A note on absolute summability factors of infinite series (Q5895384)

Let \(\sum a_ n\) be a given infinite series with the sequence of partial sums \(\{s_ n\}\). Let \(\{p_ n\}\) be a sequence of positive real constants such that \(P_ n=\sum^{n}_{m=0}p_ m\to \infty\) as \(n\to \infty\). The series \(\sum a_ n\) is said to be summable \(| \bar N,p_ n|_ k\), \(k\geq 1\) if \(\sum^{\infty}_{n=1}(P_ n/p_ n)^{k-1}| t_ n-t_{n-1}|^ k<\infty,\) where \(t_ n=P_ n^{-1}\sum^{n}_{m=0}p_ ms_ m\). Clearly \(| \bar N,p_ n|_ 1\) is the same as \(| \bar N,p_ n|\). In the special case \(p_ n=1/(n+1)\) and \(k=1\), \(| \bar N,p_ n|_ 1\) is equivalent to \(| R,\log n,1|\). The author proves the following theorem. If \(\{\lambda_ n\}\) is a convex sequence such that \(\sum p_ n\lambda_ n\) is convergent, where \(\{p_ n\}\) is a sequence of positive real constants such that \(P_ n\to \infty\) and the sequence \(\{s_ n\}\) is bounded then the series \(\sum a_ nP_ n\lambda_ n\) is summable \(| \bar N,p_ n|_ k\), \(k\geq 1\). The result proved extends a result of the reviewer [Proc. Nat. Inst. Sci. India, Part A 26, 69-73 (1960; Zbl 0098.044)] for \(| R,\log n,1|\) summability.