A summability factor theorem for absolute summability involving quasi-monotone sequences (Q1004648)

Starting from two sequences \(\{ a_n\} \), \(\{ \lambda_n\}\), let \(s_n=\sum_{i=0}^{n}a_i\lambda_i\). Let \(A=(a_{i,j}) \) denote a lower triangular matrix and consider the new sequence \( A_n=\sum_{i=0}^{n}a_{n,i}s_i\). The series \(\{s_n\}\) is summable \(|A|_k\), \(k\geq 1\) if \[ \sum_{n=1}^{\infty}n^{k-1}|A_{n}-A_{n-1}|^k <\infty. \] The author provides sufficient conditions to ensure that \(\{s_n\}\) is summable \(|A|_k\), \(k\geq 1\). It is not clear where this summability method can be used.