get to obtain Applying the integration by parts formula on the term with uðϑÞ = ðσðrÞ − ϑÞ and v Δ ðϑÞ = ϕ β ½λðϑÞð Ð σðϑÞ w λðτÞΔτÞ p−1 , we get where vðϑÞ = Ð ϑ w ϕ β ½λðθÞð Ð σðθÞ w λðτÞΔτÞ p−1 Δθ and then where vðwÞ = 0), Similarly, we see that