Within this paper, we derive entire series of new Hilbert and Hardy-Hilbert form integral inequalities with non-conjugate exponents on time scales. Firstly, we demonstrate and discuss two general equivalent inequalities of this type, as well as their corresponding reverse inequalities. We also extend Hilbert and Hardy-Hilbert inequalities in their most general form. The results obtained are then applied to a kernel of special Hardy form. Our results as special cases extend some of the dynamic inequalities achieved on time scales and also include some integral inequalities as special cases when $\mathbb{T}=\mathbb{R}$.

中文翻译：

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具有非共轭参数的动态Hardy型不等式

在本文中，我们推导了一系列新的Hilbert和Hardy-Hilbert形式的不等式，它们在时间尺度上具有非共轭指数。首先，我们证明和讨论这种类型的两个一般等价不等式，以及它们对应的反向不等式。我们还以最一般的形式扩展了希尔伯特和哈迪·希尔伯特不等式。然后将获得的结果应用于特殊Hardy形式的内核。我们作为特例的结果扩展了在时间尺度上实现的一些动态不等式，还包括一些积分不等式，如特例$\mathbb{Ť}=\mathbb{[R}$。