On the utility of the Telyakovskiĭ's class \(S\) (Q2782169)

A sequence \(\{a_n\}\) is said to belong to the class \(S\) if there exists a monotonic decreasing sequence \(\{A_n\}\) such that \(\sum^\infty_{n=1} A_n< \infty\) and \(|\Delta a_n|\leq A_n\) for all \(n\), where \(a_n\to 0\) as \(n\to\infty\). Recently, \textit{Z. Tomovski} [Math. Inequal. Appl. 4, No. 2, 231-238 (2001; Zbl 0986.42002)] introduced a subclass of \(S\). Thus according to him a null-sequence \(\{a_n\}\) belongs to the class \(S_r\) if there exists a monotonic decreasing sequence \(\{A^{(r)}_n\}\) such that \(\sum^\infty_{n=1} n^r A^{(r)}_n< \infty\) and \(|\Delta a_n|\leq A^{(r)}_n\) for all \(n, r= 1,2,\dots\)\ .NEWLINENEWLINENEWLINEA more general definition is due to \textit{L. Leindler} [Math. Inequal. Appl. 4, No. 4, 515-526 (2001; Zbl 1040.26009)]. A null sequence \(\{a_n\}\) is said to belong to the class \(S(\alpha)\), if there exists a monotonic decreasing sequence \(\{A^{(\alpha)}_n\}\) such that \(\sum^\infty_{n=1} \alpha_n A^{(\alpha)}_n< \infty\) and \(|\Delta a_n|\leq A^{(\alpha)}_n\) for all \(n\), where \(\alpha= \{\alpha_n\}\) is a positive monotone sequence tending to infinity.NEWLINENEWLINENEWLINEIn this note the author proves that if \(r\geq \beta> 0\) and \(\{a_n\}\) belongs to the class \(S_r\equiv S(n^r)\), then the sequence \(\{n^\beta a_n\}\) belongs to \(S_{r-\beta}\) and NEWLINE\[NEWLINE\sum^\infty_{n=1} n^{r-\beta} A^{(r-\beta)}_n\leq (\beta+ 1) \sum^\infty_{n=1} n^r A_n^{(r)}NEWLINE\]NEWLINE but the converse of this result is not true in general.