On inclusion relations for absolute weighted mean summability (Q1323074)

Let \((p_ n)_{n\in \mathbb{N}_ 0}\) be a sequence of positive numbers such that \(P_ n:= \sum_{\nu=0}^ n p_ \nu\to\infty\), \((n\to\infty)\), and let \(k\geq 1\). A sequence \((s_ n)_{n\in\mathbb{N}_ 0}\) is called \(| \overline{N}, p_ n|_ k\)-summable if \[ \sum_{n=1}^ \infty \biggl| {{p_ n} \over {P_ n}} \biggr|^{1-k} | A_ n(s)- A_{n-1}(s)|^ k<\infty \] where \(A_ n(s):= {1\over {P_ n}} \sum_{\nu=0}^ n p_ \nu s_ \nu\). The paper gives necessary and sufficient conditions in order that \(| \overline{N}, p_ n|_ k\)-summability implies \(| \overline{N}, q_ n|_ s\)-summability for the case \(1\leq k\leq s<\infty\), and so extends recent results of Bor, Thorpe and the author.