SOME INEQUALITIES ON TIME SCALES SIMILAR TO REVERSE HARDY’S INEQUALITY

. In this paper, we give new inverted Hardy inequalities on a general time scale by introducing two parameters and a new inequality with negative parameters. The main results are demonstrated using two new lemmas and an interesting proposition. We derive some new corollaries of continuous and discrete choice on time scale.


Introduction
Hardy inequality in the integral form plays an important role in the modern analysis of partial differential equations and is an indispensable tool in spectral theory of partial differential operators.
In 2012, Sulaiman presented a reverse Hardy's inequality [ 2. for 0 < p < 1, The aim of this work is to give a generalization for (1.1), (1.2) and to conclude new results using calculus on time scales.

Preliminaries
A time scale T is an arbitrary nonempty closed subset of R. The time scale T has the topology that it inherits from the real numbers with standard topology.
Definition 2.1. [3] Let T be a time scales. For t ∈ T we define the forward jump operator σ : T → T by: We put inf ∅ = sup T, sup ∅ = inf T.

Delta derivative.
Definition 2.2. Assume that f : T → R is a function and let t ∈ T k . We define f ∆ (t) to be the number, if it exists, defined as follows: for every We call f ∆ (t) the delta derivative of f at t.
3. Assume f : T → R is a function and let t ∈ T k . Then we have the following.

If f is continuous at t and t is right-scattered, then
3. If t is right-dense, then f is ∆-differentiable iff the limit exists as a finite number. In this case, For a, b ∈ T and a delta differentiable function h, the Cauchy integral of Theorem 2.4. [1, Theorem 1.1.2] Let h, ϕ ∈ C rd (T, R) be rd-continuous functions, a, b, c ∈ T and α, β ∈ R. Then, the following are true:

Chain rule.
Theorem 2.5. [3, Theorem 1.90] Let f : R → R be continuously differentiable and suppose g : T → R is delta differentiable. Then f o g : T → R is delta differentiable and the formula holds.
and suppose that the single integral exists for each x ∈ [a, b) and the single integral In this paper, we study some inverted Hardy dynamic integral inequalities on time scales and we give new ones with a negative parameter, also extend some continuous inequalities and their discrete analogues. We assume that all the integrals of the right and left side of the inequalities are convergent.

Main results
We state the following lemmas which are useful in the proof of the main theorem. Firstly, we extended the lemma [4, 3.6, p.15] for p ̸ = 0.
Proof. By Hölder integral inequality (2.1) for using the parameter Proposition 3.3. Let 0 < B < A are two positive numbers, then Proof. We can write , so for p = 1 we get equality.
, then we get t > 1 and We can rewrite the above inequalities (3.5) and (3.6) in the following form for p ≥ 1 : If ϕ is increasing function and (σ(s) − a) ∆ = 1, then • for 1 ≤ p ≤ q < ∞ : Proof. Let p ≥ 1. Using Hölder inequality for The hypothesis that ϕ is increasing yield and this gives us that Let ψ(τ ) = σ(τ ) − a, by applying (3.1) with the supposition that ψ ∆ = 1, we get Putting (3.12) in (3.11) and applying (3.4), we get and this completes the proof.
(ii) for 0 < q ≤ p < 1, by using the reverse Hölder inequality and the assumption ϕ is increasing function on using the inequality (3.7) and we take ν = 1, so the proof is similar to the proof of inequality (3.9).

If ϕ is decreasing function and
Proof. Let p < 0. By using Hölder inequality, we have .
If we put T = R in Theorem 3.4 and Theorem 3.5, we get the following Corollaries.
• for 0 < q ≤ p < 1 : The inequalities (4.4) and (4.5) are the new generalizations of the reverse Hardy inequalities.
Remark 4.6. By putting q = p in Corollary 4.5, we get for p < 0: The inequalities (4.6) and (4.7) are the new inverse Hardy inequalities for negative parameters.
If we put T = Z in Theorem 3.4 and Theorem 3.5, we get the following corollary. • for 0 < q ≤ p < 1 : If {w j } is decreasing, then for −∞ < q ≤ p < 0; (4.10) −p The inequalities (4.8), (4.9) and (4.10) are the new ones of the reverse Hardy inequalities in the discrete form.