Weighted dynamic Hardy-type inequalities involving many functions on arbitrary time scales

The objective of this paper is to prove some new dynamic inequalities of Hardy type on time scales which generalize and improve some recent results given in the literature. Further, we derive some new weighted Hardy dynamic inequalities involving many functions on time scales. As special cases, we get continuous and discrete inequalities.

In [8], the authors showed that if f (ζ ) > 0, (ζ ) ≥ 0 and f (ζ ) ≤ 0 on [0, ∞) T , α > 1 and there exist constants κ, β > 0 such that ζ /σ (ζ ) ≥ 1/κ and The purpose of this manuscript is to establish some new Hardy-type inequalities on time scales T involving many functions which generalize and improve some results in [4]. The following is the format of the paper: In Sect. 2, we begin with some background information about the delta derivative on T. Our main findings are obtained in Sect. 3.

Basic principles
A time scale T is an arbitrary nonempty closed subset of R. We define the forward jump operator σ : T → T by σ (ζ ) = inf{s ∈ T : s > ζ } and define the backward jump operator ρ : A function f : T → R is a right-dense continuous (rd-continuous) if f is continuous at right-dense points and its left-hand limits are finite at left-dense points in T.
Let f : T → R be a real-valued function on T. Then for ζ ∈ T k , we define f (ζ ) to be the number (if it exists) with the property that given any ε > 0 there is a neighborhood u of ζ such that, for all s ∈ u, we have In this case, we say that f is delta differentiable on T k provided f (ζ ) exists for all ζ ∈ T k . If f , g : T → R are delta differentiable at ζ ∈ T, then For a, b ∈ T and a delta differentiable function f , the Cauchy integral of f is defined by

a). The integration by parts formula on T is given by
Lemma 1 (Leibniz rule [9]) If f , f are continuous and u, v : T − → T are delta differentiable functions and f (ζ , s) mean the delta derivative of f (ζ , s) with respect to ζ , then v(ζ ) Lemma 2 (Chain rule [10]) Assume g : R → R is continuous, g : T → R is delta differentiable on T k , and f : R → R is continuously differentiable. Then there exists a point c in the Lemma 3 (Hölder's inequality [10]) Let a, b ∈ T. For rd-continuous functions f , g : where α > 1 and δ = α/(α -1).

Main results
Throughout this section, any time scale T is unbounded above with a, b ∈ T. We will make the assumption that the functionsŵ, u i , z i in the statements of the theorems are rd-continuous, nonnegative and increasing, and f i (ζ ) > 0 is an integrable function.
In Theorem 6, if we take T = N, then we have σ (s) = s + 1 and obtain the next corollary.
In Theorem 11, if we take T = N, then we have the following corollary.
The next corollary follows from Theorem 11 by taking Corollary 14 For any 1 ≤ i ≤ n, n > κ -1, κ ∈ N, if h i are rd-continuous functions and there exist λ i > 0 such that Remark 15 Letting T = R in Corollary 14, we have that σ (ζ ) = ζ and which agrees with [4, Corollary 2].
In Theorem 19, if we take T = N, then we obtain the following corollary. Remark 21 Clearly, for T = R, Theorem 16 reduces to [4,Theorem 4].

Corollary 22
For any 1 ≤ i ≤ n and α > 1, if h i are rd-continuous functions and