In this paper we characterize the functions that map the positive general monotone sequences to themselves, i.e., we obtain a necessary and a sufficient condition for the functions φ to satisfy

$${\sum\limits_{k = n}^{2n} {\left| {{a_{k + 1}} - {a_k}} \right| \le C{a_n} \Rightarrow \sum\limits_{k = n}^{2n} {\left| {\phi \left( {{a_{k + 1}}} \right)} \right| \le {C^\prime }\phi \left( {{a_n}} \right).} } }$$

for all \({\left\{ {{a_n}} \right\}_{n = 1}^\infty \subset {\mathbb{R}^ + }}\).

中文翻译：

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保留一般单调序列的功能

在本文中，我们描述了将正单调正向序列映射到其自身的函数的特征，即，我们获得了满足函数φ的必要和充分条件

$$ {\ sum \ limits_ {k = n} ^ {2n} {\ left | {{a_ {k + 1}}-{a_k}} \ right | \ le C {a_n} \ Rightarrow \ sum \ limits_ {k = n} ^ {2n} {\ left | {\ phi \ left（{{a_ {k + 1}}} \ right）} \ right | \ le {C ^ \ prime} \ phi \ left（{{a_n}} \ right）。}}} $$

对于所有\（{\ left \ {{{a_n}} \ right \} _ {n = 1} ^ \ infty \ subset {\ mathbb {R} ^ +}} \\）。