Some remarks on absolute summability methods (Q1374607)

Let \(\sum^{\infty}_{n=0} a_n\) be a given infinite series with partial sums \(s_n\), \(n=0,1,\dots \), and \((C,\alpha)\) means \(\sigma^{\alpha}_n\), \(\alpha >-1\), \(n=0,1,\ldots\). In what follows, we suppose \(p_k >0\), \(k=0,1,\dots \), \(\sum^\infty_{k=0} p_k = \infty \), and set \(P_n = \sum^n_{k=0} p_k\) and \(T_n = P_n^{-1} \sum^n_{k=0} p_k s_k\), \(n = 0,1,\dots\). The series \(\sum^{\infty}_{n=0} a_n\) is said to be summable \(|C,\alpha |_k\), \(\alpha >-1\), \(k\geq 1\), if \(\sum^{\infty}_{n=1} n^{k-1} \cdot |\Delta \sigma_{n-1}^{\alpha}|^k < \infty\). It is said to be summable \(|\overline {N}, p_n|_k\), \(k \geq 1\) if \(\sum^{\infty}_{n=1}(P_n/p_n)^{k-1} \cdot |\Delta T_{n-1}|^k < \infty\). Let us assume now that \(q_n >0\), \(n = 0,1,\dots\) and \(Q_n = \sum^n_{j=0} q_j \to \infty\) as \(n\to \infty\). \textit{H. Bor} and \textit{B. Thorpe} [Analysis 7, 145-152 (1987; Zbl 0639.40005)] showed that if for some \(A>0\) and \(B>0\) \[ A \cdot \frac {q_n}{Q_n} \leq \frac {p_n}{P_n} \leq B \cdot \frac {q_n}{Q_n}, \;n = 0,1,\dots,\tag{1} \] then \(|\overline {N},p_n|_k \Leftrightarrow |\overline {N},q_n|_k, \;k \geq 1\). (The symbol \(\Leftrightarrow\) is used here to denote the equivalence of two summability methods.) The author of this paper presents (in Section 1) a detailed detection of this assertion in the particular case \(q_n =1\), \(n=0,1,\dots \), i.e., for \(|C,1|_k\) summability. Further, he points out (in Section 2) that the condition (1) in the Bor-Thorpe assertion mentioned above can be somewhat relaxed. Finally, a little comment on a similar result of Bor on \(|C,\alpha |_k\) summability, \(0<\alpha \leq 1\), \(k\geq 1\) is added. All propositions of the paper are easy to prove using well-known techniques from calculus.