On the \({\bar |}N,p_ n|\) summability factors of infinite series (Q909163)

The author extends a theorem of \textit{M. P. Chen} [Hung-Ching Chow, 65th Annivers. Vol. 65, 114-120 (1967; Zbl 0162.080)] to obtain Theorem. Let \(\sum a_ n\) be bounded \([\bar N,p_ n]\). Let \(\{p_ n\}\) be a positive sequence such that \(P_ n\to \infty\) and there be sequences \(\{\beta_ n\}\) and \(\{\lambda_ n\}\) such that \(| \Delta \lambda_ n| \leq \beta_ n\), \(\beta_ n\to 0\) as \(n\to \infty\), \(\sum^{\infty}_{n=1}nP_ n| \Delta \beta_ n| <\infty,\) \(P_ n| \lambda_ n| =O(1)\) as \(n\to \infty\). If \(\{p_ n\}\) satisfies \(1/n=O(p_ n)\), then the series \(\sum a_ n\lambda_ n\) is summable \(| \bar N,p_ n|\). Here \(P_ n\) is the partial sum of the series \(\sum p_ n\).