We establish sufficient conditions for summability of integrals with multiplier by $|\overline{W}, p(y)|_k$, $k \geqslant 1$, method.

Rhoades and Sava¸s [6],[11] established necessary for inclusions of the absolute matrix summabilities under additional conditions. In this paper we determine or su¢ cient conditions some classes in…nite matrices, using get more general applied to all matrices.

A positive sequence (bn) is said to be almost increasing if there exists a positive increasing sequence cn and two positive constants A and B such that Acn ≤ bn ≤ Bcn (see [1]). Obviously every increasing sequence is almost increasing but the converse need not be true as can be seen from the example bn = ne(−1) n . Let ∑ an be a given infinite series with partial sums (sn). We denote by tn n-th...

Mazhar (1971) gave the characterization for the series ∑ an n 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 { n } in order to have ∑ an n summable |N̄,pn|k whenever ∑ an is summable |C,α|k, k≥ 1. 2000 Mathematics Subject Classification. Primary 40G. L...