A short note on weighted mean summability (Q1346447)

Let \(\sum \alpha_ n\) be an infinite series with a sequence of its partial sums \((S_ n)\) and let \(A= (\alpha_{n\nu})\) be a weighted mean (i.e. Riesz) matrix defined by \(\alpha_{n\nu}= p_ \nu/ P_ n\) for \(0\leq\nu\leq n\), and \(\alpha_{n\nu}=0\) for \(\nu> n\), where \((P_ n)\) is a sequence of positive real numbers and \(P_ n= p_ 0+ p_ 1+ \dots+ p_ n\), \(P_{-1}= 0\). Suppose that the sequence \((T_ n)\) is defined by \[ T_ n= {1\over P_ n} \sum_{\nu=0}^ n p_ \nu s_ \nu, \qquad n=0,1,2,\dots\;. \] If \((T_ n)\) is convergent, then \((s_ \nu)\) (or the series \(\sum \alpha_ n\)) is said to be \(R_ p\)-summable. If \[ \sum_{n=1}^ \infty n^{k-1} | T_ n- T_{n-1}|^ k <\infty, \] then \(\sum \alpha_ n\) is called \(| R_ p |_ k\), \(k\geq 1\), summable. \((p_ n)\) and \((q_ n)\) are sequences of positive real numbers such that \[ P_ n= p_ 0+ p_ 1+ \dots+ p_ n\to \infty, \quad \text{as }n\to \infty, \qquad Q_ n= q_ 0+ q_ 1+ \dots+ q_ n\to \infty, \quad \text{as }n\to \infty. \] In this paper the author presents conditions under which \(R_ p\subseteq R_ q\) implies \(| R_ p| \Rightarrow | R_ q|\) and vice versa.