On the generalized absolute summability factor theorem (Q880277)

Let \(A\) denote a lower triangular infinite matrix with nonnegative entries. With \(A_n= \sum_{i=0}^na_{ni}s_i\), a series \(\sum a_n\), with partial sums \(\{s_n\}\), is said to be summable \(| A| _k, k \geq 1,\) if \(\sum_{n=1}^{\infty}n^{k-1}| A_n - A_{n-1}| ^k < \infty\). (This definition was first formulated by \textit{T. M. Flett} [Proc. Lond. Math. Soc., III. Ser. 7, 113--141 (1957; Zbl 0109.04402)].) The present author establishes sufficient conditions for the series \(\sum\lambda_na_n\) to be summable \(| A| _k\), where \(\{\lambda_n\}\) satisfies certain conditions involving an almost increasing sequence.