Some new dynamic inequalities with several functions of Hardy type on time scales

The aim of this article is to prove some new dynamic inequalities of Hardy type on time scales with several functions. Our results contain some results proved in the literature, which are deduced as limited cases, and also improve some obtained results by using weak conditions. In order to do so, we utilize Hölder’s inequality, the chain rule, and the formula of integration by parts on time scales.


Introduction
Let z > 0, α > 0 be real numbers. If a function f is nonnegative, integrable over finite interval (0, z), and the integral of f α over (0, ∞) converges, then Hardy's inequality [1] is given as follows: and the equality holds iff f = 0 almost everywhere. On the other hand, the constant ( α α-1 ) α is optimal. Hardy proved this inequality in 1925 [1] and the discrete version in 1920 [2]. The discrete version of (1) is given by These two inequalities are known in the literature as Hardy-Hilbert type inequalities. Since the invention of these inequalities, plenty of papers containing new proofs, various extensions, and generalizations have appeared. Inequality (1) was extended in [3], where it was proved that, if α > 1 and f > 0 are integrable on (0, ∞), then The study of Hardy's inequalities (discrete and continuous) focused on the investigations of new inequalities with weighted functions. These results are of interest and importance in analysis because the size of weight classes cannot be improved and the weight conditions themselves are interesting. These inequalities have applications in diverse fields of mathematics (spectral theory, PDEs theory, ODEs theory, etc.). These inequalities lead to a large number of impressive connections between different branches of mathematics. This explored area of mathematical analysis generates the publications of various monographs and research papers. We refer the reader to [4][5][6][7][8][9][10][11][12][13] and the references therein.
Over the last decades a lot of considerable effort has been devoted to improve and generalize Hardy's inequalities (1) and (2). In what follows, we introduce some of these improvements that motivated the content of this paper. Levinson in [14] expanded inequality (1) using Jensen's inequality. Under the following conditions: • λ, f are positive functions; • there exists a constant K > 0 having the following property: In [15] Copson showed that if 1 < γ , 1 ≤ α, then Hwang and Yang, in [16], extended inequality (5) and derived that if λ, q, f are nonnegative functions, α > 1, and K is a positive constant having the following propriety: where The authors in [17] proved that, for i = 1, . . . , n, if α i > β i > 0, m i > β i are real numbers with n i=1 β i = 1 and m = n i=1 m i , and if for any constant C i > 0, where A time scale is a closed subset of real numbers denoted by T. The main objective is to demonstrate some results in dynamic inequalities where the involved functions are defined on an arbitrary time scale T domain. These results involve the classical discrete and continuous inequalities ( T = N, T = R) and can be expanded to different inequalities on different time scales like T = q N for q > 1, T = hN, h > 0, etc.
For wholeness, the main results of dynamic inequalities inspiring the subject of this article are mentioned. Using Elliott's technique [18], Řehak in [19] found the time scale version of Hardy's inequality. Particularly, Řehak derived, for α > 1 and f a positive function such that Additionally, if ν(s)/s → 0 as s → ∞, then ( α α-1 ) α is the optimal constant. Nevertheless, to determine whether the constant in inequality (10) is optimal also on all time scales or just those fulfilling the condition lim s→∞ (ν(s)/s) = 0 is still an open problem.
Özkan and Yildirim [20] found a novel inequality with weight functions that can be thought of as a time scale Hardy-Knopp type inequality proved by Kaijser et al. in [21] of the form where is a convex function on (0, ∞). Authors of [22] derived the time scale analogue of (3), that is, where γ > 1, α > 1 with the existence of a positive constant K having the following propriety: 1/K ≤ š σ (s) for s ∈ T. The authors in [23] generalized inequality (10) and showed that if γ , α > 1, then Saker et al. [24] proved Copson inequalities (6) on time scales. In particular, it has been proved that if γ , α > 1, then where In addition, some generalizations of the inequalities of Bennett and Leindler type on time The aim of this article is to prove some new Hardy-type inequalities on time scales involving many functions which generalize and improve some of the above results and also improve some other already proved results in [25]. The manuscript is arranged as follows: In the preliminaries section, we recall a few elementary results and definitions concerning the delta calculus on time scale. In the main results section, we prove our results that cover a wide spectrum of previously proved inequalities.

Preliminaries
This section is devoted to presenting some basic definitions as well as some basic results on delta calculus on time scales that will be used in the sequel; for more details, see [26]. The backward jump operator and the forward jump operator are defined by • left-scattered if s > (s). u : T → R is a right-dense continuous (noted rd-continuous) function if u is continuous at right-dense points and its left-hand limits are finite at left-dense points in T. We denote by C rd (T) the set of rd-continuous functions.
Without loss of generality, we assume that sup T is equal to ∞. We note [a, b] T := [a, b] ∩ T the time scale interval. Throughout this paper, T is provided with the topology induced by the standard topology on R (see, for instance, [26]).
If u is defined on T, then as an abbreviation u(σ (t)) = uσ (t). The derivative of UV and U/V of two delta-differentiable functions u and v is given by On the other hand, the -integral on T is characterized by the following: The chain rule for functions U : R → R, which is continuously differentiable, and V : T → R, which is delta-differentiable, is given by and this rule leads to the useful form Another formula to the chain rule is given by which provides us with the following useful form: For U, V ∈ C rd (T) and t 1 , t 2 ∈ T, the following expression is known as the integration by parts formula. Hölder's inequality on time scales is written as follows: where α > 1 and 1 α + 1 β = 1. Throughout this paper, we make the following assumptions:

Main results
First, we prove some new Hardy-type inequalities with several functions which cover a wide spectrum of previously proved inequalities. Then, we improve some inequalities showed in [25] by removing the imposed conditions on the functions. It will be convenient to use the convention 0.∞ = 0 and 0/0 = 0. To attain the first objective in this paper, we define the operators and φ by Theorem 1 Let h be nondecreasing on [a, ∞) T and r > 1, α > β > 0 be real numbers. If there exists a positive constant κ having the following propriety: Using the formula of integration by parts (18), φ(a) = 0, and v(∞) = 0, we get By utilizing the chain rule (17), we observe that and 1-r Apply the derivative of the product formula (16) on φ(ξ ) to obtain that Employing the assumption h is nondecreasing, we conclude that φ (ξ ) ≥ 0, and thus In addition, as (ξ ) = λ(ξ ) is positive, we get 1-r integrating both sides gives that By combining (22), (23), and (25), we find that Utilizing assumption (19) Applying Hölder s inequality with exponents α β and α (α-β) produces Remark 1 Let T = R. In Theorem 1, setting h(ξ ) = λ(ξ ) = g(ξ ) = 1, β = 1, κ = 1, a = 0 leads to inequality (1).
In order to prove our next result, which is a new generalization of a Copson-type inequality, we define where T is a time scale, and assume that there exists m ≥ 1 such that where (ξ ) := (ξ ) (ξ ) and is defined as in Theorem 1.
Proof For any ξ ∈ [a, b] T , we define u, v by

Lemma 1
Let h be a nondecreasing function on [a, ∞) T . If there exist a positive constant κ and real numbers r > 1, α > β > 0 such that
Proof We define the function ψ i by Then, by Theorem 1, we find For some constants C i > 0, and by using the arithmetic-geometric inequality [27], we get Remark 5 In Theorem 4, if n = 1, α i = α, β i = 1, λ i = 1 with h(t) = g(t) = 1, then we get where Then we have as stated in relation (12).

Theorem 5
Suppose that α i > β i > 0, m i > β i for i = 1, . . . , n are real numbers such that n i β i = 1. Furthermore, assume that h i (t), f i (t), g i (t) are nonnegative functions and h i (t) is nondecreasing for i = 1, . . . , n, and define If there exist positive constants κ i satisfying > 0 and C i > 0, ∀i = 1, . . . , n.
There are many special cases that can be derived from Theorems 4 and 5. For instance, we can deduce inequality (9) from Theorem 5 by taking the time scale equals R, ϑ i (t) = 1 (this leads to i (x) = x), h i (x) = 1/f i (x), and replacing g i (x) by g i (x)/x.