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More on Hölder’s Inequality and It’s Reverse via the Diamond-Alpha Integral

: In this paper, we investigate some new generalizations and reﬁnements for Hölder’s inequality and it’s reverse on time scales through the diamond- α dynamic integral, which is deﬁned as a linear combination of the delta and nabla integrals, which are used in various problems involving symmetry. We develop a number of those symmetric inequalities to a general time scale. Our results as special cases extend some integral dynamic inequalities and Qi’s inequalities achieved on time scales and also include some integral disparities as particular cases when T = R .


Introduction
Hölder's inequality is one of the greatest inequalities in pure and applied mathematics. As is well known, Hölder's inequality plays a very important role in different branches of modern mathematics, such as linear algebra, classical real and complex analysis, probability and statistics, qualitative theory of differential equations and their applications. A large number of papers dealing with refinements, generalizations and applications of Hölder's integral inequalities and their series symmetry in different areas of mathematics have appeared (see [1][2][3][4] and the references therein).
The structure of this paper is listed below. Section 2, presents the fundamental concepts of the time scale calculus in terms of delta, nabla and diamond-α calculus. Section 3, is devoted to main results, which are to generalize the inequalities (10), (12) and (13) for diamond-α time scale calculus.

Preliminaries
In this section, the fundamental theories of time scale delta and time scale nabla calculi will be presented. Time scale calculus whose detailed information can be found in [29,30] has been invented in order to unify continuous and discrete analysis.
A nonempty closed subset of R is named a time scale which is signified by T. For ϑ ∈ T, if inf ∅ = sup T and sup ∅ = inf T, then the forward jump operator σ : T → T and the backward jump operator ρ : T → T are defined as σ(ϑ) = inf(ϑ, ∞) T and ρ(ϑ) = sup(−∞, ϑ) T , respectively. From the above two concepts, it can be mentioned that a point The ∆-derivative of ψ : T → R at ϑ ∈ T k = T/(ρ(sup T), sup T], indicated by ψ ∆ (ϑ), is the number that enjoys the property that ∀ε > 0 where there is a neighborhood U of ϑ ∈ T k , such that A function ψ : T → R is rd-continuous if it is continuous at each right-dense points in T and lim s→ϑ − ψ(s) exists as a finite number for all left-dense points in T. The set C rd (T, R) represents the class of real, rd-continuous functions defined on T. If ψ ∈ C rd (T, R), then there exists a function Ψ(ϑ) such that Ψ ∆ (ϑ) = ψ(ϑ) and the delta integral of ψ is defined by

is continuous at each left-dense points in T and lim
s→ϑ + ψ(s) exists as a finite number for all right-dense points in T. The set C ld (T, R) represents the class of real, ld-continuous functions defined on T. If ψ ∈ C ld (T, R), then there exists a function Ψ(ϑ) such that Ψ ∇ (ϑ) = ψ(ϑ) and the nabla integral of ψ is defined by Now, we briefly introduce short introduction of diamond-α derivative and integrals [17,31]. For ϑ ∈ T, we define the diamond-α dynamic derivative ψ ♦ α (ϑ) by Thus, ψ is diamond-α differentiable if and only if ψ is ∆ and ∇ differentiable. The diamond-α derivative reduces to the standard ∆-derivative for α = 1, or the standard ∇-derivative for α = 0.
Moreover, the diamond-α derivatives offer a centralized derivative formula on any uniformly discrete time scale T when α = 1/2.
On the other hand, let r, ϑ ∈ T and w : T → R. Then the diamond-α integral of w is defined by We may note that the ♦ α -integral is an integral combination of ∆ and ∇. Generally speaking, It is clear that the diamond-α integral of w exists when w is a continuous function. Within the following, we display some basic properties for diamond-α calculus that play a key role in inaugurating the major findings of this paper.

Main Results
In this section, we prove new diamond-α inequalities. As particular cases we get ∆-inequalities on time scales for α = 1 and ∇-inequalities on time scales when α = 0. In the sequel, we will suppose that the functions (without mentioning) are non-negative continuous functions and the left hand side of the inequalities exists if the right hand side exists. In what follows, we will present the diamond α-version of Hölder's inequality (12) with a weight function by applying the diamond α-Hölder inequality (14).
Theorem 3. Let r, s ∈ T with r < s and u, v ∈ C([r, s] T , R), w a weight function (measurable and positive) on [r, s] T , such that Then for λ > 1, and hence, we get Proof. From Hölder's inequality (14), we obtain that is, Hence, the inequality (16) is proven.
The inequality (17) follows from substituting the following into (19). The evidence is complete.
As a specific case of Theorem 3, when α = 1 and α = 0, we get the following findings.

Corollary 1.
Let r, s ∈ T with r < s and u, v ∈ C rd ([r, s] T , R), w a weight function (measurable and positive) on [r, s] T , such that Then for λ > 1 and µ > 1 with 1/λ + 1/µ = 1, we have and then which is the delta version of (16) and (17).

Corollary 2.
Let r, s ∈ T with r < s and u, v ∈ C ld ([r, s] T , R), w a weight function (measurable and positive) on [r, s] T , such that Then for λ > 1 and µ > 1 with and then which is the nabla version of (16) and (17).
As an application of (17) in Theorem 3, we get the next theorem.
Theorem 4. Let r, s ∈ T such that r < s and u ∈ C([r, s] T , R). If Proof.
As a specific case of Theorem 4 when α = 1 and α = 0, we get the following findings.

Corollary 3.
Let r, s ∈ Tn such that r < s and u ∈ C rd ([r, s] T , R). If which is the delta version of (24), see [16] [Lemma 2.10].
which is the nabla version of (24).
The following theorems include the reverse Hölder form on time-scales.
As a specific case of Theorem 5 when α = 1 and α = 0, we get the following findings.

Corollary 7.
Let λ > 1 and µ > 1 with As a specific case of Theorem 6 and Corollary 7 when α = 1 and α = 0, we get the following findings.
As a specific case of Theorem 7 and Corollary 10 when α = 1 and α = 0, we get the following findings.
and hence, we get Remark 12. For the particular case T = R, Corollary 13 coincides with Corollary 2.4 in [13].
Now, we present a refinement of inequality (7) on time scales.
As a specific case of Theorem 8 when λ = µ = γ = 1, we get the following findings.
As a specific case of Theorem 8 and Corollary 10 when α = 1 and α = 0, we get the following findings.
which is the delta version of (65).

Conclusions and Future Work
The study of dynamic inequalities depends on the diamond-α integral on time scales. Hence, in the context of this article, we presented generalizations of symmetrical form for Hölder's inequality and it is reverse by means of the diamond-α integral, which is deflated as a linear combination of the delta and nabla integrals. Within this paper, we generalize certain delta and nabla-integrals inequalities on time scales to diamond-α integrals. Inequalities are considered in rather general forms and contain several special integral and discrete inequalities. The technique is based on the applications of well-known inequalities and new tools from time scale calculus. For future work, we can present such diamond-α integrals inequalities by using Riemann-Liouville type fractional integrals and fractional derivatives on time scales. It will also be very interesting to present such diamond-α integrals inequalities on quantum calculus.
Author Contributions: All authors contributed equally to the writing of this manuscript and all authors have read and agreed to the published version of the manuscript.