On a new absolute summability method (Q1392717)

In the paper [Proc. Am. Math. Soc. 115, No. 2, 313-317 (1992; Zbl 0756.40006)] the author defined for \(k\geq 1\) the \(|\overline N, p_n, \varphi_n|_k\) summability as follows: Let \(\{\varphi_n\}\) be any sequence of positive numbers. Then the series \(\sum a_n\) is said to be summable \(|\overline N, p_n, \varphi_n|_k\) if \[ \sum^\infty_{n=1} \varphi^{k-1}_n| t_n- t_{n-1}|^k< \infty. \] In the present paper the author proves for \(k\geq 1\) a factor theorem for \(|\overline N, q_n,\alpha_n|_k\) and the inclusion theorem saying that \(|\overline N, p_n,\beta_n|_k\) implies \(|\overline N, q_n,\alpha_n|_k\), and he deduced some known results of H. Bor and of H. Bor and B. Thorpe from the above inclusion relation.