In this paper, we establish some necessary and sufficient conditions for the validity of a generalized dynamic Hardy-type inequality with higher-order derivatives with two different weighted functions on time scales. The corresponding continuous and discrete cases are captured when $\mathbb{T=R}$ and $\mathbb{T=N}$ , respectively. Finally, some applications to our main result are added to conclude some continuous results known in the literature and some other discrete results which are essentially new.

中文翻译：

---

具有高阶导数的加权动态哈代型不等式的表征

在本文中，我们为具有时间尺度上两个不同加权函数的高阶导数的广义动态哈代型不等式的有效性建立了一些充要条件。分别在 $\mathbb{T=R}$ 和 $\mathbb{T=N}$ 时捕获相应的连续和离散情况。最后，对我们的主要结果添加了一些应用，以得出文献中已知的一些连续结果和一些本质上是新的离散结果。