In this article, by using dynamic Jensen’s inequality and its reverse, we prove some forms of dynamic inequalities involving the dynamic Hardy’s operator. As special cases of our results, we obtain refinements of some well-known Hardy-type inequalities and Hardy-Hilbert’s inequality for double integrals. Our findings are the generalization of some results in the literature.

In this paper, we obtain some new Hardy-type inequalities with negative exponents on time scales. These inequalities, as a special case, when the time scale and contain some new integral and discrete inequalities with negative exponents. To the best of the authors' knowledge, the findings presented in this paper are assumed to be new in literature.

This paper investigates refined Hardy-type inequalities on time scales associated with nonnegative kernels and superquadratic functions. Starting from a generalized Hardy operator, we derive an inequality involving explicit remainder terms through a combination of superquadratic Jensen-type estimates and an algebraic power inequality. The resulting estimate is formulated within a time-scale measure framework and is subsequently specialized to power and exponential-type functions. Several consequences, including Hardy–Hilbert- and Pólya–Knopp-type forms, are obtained under appropriate choices of the kernel and the weight. The relation between the proposed inequality and selected results from the existing literature is examined through continuous and time-scale specializations.