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} $toimply$|\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 $\mathrm{\mid }\bar{N},p_n{\mathrm{\mid }}_k$ and $\mathrm{\mid }\bar{N},q_n{\mathrm{\mid }}_s$.