N-dimension for dynamic generalized inequalities of H¨older and Minkowski type on diamond alpha time scales

: Expanding on our research, this paper introduced novel generalizations of H¨older’s and Minkowski’s dynamic inequalities on diamond alpha time scales. Specifically, as particular instances of our findings, we replicated the discrete inequalities established when T = N . Furthermore, our investigation extended to the continuous case with T = R , revealing additional inequalities that are both new and valuable for readers seeking a comprehensive understanding of the topic


Introduction
In 1889, Hölder [1] proved that where {x k } n k=1 and {y k } n k=1 are positive sequences, p and q are two positive numbers, such that 1/p + 1/q = 1.
Expanding upon this trend, the present paper aims to broaden the scope of the inequalities (1.4) and (1.5) on diamond alpha time scales.We will establish the generalized form of Hölder's inequality and Minkowski's inequality for the forward jump operator and the backward jump operator in the discrete calculus.In addition, we prove these inequalities in diamond −α calculus, where for α = 1, we can get the inequalities for the forward operator, and when α = 0, we get the inequalities for the backward jump operator.Also, we can get the special cases of our results in the discrete and continuous calculus.
The paper is organized as follows: In Section 2, we present some definitions, theorems, and lemmas on time scales, which are needed to get our main results.In Section 3, we state and prove new dynamic inequalities and present their special cases in different (continuous, discrete) calculi.

Preliminaries and basic lemmas
In 2001, Bohner and Peterson [28]  (The Delta derivative [28]) Assume that φ: T → R is a function and let ξ ∈ T. We define φ ∆ (ξ) to be the number, provided it exists, as follows: for any ϵ > 0, there is a neighborhood for some δ > 0, such that In this case, we say φ ∆ (ξ) is the delta or Hilger derivative of φ at ξ. Definition 2.2.(The nabla derivative [29]) A function λ: T → R is said to be ∇-differentiable at ξ ∈ T, if ψ is defined in a neighborhood U of ξ and there exists a unique real number ψ ∇ (ξ), called the nabla derivative of ψ at ξ, such that for each ϵ > 0, there exists a neighborhood N of ξ with N ⊆ U and for all s ∈ N.
Definition 2.3.([6]) Let T be a time scale and Ξ(ξ) be differentiable on T in the ∆ and ∇ sense.For ξ ∈ T, we define the diamond −α derivative Ξ ♢ α (ξ) by Thus, Ξ is diamond −α differentiable if, and only if, Ξ is ∆ and ∇ differentiable.For α = 1, we get that and for α = 0, we have that (2) Ξ.Ω: Theorem 2.2.( [6]) Let Ξ, Ω: T → R be diamond-α differentiable at ξ ∈ T, then the following holds Theorem 2.3.( [6]) Let a, ξ ∈ T and h: T → R, then the diamond−α integral from a to ξ of h is defined by provided that there exist delta and nabla integrals of h on T.

It is known that
so that the equality above holds only when and Ξ and Ω be continuous functions on [a, b] ∪ T, then the following properties hold. (

Main results
In the manuscript, we will operate under the assumption that the considered integrals are presumed to exist.Also, we denote and ϖ, Θ: Proof.Applying (2.4) with and then (note that 1 sp + 1 sq = 1) and ϖ, Θ: R N → R + are continuous functions, then , and ϖ, Θ are positive sequences, then Using the above assumptions, we observe that Also, we can get and so From (3.4) and (3.5) (note 1 sp + 1 sq = 1), we have that .
The proof is complete.
In the following, we establish the reversed form of inequality (3.15).

Conclusions and future work
In this paper, we present novel generalizations of Hölder's and Minkowski's dynamic inequalities on diamond alpha time scales.These inequalities give us the inequalities on delta calculus when α = 1 and the inequalities on nabla calculus when α = 0. Also, we introduced some of the continuous and discrete inequalities as special cases of our results.In addition, we added an example in our results to indicate the work.
In the future, we will establish some new generalizations of Hölder's and Minkowski's dynamic inequalities on conformable delta fractional time scales.Also, we will prove some new reversed versions of Hölder's and Minkowski's dynamic inequalities on time scales.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.