On absolute summability factors (Q5917542)

\textit{T. M. Flett} [Proc.~Lond. Math.~Soc., III. Ser. 7, 113--141 (1957; Zbl 0109.04402)] has defined absolute summability of order \(k\) in the following manner:\ Let \(\{s_n\}\) be the sequence of partial sums of the infinite series \(\sum a_n\), and let \(\sigma_n^\alpha\) denote the \(n\)th Cesàro mean, of order \(\alpha>-1\), of \(\{s_n\}\). Then \(\sum a_n\) is termed summable \(|C,\alpha|_k\), \(k\geq 1\), \(\alpha>-1\), if \(\sum^\infty_{n=1} n^{k-1}|\Delta\sigma_n^\alpha|^k < \infty\), where, for a given sequence \(\{b_n\}\), \(\Delta b_n=\colon b_n-b_{n-1}\). Here, the author establishes the following result concerning absolute, weighted-mean summability:\ Let \(T\) be a lower-triangular matrix with nonzero entries and row sums 1; let \(\overline{T}_{n\nu}=\sum^n_{i=\nu} t_{\nu i}\), \(0\leq\nu\leq n\), \(n=0,1,\ldots\); let \(\hat t_{n\nu}=\overline{T}_{n\nu}-\overline{T}_{n-1,\nu},\;n=1,2,\ldots\); let be \(1< k\leq s <\infty\); and suppose that \(\{p_n\}\) is a positive sequence such that \(\lim_n P_n=\infty\), and \(\sum^\infty_{n=\nu+1} n^{k-1} ({p_n\over P_n P_{n-1}})^k=0({1\over p_\nu})^k\). If \(\sum a_n\lambda_n\) is summable \(|T|_s\), whenever \(\sum a_n\) is summable \(|\overline{N},p_n|_k\), then: \[ t_{\nu\nu}\lambda_\nu=0(({p_\nu\over P_\nu})\nu^{{1\over s}-{1\over k}}),\quad\text{and}\quad \sum^\infty_{n=\nu+1} n^{s-1}|\Delta_\nu (\hat t_{n\nu}\lambda_\nu)|^s=0(({p_\nu\over P_\nu})^s\nu^{s-{s\over k}}). \] In conclusion, the author obtains, as corollaries of this theorem, some inclusion theorems for pairs of weighted-mean, matrix summability methods.