Generalization of some absolute summability methods

Abstract

Mazhar (1971) gave the characterization for the series ∑anϵn to be summable |N,pn| whenever ∑an is summable |C,α|k, α≥0, k≥1. Here we prove two theorems, the first concerns the sufficient conditions and the second the necessary conditions satisfied by {ϵn} in order to have ∑anϵn summable |N¯,pn|k whenever ∑an is summable |C,α|k, k≥1.