Necessary and sufficient conditions for inclusion relations for absolute summability

We obtain a set of necessary and sufficient conditions for $| \bar{N}, p_{n} |_{k} $ to imply $|\bar{N}, q_{n} |_{s}$ for $1 < k \leq s < \infty$. Using this result we establish several inclusion theorems as well as conditions for the equivalence of $| \bar {N}, p_{n} |_{k}$ and $|\bar{N},q_{n} |_{s}$.