On almost increasing sequences for generalized absolute summability (Q2895092)

Let \(\sum a_n\) be an infinite series with partial sums \((s_n)\), and let \(A\) denote a lower triangular matrix. The series \(\sum a_n\) is said to be absolutely \(A\)-summable of order \(k\geq 1\) if NEWLINE\[NEWLINE \sum_{n=1}^{\infty}n^{k-1} |T_n-T_{n-1}|^k<\infty , NEWLINE\]NEWLINE where \(T_n=\sum_{v=0}^{n}a_{nv}s_{v}.\) The series \(\sum a_n\) is summable \(| A,\delta | _{k}\), \(k\geq 1\), \(\delta \geq 0,\) if NEWLINE\[NEWLINE \sum_{n=1}^{\infty }n^{\delta k+k-1}| T_{n}-T_{n-1}| ^{k}<\infty .NEWLINE\]NEWLINE A positive sequence \(\gamma =( \gamma _n) \) is said to be a quasi-\(\beta \)-power increasing sequence if there exists a constant \( K=K( \beta ,\gamma ) \geq 1\) such that NEWLINE\[NEWLINE Kn^{\beta }\gamma _{n}\geq m^{\beta }\gamma _{m} NEWLINE\]NEWLINE holds for all \(n\geq m\geq 1.\) It may be mentioned that every almost increasing sequence is a quasi-\(\beta \)-power increasing sequence for any nonnegative \(\beta ,\) but the converse need not be true.NEWLINENEWLINETwo lower triangular matrices \(\bar{A}\) and \(\hat{A}\) are associated with \(A\) as follows NEWLINE\[NEWLINE \bar{a}_{nv}=\sum_{r=v}^{n}a_{nr}, \quad n,v=0,1,\dots, NEWLINE\]NEWLINE NEWLINE\[NEWLINE \hat{a}_{nv}=\bar{a}_{nv}-\bar{a}_{n-1,v}, \quad n=1,2,\dots,n,\quad\hat{a}_{00}=\bar{a} _{00}=a_{00}. NEWLINE\]NEWLINE In this paper, a general result concerning absolute summability of infinite series by quasi-power increasing sequence is proved.NEWLINENEWLINE Reviewer's remak: Remark 1 is wrong. The reviewer suggests the author to read Theorem 2.1 and Theorem 2.2 in [\textit{E. Savaş} and \textit{H. Şevli}, J. Inequal. Appl. 2009, Article ID 675403 (2009; Zbl 1185.40002)].