On the relationships of seven numerical sequences
We analyze the relationships of four classical and three recently defined classes of numerical sequences. We shall consider sequences with nonnegative terms. For a sequence \(c:=\{c_n\}\), let \(\Delta c_n:=c_n-c_{n+1}\), positive constants \(K\) are not necessarily the same at each occurrence. 1. A sequence \(c:=\{c_n\}\) is said to be quasimonotone decreasing, in brief \(c\in QMS\), if for any \(n\geqq m\) \[ c_n\leqq K(c)c_m \leqno{(2.1)} \] holds. If (2.1) holds with \(K(c)=1\) we get the monotone sequences (MS). 2. We say that a sequence \(c\) is locally quasimonotone (LQMS) if (2.1) holds with \(m\leqq n\leqq 2m\), \(m=1,2,\dots \). The classical quasimonotone sequences (CQMS) will be defined here by \(0<\alpha \leqq 1\) and \[ c_{n+1}\leqq c_n\left(1+\frac{\alpha}{n}\right), \quad n=1,2,\dots \,. \leqno{(2.2)} \] 4. A sequence \(c\) is called quasi \(\beta\)-power monotone (Q\(\beta\)MS) if for some positive \(\beta\) and \(n\geqq m\) \[ n^\beta c_n \leqq K(c)m^\beta c_m \leqno{(2.3)} \] holds. 5. A null-sequence \((c_n\to 0)\) satisfying the inequalities \[ \sum^\infty_{n=m}| \Delta c_n| \leqq K(c)c_m,\quad m=1,2,\dots\, \leqno{(2.4)} \] is said to be a sequence of rest bounded variation (RBVS). 6. If there exists a natural number \(N\) such that \[ \sum^{2m}_{n=m} | \Delta c_n| \leqq K(c) \max_{m\leqq n < m +N}=: K(c)c^*_m \leqno{(2.5)} \] holds for all \(m\), then we say that the sequence \(c\) belongs to class GBVS. We would call such a \(c\), a sequence of Group Bounded Variation. These sequences create classes. If two classes of sequences \(A\) and \(B\) are not comparable we shall denote this fact by \(A\nsim B\). In Chapter 3 the author researches embedding relations. In Chapter 5 the author gives proofs on all facts in Chapter 3.
scientific article; zbMATH DE number 5150265