A summability factor theorem for absolute summability involving almost increasing sequences

A triangle is a lower triangular matrix with all principal diagonal entriesbeing nonzero. Given an almost increasing sequence {Xn} and a sequence {λn} satisfying certain conditions, we obtain sufficient conditions for the series ∑ anλn to be absolutely summable of order k ≥ 1 by a triangle T . As a corollary we obtain the corresponding result when T is a weighted mean matrix. Theorem 1 of this paper is an example of a summability factor theorem. There is a large literature dealing with summability factor theorems. For example, MathSciNet lists over 500 papers dealing with this topic. For some other papers treating absolute summability factor theorems (of order k≥ 1) the reader may wish to consult [3, 4, 5]. Let T be a lower triangular matrix,and {sn} a sequence. Then Tn := n ∑ ν=0 tnνsν . (1)