Diamond Alpha Hilbert-Type Inequalities on Time Scales

: In this article, we will prove some new diamond alpha Hilbert-type dynamic inequalities on time scales which are deﬁned as a linear combination of the nabla and delta integrals. These inequalities extend some known dynamic inequalities on time scales, and unify and extend some continuous inequalities and their corresponding discrete analogues. Our results will be proven by using some algebraic inequalities, diamond alpha Hölder inequality, and diamond alpha Jensen’s inequality on time scales.


Introduction
Over the past decade, a great number of dynamic Hilbert-type inequalities on time scales has been established by many researchers who were motivated by various applications; see the papers [1][2][3][4].
Points that are simultaneously right-dense and left-dense are said to be dense points, whereas points that are simultaneously right-scattered and left-scattered are said to be isolated points.
The sets T κ , T κ and T κ κ are introduced as follows: if T has a left-scattered maximum ζ 1 , then T κ = T − {ζ 1 }, otherwise T κ = T. If T has a right-scattered minimum ζ 2 , then We define the open intervals and half-closed intervals similarly. Assume : T → R is a function and ζ ∈ T κ . Then ∆ (ζ) ∈ R is said to be the delta derivative of at ζ if for any ε > 0 there exists a neighborhood U of ζ such that, for every s ∈ U, we have Moreover, is said to be delta differentiable on T κ if it is delta differentiable at every Similarly, we say that ∇ (ζ) ∈ R is the nabla derivative of at ζ if, for any ε > 0, there is a neighborhood V of ζ, such that for all s ∈ V Furthermore, is said to be nabla differentiable on T κ if it is nabla differentiable at each ζ ∈ T κ .
A function : T → R is said to be right-dense continuous (rd-continuous) if is continuous at all right-dense points in T and its left-sided limits exist at all left-dense points in T.
In a similar manner, a function : T → R is said to be left-dense continuous (ldcontinuous) if is continuous at all left-dense points in T and its right-sided limits exist at all right-dense points in T.
The delta integration by parts on time scales is given by the following formula whereas the nabla integration by parts on time scales is given by The following relations will be used.
where ∆ and ∇ are the forward and backward difference operators, respectively. Now we will introduce the diamond-α calculus on time scales, and we refer the interested reader to [8,9] for further details on the definitions of nabla and delta integrals and derivatives.
If T is a time scale, and is a function that is delta and nabla differentiable on T, then, for any ζ ∈ T, the diamond-α dynamic derivative of at ζ, denoted by ♦ α (ζ), is defined by We conclude from the last relation that a function is diamond-α differentiable if and only if it is both delta and nabla differentiable. For α = 1, the diamond-α derivative boils down to a delta derivative, and for α = 0 it boils down to a nabla derivative.
Assume , g : T → R are diamond-α differentiable functions at ζ ∈ T, and let k ∈ R. Then Next, we write Hölder's inequality and Jensen's inequality on time scales.
This inequality is reversed if 0 < p < 1 and if p < 0 or q < 0.
In this paper, we extend some generalizations of the integral Hardy-Hilbert inequality to a general time scale using diamond alpha calculus. As special cases of our results, we will recover some dynamic integral and discrete inequalities known in the literature. Now we are ready to state and prove our main results.

Main Results
First, we enlist the following assumptions for the proof of our main results: and , respectively. (S 4 ) All functions used in this section are integrable according to ♦ α sense.
Now, we are ready to state and prove the main results that extend several results in the literature. Theorem 1. Let S 1 , S 4 , S 5 , S 14 , S 6 , S 15 , and S 8 be satisfied. Then for S 10 , S 18 and S 20 we find that Proof. From the hypotheses of Theorem 1, S 14 , S 15 , and S 8 , it is easy to observe that By using inverse Jensen dynamic inequality, we obtain that Applying inverse Hölder's inequality on the right hand side of (21) with indices γ and β , it is easy to observe that By using the following inequality on the term ( we get that Integrating both sides of (24) over s , from 0 to ϑ , ς ( = 1, . . . , n), we obtain that Applying inverse Hölder's inequality on the right hand side of (25) with indices γ and β , it is easy to observe that Using Fubini's theorem, we observe that By using the fact ϑ ρ(ϑ ), and ς ρ(ς ), we get that This completes the proof. As a special case of Theorem 1, when T = Z, α = 1 we have ρ(n) = n − 1, we get the following result.

Remark 3.
In Corollary 3, if we take T = R, then the inequality (29) changes to ϑ 1 0 This is an inverse of the inequality (6) which was proved by Pachpatte [7].

Corollary 4.
In Corollary 2, if we take β = n−1 n the inequality (28) becomes Theorem 2. Let S 1 , S 4 , S 5 , S 6 , S 9 , S 15 , and S 16 be satisfied. Then for S 10 , S 18 and S 20 we have that Proof. From the hypotheses of Theorem 2, and by using inverse Jensen dynamic inequality, we have Applying inverse Hölder's inequality on the right hand side of (32) with indices γ and β , it is easy to observe that By using the inequality (23), on the term (s − 0 )( − 0 ) 1 γ we get that Integrating both sides of (33) over s , from 0 to ϑ , ς ( = 1, . . . , n), we get that Applying inverse Hölder's inequality on the right hand side of (34) with indices γ and β , it is easy to observe that By using Fubini's theorem, we observe that By using the fact ϑ ρ(ϑ ), and ς ρ(ς ), we get that This completes the proof. As a special case of Theorem 2, when T = Z, α = 1 we have ρ(n) = n − 1, we get the following result.
respectively, and with suitable changes, we have the following new corollary: Corollary 6. Let S 22 , S 23 , S 26 , S 27 and S 28 be satisfied. Then, for S 18 , S 20 and S 25 , we have that Corollary 7. In Corollary 6, if we take n = 2, β = 1 2 then the inequality (28) changes to (37) Remark 6. In Corollary 7, if we take T = R, then the inequality (37) changes to This is an inverse of the inequality (7), which was proven by Pachpatte [7].
Corollary 9. In Corollary 6, if we take β = n−1 n ( = 1, . . . , n) the inequality (36) Theorem 3. Let S 1 , S 4 , S 2 , S 9 , S 11 , S 7 , S 13 , S 3 , S 12 , S 8 and S 17 be satisfied. Then, for S 10 we have that Proof. From the hypotheses of Theorem 3, we obtain From (41) and S 8 , it is easy to observe that By using inverse Jensen's dynamic inequality, we get that Applying inverse Hölder's inequality on the right hand side of (43) with indices 1/γ and 1/γ , we obtain Using the following inequality on the term (s − 0 )( − 0 ) γ where γ < 0 and We obtain that From (46), we have that Integrating both sides of (47) over s , from 0 to ϑ , ς ( = 1, . . . , n), we get that Applying inverse Hölder's inequality on the right hand side of (48) with indices 1/γ and 1/γ , we obtain By using Fubini's theorem, we observe that By using the fact ϑ ρ(ϑ ), and ς ρ(ς ), we get that This completes the proof.

Remark 12.
In Corollary 10, if we take T = R, α = 1 we get an inverse form of inequality (4), which was given by Handley et al.

Remark 14.
If we take T = Z, α = 1 the inequality (52) is an inverse of inequality of (1), which was given by Pachpatte.
Remark 15. If we take T = R, α = 1 the inequality (52) is an inverse of inequality of (2), which was given by Pachpatte.

Conclusions
In this work, by applying ♦ α calculus, defined as a linear combination of the nabla and delta integrals, we introduced some novel results of Hardy-Hilbert-type inequalities on a general time-scale. Furthermore, we gave the multidimensional generalization for these inequalities to time scales. We also applied our inequalities to discrete and continuous calculus to obtain some new inequalities as special cases.