DYNAMIC HARDY-TYPE INTEGRAL INEQUALITIES ON TIME SCALES VIA NABLA CALCULUS

In this paper, we derive dynamic Hardy-type integral inequalities on time scales using nabla calculus.By utilizing the chain rule, Hlder's inequality, and certain properties of multiple integrals within nabla calculus, we have derived novel integral inequalities that unify and extend classical results to both continuous and discrete settings.The results include, as special cases, the time scales T = R (continuous case), which yield continuous inequalities, and T = Z (discrete case), which yield discrete inequalities.Here, R and Z represent the sets of real numbers and integers, respectively.Furthermore, the derived inequalities on time scales are validated through illustrative examples and various applications.