A summability factor theorem for absolute summability involving almost increasing sequences
A positive sequence \(\{b_n\}^\infty_{n=1}\) is almost increasing if and only if there are a positive, increasing sequence \(\{c_n\}^\infty_{n=1}\) and positive constants \(A\) and \(B\) such that \(A c_n\leq b_n\leq B c_n\), for all \(n\). If \(T\) be a lower triangular matrix, if \(s_n= \sum^n_{k=1} a_k\), and if \(T_n:= \sum^n_{j=0} t_{nj} s_j\), then \(\sum^\infty_{n=1} a_n\) is summable \(|T|_k\), \(k\geq 1\) if and only if \(\sum^\infty_{n=1} n^{k-1}|T_n- T_{n-1}|^k<+\infty\). The authors give sufficient conditions for absolute summability of order \(k\) for series \(\sum^\infty_{n=1} a_n\lambda_n\) given by almost increasing sequences and suitable summability factors.
scientific article; zbMATH DE number 2171819