A note on absolute summability methods (Q1308856)

The following theorem is proved: Let \(\{P_ n\}\) be a sequence of positive real constants such that, as \(n\to\infty\), \(np_ n\asymp P_ n\) (that is \(np_ n=O(P_ n)\) and \(P_ n=O(np_ n))\) and let \(T_ n\) be the \((\overline N,p_ n)\)-mean of the series \(\Sigma a_ n\). If \(\sum^ \infty_{n=1}(P_ n/p_ n)^{(2-\alpha)k-1}|\Delta T_{n-1}|^ k<\infty\), then the series \(\Sigma a_ n\) is summable \(| C,\alpha|_ k\), \(k\geq 1\), \(0<\alpha\leq 1\). The particular case \(\alpha=1\) reduces to a result proved by the author [Proc. Am. Math. Soc. 98, 81-84 (1986; Zbl 0601.40004)].