A generalization of some integral inequalities similar to Hardy inequality on time scales

Abstract

In this paper, we prove some new dynamic inequalities similar to Hardy's inequality on time scales T. The results as special cases when T=R contain continuous inequalities similar to Hardy's inequality, and when T=Z, the results contain discrete inequalities that are essentially new to the best of the authors' knowledge. R and Z denote the set of all real numbers and the set of all integers, respectively. Our main results are proved by using the Hölder inequality, Chain rule, and some properties of multiple integrals on time scales. Furthermore, some applications and examples are given to illustrate the investigated results.