On almost increasing sequences for generalized absolute summability

In this paper, we establish a summability factor theorem for summability vertical bar A, delta vertical bar(k) as defined in (2) where A is a lower triangular matrix with non-negative entries satisfying certain conditions. This paper is an extension of the main result of [3] using definition (2) below. Let A be a lower triangular matrix, f {s(n)} a sequence. Then [GRAPHICS] A series Sigma a(n) is said to be summable vertical bar A vertical bar(k), k >= 1 if [GRAPHICS] and it is said to be summable vertical bar A, delta vertical bar(k), k >= 1 and delta >= 0 if (see,[1]) [GRAPHICS] We may associate with A two lower triangular matrices (A) over bar and (z) over cap defined as follows: [GRAPHICS] and (a) over cap (nv) = (a) over bar (nv) - (a) over bar (n-1,v), n = 1,2,3,... We may write [GRAPHICS]