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In this paper, we prove some reverse discrete inequalities with weights of Muckenhoupt and Gehring types and use them to prove some higher summability theorems on a higher weighted space $l_{w}^{p}({\open N})$ form summability on the weighted space $l_{w}^{q}({\open N})$ when p > q. The proofs are obtained by employing new discrete weighted Hardy's type inequalities and their converses for non-increasing sequences, which, for completeness, we prove in our special setting. To the best of the authors' knowledge, these higher summability results have not been considered before. Some numerical results will be given for illustration. [ABSTRACT FROM AUTHOR]

In this paper, we prove some new reverse dynamic inequalities of Renaud- and Bennett-type on time scales. The results are established using the time scales Fubini theorem, the reverse Hölder inequality and a time scales chain rule. [ABSTRACT FROM AUTHOR]